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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
2
high resolution water consumption data
interaction with customers for socio-psychographic data gathering
management strategies: dynamic pricingrewards
The project
3
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
3
Sparse optimization approach
Assumptions (appliance level)Piece-wise constant consumption profiles
Finite number of operating modesKnowledge of water consumption at each operating mode
𝑦"(𝑘) = 𝐵((") … 𝐵*"
(")𝜃((")(𝑘)⋮
𝜃*"(")(𝑘)
= 𝐵(")-𝜃(")(𝑘)
𝜃(")(𝑘): unknown, sparse (only one component equal to 1)
4
Sparse optimization approach
Minimizing fitting error (least-squares)
min1 2 3
4 𝑦 𝑘 −4𝐵(")-𝜃(")(𝑘)
𝑦"(𝑘)
6
"7(
89
37(
Not unique solution (solution not reliable)
5
Sparse optimization approach
Adding regularization
min1 2 3
4 𝑦 𝑘 −4𝐵(")-𝜃(")(𝑘)
𝑦"(𝑘)
6
"7(
8
+ 𝛾( 44 𝜃(")(𝑘) <
6
"7(
9
37(
9
37(
Ø l0-norm enforces sparsity in the vector 𝜃(")(𝑘)
Ø balances the tradeoff between fitting and sparsity𝛾(
non-convex optimization problem
𝑠. 𝑡. 𝜃 " 𝑘 ≥ 0, 𝜃(" 𝑘 + …+ 𝜃*"
" 𝑘 = 1
6
Sparse optimization approach
Adding regularization (l1-norm)
min1 2 3
4 𝑦 𝑘 −4𝐵(")-𝜃(")(𝑘)
𝑦"(𝑘)
6
"7(
8
+ 𝛾( 44 𝜃(")(𝑘) (
6
"7(
9
37(
9
37(
Ø replace l0-norm with l1-norm
Ø l1-norm still promotes sparsity
convex optimization problem
𝑠. 𝑡. 𝜃 " 𝑘 ≥ 0, 𝜃(" 𝑘 + …+ 𝜃*"
" 𝑘 = 1
7
Sparse optimization approach
Adding regularization (l1-norm)
min1 2 3
4 𝑦 𝑘 −4𝐵(")-𝜃(")(𝑘)
𝑦"(𝑘)
6
"7(
8
+ 𝛾( 44 𝜔 " (𝑘)⊙ 𝜃(")(𝑘) (
6
"7(
9
37(
9
37(
Ø replace l0-norm with l1-norm
Ø l1-norm still promotes sparsity
convex optimization problem
Ø fixed weights take into time-of-the-day probability 𝜔 " (𝑘)
𝑠. 𝑡. 𝜃 " 𝑘 ≥ 0, 𝜃(" 𝑘 + …+ 𝜃*"
" 𝑘 = 1
8
Sparse optimization approach
Enforce piece-wise constant consumption profiles
min1 2 3
4 𝑦 𝑘 −4𝐵(")-𝜃(")(𝑘)
𝑦"(𝑘)
6
"7(
8
+ 𝛾( 44 𝜔 " (𝑘)⊙ 𝜃(")(𝑘) (
6
"7(
+ 𝛾8 44𝑘"𝜃((") 𝑘 − 𝜃(
(")(𝑘 − 1)⋮
𝜃*"(") 𝑘 − 𝜃*"
(")(𝑘 − 1)F
6
"7(
9
378
9
37(
9
37(
Ø penalize time variation of the vector
Ø only the largest variation is penalized
convex optimization problem
𝜃(")(𝑘)
Ø fixed weights to more penalize rarely time varying appliances𝑘"
𝑠. 𝑡. 𝜃 " 𝑘 ≥ 0, 𝜃(" 𝑘 + …+ 𝜃*"
" 𝑘 = 1
9
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
Ø New convex optimization based algorithm for end-use characterization
Ø Main assumption: piecewise constant consumption profiles (requires high-resolution consumption readings)
Conclusions
Ø Development of final-refinements to deal with low-resolution data
Ø Development of tailored numerical solvers
Future works
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consortium cluster
thank you
http://www.smarth2o-fp7.eu/
@smartH2Oproject #SmartH2O
Andrea [email protected]
Politecnico di MilanoDepartment of Electronics,
Information and Bioengineering
