On active constraints in optimal power flow · 2019-06-06 · On active constraints in optimal power flow Learning optimal solutions and identifying important constraints Line A

On active constraints in optimal power flow Learning optimal solutions and identifying important constraints

Line A Roald University of Wisconsin ndash Madison

with Sidhant Misra (LANL) Yee Sian Ng (MIT)

and Daniel K Molzahn (Georgia Tech)

NREL Workshop April 12 2019

Transmission System Operation

41619 | 2


Higher and frequently Increased uncertainty changing power flows

41619 | 3


Transmission System Operator


41619 | 4

Impact of uncertainty

Non-Linear Network

How to maintain grid security

Chance-constrained robust Adapt to uncertainty stochastic optimization in real time

41619 | 5

The Optimal Power Flow Problem

41619 | 6

Optimal Power Flow

Goal Low cost operation while enforcing technical limits

)min sumisin ()+ + ( +

st 2 3 4 + 5 = 0

+8 9lt= gt isin 9 le +8 le +858 9lt= gt isin 9 le 58 le 58

9 le 4 le 49lt=4 isin ℬ

ABC (3 4 + 5) le ABC9lt= F isin ℒ

Minimize generation cost

HI

HJ HLNon-Linear AC Power Flow

Generation constraints

HJHIVoltage

constraints HK

Transmission constraints

HJ

HK

41619 | 7

Optimal Power Flow



st 2 3 4 + 5 = 0



9lt=ABC (3 4 + 5) le ABC F isin ℒ


Non-Linear AC Power Flow


Voltage constraints


Observation 1

Typically only very few transmission constraints are active at optimum

Can be exploited

algorithmically

Eg constraint generation [Bienstock Harnett and Chertkov

SIAM Review 2013]

hellip

41619 | 8

Optimal Power Flow

Impact of renewable energy variationsload uncertainty amp

min sum+isin- +12+ amp + 4+ 12+(amp)

st (amp) lt(amp) 1(amp) =(amp) = 0

12 ADE F isin -A+B le 12(amp) le 12=2 ADE F isin -A+B le =2(amp) le =2

ADElt+A+B le lt+(amp) le lt+ G isin ℬ

ADEGIJ (amp) le GIJ K isin ℒ




Voltage constraints

$


$

41619 | 9

Optimal Power Flow

Impact of renewable energy variationsload uncertainty

min sumampisin( )amp-amp + )amp -amp()

st 5 6() 7() () 8() = 0

- lt A isin ( ltamp= le -() le -8- lt A isin ( ltamp= le 8-() le 8-

7ampltamp= le 7amp() le 7amplt B isin ℬ

ltBDE () le BDE F isin ℒ




Voltage constraints


Observation 2 Typically only very few transmission constraints are ever active even for different parameters

The topic of this talk Is this observation true How can we use it

41619 | 10

1 Learning solutions to (power system) optimization problems through optimal active sets

2 Identifying potentially active constraints

41619 | 11

Learning solutions to (power system) optimization problems

Yee Sian Ng Sidhant Misra (MIT) (LANL)

Yee Sian Ng Sidhant Misra Line Roald and Scott Backhaus laquoStatistical Learning for DC Optimal Power Flowraquo Power System Computation Conference (PSCC) 2018

Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639

41619 | 12

Optimal Power Flow










Voltage constraints

$


$

41619 | 13

Optimal Power Flow

lowastResolve problem every 5-15 min For each 2 obtain 01 2

min sumisin -01 2 + -4 01(2)

st (2) lt(2) 0(2) =(2) = 0

01 ADE F isin AB le 01(2) le 01=1 ADE F isin AB le =1(2) le =1

ADEltAB le lt(2) le lt G isin ℬ

ADEGIJ (2) le GIJ K isin ℒ




Voltage constraints

$


$

41619 | 14

Repeated solution process

OPF at rarr amplowast () OPF at rarr amplowast () OPF at + + rarr amplowast (+)

hellip

Can we use learning to speed up the solution process by using information from past solutions - 0

lowast -

41619 | 15

Learning for optimization

Renewable Prediction method

Optimal energy dispatch

lowast$ $

Can we use learning to speed up the solution process by using information from past solutions $

lowast $

41619 | 16

First attempt

Train a neural net

41619 | 17

First Attempt ndash Train a Neural Net

Renewable Dispatchenergy $

41619 | 18



bull This didnrsquot work well

(DISCLAIMER I will admit that we gave up quite fast)

41619 | 19



bull This didnrsquot work well ndash Hard to satisfy safety constraints

In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639

41619 | 20




ndash Projection back onto feasible space cause suboptimalityhellip


41619 | 21



bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output


41619 | 22




bull This can work well under some circumstances Wide enough and deep enough and with enough data [Karg and Lucia 2018]

41619 | 23



We have a mathematical optimization problem

ndash can we use more information about the problem structure

41619 | 24

Think again

How can we leverage pre-existing knowledge

about the solution

41619 | 25

Think again


about the solution

New idea Learn the optimal set of active constraints

41619 | 26

Optimal set of active constraints

Set of constraints that are active at optimum

bull Equality (power flow) constraints are always active

bull Only very few of the inequality constraints are active ndash Generation constraints ndash Voltage constraints ndash Transmission constraints

41619 | 27

Learn optimal set of active constraints

Renewable Predict optimal active set

Predictrecover optimal solution

Optimal active set amplowast $

Optimal

$energy dispatch

lowast $

bull Why ndash Optimal active set is the ldquominimalrdquo information we need to recover optimal solution

ndash Inherently encodes information about physical constraints and technical limits

ndash Finite low dimensional object

ndash Nice physical interpretation (power system operational pattern)

