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Modelling and forecasting daily electricity load curves: a hybrid approach
Haeran Cho, Yannig Goude, Xavier Brossat, Qiwei Yao
TL;DR
Electricity providers need accurate short-term load forecasts for scheduling, but the loads have trend and seasonal structure as well as dependence across days. The paper combines weekly-average trend modelling with curve regression and reports better accuracy than conventional methods, though EDF's operational model is more accurate.
Problem
Electricity providers need accurate short-term load forecasts because electricity can be stored or discharged only at extra cost, and forecasts inform power-system scheduling.
Source: §1 · p. 1Method
A generalised additive model captures trends in weekly averages, while Hilbert-space singular value decomposition reduces daily curve regression to scalar regressions.
Source: §1 · p. 3Results
The hybrid method is more accurate than conventional seasonal ARIMA or exponential-smoothing methods, while EDF's operational model is more accurate.
Source: §1 · p. 3Takeaways & Limitations
The authors describe the approach as potentially more adaptive to changing electricity-consumption environments.
Source: §5 · p. 26The authors note scope to improve the hybrid approach by improving the daily-level linear regression fit.
Source: §4.2 · p. 23
Abstract
from arXiv · showhide
We propose a hybrid approach for the modelling and the short-term forecasting of electricity loads. Two building blocks of our approach are (i) modelling the overall trend and seasonality by fitting a generalised additive model to the weekly averages of the load, and (ii) modelling the dependence structure across consecutive daily loads via curve linear regression. For the latter, a new methodology is proposed for linear regression with both curve response and curve regressors. The key idea behind the proposed methodology is the dimension reduction based on a singular value decomposition in a Hilbert space, which reduces the curve regression problem to several ordinary (i.e. scalar) linear regression problems. We illustrate the hybrid method using the French electricity loads between 1996 and 2009, on which we also compare our method with other available models including the EDF operational model.
1 Introduction
The paper proposes a hybrid model for daily electricity-load forecasting and reports better accuracy than conventional methods, while the EDF operational model remains more accurate. Its approach separates weekly trend modelling from dependence modelling across daily load curves.
1 Introduction
- Accurate short- and middle-term load forecasts matter because electricity storage or discharge entails extra costs and forecasts inform power-system scheduling.Improved forecasts can also reduce production costs and increase trading advantages, especially during peak periods.Source: §1 · pp. 1–2
- French electricity loads show an overall increasing trend and annual seasonality; daily patterns arise from changing demand within the day but are not visible at the full-period scale.Source: §1 · pp. 2–3
- A generalised additive model fitted to weekly averages models long-term trends, while curve linear regression models dependence among consecutive daily loads.The trend model includes temperature and other meteorological factors as explanatory variables.Source: §1 · p. 3
- A Hilbert-space singular value decomposition reduces curve regression to scalar regressions, while treating loads as curves retains continuity and embeds daily patterns in a stationary framework.Source: §1 · p. 3
- The hybrid method outperforms conventional methods such as seasonal ARIMA and exponential smoothing, but EDF's operational model is more accurate despite being more complex and knowledge-intensive.The authors characterize their approach as simpler while retaining competitive prediction capacity.Source: §1 · p. 3
2 Modelling weekly averages
The weekly-average GAM models long-term trend and seasonality while preserving daily-load dependence for a later stage. Adding interactions with the week index improves the fitted trend and weekly and daily forecasts relative to the initial model.
2 Modelling weekly averages
- In model (1), the fitted curve explained 98.7% of weekly-average data, with MAPE 1.63% and RMSE 1014MW.The GAM smooths time, seasonal offset, lagged load, temperature, lagged temperature, and cloud cover.Source: §2 · pp. 5–6
- The low-cloud-cover effect may be an artifact because few observations there produce high fitted-curve variance.Cloud cover differs significantly from zero at large values, where heavy cloud cover is associated with increased lighting use.Source: §2 · p. 7
- Model (3) adds the weekly index to the lagged-load, temperature, lagged-temperature, and cloud-cover smooths, allowing those effects to vary through the year.The approach treats trend and seasonality as constant within each week, preserving daily-load dependence while discarding little from gradual within-week trend changes.Source: §2 · pp. 4, 7
- Model (3) lets the lagged-load relationship vary by week, remaining threshold-shaped but stronger in winter than summer.Temperature shifts smoothly from winter heating effects to summer cooling effects across the year.Source: §2 · p. 7
- With model (3), the fitted trend explained 99.2% of the data, with MAPE 1.28%, RMSE 801MW, and GCV score 8.4 × 10^5.Model (1) explained 98.7% of the data, with MAPE 1.63% and RMSE 1014MW; its comparison GCV score was 1.2 × 10^6.Source: §2 · pp. 6–7
- Daily 2009 forecasts from model (3) had MAPE 1.35% and RMSE 869MW, versus 1.41% and 901MW for model (1).For weekly-average forecasts in 2009, model (3) also had MAPE 1.72% and RMSE 1250MW, versus 2.15% and 1532MW for model (1).Source: §2 · p. 9
3 Regression of daily load curves
This section develops curve-on-curve regression for detrended daily load residuals, using SVD-based dimension reduction and estimation. It illustrates forecasting a daily load curve, compares the hybrid predictor with oracle and baseline alternatives, and describes an extension to multi-step prediction.
