On the Probability Density Function of an Air Conditioner Duty Cycle

In this paper the probability density function of the duty cycle of a residential cooling unit is derived. The indoor temperature is modeled via a stochastic differential equation driven by a white Gaussian process. The state of the model is a hybrid state. The continuous part represents the indoor temperature and the discrete part of the model state represents the state of the thermostat switch (ON or OFF). The hybrid-state equation models the thermodynamics of a residential dwelling as seen by the thermostat. The density function of the duty cycle is a function of the house parameters, outside temperature, and the statistics of the driving process. It is shown that the density function of the duty cycle is a modified Bessel function of the second kind of order one. This density function reduces to a Dirac delta function whose argument is the deterministic duty cycle in the absence of the driving process.