41619 | 28





Optimal

$energy dispatch

lowast $

bull Related to explicit MPC ndash Explicit MPC ndash each optimal active set corresponds to an optimal affine control policy [Bemporad et al 2002] [Pannochia Rawlings Wright 2007] [Zeilinger et al 2011] [Karg and Lucia 2018] hellip

Look only at specific classes of problems Not very scalable Do not consider input distribution over

41619 | 29

Ensemble Policy

$

amp() amp() amp+()

Realization

Candidates for optimal active set

Solve problem given the active set

rarr Solve a reduced problem with fewer constraints

rarr Solve a set of linear equations (linear problem)

Easier than solving the full optimal power flow problem

41619 | 30

Ensemble Policy

Realization


Solve problem given the optimal active set

Feasible Feasible Evaluate cost and feasibility Infeasible Low cost High cost

$

amp() amp() amp+()

41619 | 31

Ensemble Policy

Realization

$

amp() amp() amp+()

Infeasible Feasible Low cost

Feasible High cost



Evaluate cost and feasibility

Pick best (optimal) solution

41619 | 32

Limits of the Approach

Realization Works well if the number of

active sets + is small

Total number of possible

active sets is exponential bull

Maybe only a few are

practically relevant bull

$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost

How to identify the collection of

relevant active sets

High probability that one of the active sets is optimal for a

new realization

Do NOT search entire parameter space 41619 | 34

Using Sampling to Learn Important Active Sets

41619 | 35

Learning Collection of Optimal Active Sets

hellip $amp $

$ $amp+

$

$amp hellip$+ hellip

$amp() hellip

samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets

41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $

Collection of $ $amp+ observed active sets

$

$amp hellip - =$+ hellip

$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$

Collection of$ $amp+ observed active sets

$


$amp()

$ $

hellip


41619 | 41


hellip

$amp$$ Collection of$ $amp+

observed active sets$


$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp


$amp hellip - =$$++ hellip

$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip


When do I stop 41619 | 46

Streaming Algorithm to Learn

Collection of Optimal Active Sets

41619 | 47


Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set

Goal Find a active sets that together have a high probability of being optimal

41619 | 48


1 Observe optimal active sets for M samples

lowast$lowast lowasthellip

Collection of observed active sets

41619 | 49



lowast lowast hellip lowast $


2 Check ldquorate of discoveryrdquo for W samples

lowast lowast lowast $ hellip

How frequently do we observe sets we have not seen before

ABCDEFGHIGJRate of discovery = A

where ( = + maxlog 5 log 5 -

+ gt= 5 = 1 + =

+lt

41619 | 50


1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples

lowast lowast hellip lowast lowast lowast hellip lowast))))-))

Collection of observed active sets How frequently do we observe sets we have not seen before

G- = HIJKLMNOPNQRate of discovery

H

bull If the rate of discovery is below the threshold $ le amp minus ( stop

where 0 = 23 maxlog = log =

3 EFD= = 1

45

+ A 3BC

D

performance guarantee

41619 | 51



lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0


bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination

[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples

41619 | 52







Guaranteed to converge

41619 | 53







[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]

41619 | 54

Practicability of the approach

Realization

$

amp()

Feasible Low cost

Simultaneously establishes

- Collection of optimal active sets

- Practicability of the approach

No assumptions on distribution

No assumptions on problem structure

Guaranteed to converge fast for low number of optimal active sets

41619 | 55

Results for the (linear)

DC Optimal Power Flow Problem

41619 | 56

Streaming Algorithm Results ndash PGLib-OPF v 1708

Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001

Max difference = 004

Confidence level = 001

Hyperparameter 0 = 2

H Initial W 6 = 13L259 8 IJH 6 =

78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)

41619 | 57








Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]

41619 | 58

Example ndash RTE 1951 bus test case

Number of active sets 5

Rate of discovery 00069

0 10 20 30 40 M=47 W=13rsquo259

When there are few relevant active sets the algorithm terminates fast

41619 | 59

Example ndash PSERC 200 bus test case

06

Number of active sets 05 5 04

03


00069 01

00 0 10 20 30 40 M=47

W=13rsquo259 Probability of observed sets


41619 | 60


012 Number of active sets

010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602


When there are many relevant active sets the algorithm terminates slowly

41619 | 61


Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001

System sizes ranging from 3 to 1951 nodes

41619 | 62



Normal distribution ~ 0 0 1 = 0033

Uniform distribution with support

isin [minus0093 0093]

41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast

High probability of optimal solutions

41619 | 67



Few active sets

Terminates fast


Not always Large number of active sets

41619 | 68



Few active sets

Terminates fast


Not always Large number of active sets Slow to terminate

41619 | 69



Few active sets

Terminates fast


Not always Large number of active sets Slow to terminate Lower probability of optimal solutions

41619 | 70

Practical Implications for Power Systems Operation

41619 | 71

IEEE 300 bus system with normally distributed load

Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load

Number of unique active sets

Increasing parameter uncertainty = Increasing number of optimal active sets

41619 | 72





laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo

General perception among system operators

41619 | 73







What does this imply for system risk Price stability

41619 | 74

Summary

1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes

2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets

3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems

bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets

41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification

[Deka and Misra 2019]

41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification

Efficient solution Active set solver local approximation hellip

41619 | 77

Preliminary results for the (non-linear non-convex)

