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Shape-Based Approach to Household Load Curve Clustering and Prediction

Thanchanok Teeraratkul, Daniel O'Neill, Sanjay Lall

arXiv:1702.01414v1stat.ML

TL;DR

Household demand response needs stable consumer classification and accurate prediction despite variable timing in daily device use. The paper uses DTW-based shape clustering with Markov prediction and extends it to device-level usage estimation, reducing representative groups by 50% and lowering prediction error against selected techniques.

  • Problem

    Household load curves vary in device-use timing, causing traditional clustering to produce many clusters and complicating stable classification and individual prediction.

  • Method

    The paper clusters 24-hour household load curves by DTW shape similarity, predicts next-day prototypes with a Markov model, and estimates device use through Power Level Decomposition.

  • Results

    50% fewer 24-hour energy clusters result from DTW clustering, while the DTW-Markov model achieves mean DTWE 0.182 and reduces DTWE by 15%, 27%, 10%, and 31% versus models A, B, C, and D.

  • Takeaways & Limitations

    DTW-based prototypes provide a smaller grouping of household consumption shapes for prediction, and the approach extends to estimating which devices are used and when.

Abstract

from arXiv · show

Consumer Demand Response (DR) is an important research and industry problem, which seeks to categorize, predict and modify consumer's energy consumption. Unfortunately, traditional clustering methods have resulted in many hundreds of clusters, with a given consumer often associated with several clusters, making it difficult to classify consumers into stable representative groups and to predict individual energy consumption patterns. In this paper, we present a shape-based approach that better classifies and predicts consumer energy consumption behavior at the household level. The method is based on Dynamic Time Warping. DTW seeks an optimal alignment between energy consumption patterns reflecting the effect of hidden patterns of regular consumer behavior. Using real consumer 24-hour load curves from Opower Corporation, our method results in a 50% reduction in the number of representative groups and an improvement in prediction accuracy measured under DTW distance. We extend the approach to estimate which electrical devices will be used and in which hours.

I. INTRODUCTION

Household demand response requires stable classification and prediction despite substantial day-to-day timing variation in device use. The paper motivates shape-based clustering with DTW to group similar consumption patterns and support household-level prediction and device-use estimation.

  • Motivation: Household DR seeks to classify consumers, predict next-day consumption, and estimate device use to identify households worth incentivizing.These tasks support estimating the potential benefit of incentivizing particular customers.
  • Motivation: A household load curve records 24 hours of energy consumption and reflects the usage patterns of household electrical devices.
  • Contributions: 50% fewer 24-hour energy clusters result from DTW clustering compared with traditional K-means and Gaussian-based E&M.
  • Motivation: Day-to-day variation in device timing can produce many clusters for one consumer, complicating stable classification and individual load prediction.The variation can occur even when underlying behavioral patterns are consistent.
  • Shape-based approach: DTW aligns load curves by non-linearly stretching and contracting their time axes, grouping curves with similar shapes despite differing values at particular times.This makes timing differences less important when identifying common device-usage patterns.
  • Shape-based approach: A DTW warping path permits up to about 1 hour of device-timing variation by comparing each point with at most two earlier or later points.The method applies stretching and contracting within each 2-hour period.

A. Data Description

The clustering experiment uses 23,254 normalized 24-hour load curves from 1,057 households and evaluates DTW against K-means and E&M using cluster compactness, diversity, and their ratio.

  • Evaluation: Three metrics evaluate clustering quality: within-cluster compactness WC, between-center diversity WB, and the within-to-between variation ratio WCBCR.Lower WC and WCBCR and larger WB indicate better clustering.
  • Results: DTW clustering achieves lower WC and WCBCR and larger WB than the comparison methods.These metrics indicate more compact and diverse clusters under the paper’s evaluation criteria.
  • Results: 8 to 10 clusters are selected for DTW, compared with 20 for K-means and 14 to 16 for E&M using the WCBCR knee rule.The cluster counts are estimated graphically from the WCBCR plot.

