LAB · 05 / 09

Heat and electricity load

How much each degree drives electricity demand and a day-ahead forecast with 4% error. ONS · INMET · gradient boosting

How it works

Visual mode: shadows, ambient occlusion, depth of field and quality that adapts to your device. Light mode: no shadows or post-processing. Each bar is one cell of the data; the full analysis is right below.

Sheet of each analysis

Objective, why, target variable, predictors, method, metric, validation and whether the objective was reached.

The shape of a day

Objective: Describe the daily load curve and the calendar effect.
Why: Without separating working days from holidays, any model blames weather for what is calendar.
Target variable: Mean load per hour (MW).
Predictors / explanatory variables: Hour, working day × weekend/holiday.
Method: Conditional means.
Metric: MW; difference per hour.
Validation: Official 2025 national holidays + Carnival.
Objective reached?: Yes: mean peak at 7 pm; largest gap at 2–3 pm.

The hockey-stick curve

Objective: Measure how much each degree adds to load and from what temperature.
Why: A heat wave is the biggest operational error risk; knowing the slope above the break point turns a weather forecast into MW.
Target variable: Daily mean load (working days).
Predictors / explanatory variables: Daily maximum temperature (mean of 2 stations).
Method: Segmented regression with break point searched between 20 and 32 °C.
Metric: MW per °C above and below the break; R².
Validation: Station coverage of 99.5% and 99.7% of hours.
Objective reached?: Yes: break at 23.5 °C, +764 MW/°C above, R² = 0.52.

Forecasting tomorrow

Objective: Forecast next-day load with less error than the naive method, hot days included.
Why: Each point of error becomes unnecessary generation or expensive short-term purchase; hot days are where error costs most.
Target variable: Hourly load (MW).
Predictors / explanatory variables: Hour, weekday, holiday, working day, temperature, degrees above 23.5 °C, load 2 and 7 days earlier.
Method: Six models, from naive to gradient boosting (HistGradientBoosting).
Metric: MAPE overall and on the hottest 10% of days; permutation importance.
Validation: Train Jan–Sep, test Oct–Dec (in the future); observed temperature as an upper bound of a weather forecast.
Objective reached?: Yes: MAPE 4.0% (6.7% naive) and 6.1% on hot days (8.4%).

Load and weather, hour by hour

Hourly load of the Southeast/Centre-West subsystem (ONS) and temperature from two INMET stations in São Paulo, all of 2025.

Six models, one test in the future

From "same hour last week" to gradient boosting with temperature and recent load. Training January to September, testing October to December.

MAPE = média(|real − previsto| / real)

Heat and electricity load in data

A full year, hour by hour, of electricity load in Brazil’s Southeast/Centre-West subsystem, crossed with the temperature of two INMET stations in São Paulo. The question is that of whoever operates or buys power: how much does heat drive consumption, and can tomorrow’s load be forecast with small error?

Real-world use

Day-ahead scheduling and energy purchasing: each percentage point of forecast error becomes generation dispatched for nothing or power bought short-term, at a higher price.

8,583hours with load and temperature (2025)

44.3 GWmean subsystem load

+764 MWper degree above 23.5 °C

4.0%day-ahead forecast error

The shape of a day

QuestionHow is load spread over the day, and does the weekend change it?

Decision and whyI separated working days from weekends and holidays (2025 national holidays plus Carnival). Without that split, any model confuses "Sunday" with "cold day".

ResultOn working days load rises from 6 am, plateaus through business hours and has its mean peak at 7 pm, when lighting and household use add up. On weekends the whole curve is lower, and the gap is largest at 2 and 3 pm (almost 9 GW): it is commercial and industrial demand that disappears.

Source: ONS · curva de carga horária 2025 (SE/CO)

Analysis technical sheet

Objective: Describe the daily load curve and the calendar effect.
Why: Without separating working days from holidays, any model blames weather for what is calendar.
Target variable: Mean load per hour (MW).
Predictors / explanatory variables: Hour, working day × weekend/holiday.
Method: Conditional means.
Metric: MW; difference per hour.
Validation: Official 2025 national holidays + Carnival.
Objective reached?: Yes: mean peak at 7 pm; largest gap at 2–3 pm.