AC Optimal Power Flow Problem

41619 | 78

Rate of discovery of new optimal active sets

AC Optimal Power Flow


RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets

0039 0967

(did not terminate)

41619 | 79



RTE 1951 bus test case PSERC 200 bus test case

0967 0039

Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets

(did not terminate)Rate of discovery of new active constraints

00035

Rate of discovery of new active constraints

00001

41619 | 80

(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints




Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active

00035 00001

41619 | 81



41619 | 82

Identifying potentially active constraints

Daniel K Molzahn

Georgia Tech

Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019

Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757

41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0

01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt

le (2) le BC E isin ℬ

BCEGH (2) le EGH I isin ℒ

Before we learned constraints that are likely to be active

Now we want to understand which constraints can possibly be active

Feasible set in the direction of the cost function

The full feasible set

41619 | 84

Optimization-based constraint screening

Main idea Minimizemaximize the value of the constraints

min $ max $

st + $ = 0

23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23

$45 le $ le $478 lt isin ℬ

gt ( $ ) le gt478 C isin ℒ

Minimizemaximizevoltage currents hellip



Voltage constraints


If max $ le $478

and min $ ge $45

Voltage will never go out of bounds

41619 | 85



min $ max $

st + $ = 0







Voltage constraints


bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]

bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]

bull Our interest - Large ranges of load - AC OPF (distribution grids)

Distribution grids ndash AC Optimal Power Flow

- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol

min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367

Minimizemaximizevoltage


8 isin Load 8 isin variations

Voltage constraints 8 isin on nodes with measurementscontrol

Valid bounds

Use convex relaxation

QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]

Challenging

- non-standard objective

(relaxation is weak)

- low-voltage solutions

- hellip

Can we certify safe operations

IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]

Start out assuming only one node with controlled voltage

41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation

50 load variability What to do about violations

Increasing load variability increasing voltage range 10 085 090 095

Add more controllable nodes and tighten the voltage limits 41619 | 89

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation

Bus 320905 le amp le 1095

50 load variability

085 090 095 10

Added controllability and tighter voltage range on Bus 32

41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation

Bus 11 092 le amp le 109

50 load variability

085 090 095 10


41619 | 91

Redundant constraints in DC optimal power flow

How many constraints can ever be active in DC optimal power flow

min amp() Minimizemaximize constraints $

st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant

lt 8 le ) le ) Generation constraints Non-redundant

minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant

Allow power demand 34(5) to vary plusmn A where 0 le A le 100

Relax generator lower bounds to 8

Results on PGLib-OPF test cases

Percentage of line flow Remaining

constraints non-redundant constraints

plusmn Load variation

41619 | 93

Results on standard test cases

41619 | 94


bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)

ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints

bull Works really well for power flow optimization

bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip

41619 | 95

THANK YOU

Line Roald roaldwiscedu

41619 | 96

Summary of streaming algorithm results

1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion

Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of

If the window size() is defined as unobserved active sets 8

98 5 confidence level = amp

() maxlog log with = 1 + 5 67

G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F

2 Guaranteed to terminate fast for benign systems

Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than

1 = HI log 2 + log

1

J minus JI FI

41619 | 97


41619 | 2



41619 | 3




41619 | 4


Non-Linear Network



41619 | 5


41619 | 6

Optimal Power Flow



st 2 3 4 + 5 = 0





HI



HJHIVoltage

constraints HK


HJ

HK

41619 | 7

Optimal Power Flow



st 2 3 4 + 5 = 0







Voltage constraints


Observation 1


Can be exploited

algorithmically


SIAM Review 2013]

hellip

41619 | 8

Optimal Power Flow










Voltage constraints

$


$

41619 | 9

Optimal Power Flow



st 5 6() 7() () 8() = 0







Voltage constraints




41619 | 10



41619 | 11





41619 | 12

Optimal Power Flow










Voltage constraints

$


$

41619 | 13

Optimal Power Flow


min sumisin -01 2 + -4 01(2)

st (2) lt(2) 0(2) =(2) = 0







Voltage constraints

$


$

41619 | 14



hellip


lowast -

41619 | 15




lowast$ $


lowast $

41619 | 16

First attempt

Train a neural net

41619 | 17



41619 | 18





41619 | 19





41619 | 20






41619 | 21





41619 | 22





41619 | 23





41619 | 24

Think again


about the solution

41619 | 25

Think again


about the solution


41619 | 26





41619 | 27





Optimal

$energy dispatch

lowast $





41619 | 28





Optimal

$energy dispatch

lowast $



41619 | 29

Ensemble Policy

$

amp() amp() amp+()

Realization






41619 | 30

Ensemble Policy

Realization




$

amp() amp() amp+()

41619 | 31

Ensemble Policy

Realization

$

amp() amp() amp+()


Feasible High cost





41619 | 32








$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97



41619 | 3




41619 | 4


Non-Linear Network



41619 | 5


41619 | 6

Optimal Power Flow



st 2 3 4 + 5 = 0





HI



HJHIVoltage

constraints HK


HJ

HK

41619 | 7

Optimal Power Flow



st 2 3 4 + 5 = 0







Voltage constraints


Observation 1


Can be exploited

algorithmically


SIAM Review 2013]

hellip

41619 | 8

Optimal Power Flow










Voltage constraints

$


$

41619 | 9

Optimal Power Flow



st 5 6() 7() () 8() = 0







Voltage constraints




41619 | 10



41619 | 11





41619 | 12

Optimal Power Flow










Voltage constraints

$


$

41619 | 13

Optimal Power Flow


min sumisin -01 2 + -4 01(2)

st (2) lt(2) 0(2) =(2) = 0







Voltage constraints

$


$

41619 | 14



hellip


lowast -

41619 | 15




lowast$ $


lowast $

41619 | 16

First attempt

Train a neural net

41619 | 17



41619 | 18





41619 | 19





41619 | 20






41619 | 21





41619 | 22





41619 | 23





41619 | 24

Think again


about the solution

41619 | 25

Think again


about the solution


41619 | 26





41619 | 27





Optimal

$energy dispatch

lowast $





41619 | 28





Optimal

$energy dispatch

lowast $



41619 | 29

Ensemble Policy

$

amp() amp() amp+()

