Message passing for integrating and assessing renewable generation

in a redundant power grid

Lenka Zdeborová (Los Alamos Natl. Lab.)

in collaboration with Scott Backhaus (LANL)Misha Chertkov (LANL)

Saturday, January 30, 2010

LDRD project 2010 - 2012

http://cnls.lanl.gov/~chertkov/SmarterGrids/ (or google “Chertkov” and follow “smart grid” link)

Information science foundations for the Smart Grid

Saturday, January 30, 2010

Saturday, January 30, 2010

design

control

stability

Saturday, January 30, 2010

design

control

stability

Saturday, January 30, 2010

Assessing renewable generation

Intermittent renewable-sources-based generation destabilizes the grid. How to improve grid control schemes?

If renewable sources produce power x, how much can be saved on the level of the firm generation?

Saturday, January 30, 2010

Improvement trough redundancy

Build additional power lines and introduce switches (on / off = power line connected to / disconnected from the network)

Redundancy must help to optimize both stability and efficiency - larger space to optimize over. But how much does redundancy help?

Saturday, January 30, 2010

Methodology:Approach A: Take a realistic power grid model and several computers and run simulations. Do again when details change ...

Approach B (probabilistic + physicist way): Study behavior of simple abstract models that facilitate the analysis, and look for universal properties, dependencies and behavior. Model choice criteria (in physics): The simpler and richer the better.

Saturday, January 30, 2010

Our power grid modelM producers, N=DM consumers

Out of every D consumers R have auxiliary lines

M=4, N=12, D=3

Consumer “i” consumes xi

ziproduces

yaProducer “a” capability

Saturday, January 30, 2010

SettingSwitch variables for power lines:

!ia = 0/!ia = 1

Constraints!

a!!i

!ia = 1

!

i!!a

!ia(xi ! zi) " ya

Every consumer one connection

Producers not overloaded

Each consumer has exactly one line on.

Saturday, January 30, 2010

0R

1R

2R

3R

M 1R=

=

=

=

=

Saturday, January 30, 2010

0R

1R

2R

3R

M 1R=

=

=

=

=

Note that the final topology is a tree, hence the Kirchhoff’s laws satisfied.

However, general power flow optimum cannot be worse than the tree case!

Saturday, January 30, 2010

QuestionsGiven can all the constraints be simultaneously satisfied? (Nobody overloaded.)

If yes, then how many satisfying configurations of the switches are there? Is it easy to find one?

{xi}, {zi}, {ya}

Answer: via Belief Propagation

Saturday, January 30, 2010

How does BP work?

!a!i!ia

!i!a!ia

Prob. that line “ia” is in state conditioned !ia

constraint on “i” is missing

constraint on “a” is missing

Iterative “message passing” scheme

!

Saturday, January 30, 2010

Belief PropagationDistributed approximative way of:

(a) computing the probability that a given switch is on or off.

(b) estimating number of valid (not overloading) configurations.

For large number of customers and producers (thermodynamic limit) - average analysis solvable.

Saturday, January 30, 2010

Fraction 1/3 of consumers produce amount zEvery consumer consumes random number in (0.9,1.1)

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2

firm generation per customer, y/D

renewable generation capability, z

D=3, x=1, !=0.2, f=1/3

R=0,R=1R=2R=3

Example n. 1# producers

M !"

# consumersN = 3M

y/3 = 1! z/3

generation > consumption

y/3 = 1.1

somebody must serve D-R+1 fully demanding consumers

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Example n. 2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1firm generation per customer, y/D

avg. renewables per customer, z/2

D=3, x=1, !=0.2

R=0, R=1R=2R=3

Every consumer produced a random number between (0,z)

Saturday, January 30, 2010

Example n. 3

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

firm generation per customer, y/D

coverage of renewables, f

D=3, R=3, x=1, !=0.2

z=0.5

z=1.0

z=2.0

amount z is produced by fraction f of consumers

produced > consumed

y/3 > 1! fz

Saturday, January 30, 2010

Conclusions and Perspectives

Existence of SAT/UNSAT phase transition and regimes where higher penetration useful or futile.

Redundancy + switches help renewable integrations. Belief propagation a tool of analysis but also distributed control algorithm.

In physics: Study of toy models (and phase transitions) leads to qualitative understanding. Is that true also for the Smart Grid?

Combine belief propagation with DC or AC power flow rules on a non-tree topology.

Saturday, January 30, 2010

ReferencesL. Zdeborová, A. Decelle, M. Chertkov; Phys. Rev. E 90, 046112 (2009).

L. Zdeborová, S. Backhaus, M. Chertkov; in HICSS 43.

Saturday, January 30, 2010

Belief Propagation Equations

!i!a1 =

1Zi!a

!

b"!i\a

"b!i0

!i!a0 =

1Zi!a

"

b"!i\a

"b!i1

!

c"!i\a,b

"c!i0

"a!i1 =

1Za!i

"

"!a\ia

#(ya ! wi !"

j"!a\i

$jawj)!

j"!a\i

!j!a"ja

"a!i0 =

1Za!i

"

"!a\ia

#(ya !"

j"!a\i

$jawj)!

j"!a\i

!j!a"ja

Saturday, January 30, 2010

Example n. 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.05 0.1 0.15 0.2 0.25 0.3

distribution width

mean

D=4, !=0.1

R=4

R=3

R=2

R=1

R=0

ya = 1 !a

zi = 0 !i

Everybody consuming random number between (mean-width/2) and (mean+width/2), epsilon - fraction of consumers with no demand.

Saturday, January 30, 2010

Asymptotic, but also algorithmic solution

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

distribution width

mean

D=3, R=2, !=0

BP theoryWalkGrid

Saturday, January 30, 2010