I would form a daily model between consumption (the k values ) and weather . Arima models that include predictor variables both user-specified and identified from the data can be very useful ... they are sometimes referred to ARMAX models or Transfer Function Models or Dynamic Regression models. Data analysis might suggest monthly effects , holiday effects , day-of-the-week effects , day-of-the-month effects , week=of-the-month effects , long-weekend effects et al. and perhaps some memory effects.
I would search for "daily data" in SE for example Simple method of forecasting number of guests given current and historical data to get a flavor of this kind of approach