Realization






41619 | 30

Ensemble Policy

Realization




$

amp() amp() amp+()

41619 | 31

Ensemble Policy

Realization

$

amp() amp() amp+()


Feasible High cost





41619 | 32








$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97




41619 | 4


Non-Linear Network



41619 | 5


41619 | 6

Optimal Power Flow



st 2 3 4 + 5 = 0





HI



HJHIVoltage

constraints HK


HJ

HK

41619 | 7

Optimal Power Flow



st 2 3 4 + 5 = 0







Voltage constraints


Observation 1


Can be exploited

algorithmically


SIAM Review 2013]

hellip

41619 | 8

Optimal Power Flow










Voltage constraints

$


$

41619 | 9

Optimal Power Flow



st 5 6() 7() () 8() = 0







Voltage constraints




41619 | 10



41619 | 11





41619 | 12

Optimal Power Flow










Voltage constraints

$


$

41619 | 13

Optimal Power Flow


min sumisin -01 2 + -4 01(2)

st (2) lt(2) 0(2) =(2) = 0







Voltage constraints

$


$

41619 | 14



hellip


lowast -

41619 | 15




lowast$ $


lowast $

41619 | 16

First attempt

Train a neural net

41619 | 17



41619 | 18





41619 | 19





41619 | 20






41619 | 21





41619 | 22





41619 | 23





41619 | 24

Think again


about the solution

41619 | 25

Think again


about the solution


41619 | 26





41619 | 27





Optimal

$energy dispatch

lowast $





41619 | 28





Optimal

$energy dispatch

lowast $



41619 | 29

Ensemble Policy

$

amp() amp() amp+()

Realization






41619 | 30

Ensemble Policy

Realization




$

amp() amp() amp+()

41619 | 31

Ensemble Policy

Realization

$

amp() amp() amp+()


Feasible High cost





41619 | 32








$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97


Non-Linear Network



41619 | 5


41619 | 6

Optimal Power Flow



st 2 3 4 + 5 = 0





HI



HJHIVoltage

constraints HK


HJ

HK

41619 | 7

Optimal Power Flow



st 2 3 4 + 5 = 0







Voltage constraints


Observation 1


Can be exploited

algorithmically


SIAM Review 2013]

hellip

41619 | 8

Optimal Power Flow










Voltage constraints

$


$

41619 | 9

Optimal Power Flow



st 5 6() 7() () 8() = 0







Voltage constraints




41619 | 10



41619 | 11





41619 | 12

Optimal Power Flow










Voltage constraints

$


$

41619 | 13

Optimal Power Flow


min sumisin -01 2 + -4 01(2)

st (2) lt(2) 0(2) =(2) = 0







Voltage constraints

$


$

41619 | 14



hellip


lowast -

41619 | 15




lowast$ $


lowast $

41619 | 16

First attempt

Train a neural net

41619 | 17



41619 | 18





41619 | 19





41619 | 20






41619 | 21





41619 | 22





41619 | 23





41619 | 24

Think again


about the solution

41619 | 25

Think again


about the solution


41619 | 26





41619 | 27





Optimal

$energy dispatch

lowast $





41619 | 28





Optimal

$energy dispatch

lowast $



41619 | 29

Ensemble Policy

$

amp() amp() amp+()

Realization






41619 | 30

Ensemble Policy

Realization




$

amp() amp() amp+()

41619 | 31

Ensemble Policy

Realization

$

amp() amp() amp+()


Feasible High cost





41619 | 32








$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97


41619 | 6

Optimal Power Flow



st 2 3 4 + 5 = 0





HI



HJHIVoltage

constraints HK


HJ

HK

41619 | 7

Optimal Power Flow



st 2 3 4 + 5 = 0







Voltage constraints


Observation 1


Can be exploited

algorithmically


SIAM Review 2013]

hellip

41619 | 8

Optimal Power Flow










Voltage constraints

$


$

41619 | 9

Optimal Power Flow



st 5 6() 7() () 8() = 0







Voltage constraints




41619 | 10



41619 | 11





41619 | 12

Optimal Power Flow










Voltage constraints

$


$

41619 | 13

Optimal Power Flow


min sumisin -01 2 + -4 01(2)

st (2) lt(2) 0(2) =(2) = 0







Voltage constraints

$


$

41619 | 14



hellip


lowast -

41619 | 15




lowast$ $


lowast $

41619 | 16

First attempt

Train a neural net

41619 | 17



41619 | 18





41619 | 19





41619 | 20






41619 | 21





41619 | 22





41619 | 23





41619 | 24

Think again


about the solution

41619 | 25

Think again


about the solution


41619 | 26





41619 | 27





Optimal

$energy dispatch

lowast $





41619 | 28





Optimal

$energy dispatch

lowast $



41619 | 29

Ensemble Policy

$

amp() amp() amp+()

Realization






41619 | 30

Ensemble Policy

Realization




$

amp() amp() amp+()