3 Regression of daily load curves
- After removing the fitted long-term trend, the model treats each day's residual as a curve response and uses the previous day's residual curve or a curve combining load and temperature as a regressor.The response and regressor curves may be defined on different index sets, with a coefficient function over their product and mean-zero noise.Source: §3 · p. 9
3.1 Curve linear regression via dimension reduction
- The proposed SVD isolates directions in which response curves are most correlated with regressor curves, reducing an r-dimensional curve regression to r scalar regressions.Unlike canonical correlation analysis, the method regresses Yi on Xi and does not treat the two curves symmetrically.Source: §3 · p. 10, §3.1 · p. 12
3.2 Estimation
- The SVD-based estimator of correlation dimension is consistent under the stated conditions, and simulations confirmed good performance across a range of dimensions.The information criteria select the true dimension with probability tending to one; the ratio estimator is motivated by adjacent estimated singular-value ratios separating at the true dimension.Source: §3.2 · pp. 14–15
3.3 An illustration
- For the 2 April 2009 forecast, q = 4 minimized IC2(q) over the longest interval of τ2 across 100 settings, yielding estimator br = 4.The regressor joined the detrended Tuesday, 1 April 2009 load curve and 2 April temperature curve; model fitting used Tuesday–Wednesday April curve pairs from 1996–2008 (n = 53).Source: §3.3 · pp. 15–16
- The daily curve-regression models used the first 10 singular value components of the regressor because including more did not dramatically improve forecasting.The regression estimators were used to obtain predictors for the daily load curve.Source: §3.3 · p. 18
- MAPE and RMSE were (0.91%, 634MW) for the hybrid predictor, (0.60%, 420MW) for the oracle, and (3.14%, 1911MW) for the baseline.The hybrid predictor improved on the baseline by using dependence across days; the oracle performed better but is impractical because its required de-trended load is unavailable.Source: §3.3 · pp. 18–19, Figure 9 · p. 19
- For multi-step forecasts, weekly GAM forecasts use that week’s average temperature and cloud cover, while daily regression forecasts are iterated using predicted loads as regressors.The next-day predicted load replaces part of the regressor curve, and this is repeated until the desired horizon is reached.Source: §3.3 · p. 19
- Using identical steps described in this section, the forecast achieves MAPE 1.06% and RMSE 657MW; multi-step-ahead performance is generally worse than one-day-ahead because errors carry over.Source: §3.3 · p. 19
4 Predicting daily loads in 2009
The study forecasts each day of 2009 using data available through the previous day, classifying daily curves by weekday and season. Across comparisons, the hybrid methods are competitive with the EDF operational model in some warmer months, while forecast accuracy varies by season and day type.
4 Predicting daily loads in 2009
- For each day in 2009, models use data from 1 January 1996 through the preceding day, first predicting the trend with a GAM and then modelling residual daily curves.The comparison focuses on working days and assumes forecasts of next-day temperature and the following week’s average temperature are available.Source: §4 · p. 20, §4.2 · p. 22
4.1 Classification of daily curves
- Daily curves are classified by weekday and expert-defined day type, with holidays separated; the year is split into nine segments based on changes in principal-component decompositions.The authors note that this classification lacks rigorous statistical grounding, though its prediction model performs well in practice.Source: §4.1 · p. 20
4.2 Prediction comparisons
- SARIMA and GSARIMA are consistently outperformed by other methods; all models forecast more accurately in summer than winter, and the oracle has the lowest errors except in December.The oracle’s de-trended load curve is unavailable in practice.Source: §4.2 · pp. 22–23
- H2 and H4, which use more temperature information than H1 and H3, generally have considerably smaller errors, but this pattern does not hold throughout the year.The authors suggest that choosing the regressor adaptively by temperature level may improve performance.Source: §4.2 · p. 23
- Half-day regressors outperform their full-day counterparts in some colder months, while the reverse holds in warmer months; the authors relate this pattern to greater winter curve variability.The colder months named are February–April and October–November.Source: §4.2 · p. 23
- H1 and H2 errors are comparable to or smaller than the EDF operational model’s in June, July, and September, although its errors are smaller on average.The hybrid methods’ gap from the operational model narrows from May to September.Source: §4.2 · p. 23
- The evaluation excludes factors such as EDF’s special winter tariff days, which may affect loads on those days and adjacent days and could improve winter forecasts if incorporated.The comparison is restricted to working days.Source: §4.2 · pp. 22–23
5 Conclusions
The paper combines weekly trend and seasonal modelling with daily curve regression to forecast electricity loads. On the French data, the method retains competitive prediction capacity relative to EDF’s operational model, while leaving scope to incorporate additional influential factors and improve daily regression.
5 Conclusions
- The hybrid approach models weekly trend and seasonality with a GAM and successive-day dependence through curve linear regression.The regression assumes a linear relationship between consecutive daily curves and uses dimension reduction to obtain scalar linear regression problems.Source: §5 · pp. 23, 26
- Compared with EDF’s operational model, the method is more model-centred and uses less accumulated EDF-specific knowledge while retaining competitive prediction capacity.The authors describe it as potentially more adaptive and applicable to a wider range of problems without much human intervention.Source: §5 · p. 26
- Some influential factors were not incorporated, which the authors say could have worsened performance on winter days relative to the operational model.They identify adaptive regressor selection as a possible avenue for improvement.Source: §5 · p. 26