A convex optimization approach for automated water and energy end use disaggregation

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.

10

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