41619 | 31

Ensemble Policy

Realization

$

amp() amp() amp+()


Feasible High cost





41619 | 32








$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97

Optimal Power Flow



st 2 3 4 + 5 = 0





HI



HJHIVoltage

constraints HK


HJ

HK

41619 | 7

Optimal Power Flow



st 2 3 4 + 5 = 0







Voltage constraints


Observation 1


Can be exploited

algorithmically


SIAM Review 2013]

hellip

41619 | 8

Optimal Power Flow










Voltage constraints

$


$

41619 | 9

Optimal Power Flow



st 5 6() 7() () 8() = 0







Voltage constraints




41619 | 10



41619 | 11





41619 | 12

Optimal Power Flow










Voltage constraints

$


$

41619 | 13

Optimal Power Flow


min sumisin -01 2 + -4 01(2)

st (2) lt(2) 0(2) =(2) = 0







Voltage constraints

$


$

41619 | 14



hellip


lowast -

41619 | 15




lowast$ $


lowast $

41619 | 16

First attempt

Train a neural net

41619 | 17



41619 | 18





41619 | 19





41619 | 20






41619 | 21





41619 | 22





41619 | 23





41619 | 24

Think again


about the solution

41619 | 25

Think again


about the solution


41619 | 26





41619 | 27





Optimal

$energy dispatch

lowast $





41619 | 28





Optimal

$energy dispatch

lowast $



41619 | 29

Ensemble Policy

$

amp() amp() amp+()

Realization






41619 | 30

Ensemble Policy

Realization




$

amp() amp() amp+()

41619 | 31

Ensemble Policy

Realization

$

amp() amp() amp+()


Feasible High cost





41619 | 32








$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97

Optimal Power Flow



st 2 3 4 + 5 = 0







Voltage constraints


Observation 1


Can be exploited

algorithmically


SIAM Review 2013]

hellip

41619 | 8

Optimal Power Flow










Voltage constraints

$


$

41619 | 9

Optimal Power Flow



st 5 6() 7() () 8() = 0







Voltage constraints




41619 | 10



41619 | 11





41619 | 12

Optimal Power Flow










Voltage constraints

$


$

41619 | 13

Optimal Power Flow


min sumisin -01 2 + -4 01(2)

st (2) lt(2) 0(2) =(2) = 0







Voltage constraints

$


$

41619 | 14



hellip


lowast -

41619 | 15




lowast$ $


lowast $

41619 | 16

First attempt

Train a neural net

41619 | 17



41619 | 18





41619 | 19





41619 | 20






41619 | 21





41619 | 22





41619 | 23





41619 | 24

Think again


about the solution

41619 | 25

Think again


about the solution


41619 | 26





41619 | 27





Optimal

$energy dispatch

lowast $





41619 | 28





Optimal

$energy dispatch

lowast $



41619 | 29

Ensemble Policy

$

amp() amp() amp+()

Realization






41619 | 30

Ensemble Policy

Realization




$

amp() amp() amp+()

41619 | 31

Ensemble Policy

Realization

$

amp() amp() amp+()


Feasible High cost





41619 | 32








$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97

Optimal Power Flow










Voltage constraints

$


$

41619 | 9

Optimal Power Flow



st 5 6() 7() () 8() = 0







Voltage constraints




41619 | 10



41619 | 11





41619 | 12

Optimal Power Flow










Voltage constraints

$


$

41619 | 13

Optimal Power Flow


min sumisin -01 2 + -4 01(2)

st (2) lt(2) 0(2) =(2) = 0







Voltage constraints

$


$

41619 | 14



hellip


lowast -

41619 | 15




lowast$ $


lowast $

41619 | 16

First attempt

Train a neural net

41619 | 17



41619 | 18





41619 | 19





41619 | 20






41619 | 21





41619 | 22





41619 | 23





41619 | 24

Think again


about the solution

41619 | 25

Think again


about the solution


41619 | 26





41619 | 27





Optimal

$energy dispatch

lowast $





41619 | 28





Optimal

$energy dispatch

lowast $



41619 | 29

Ensemble Policy

$

amp() amp() amp+()

Realization






41619 | 30

Ensemble Policy

Realization




$

amp() amp() amp+()

41619 | 31

Ensemble Policy

Realization

$

amp() amp() amp+()


Feasible High cost





41619 | 32








$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97

Optimal Power Flow



st 5 6() 7() () 8() = 0







Voltage constraints




41619 | 10



41619 | 11





41619 | 12

Optimal Power Flow










Voltage constraints

$


$

41619 | 13

Optimal Power Flow


min sumisin -01 2 + -4 01(2)

st (2) lt(2) 0(2) =(2) = 0







Voltage constraints

$


$

41619 | 14



hellip


lowast -

41619 | 15




lowast$ $


lowast $

41619 | 16

First attempt

Train a neural net

41619 | 17



41619 | 18





41619 | 19





41619 | 20






41619 | 21





41619 | 22





41619 | 23





41619 | 24

Think again


about the solution

41619 | 25

Think again


about the solution


41619 | 26





41619 | 27





Optimal

$energy dispatch

lowast $





41619 | 28





Optimal

$energy dispatch

lowast $



41619 | 29

Ensemble Policy

$

amp() amp() amp+()

Realization






41619 | 30

Ensemble Policy

Realization




$

amp() amp() amp+()

41619 | 31

Ensemble Policy

Realization

$

amp() amp() amp+()


