chertkov turbulence kitp

8/13/2019 Chertkov Turbulence KITP

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IntroductionPredicting Failures (Static Overloads) in Power Grids

Control of Reactive Flows in Distribution NetworksAn Optimization Approach to Design of Transmission Grids

Models, Optimization and Control of CollectivePhenomena in Power Grids

Michael (Misha) Chertkov

Center for Nonlinear Studies & Theory Division,Los Alamos National Laboratory

& New Mexico Consortium

KITP/UCSB, Apr 7, 2011

Michael (Misha) Chertkov [email protected] http://cnls.lanl.gov/chertkov/SmarterGrids/
http://find/http://goback/


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Outline

1 IntroductionSo what?Smart Grid Project (LDRD DR) at LANLPreliminary Technical Remarks. Scales.Technical Intro: Power Flows

2 Predicting Failures (Static Overloads) in Power GridsModel of Load SheddingError Surface & InstantonsInstantons for Wind Generation

3 Control of Reactive Flows in Distribution NetworksLosses vs Quality of VoltageControl & Compromises

4 An Optimization Approach to Design of Transmission GridsMotivational Example

Network OptimizationMichael (Misha) Chertkov [email protected] http://cnls.lanl.gov/chertkov/SmarterGrids/


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So what?Smart Grid Project (LDRD DR) at LANLPreliminary Technical Remarks. Scales.Technical Intro: Power Flows

So What? Impact! Savings!

30b$ annually is the cost of power losses10% efficiency improvement - 3b$ savings

cost of 2003 blackout is 7 10b$80b$ is the total cost of blackouts annually in US

further challenges (more vulnerable, cost of not doingplanning, control, mitigation)

Grid is being redesigned[stimulus]

The research is timely:

2T$ in 20 years (at least) in US

Renewables - Desirable but difficult to handle

Integration within itself, but also with Other Infrastructures,e.g. Transportation (Electric Vehicles)

Tons of Interesting (Challenging) ResearchProblems!

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I t d ti S h t?


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Introduction So what?
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Preliminary Remarks

The power grid operates according to the laws of electrodynamics

Transmission Grid (high voltage) vs Distribution Grid (lowvoltage)

Alternating Current (AC) flows ... often considered inlinearized (DC) approximation

No waiting periods power constraints should be satisfiedimmediately. Many Scales.

Loads and Generators are players of two types (distributed

renewable will change the paradigm)At least some generators are adjustable - to guarantee that ateach moment of time the total generation meets the total load

The grid is a graph ... but constraints are (graph-) global




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Many Scales InvolvedPower & Voltage

1KW - typical household;103KW = 1MW- consumption of a medium-to-largeresidential, commercial building;106KW = 1GW-large unit of a Nuclear Powerplant (30GW is the installed wind capacity of Germany =8% of total, US windpenetration is 5%- [30% by 2030?]); 109KW = 1TW - US capacity

Distribution -4 13KV. Transmission -100 1000KV.

Spatial Scales

1mm 103km; US grid = 3 106km lines (operated by 500 companies)

Temporal Scales[control is getting faster]

17ms-AC (60Hz) period, target for Phasor Measurement Units sampling rate(10-30 measurements per second)

1s- electro-mechanical wave [motors induced] propagates 500km

2-10s- SCADA delivers measurements to control units

1 min- loads change (demand response), wind ramps, etc (toughest scale tocontrol)

5-15min- state estimations are made (for markets), voltage collapse

up to hours- maturing of a cascading outage over transmissiongridsMichael (Misha) Chertkov [email protected] http://cnls.lanl.gov/chertkov/SmarterGrids/

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Basic AC Power Flow Equations (Static)The Kirchhoff Laws(linear)

a G0 :

baJab= Ja for currents(a, b) G1 : Jabzab=Va Vb for potentials

(a, b) G1 : Ja =

bG0YabVb

Y = (Yab|a, b G0), {a, b}: Yab=

0, a=b, a byab, a=b, a b

cac=a yac, a= b.{a, b}: yab=gab+iab= (zab)1, zab=rab+xab

Complex Power Flows [balance of power, nonlinear]

a G0 : Pa =pa+ iqa =VaJa =Va

baJab

= Va

baVa V

bz

ab

= baexp(2a)exp(a+b+iaib)

zab

Flows on graphs, but very different from transportation networksNonlinear in terms of Real and Reactive powersReactive Power needs to be injected to maintain reasonably stable voltageQuasi-static (transients may be relevant on the scale of seconds and less)Different (injection/consumption/control) conditions on generators (p, V) andloads (p, q)(, ) are conjugated (Lagrangian multipliers) to (p, q),energylandscape


