A convex optimization approach for automated water and energy end use disaggregation
- 1. A convex optimization approach for automated water and energy end use disaggregation Dario Piga, Andrea Cominola, Matteo Giuliani, Andrea Castelletti, Andrea Emilio Rizzoli
- 2. The project 2 high resolution water consumption data interaction with customers for socio-psychographic data gathering management strategies: dynamic pricing rewards
- 3. The project 3 SMART METERS USER MODEL WDMS customized feedbacks dynamic pricing Toilet Shower Dishwasher Washing machine Garden Swimming pool GAMIFICATION | ONLINE BILL GAMIFICATION | ONLINE BILL
- 4. Water consumption disaggregation into end uses Toilet Shower Dishwasher Washing machine Garden Swimming 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 consumption Information on potential saving in deferring to peak-off hours Leak detection Customized WDMS 3
- 5. Sparse optimization approach Assumptions (appliance level) Piece-wise constant consumption profiles Finite number of operating modes Knowledge 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
- 6. Sparse optimization approach Minimizing fitting error (least-squares) min 1 2 3 4 𝑦 𝑘 − 4 𝐵(")- 𝜃(") (𝑘) 𝑦"(𝑘) 6 "7( 89 37( Not unique solution (solution not reliable) 5
- 7. Sparse optimization approach Adding regularization min 1 2 3 4 𝑦 𝑘 − 4 𝐵(")- 𝜃(") (𝑘) 𝑦"(𝑘) 6 "7( 8 + 𝛾( 4 4 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
- 8. Sparse optimization approach Adding regularization (l1-norm) min 1 2 3 4 𝑦 𝑘 − 4 𝐵(")- 𝜃(") (𝑘) 𝑦"(𝑘) 6 "7( 8 + 𝛾( 4 4 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
- 9. Sparse optimization approach Adding regularization (l1-norm) min 1 2 3 4 𝑦 𝑘 − 4 𝐵(")- 𝜃(") (𝑘) 𝑦"(𝑘) 6 "7( 8 + 𝛾( 4 4 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 probability𝜔 " (𝑘) 𝑠. 𝑡. 𝜃 " 𝑘 ≥ 0, 𝜃( " 𝑘 + …+ 𝜃*" " 𝑘 = 1 8
- 10. Sparse optimization approach Enforce piece-wise constant consumption profiles min 1 2 3 4 𝑦 𝑘 − 4 𝐵(")- 𝜃(") (𝑘) 𝑦"(𝑘) 6 "7( 8 + 𝛾( 4 4 𝜔 " (𝑘) ⊙ 𝜃(") (𝑘) ( 6 "7( + 𝛾8 4 4 𝑘" 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
- 11. 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
- 12. 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 11
- 13. 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 12
- 15. thank you http://www.smarth2o-fp7.eu/ @smartH2Oproject #SmartH2O Andrea Cominola andrea.cominola@polimi.it Politecnico di Milano Department of Electronics, Information and Bioengineering