Feasible High cost





41619 | 32








$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97



41619 | 11





41619 | 12

Optimal Power Flow










Voltage constraints

$


$

41619 | 13

Optimal Power Flow


min sumisin -01 2 + -4 01(2)

st (2) lt(2) 0(2) =(2) = 0







Voltage constraints

$


$

41619 | 14



hellip


lowast -

41619 | 15




lowast$ $


lowast $

41619 | 16

First attempt

Train a neural net

41619 | 17



41619 | 18





41619 | 19





41619 | 20






41619 | 21





41619 | 22





41619 | 23





41619 | 24

Think again


about the solution

41619 | 25

Think again


about the solution


41619 | 26





41619 | 27





Optimal

$energy dispatch

lowast $





41619 | 28





Optimal

$energy dispatch

lowast $



41619 | 29

Ensemble Policy

$

amp() amp() amp+()

Realization






41619 | 30

Ensemble Policy

Realization




$

amp() amp() amp+()

41619 | 31

Ensemble Policy

Realization

$

amp() amp() amp+()


Feasible High cost





41619 | 32








$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97





41619 | 12

Optimal Power Flow










Voltage constraints

$


$

41619 | 13

Optimal Power Flow


min sumisin -01 2 + -4 01(2)

st (2) lt(2) 0(2) =(2) = 0







Voltage constraints

$


$

41619 | 14



hellip


lowast -

41619 | 15




lowast$ $


lowast $

41619 | 16

First attempt

Train a neural net

41619 | 17



41619 | 18





41619 | 19





41619 | 20






41619 | 21





41619 | 22





41619 | 23





41619 | 24

Think again


about the solution

41619 | 25

Think again


about the solution


41619 | 26





41619 | 27





Optimal

$energy dispatch

lowast $





41619 | 28





Optimal

$energy dispatch

lowast $



41619 | 29

Ensemble Policy

$

amp() amp() amp+()

Realization






41619 | 30

Ensemble Policy

Realization




$

amp() amp() amp+()

41619 | 31

Ensemble Policy

Realization

$

amp() amp() amp+()


Feasible High cost





41619 | 32








$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97

Optimal Power Flow










Voltage constraints

$


$

41619 | 13

Optimal Power Flow


min sumisin -01 2 + -4 01(2)

st (2) lt(2) 0(2) =(2) = 0







Voltage constraints

$


$

41619 | 14



hellip


lowast -

41619 | 15




lowast$ $


lowast $

41619 | 16

First attempt

Train a neural net

41619 | 17



41619 | 18





41619 | 19





41619 | 20






41619 | 21





41619 | 22





41619 | 23





41619 | 24

Think again


about the solution

41619 | 25

Think again


about the solution


41619 | 26





41619 | 27





Optimal

$energy dispatch

lowast $





41619 | 28





Optimal

$energy dispatch

lowast $



41619 | 29

Ensemble Policy

$

amp() amp() amp+()

Realization






41619 | 30

Ensemble Policy

Realization




$

amp() amp() amp+()

41619 | 31

Ensemble Policy

Realization

$

amp() amp() amp+()


Feasible High cost





41619 | 32








$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97

Optimal Power Flow


min sumisin -01 2 + -4 01(2)

st (2) lt(2) 0(2) =(2) = 0







Voltage constraints

$


$

41619 | 14



hellip


lowast -

41619 | 15




lowast$ $


lowast $

41619 | 16

First attempt

Train a neural net

41619 | 17



41619 | 18





41619 | 19





41619 | 20






41619 | 21





41619 | 22





41619 | 23





41619 | 24

Think again


about the solution

41619 | 25

Think again


about the solution


41619 | 26





41619 | 27





Optimal

$energy dispatch

lowast $





41619 | 28





Optimal

$energy dispatch

lowast $



41619 | 29

Ensemble Policy

$

amp() amp() amp+()

Realization






41619 | 30

Ensemble Policy

Realization




$

amp() amp() amp+()

41619 | 31

Ensemble Policy

Realization

$

amp() amp() amp+()


Feasible High cost





41619 | 32








$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97



hellip


lowast -

41619 | 15




lowast$ $


lowast $

41619 | 16

First attempt

Train a neural net

41619 | 17



41619 | 18





41619 | 19





41619 | 20






41619 | 21





41619 | 22





41619 | 23





41619 | 24

Think again


about the solution

41619 | 25

Think again


about the solution


41619 | 26





41619 | 27





Optimal

$energy dispatch

lowast $





41619 | 28





Optimal

$energy dispatch

lowast $



41619 | 29

Ensemble Policy

$

amp() amp() amp+()

Realization






41619 | 30

Ensemble Policy

Realization




$

amp() amp() amp+()

41619 | 31

Ensemble Policy

Realization

$

amp() amp() amp+()


Feasible High cost





41619 | 32








$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97




lowast$ $


lowast $

41619 | 16

First attempt

Train a neural net

41619 | 17



41619 | 18





41619 | 19





41619 | 20






41619 | 21





41619 | 22





41619 | 23





41619 | 24

Think again


about the solution

41619 | 25

Think again


about the solution


41619 | 26





41619 | 27





Optimal

$energy dispatch

lowast $





41619 | 28





Optimal

$energy dispatch

lowast $



41619 | 29

Ensemble Policy

$

amp() amp() amp+()

Realization






41619 | 30

Ensemble Policy

Realization




$

amp() amp() amp+()

41619 | 31

Ensemble Policy

Realization

$

amp() amp() amp+()