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So hat?Smart Grid Project (LDRD DR) at LANLPreliminary Technical Remarks. Scales.Technical Intro: Power Flows

Energy Functional Landscape (Static)

Transmission Networks(resistance is much smaller than inductance, rab xab)Q(,) =

{a,b}G1

exp(2a)+exp(2b)2 exp(a+b) cos(ab)2xab

aG0

apa

aGloads

aqa

Single Load (p1, q1)and Slack Bus (0 =0 = 0)

Q= 1+exp(21 )2 exp(1) cos(1 )2x

1p1 1q1


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Predicting Failures (Static Overloads) in Power GridsControl of Reactive Flows in Distribution Networks

An Optimization Approach to Design of Transmission Grids

Smart Grid Project (LDRD DR) at LANLPreliminary Technical Remarks. Scales.Technical Intro: Power Flows

DC [linearized] approximation (for AC power flows)

(0) The amplitude of the complex potentials are all fixed to the same number(unity, after trivial re-scaling): a: a = 0.

(1) {a, b}: |a b| 1 - phase variation between any two neighbors on thegraph is small

(2) {a, b}: rabxab - resistive (real) part of the impedance is much smallerthan its reactive (imaginary) part. Typical values for the r/x is in the1/27 1/2 range.

It leads to

Linearized relation between powers and phases (at the nodes):

a G0 : pa =

ba

ab

xab

Losses of real power are zero in the network (in the leading order)

apa = 0

Reactive power needs to be injected (lines are inductances - only consumereactive power=accumulate magnetic energy per cycle)


IntroductionModel of Load Shedding
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Model of Load SheddingError Surface & InstantonsInstantons for Wind Generation

Our Publications onGrid Stability

21. M. Chertkov, M. Stepanov, F. Pan, and R. Baldick , Exact and EfficientAlgorithm to Discover Stochastic Contingencies in Wind Generation overTransmission Power Grids, invited session on Smart Grid Integration ofRenewable Energy: Failure analysis, Microgrids, and Estimation at CDC/ECC2011.

16. P. van Hentenryck, C. Coffrin, and R. Bent , Vehicle Routing for the LastMile of Power System Restoration, submitted to PSCC.

15. R. Pfitzner, K. Turitsyn, and M. Chertkov , Statistical Classification ofCascading Failures in Power Grids , arxiv:1012.0815, accepted for IEEE PES2011.

14. S. Kadloor and N. Santhi , Understanding Cascading Failures in Power Grids, arxiv:1011.4098 submitted to IEEE Transactions on Smart Grids.

13. N. Santhi and F. Pan , Detecting and mitigating abnormal events in largescale networks: budget constrained placement on smart grids , proceedings ofHICSS44, Jan 2011.

8. M. Chertkov, F. Pan and M. Stepanov, Predicting Failures in Power Grids,arXiv:1006.0671, IEEE Transactions on Smart Grids 2, 150 (2010).


Introduction Model of Load Shedding


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Outline

1

IntroductionSo what?Smart Grid Project (LDRD DR) at LANLPreliminary Technical Remarks. Scales.Technical Intro: Power Flows


3 Control of Reactive Flows in Distribution Networks

Losses vs Quality of VoltageControl & Compromises

4 An Optimization Approach to Design of Transmission GridsMotivational Example

Network OptimizationMichael (Misha) Chertkov [email protected] http://cnls.lanl.gov/chertkov/SmarterGrids/

IntroductionP di i F il (S i O l d ) i P G id

Model of Load Shedding
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MC, F. Pan (LANL) and M. Stepanov (UA Tucson)

Predicting Failures in Power Grids:The Case of Static Overloads, IEEETransactions on Smart Grids2, 150(2010).

MC, FP, MS & R. Baldick (UT Austin)

Exact and Efficient Algorithm toDiscover Extreme StochasticEvents in Wind Generation overTransmission Power Grids, invitedsession on Smart Grid Integrationof Renewable Energy at CDC/ECC

2011.Michael (Misha) Chertkov [email protected] http://cnls.lanl.gov/chertkov/SmarterGrids/

IntroductionP di ti F il (St ti O l d ) i P G id



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S gError Surface & InstantonsInstantons for Wind Generation

Normally the grid is ok (SAT) ... but sometimes failures

(UNSAT) happensHow to estimate a probability of a failure?

