Extraction of Basic Patterns of Household Energy Consumption

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1 ,, ( ),,,,.,., KL. Annex42, Extraction of Basic Patterns of Household Energy Consumption Solar power, wind power, and co-generation (combined heat and power) systems are possible candidate for household power generation. These systems have their advantages and disadvantages. To propose the optimal combination of the power generation systems, the extraction of basic patterns of energy consumption of the house is required. In this study, energy consumption patterns are modeled by mixtures of Gaussian distributions. Then, using the symmetrized Kullback-Leibler divergence as a distance measure of the distributions, the basic pattern of energy consumption is extracted by means of hierarchical clustering. By an experiment using the Annex 42 dataset, it is shown that the proposed method is able to extract typical energy consumption patterns. 1 School of Science and Engineering, Department of Electrical Engineering and Bioscience, Waseda University 1.,,,.,,. 3), 4), 5), 1),2).,,.,, 6).,,,,,,,,,,, 7) 9),,,,,,, 2., 1, EM 2.1,,., 1 c 2011 Information Processing Society of Japan

2 , 1 φ(x; µ, Σ) = ( (2π)d Σ exp 1 ) 2 (x µ)t Σ 1 (x µ), (1)., x R d, µ R d Σ R d d, A A. M p(x; θ) = M π m φ(x; µ m, Σ m ). (2) m=1, π m, : M π m = 1, π m 0; (m = 1,..., M). (3) m=1, m, m = 1,..., M. EM 10). EM,, E M 2. E,, M, E {x 1,..., x n }, θ = {π m, µ m, Σ m } M m=1, π m R, µ m R d, Σ m R d d. t θ (t) θ, EM, E-step: M-step: Q : n M Q(θ, θ (t) ) = p(m x i ; θ (t) ) log p(x i, m; θ). (4) i=1 m=1 Q θ (t+1) : θ (t+1) = arg max Q(θ, θ (t) ). (5) θ,, 11),. π (t+1) m µ (t+1) = Σn i=1w m i n, (6) m = Σn i=1wi m x i, (7) Σ n i=1 wm i Σ (t+1) m = Σn i=1w m i x i x T i Σ n i=1 wm i 3. µ (t+1) m (µ (t+1) m ) T. (8) w m i = p(m x i; θ (t) ) (9), 3.1 KL Kullback-Leibler (KL ) 12) 2, KL, p q, p q KL D(p, q) = p(x) log p(x) dx. (10) q(x) KL,, KL 13),14),, KL : D g (p, q) = 1 Σq [log 2 Σ + p Tr(Σ 1 q Σ p ) d + (µ p µ q ) T Σ 1 q (µ p µ q )]. (11), µ p µ q p(x) q(x), Σ p Σ q 2 p(x; θ), q(x; θ ), θ θ, KL : M M m D gm(p, q) = π m log π =1 m exp D g(φ m,ψ m ) M ω. (12) m =1 m exp D g(φ m,ψ m ) m=1 {π m } M m=1 {ω m } M m =1, p(x; θ) q(x; θ ) φ m(x) ψ m (x) D g(φ m, ψ m ) (11), 2 c 2011 Information Processing Society of Japan

3 ,., KL.,, p V (p) q V (q) 2 p(x) = V p p(x), q(x) = V q q(x),, KL 13) : ( D( p, q) = p(x) log p(x) ) q(x) p(x) + q(x) dx, x V pd gm(p, q) + V p log Vp V q V p + V q. D gm (p, q) (12) 3.2,,, 15),16)., 2.,,, 2, KL 17) D( p, q) + D( q, p) d KL ( p, q) = 2 18). 4.,, Annex 42, The Simulation of Building-Integrated Fuel Cell and Other Cogeneration Systems Generated Canadian Occupantdriven Electrical Load Profiles(EXCEL format, low electricity consumption) 19). (13) 3, 5, ( 1 Examples of energy consumption data of a household in a day. W), 2, 1.,,,, 1, 1, 1 EM., R 20) mclust Mclust,,,,,, 3., BIC 21) 1 3,, (13) 2,. 3 c 2011 Information Processing Society of Japan