Feasible High cost





41619 | 32








$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97

First attempt

Train a neural net

41619 | 17



41619 | 18





41619 | 19





41619 | 20






41619 | 21





41619 | 22





41619 | 23





41619 | 24

Think again


about the solution

41619 | 25

Think again


about the solution


41619 | 26





41619 | 27





Optimal

$energy dispatch

lowast $





41619 | 28





Optimal

$energy dispatch

lowast $



41619 | 29

Ensemble Policy

$

amp() amp() amp+()

Realization






41619 | 30

Ensemble Policy

Realization




$

amp() amp() amp+()

41619 | 31

Ensemble Policy

Realization

$

amp() amp() amp+()


Feasible High cost





41619 | 32








$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97



41619 | 18





41619 | 19





41619 | 20






41619 | 21





41619 | 22





41619 | 23





41619 | 24

Think again


about the solution

41619 | 25

Think again


about the solution


41619 | 26





41619 | 27





Optimal

$energy dispatch

lowast $





41619 | 28





Optimal

$energy dispatch

lowast $



41619 | 29

Ensemble Policy

$

amp() amp() amp+()

Realization






41619 | 30

Ensemble Policy

Realization




$

amp() amp() amp+()

41619 | 31

Ensemble Policy

Realization

$

amp() amp() amp+()


Feasible High cost





41619 | 32








$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97





41619 | 19





41619 | 20






41619 | 21





41619 | 22





41619 | 23





41619 | 24

Think again


about the solution

41619 | 25

Think again


about the solution


41619 | 26





41619 | 27





Optimal

$energy dispatch

lowast $





41619 | 28





Optimal

$energy dispatch

lowast $



41619 | 29

Ensemble Policy

$

amp() amp() amp+()

Realization






41619 | 30

Ensemble Policy

Realization




$

amp() amp() amp+()

41619 | 31

Ensemble Policy

Realization

$

amp() amp() amp+()


Feasible High cost





41619 | 32








$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97





41619 | 20






41619 | 21





41619 | 22





41619 | 23





41619 | 24

Think again


about the solution

41619 | 25

Think again


about the solution


41619 | 26





41619 | 27





Optimal

$energy dispatch

lowast $





41619 | 28





Optimal

$energy dispatch

lowast $



41619 | 29

Ensemble Policy

$

amp() amp() amp+()

Realization






41619 | 30

Ensemble Policy

Realization




$

amp() amp() amp+()

41619 | 31

Ensemble Policy

Realization

$

amp() amp() amp+()


Feasible High cost





41619 | 32








$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97






41619 | 21





41619 | 22





41619 | 23





41619 | 24

Think again


about the solution

41619 | 25

Think again


about the solution


41619 | 26





41619 | 27





Optimal

$energy dispatch

lowast $





41619 | 28





Optimal

$energy dispatch

lowast $



41619 | 29

Ensemble Policy

$

amp() amp() amp+()

Realization






41619 | 30

Ensemble Policy

Realization




$

amp() amp() amp+()

41619 | 31

Ensemble Policy

Realization

$

amp() amp() amp+()


Feasible High cost





41619 | 32








$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97





41619 | 22





41619 | 23





41619 | 24

Think again


about the solution

41619 | 25

Think again


about the solution


41619 | 26





41619 | 27





Optimal

$energy dispatch

lowast $





41619 | 28





Optimal

$energy dispatch

lowast $



41619 | 29

Ensemble Policy

$

amp() amp() amp+()

Realization






41619 | 30

Ensemble Policy

Realization




$

amp() amp() amp+()

41619 | 31

Ensemble Policy

Realization

$

amp() amp() amp+()


Feasible High cost





41619 | 32








$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97





41619 | 23





41619 | 24

Think again


about the solution

41619 | 25

Think again


about the solution


41619 | 26





41619 | 27





Optimal

$energy dispatch

lowast $





41619 | 28





Optimal

$energy dispatch

lowast $



41619 | 29

Ensemble Policy

$

amp() amp() amp+()

Realization






41619 | 30

Ensemble Policy

Realization




$

amp() amp() amp+()

41619 | 31

Ensemble Policy

Realization

$

amp() amp() amp+()


Feasible High cost





41619 | 32








$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97





41619 | 24

Think again


about the solution

41619 | 25

Think again


about the solution


41619 | 26





41619 | 27





Optimal

$energy dispatch

lowast $





41619 | 28





Optimal

$energy dispatch

lowast $



41619 | 29

Ensemble Policy

$

amp() amp() amp+()

Realization






41619 | 30

Ensemble Policy

Realization




$

amp() amp() amp+()

41619 | 31

Ensemble Policy

Realization

$

amp() amp() amp+()


Feasible High cost





41619 | 32








$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97

Think again


about the solution

41619 | 25

Think again


about the solution


41619 | 26





41619 | 27





Optimal

$energy dispatch

lowast $





41619 | 28





Optimal

$energy dispatch

lowast $



41619 | 29

Ensemble Policy

$

amp() amp() amp+()

Realization






41619 | 30

Ensemble Policy

Realization




$

amp() amp() amp+()

41619 | 31

Ensemble Policy

Realization

$

amp() amp() amp+()


Feasible High cost





41619 | 32








$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97

Think again


about the solution


41619 | 26





41619 | 27





Optimal

$energy dispatch

lowast $





41619 | 28





Optimal

$energy dispatch

lowast $



41619 | 29

Ensemble Policy

$

amp() amp() amp+()

Realization






41619 | 30

Ensemble Policy

Realization




$

amp() amp() amp+()

41619 | 31

Ensemble Policy

Realization

$

amp() amp() amp+()