How to predict (anticipate and hopefully) prevent the systemfrom going towards a failure?

Phase space of possibilities is huge (finding the needle in thehaystack)



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gError Surface & InstantonsInstantons for Wind Generation

Model of Load Shedding [MC, F.Pan & M.Stepanov 10]

Minimize Load Shedding = Linear Programming for DC

LPDC(d|G; x; u; P) = minf,,p,s

aGd

sa

COND(f,,p,d,s|G;x;u;P)

COND=CONDflow CONDDC CONDedge CONDpower CONDover

CONDflow =

a:

ba

fab=

pa, a Gp

da+ sa, a Gd0, a G0\ (Gp Gd)

CONDDC=

{a, b}: a b+xabfab= 0

, CONDedge=

{a, b}: uab fab uab

CONDpower=

a: 0 pa Pa

, CONDover=

a: 0 sa da

-phases; f -power flows through edges; x - inductances of edges



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Error Surface & InstantonsInstantons for Wind Generation

Instanton Search Algorithm[Sampling]

Borrowed (with modifications) from Error-Correction studies:analysis of error-floor [MC, M.Stepanov, et al 04-10]

ConstructQ(d) =

P(d), LPDC(d)> 00 , LPDC(d) = 0Generate a simplex (N+1points) of UNSAT points

Use Amoeba-Simplex

[Numerical Recepies] tomaximizeQ(d)Repeat multiple times(sampling the space ofinstantons)

Point at the Error Surfaceclosest to normal operational point

normal operational point

demand1

demand2

demand...

Error Surface

load sheddingload sheddingload shedding

no load sheddingno load sheddingno load shedding



Model of Load SheddingE S f & I
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Example of Guam [MC, F.Pan & M.Stepanov 10]

The instantons are sparse (localized ontroubled nodes)

The troubled nodes are repetitive inmultiple-instantons

Instanton structure is not sensitive tosmall changes in D and statistics ofdemands



Model of Load SheddingE S f & I t t
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g ( )Control of Reactive Flows in Distribution Networks



Example of IEEE RTS96 system [MC, F.Pan & M.Stepanov 10]

The instantons are well localized (but stillnot sparse)

The troubled nodes and structures arerepetitive in multiple-instantons

Instanton structure is not sensitive tosmall changes in D and statistics ofdemands



Model of Load SheddingError Surface & Instantons
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g ( )Control of Reactive Flows in Distribution Networks



Triangular Example (illustrating a paradox)

lowering demand may betroublesome [SAT UNSAT]develops when a cycle contains a

weak linksimilar observation was made inother contexts before, e.g. by S.Oren and co-authors

the problem is typical in realexamples

consider fixing it with extrastorage [future project]



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Instantons for Wind Generation

SettingRenewables is the source of fluctuations

Loads are fixed (5 min scale)

Standard generation is adjusted according to a droop control

(low-parametric, linear)

Results

The instanton algorithm discovers most probable extremestatistics events

The algorithm is EXACT and EFFICIENT (polynomial)

Illustrate utility and performance on IEEE RTS-96 exampleextended with additions of 10%, 20% and 30% of renewablegeneration.



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Simulations: IEEE RTS-96 + renewables

10% of penetration -localization, longcorrelations

20% of penetration -worst damage, leadinginstanton is delocalized

Instanton1

Instanton2

Instanton3

30% of penetration -spreading anddiversifying decreasesthe damage, instantonsare localized

Instanton1

Instanton2

Instanton3



C l f R i Fl i Di ib i N k

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Path Forward (for predicting failures)

Path Forward

Many large-scale practical tests, e.g. ERCOT wind integration

The instanton-amoeba allows upgrade to other (than LPDC)

network stability testers, e.g. for AC flows and transients

Instanton-search can be accelerated, utilizing LP-structure of thetester (exact & efficient for example of renewables)

This is an important first step towards exploration of next level

problems in power grid, e.g. on interdiction [Bienstock et. al 09],optimal switching [Oren et al 08], cascading outages/extremes[Dobson et al 06], and control of the outages [Ilic et al 05,Bienstock 11]



C t l f R ti Fl i Di t ib ti N t kLosses vs Quality of VoltageC t l & C i
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Control & Compromises

Our Publications onGrid Control

20. K. Turitsyn, S. Backhaus, M. Ananyev and M. Chertkov , Smart Finite State Devices: A ModelingFramework for Demand Response Technologies, invited session on Demand Response at CDC/ECC 2011.