4 (a) cluster 1 (b) cluster 2 (c) cluster 3 (d) cluster 4 (e) cluster 5 (f) cluster 6 (g) cluster 7 (h) cluster dimensional plots of energy consumption data within each cluster in a household. 2, 8 (a) (h) ( W),. 3. 2, 3, : 1, , 5 6, 7,.,,. 1, 1 Number of elements in each cluster Cluster # of elements , 2, c 2011 Information Processing Society of Japan

5 (a) cluster 1 (b) cluster 2 (c) cluster 3 (d) cluster 4 (e) cluster 5 (f) cluster 6 (g) cluster 7 (h) cluster 8 3 Average and one standard deviations of the basic energy consumption patterns in eight clusters extracted by the proposed method. Most of plots show clear peaks which characterizes the basic pattern., 5, 6,,. 5.,,.,, 8,,,,,,,,, Annex42, 5 c 2011 Information Processing Society of Japan

6 ,,.,,,,,. Acknowledgment, ( ) No , No.S ) A.Sayigh, Worldwide progress in renewable energy, vol.3, no.4, 2008, pp ) E. Martinot, A. Chaurey, D. Lew, J. R. Moreira, and N. Wamukonya, Renewable energy markets in developing countries, Annual Review of Energy and the Environment, vol.27, no.1, pp , ) W.Grossmann, I.Grossmann, and W.K. Steininger, Indicators to determine winning renewable energy technologies with an application to photovoltaics, Environmental Science & Technology, ) C.Samaras, Ed., Learning from Wind: A framework for effective low-carbon energy diffusion, World Renewable Energy Congress. Elsevier, August ) 10th International Conference on Intelligent Systems Design and Applications, ISDA 2010, November 29 - December 1, 2010, Cairo, Egypt. IEEE, ) Vringer, Kees, Theo Aalbers and Kornelis Blok, Household energy requirement and value patterns, Energy Policy, 35 (1) p.p , ) M.L. Baughman, N.A. Eisner, and P.S. Merrill, Optimizing combined cogeneration and thermal storage systems: An engineering economics approach, Computer Engineering, vol.4, no.3, ) N.W. Alnaser, R.Flanagan, and W.E. Alnaser, A comprehensive model for accelerating the building integrated photovoltaic (bipv) / wind turbine (biwt) construction projects in the kingdom of bahrain, The Open Construction and Building Technology Journal, vol.3, pp. 1 11, ) A.B. Kanas-Patill, R.P. Saini, and M.P. Shama, Sizing of integrated renewable energy system based on load profiles and reliability index for the state of uttarakhand in india, Renewable Energy, vol.36, pp , ) A.P. Dempster, N.M. Laird, and D.B. Rubin, Maximum likelihood from incomplete data via the EM algorithm, Journal of the Royal Statistical Society Series. B, vol.39, ) C.M. Bishop, Pattern Recognition and Machine Learning. Springer, ) T.M. Cover and J.A. Thomas, Elements of information theory. John Wiley and Sons, Inc., ) J.R. Hershey and P.A. Olsen, Approximating the Kullback Leibler Divergence Between Gaussian Mixture Models, in IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 2007., vol.4, 2007, pp. IV 317 IV ) M.N. Do, Fast approximation of Kullback-Leibler distance for dependence trees and hidden Markov models, Signal Processing Letters, IEEE, vol. 10, no. 4, pp , March ) A.K. Jain and R.C. Dubes, Algorithms for Clustering Data. Prentice Hall, ) A. Jain, M. Murty, and P. Flynn, Data clustering: A review, ACM Computing Surveys, vol.31, no.3, pp , ) D. H. Johnson and S. Sinanovic, Symmetrizing the kullback-leibler distance, Technical Report, Rice University., Tech. Rep., ) J.H. Ward, Hierarchical Grouping to Optimize an Objective Function, Journal of the American Statistical Association, vol.58, no. 301, pp , March ) I. Beausoleil-Morrison, An experimental and simulation-based investigation of the performance of small-scale fuel cell and combustion-based cogeneration devices serving residential buildings, [Online]. Available: 42 Final Report.pdf 20) R Development Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, ) G. Schwarz, Estimating the Dimension of a Model, The Annals of Statistics, vol.6, no.2, pp , c 2011 Information Processing Society of Japan

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