Feasible High cost





41619 | 32








$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97





41619 | 27





Optimal

$energy dispatch

lowast $





41619 | 28





Optimal

$energy dispatch

lowast $



41619 | 29

Ensemble Policy

$

amp() amp() amp+()

Realization






41619 | 30

Ensemble Policy

Realization




$

amp() amp() amp+()

41619 | 31

Ensemble Policy

Realization

$

amp() amp() amp+()


Feasible High cost





41619 | 32








$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97





Optimal

$energy dispatch

lowast $





41619 | 28





Optimal

$energy dispatch

lowast $



41619 | 29

Ensemble Policy

$

amp() amp() amp+()

Realization






41619 | 30

Ensemble Policy

Realization




$

amp() amp() amp+()

41619 | 31

Ensemble Policy

Realization

$

amp() amp() amp+()


Feasible High cost





41619 | 32








$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97





Optimal

$energy dispatch

lowast $



41619 | 29

Ensemble Policy

$

amp() amp() amp+()

Realization






41619 | 30

Ensemble Policy

Realization




$

amp() amp() amp+()

41619 | 31

Ensemble Policy

Realization

$

amp() amp() amp+()


Feasible High cost





41619 | 32








$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97

Ensemble Policy

$

amp() amp() amp+()

Realization






41619 | 30

Ensemble Policy

Realization




$

amp() amp() amp+()

41619 | 31

Ensemble Policy

Realization

$

amp() amp() amp+()


Feasible High cost





41619 | 32








$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97

Ensemble Policy

Realization




$

amp() amp() amp+()

41619 | 31

Ensemble Policy

Realization

$

amp() amp() amp+()


Feasible High cost





41619 | 32








$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97

Ensemble Policy

Realization

$

amp() amp() amp+()


Feasible High cost





41619 | 32








$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97








$

amp()

Feasible Low cost

41619 | 33


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97


Realization

$

amp()

Feasible Low cost




new realization



41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97


41619 | 35


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 36


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 37


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97


hellip $amp $

$ $amp+

$


$amp() hellip


41619 | 38


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97


hellip $amp $


$


$amp()

$

hellip


41619 | 39


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97


hellip $amp $


$


$amp()

$

hellip


41619 | 40


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97


hellip $amp$$


$


$amp()

$ $

hellip


41619 | 41


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97


hellip




$amp()

$ $

hellip


41619 | 42


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97


$$

$

hellip $amp



$amp()

$ $

hellip


41619 | 43


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97


$$

$

hellip $amp



$amp()

$ $ $

hellip


41619 | 44


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97


hellip $amp $



$amp()

$ $ hellip $

hellip


$ hellip hellip

hellip

41619 | 45


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97


$ helliphellip

hellip $amp

hellip

$



$amp()

$ $ hellip $

hellip





41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97



41619 | 47




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97




41619 | 48





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97





41619 | 49










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97










+ gt= 5 = 1 + =

+lt

41619 | 50






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97






H



3 EFD= = 1

45

+ A 3BC

D


41619 | 51







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97







41619 | 52








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97








41619 | 53








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97








41619 | 54


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97


Realization

$

amp()

Feasible Low cost







41619 | 55



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97



41619 | 56








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97








41619 | 57









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97









41619 | 58




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97




0 10 20 30 40 M=47 W=13rsquo259


41619 | 59


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97


06


03


00069 01

00 0 10 20 30 40 M=47



41619 | 60



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97



010 163 008

006


00099 002

00 0 500 1000 1500 2000 M=2602



41619 | 61




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97




41619 | 62






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97






41619 | 63



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97



41619 | 64



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97



Few active sets

41619 | 65



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97



Few active sets

Terminates fast

41619 | 66



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97



Few active sets

Terminates fast


41619 | 67



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97



Few active sets

Terminates fast



41619 | 68



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97



Few active sets

Terminates fast



41619 | 69



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97



Few active sets

Terminates fast



41619 | 70


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97


41619 | 71





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97





41619 | 72







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97







41619 | 73








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97








41619 | 74

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97

Summary





41619 | 75

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 76

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97

Outlook

Realization

$

amp()

Feasible Low cost

Classification


41619 | 77



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97



41619 | 78





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97





0039 0967

(did not terminate)

41619 | 79




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97




0967 0039



00035


00001

41619 | 80






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97






00035 00001

41619 | 81



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97



41619 | 82


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97


Daniel K Molzahn

Georgia Tech



41619 | 83

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97

Optimal Power Flow

-01 2 + -4 01(2)min sumisin $()

st 8 9(2) (2) 0(2) (2) = 0








41619 | 84



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97



min $ max $

st + $ = 0







Voltage constraints


If max $ le $478

and min $ ge $45


41619 | 85



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97



min $ max $

st + $ = 0







Voltage constraints







min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97



min $ max $

st + $ = 0

36734 le 2 le 2236734 le 2 le 22

$34 le $ le $367





Valid bounds



Challenging




- hellip




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97




41619 | 88

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation




001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 90

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97

001

0203

0405

0607

0809

10

01

02

03

04

05

06

07

08

09 1

085

090

095

100


Substation


50 load variability

085 090 095 10


41619 | 91













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97













41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97





41619 | 93


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97


41619 | 94






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97






41619 | 95

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97

THANK YOU


41619 | 96










1 = HI log 2 + log

1

J minus JI FI

41619 | 97










1 = HI log 2 + log

1

J minus JI FI

41619 | 97

Documents

On active constraints in optimal power flow · 2019-06-06 · On active constraints in optimal power flow Learning optimal solutions and identifying important constraints Line A