Mean load by hour (MW)

working day
weekend and holiday

The hockey-stick curve

QuestionHow much does each extra degree raise consumption?

Decision and whyI plotted the daily maximum temperature against mean load, on working days only. Instead of one line I fitted two lines with a break point, searching for the point that minimises error between 20 and 32 °C. That is the expected shape: below some temperature nobody turns on the air conditioning.

ResultThe break point landed at 23.5 °C. Below it load changes 273 MW per degree; above it, 764 MW per degree, almost 3 times more. Temperature alone explains 52% of the daily variation; the rest is calendar and trend. The year’s peak (62,150 MW) came at 14:00 on 2025-02-18, with 29.3 °C in São Paulo.

Source: ONS × INMET A701 e A771 · analises/energia.py

Analysis technical sheet

Objective: Measure how much each degree adds to load and from what temperature.
Why: A heat wave is the biggest operational error risk; knowing the slope above the break point turns a weather forecast into MW.
Target variable: Daily mean load (working days).
Predictors / explanatory variables: Daily maximum temperature (mean of 2 stations).
Method: Segmented regression with break point searched between 20 and 32 °C.
Metric: MW per °C above and below the break; R².
Validation: Station coverage of 99.5% and 99.7% of hours.
Objective reached?: Yes: break at 23.5 °C, +764 MW/°C above, R² = 0.52.

Daily maximum temperature × mean load

working day
weekend and holiday

max temperature (°C) × mean load (MW)

Dashed line: the two-line fit, using working days only.

Forecasting tomorrow

QuestionWhich model best forecasts next-day load, and does temperature help?

Decision and whyTraining from 2025-01-08 to 2025-09-30 and testing from 2025-10-01 to 2025-12-31, always in the future relative to training. I compared six models, from the simplest (repeat the same hour last week) to gradient boosting with temperature and the load from two days earlier, which is already known when the forecast is made. Metric: mean absolute percentage error (MAPE), overall and on the hottest 10% of days.

ResultThe best model errs by 4.0% on average, 40% less than repeating last week (6.7%). On hot days, where error costs most, it also wins: 6.1% versus 8.4%. Lesson: the temperature line, which looked like the cleanest explanation, got worse out of sample (11.7% on hot days), because the relationship learned from January to September did not hold at year end. The recent load level corrects that shift.

Source: ONS × INMET · analises/energia.py (scikit-learn)

Analysis technical sheet

Objective: Forecast next-day load with less error than the naive method, hot days included.
Why: Each point of error becomes unnecessary generation or expensive short-term purchase; hot days are where error costs most.
Target variable: Hourly load (MW).
Predictors / explanatory variables: Hour, weekday, holiday, working day, temperature, degrees above 23.5 °C, load 2 and 7 days earlier.
Method: Six models, from naive to gradient boosting (HistGradientBoosting).
Metric: MAPE overall and on the hottest 10% of days; permutation importance.
Validation: Train Jan–Sep, test Oct–Dec (in the future); observed temperature as an upper bound of a weather forecast.
Objective reached?: Yes: MAPE 4.0% (6.7% naive) and 6.1% on hot days (8.4%).

Day-ahead forecast error (MAPE)

all days
hottest 10%

Hottest test week, from 2025-12-23 (MW)

actual
best model
last week

The week includes Christmas. On the 25th the model errs most (about 9% above actual): holidays are few in training and each has its own shape.

What the best model uses (permutation importance)

How much MAPE rises when the variable is shuffled in the test set.

What each dataset contributed

ONS hourly load: the shape of the day and the weight of the calendar (working day, holiday).
INMET temperature: the air-conditioning break point and what each degree costs in MW.
Recent load: the most important variable, following level shifts the calendar cannot see.
Testing in the future: showed that the nicest model in training was not the best outside it.

Limits of this analysis

I use the observed temperature as if it were the weather forecast: it is the upper bound of the gain. With a real forecast, error would be somewhat higher.
Two São Paulo stations stand in for a subsystem stretching from Minas Gerais to the Centre-West.
One year of data: it captures neither multi-year trend nor drought years.

Analysis code: analises/energia.py