19. S. Kundu, N. Sinitsyn, S. Backhaus, and I. Hiskens, Modeling and control of thermostaticallycontrolled loads, submitted to 17th Power Systems Computation Conference 2011, arXiv:1101.2157.

16. P. van Hentenryck, C. Coffrin, and R. Bent , Vehicle Routing for the Last Mile of Power SystemRestoration, submitted to PSCC.

12. P. Sulc, K. Turitsyn, S. Backhaus and M. Chertkov , Options for Control of Reactive Power byDistributed Photovoltaic Generators, arXiv:1008.0878, to appear in Proceedings of the IEEE, special issue

on Smart Grid (2011).11. F. Pan, R. Bent, A. Berscheid, and D. Izrealevitz , Locating PHEV Exchange Stations in V2G,arXiv:1006.0473, IEEE SmartGridComm 2010

10. K. S. Turitsyn, N. Sinitsyn, S. Backhaus, and M. Chertkov, Robust Broadcast-Communication Controlof Electric Vehicle Charging, arXiv:1006.0165, IEEE SmartGridComm 2010

9. K. S. Turitsyn, P. Sulc, S. Backhaus, and M. Chertkov, Local Control of Reactive Power by DistributedPhotovoltaic Generators, arXiv:1006.0160, IEEE SmartGridComm 2010

7. K. S. Turitsyn, Statistics of voltage drop in radial distribution circuits: a dynamic programming

approach, arXiv:1006.0158, accepted to IEEE SIBIRCON 20105. K. Turitsyn, P. Sulc, S. Backhaus and M. Chertkov, Distributed control of reactive power flow in aradial distribution circuit with high photovoltaic penetration, arxiv:0912.3281 , selected for super-session atIEEE PES General Meeting 2010.

2. L. Zdeborova, S. Backhaus and M. Chertkov, Message Passing for Integrating and Assessing RenewableGeneration in a Redundant Power Grid, presented at HICSS-43, Jan. 2010, arXiv:0909.2358

1. L. Zdeborova, A. Decelle and M. Chertkov, Message Passing for Optimization and Control of PowerGrid: Toy Model of Distribution with Ancillary Lines, arXiv:0904.0477, Phys. Rev. E80, 046112(2009)



Control of Reactive Flows in Distribution NetworksLosses vs Quality of VoltageControl & Compromises
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Outline

1 IntroductionSo what?Smart Grid Project (LDRD DR) at LANLPreliminary Technical Remarks. Scales.Technical Intro: Power Flows




4 An Optimization Approach to Design of Transmission GridsMotivational ExampleNetwork Optimization



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K. Turitsyn (MIT), P. Sulc (NMC), S. Backhaus and M.C.

Optimization of Reactive Power by Distributed PhotovoltaicGenerators, to appear in Proceedings of the IEEE, special issueon Smart Grid (2011), http://arxiv.org/abs/1008.0878

Local Control of Reactive Power by Distributed Photovoltaic

Generators, proceedings of IEEE SmartGridComm 2010,http://arxiv.org/abs/1006.0160

Distributed control of reactive power flow in a radial

distribution circuit with high photovoltaic penetration, IEEEPES General Meeting 2010 (invited to a super-session),

http://arxiv.org/abs/0912.3281



http://arxiv.org/abs/1008.0878http://arxiv.org/abs/1006.0160http://arxiv.org/abs/0912.3281http://arxiv.org/abs/0912.3281http://arxiv.org/abs/1006.0160http://arxiv.org/abs/1008.0878http://find/


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Setting & Question & Idea

Distribution Grid (old rules, e.g.voltage is controlled only at thepoint of entrance)

Significant Penetration of

Photovoltaic (new reality)How to controlswinging/fluctuating voltage(reactive power)?

Idea(s)

Use Inverters.

Control Locally.Michael (Misha) Chertkov [email protected] http://cnls.lanl.gov/chertkov/SmarterGrids/


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An Optimization Approach to Design of Transmission Gridsp





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An Optimization Approach to Design of Transmission Gridsp





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Control of Reactive Flows in Distribution NetworksO f G

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IntroductionPredicting Failures (Static Overloads) in Power GridsControl of Reactive Flows in Distribution Networks

A O ti i ti A h t D i f T i i G id

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An Optimi ation Approach to Design of Transmission Grids

Motivational ExampleNetwork Optimization
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Our Publications onGrid Planning

18. R. Bent, A. Berscheid, and L. Toole , Generation and TransmissionExpansion Planning for Renewable Energy Integration, submitted to PowerSystems Computation Conference (PSCC).