C. Consumer Variability

The paper evaluates household variability with entropy and describes a DTW-Markov procedure that encodes load-shape clusters to predict the next day, either as a full curve or by periods.

  • Consumer Variability: After clustering into 10 clusters, average entropy is 0.5 for DTW, 0.7 for K-means, and 0.64 for E&M.Smaller entropy corresponds to greater household consistency.
  • Consumer Variability: Figure 6 contrasts one DTW cluster with three K-means clusters for a single individual’s load curves.The comparison illustrates different household classifications under the two clustering methods.
  • Day Ahead Load Shape Selection: The prediction procedure selects a next-day cluster prototype conditionally on the current load curve, then uses that prototype as the predicted 24-hour shape.Past curves are encoded by their nearest DTW cluster and modeled as a stationary Markov process.
  • Day Ahead Load Shape Selection: For period-level prediction, each day is divided into equal periods, each period is clustered separately, and conditional transitions iteratively select prototypes from period 1 through period np.Separate weekday and weekend Markov models can reflect calendar effects.
  • Day Ahead Load Shape Selection: The procedure includes cluster encoding, transition-probability calculation, and prototype selection across historical days and periods.These operations are represented in Algorithms 1–3 and the accompanying transition-probability example.

B. Adjustment on Predicted Load Shape

Predicted load-shape prototypes are normalized, so the method rescales them using historical forecast errors; when history is insufficient, it uses a naive estimate.

  • Adjustment on Predicted Load Shape: 14-period prototypes are rescaled after clustering to produce the final forecast for each period.The additional scaling step compensates for normalization during clustering.
  • Adjustment on Predicted Load Shape: The scaling coefficient αD+1,p minimizes a weighted sum of squared errors from past forecasts.The weighting factor β de-emphasizes older data.
  • Adjustment on Predicted Load Shape: When past forecasts are unavailable or too limited for accurate recursive prediction, the method uses a naive estimate.Under this estimate, the final predicted curve’s total equals the average total of past period-p load curves.

C. Load Curve Prediction Experiment

The experiment predicts next-day household load curves from empirical Opower data using DTW clusters and separate Markov models for weekdays and weekends.

  • Load Curve Prediction Experiment: Next-day load curves are predicted using empirical Opower data and clusters derived in the clustering section.The load curves are encoded before constructing the prediction models.
  • Load Curve Prediction Experiment: Two Markov-based models are constructed: one for weekdays and one for weekends.This separates prediction according to the calendar grouping used in the experiment.
  • Load Curve Prediction Experiment: The validation set contains 3,300 load curves from 150 low-variability households, with 22 consecutive days per household.The records span July 19, 2012 to August 9, 2012.
  • Load Curve Prediction Experiment: Leave-one-out cross-validation uses 21 days to construct Markov models and predicts the remaining day.The validation set is used for selecting prediction-model parameters, while a separate test set is used for prediction.

D. Prediction Model Parameters Selection

The DTW-Markov model selects cluster and history parameters by minimizing validation DTWE, then evaluates the selected model on household test data against existing methods. The selected configuration uses 12 clusters and two periods, achieving lower DTWE than the comparison models.

  • The model varies K from 2 to 50 and np from 1 to 3, producing 75 candidate configurations for validation.
  • K = 12 and np = 2 are selected because DTWE stabilizes near this cluster count while balancing error and model size.The selected configuration achieves validation DTWE of 0.172.
  • The selected DTW-Markov model obtains mean DTWE of 0.182 on test data from 50 Opower households.
  • DTW-Markov reduces DTWE by 15%, 27%, 10%, and 31% relative to models A, B, C, and D, respectively.
  • The prediction algorithm combines DTW-based clustering with Markov-chain modeling, with clustering performed once and subsequent prediction complexity bounded by the history length.

VI. POWER LEVEL DECOMPOSITION (PLD)

The PLD extension decomposes a household’s complete 24-hour load curve into device-related power-level usage. It estimates an appliance usage matrix and uses its structure to represent when devices contribute to total consumption.

