Appendix A:
Normal Aggregate Power Demand of a Set of Identical Heating Systems with Hysteresis
For simplification, we suppose that weather conditions remain constant and that the temperature evolution curve in all homes is a zig-zag function of time (Figure A.1): when the heating system is on, the temperature increases by X °C/hour, when it is off, it decreases by Y °C per hour. All homes are equipped with a traditional thermostat that regulates the temperature with a hysteresis of H °C: the heating system is switched on when the temperature reaches Tref, and switched off when it reaches Tref + H. In reality, these temperature evolution curves are exponential functions of time, but this linear approximation is valid if H is small compared to the Tref–Text, the temperature gradient between the inside and the outside of the house walls. The heating system therefore remains switched on for H/X hours (the part of the zig-zag during which temperature increases), and then remains switched off for H/Y hours (the part of the zig-zag during which temperature decreases).
Figure A.1 Evolution of power demand after a synchronized thermostat offset.
We further suppose that initially thermostats are not correlated: at any moment, if we sort homes according to the last time the heating system was switched on, an identical number k.δt of homes had their heating system last switched on between ...