17. R. Bent and W.B. Daniel , Randomized Discrepancy Bounded Local Searchfor Transmission Expansion Planning, accepted for IEEE PES 2011.

11. F. Pan, R. Bent, A. Berscheid, and D. Izrealevitz , Locating PHEV

Exchange Stations in V2G, arXiv:1006.0473, IEEE SmartGridComm 20106. J. Johnson and M. Chertkov, A Majorization-Minimization Approach toDesign of Power Transmission Networks, arXiv:1004.2285, 49th IEEEConference on Decision and Control (2010).

4. R. Bent, A. Berscheid, and G. Loren Toole, Transmission Network ExpansionPlanning with Simulation Optimization, Proceedings of the Twenty-Fourth AAAI

Conference on Artificial Intelligence (AAAI 2010), July 2010, Atlanta, Georgia.3. L. Toole, M. Fair, A. Berscheid, and R. Bent, Electric Power TransmissionNetwork Design for Wind Generation in the Western United States: Algorithms,Methodology, and Analysis , Proceedings of the 2010 IEEE Power EngineeringSociety Transmission and Distribution Conference and Exposition (IEEE TD2010), April 2010, New Orleans, Louisiana.






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Outline

1 Introduction





4 An Optimization Approach to Design of Transmission GridsMotivational ExampleNetwork Optimization




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Grid Design: Motivational Example

Cost dispatch only(transportation,economics)

Power flows highly approximate

Unstable solutions

Intermittency in Renewables not

accounted

An unstable grid example

Hybrid Optimization - is currentengineering solution developed atLANL: Toole,Fair,Berscheid,Bent 09extending and built on NREL 20% by2030 report for DOE

Network Optimization

Design of the Grid as a tractableglobal optimization




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Network Optimization (for fixed production/consumption p)

ming

p+ G(g)1 p minimize losses

convex overg

, Gab= 0, a

=b, a b

gab, a =b, a bcac=agac, a=b.

Discrete Graph Laplacian of conductance

Network Optimization (averaged over p)

mingp+

G(g)1

p = mingtr

G(g)1

pp+

=

ming

trG(g)1 P still convex

, P

covariance matrix of load/generation

Boyd,Ghosh,Saberi 06in the context of resistive networksalsoBoyd, Vandenberghe, El Gamal and S. Yun 01forIntegratedCircuits




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Network Optimization: Losses+Costs [J. Johnson, MC 10]

Costs need to account for

sizing lines - grows with gab, linearly or faster (convex in g)

breaking ground - l0-norm (non convex in g) but also imposesdesiredsparsity

Resulting Optimization is non-convex

ming>0

tr

G(g)

1P

+{a,b}

(abgab+ab(gab))

, (x) =

xx+

Tricks (for efficient solution of the non-convex problem)

annealing: start from large (convex) and track to 0(combinatorial)

Majorization-minimization (from Candes, Boyd 05) for current :

gt+1 = argming>0 tr(L) + .

g+.

(g

tab).

gab




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p pp g

Single-Generator Examples (I)




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Single-Generator Examples (II)




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Multi-Generator Example




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Adding Robustness

To impose the requirement that the network design should berobust to failures of lines or generators, we use the worst-casepower dissipation:

L\k

(g) = max{a,b}:zab{0,1}|

{a,b}zab=NkL(z. g))

It is tractable to compute only for small values ofk.

Note, the point-wise maximum over a collection of convex

function is convex.So the linearized problem is again a convex optimizationproblem at every step continuation/MM procedure.




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Single-Generator Examples [+Robustness] (I)




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Single-Generator Examples [+Robustness] (II)




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Multi-Generator Example [+Robustness]




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Conclusion (for the Network Optimization part)

A promising heuristic approachto design of power transmissionnetworks. However, cannot guarantee global optimum.

CDC10: http://arxiv.org/abs/1004.2285

Future Work:Applications to real grids, e.g. for 30/2030

Bounding optimality gap?

