Post on 13-Aug-2015
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A convex optimization approach for automated water and energy end use disaggregation
Dario Piga, Andrea Cominola, Matteo Giuliani, Andrea Castelletti, Andrea Emilio Rizzoli
The project
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high resolution water consumption data
interaction with customers for socio-psychographic data gathering
management strategies: dynamic pricingrewards
The project
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SMART METERS
USER MODEL
WDMScustomized feedbacks
dynamic pricing
ToiletShower
DishwasherWashing machine
GardenSwimming pool
GAMIFICATION | ONLINE BILL GAMIFICATION | ONLINE BILL
Water consumption disaggregation into end uses
ToiletShower
DishwasherWashing machine
GardenSwimming pool
ONE MEASURE MANY END USES
Need for fully automated disaggregation algorithms
overlapping, simultaneous water end uses
human-dependent vs
automatic fixtures
Personalized hints for reducing water/energy consumptionInformation on potential saving in deferring to peak-off hours
Leak detection Customized WDMS
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Sparse optimization approach
Assumptions (appliance level)Piece-wise constant consumption profiles
Finite number of operating modesKnowledge of water consumption at each operating mode
๐ฆ"(๐) = ๐ต((") โฆ ๐ต*"
(")๐((")(๐)โฎ
๐*"(")(๐)
= ๐ต(")-๐(")(๐)
๐(")(๐): unknown, sparse (only one component equal to 1)
4
Sparse optimization approach
Minimizing fitting error (least-squares)
min1 2 3
4 ๐ฆ ๐ โ4๐ต(")-๐(")(๐)
๐ฆ"(๐)
6
"7(
89
37(
Not unique solution (solution not reliable)
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Sparse optimization approach
Adding regularization
min1 2 3
4 ๐ฆ ๐ โ4๐ต(")-๐(")(๐)
๐ฆ"(๐)
6
"7(
8
+ ๐พ( 44 ๐(")(๐) <
6
"7(
9
37(
9
37(
ร l0-norm enforces sparsity in the vector ๐(")(๐)
ร balances the tradeoff between fitting and sparsity๐พ(
non-convex optimization problem
๐ . ๐ก. ๐ " ๐ โฅ 0, ๐(" ๐ + โฆ+ ๐*"
" ๐ = 1
6
Sparse optimization approach
Adding regularization (l1-norm)
min1 2 3
4 ๐ฆ ๐ โ4๐ต(")-๐(")(๐)
๐ฆ"(๐)
6
"7(
8
+ ๐พ( 44 ๐(")(๐) (
6
"7(
9
37(
9
37(
ร replace l0-norm with l1-norm
ร l1-norm still promotes sparsity
convex optimization problem
๐ . ๐ก. ๐ " ๐ โฅ 0, ๐(" ๐ + โฆ+ ๐*"
" ๐ = 1
7
Sparse optimization approach
Adding regularization (l1-norm)
min1 2 3
4 ๐ฆ ๐ โ4๐ต(")-๐(")(๐)
๐ฆ"(๐)
6
"7(
8
+ ๐พ( 44 ๐ " (๐)โ ๐(")(๐) (
6
"7(
9
37(
9
37(
ร replace l0-norm with l1-norm
ร l1-norm still promotes sparsity
convex optimization problem
ร fixed weights take into time-of-the-day probability ๐ " (๐)
๐ . ๐ก. ๐ " ๐ โฅ 0, ๐(" ๐ + โฆ+ ๐*"
" ๐ = 1
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Sparse optimization approach
Enforce piece-wise constant consumption profiles
min1 2 3
4 ๐ฆ ๐ โ4๐ต(")-๐(")(๐)
๐ฆ"(๐)
6
"7(
8
+ ๐พ( 44 ๐ " (๐)โ ๐(")(๐) (
6
"7(
+ ๐พ8 44๐"๐((") ๐ โ ๐(
(")(๐ โ 1)โฎ
๐*"(") ๐ โ ๐*"
(")(๐ โ 1)F
6
"7(
9
378
9
37(
9
37(
ร penalize time variation of the vector
ร only the largest variation is penalized
convex optimization problem
๐(")(๐)
ร fixed weights to more penalize rarely time varying appliances๐"
๐ . ๐ก. ๐ " ๐ โฅ 0, ๐(" ๐ + โฆ+ ๐*"
" ๐ = 1
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Tests on high-resolution electricity data
AMPds dataset: S. Makonin et al., AMPDs: a public dataset for load disaggregation and eco-feedback research, In Electrical Power and Energy Conference, 2013.
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Tests on water data
WEEP dataset: Heinrich, Water End Use and Efficiency Project, New Zealand, 2007
31%
37%
32%
SPARSE OPTIMIZATION
34%
36%
30%
ACTUAL
Toilet
Tap
Shower
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Conclusions and follow up
ร New convex optimization based algorithm for end-use characterization
ร Main assumption: piecewise constant consumption profiles (requires high-resolution consumption readings)
Conclusions
ร Development of final-refinements to deal with low-resolution data
ร Development of tailored numerical solvers
Future works
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