Message passing for integrating and assessing...
Transcript of Message passing for integrating and assessing...
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
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design
control
stability
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design
control
stability
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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?
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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?
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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.
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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
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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.
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0R
1R
2R
3R
M 1R=
=
=
=
=
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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!
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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
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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
!
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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.
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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)
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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
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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.
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ReferencesL. Zdeborová, A. Decelle, M. Chertkov; Phys. Rev. E 90, 046112 (2009).
L. Zdeborová, S. Backhaus, M. Chertkov; in HICSS 43.
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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
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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.
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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
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