Use non-convex continuation approach to place generators

possibly useful for graph partitioning problems

adding further constraints (e.g. dont overload lines)

extension to (exact) AC power flow?

http://arxiv.org/abs/1004.2285http://arxiv.org/abs/1004.2285http://find/


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Bottom Line

A lot of interestingcollective phenomenain the power grid settings for Applied

Math, Physics, CS/IT analysisThe research is timely (blackouts, renewables, stimulus)

Other Problems the team plans working on

Efficient PHEV charging via queuing/scheduling with and withoutcommunications and delays

Power Grid Spectroscopy (power grid as a medium, electro-mechanical wavesand their control, voltage collapse, dynamical state estimations)

Effects of Renewables (intermittency of winds, clouds) on the grid & control

Load Control, scheduling with time horizon (dynamic programming +)

Price Dynamics & Control for the Distribution Power Grid

Post-emergency Control (restoration and de-islanding)

For more info - check:

http://cnls.lanl.gov/~chertkov/SmarterGrids/

https://sites.google.com/site/mchertkov/projects/smart-grid

http://cnls.lanl.gov/~chertkov/SmarterGrids/http://cnls.lanl.gov/~chertkov/SmarterGrids/http://cnls.lanl.gov/~chertkov/SmarterGrids/https://sites.google.com/site/mchertkov/projects/smart-gridhttps://sites.google.com/site/mchertkov/projects/smart-gridhttp://cnls.lanl.gov/~chertkov/SmarterGrids/http://find/


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Statistical Classification of Cascading Failures Algorithm of the Cascade

Phase Diagram of Cascades


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Outline

5 Statistical Classification of Cascading FailuresAlgorithm of the CascadePhase Diagram of Cascades





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Rene Pfitzner (NMC), Konstantin Turitsyn (MIT) & MC

Statistical Classification of Cascading Failures in Power Grids,accepted to IEEE PES 2011,http://arxiv.org/abs/1012.0815



http://arxiv.org/abs/1012.0815http://arxiv.org/abs/1012.0815http://find/


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Objectives:

Have a realisticmicroscopicmodel of a cascade [not (!!) a

disease-spread like phenomenological model]Resolvediscrete eventsdynamics (lines tripping, overloads,islanding) explicitly

Address (first) thecurrent realityof the transmission grid

operation, e.g. automatic control on the sub-minute scaleConsider (first)fluctuations in demandas a source of cascadein the overloaded (modern) grid

Analyze the results, e.g. in terms of phases observed, onavailable power grid models [IEEE test beds]

Building on

I. Dobson, B. Carreras, V. Lynch, and D. Newman, Aninitialmodel for complex dynamics in electric power system

blackouts, HICSS-34, 2001Michael (Misha) Chertkov [email protected] http://cnls.lanl.gov/chertkov/SmarterGrids/



Al i h f h C d
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Algorithm of the Cascade

Optimum Power Flowfinds (cost)optimal distribution of generation(decided once for 15 min - in betweenstate estimations)

DC power flow is our (simplest) choice

Droop Control= equivalent (pre set for

15 min) response of all the generators tochange in loads

Identify islandswith a proper connectedcomponent algorithm(s)

Discrete time Evolution of Loads= (a)generate configuration of demand from

given distribution (our enabling example= Gaussian, White); (b) assume that theconfiguration grow from the typical one(center of the distribution) in continuoustime, t [0; 1]; (c) project next discreteevent (failure of a line or saturation of agenerator)andjumpthere

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Statistical Classification of Cascading Failures Algorithm of the CascadePhase Diagram of Cascades

General Conclusions (3 phases)


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General Conclusions (3 phases)

Phase #0 The grid is resilient against fluctuationsin demand.

Phase #1 shows tripping of demands due totripping of overloaded lines. This has aoverall de-stressing effect on the grid.

Phase #2 Generator nodes start to become tripped,

mainly due to islanding of individualgenerators. With the early tripping ofgenerators the system becomes stressedand cascade evolves much faster (withincrease in the level of demandfluctuations) when compared with arelatively modest increase observed in

Phase #1.

Phase #3 Significant outages are observed. Theyare associated with removal from the gridof complex islands, containing bothgenerators and demands.


Statistical Classification of Cascading Failures Algorithm of the CascadePhase Diagram of Cascades

Path Forward (Cascades)
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Path Forward (Cascades)

From DC solver to AC solver

Mixed models - combining fluctuations in demands andincidental line tripping

More detailed study of effect of capacity inhomogeneity (e.g.on islanding)

Towards validated (derived from micro-) phenomenologicalmodel and theory of cascades [power tails, scaling, dynamic

mechanisms]

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chertkov turbulence kitp

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