  • The method estimates device energy use for each hour from either a complete 24-hour load curve or previous days’ curves.
  • The Appliance Usage Matrix records energy used at each power level or device during each hour.
  • The unknown appliance matrix is distinguished from its complete-curve estimate and its partial-information prediction.The estimate uses a minimum Frobenius norm formulation, which the paper notes has useful properties for clustering and prediction.
  • The load vector x is the complete 24-hour household curve, while p contains device-associated power levels and α scales their units.Aijpj represents device j’s energy usage during hour i.
  • Rows of the estimated matrix with larger elements correspond to hours with higher energy usage, and Figures 10 and 11 show sparsity patterns for evening-peak and dual-peak consumers.

B. Clustering PLD matrices

The paper extends DTW-based comparison from load curves to PLD matrices and analyzes the resulting prediction-error bounds. The bounds depend on the load-prediction error, matrix perturbation structure, and the scaling of the power vector.

  • PLD-matrix DTW distance sums DTW distances across corresponding power-level columns, interpreting similarity over time at each power level.
  • Clustering PLD matrices with DTW is equivalent to clustering load curves and then converting cluster members into PLD matrices.
  • Given previous load curves, the method first predicts the next load curve with DTW-Markov and then obtains the predicted PLD matrix from that prediction.
  • The PLD prediction-error bound is formulated under a perturbation model in which the matrix error H has rank RH and maximum singular value σ1.
  • The analysis models H as a Gaussian random matrix and uses the distribution of its largest singular value to characterize upper and lower error-bound CDFs.
  • The bound is tight when the power-vector units are appropriately chosen, and smaller ∥αp∥2 narrows the gap between the bounds.

D. Sparse PLD Estimate

The sparse PLD variant adds entry-wise L1 regularization to reduce unnecessary zero or nonzero structure in the appliance estimate. This changes the optimization and prediction procedure relative to the minimum-norm formulation.

  • The sparse estimate combines Frobenius-norm and entry-wise L1 regularization to control the appliance matrix structure.
  • Increasing λ2 relative to λ1 makes the estimate more sparse.
  • When λ2 ≠ 0, the estimate lacks a closed-form solution, so DTW K-medoids clustering and prediction must be applied separately to each device column.
  • The equivalence relations and prediction-error bound established for the non-sparse case no longer hold when λ2 ≠ 0.

VII. CONCLUSION

The paper concludes with a DTW-based framework for household load-curve clustering, prediction, and device-use estimation. DTW clustering reduces class count and variability, while DTW-based prediction achieves lower average error than selected forecasting methods.

  • Shape-based clustering: DTW aligns load-curve shapes through non-linear stretching and contracting of time axes, producing relatively fewer classes and lower household variability than traditional clustering methods.The comparison is against K-means and the gaussian-based E&M algorithm.
  • Household load prediction: DTW-cluster prototypes encode each load curve, and a Markov model predicts the next-day prototype from the current encoded curve.Prediction accuracy is measured using the DTW-based prediction-error metric DTWE.
  • Household load prediction: The DTW-Markov prediction method produces lower average prediction error than selected forecasting techniques in the literature.
  • Device-use estimation: Power Level Decomposition represents a load curve as a matrix of device energy use by hour and supports prediction of the full matrix from previous days' load curves.The fine-grain prediction bound under the Frobenius norm is tight when the power vector is appropriately chosen.

VIII. APPENDIX

The appendix provides implementation details for the load-curve prediction models and records proof steps establishing an equivalence relation from prior equations and lemmas.

  • Prediction-model implementation: The appendix gives detailed implementation information for the load-curve prediction models compared with the DTW-Markov model in Table IV.
  • Prediction-model implementation: Model A is Tao's vanilla benchmark with a recency effect, implemented as a multiple linear regression model for benchmark scoring.The best average-lag pair is selected using mean DTWE across 190 households in the validation set.
  • Proof details: The appendix derives an equivalence relation using results from earlier equations and the condition H_p = 0, then applies additional equation and lemma results.
  • Proof details: The proof concludes after the preceding equivalence and lemma-based steps are established.
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