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A Gray-Box Model of a Two-Stage Heat Pump for Electrical Load Forecasting in a Single-Family Residence



Farhad Omar


Buildings as a set of electric loads and generation resources can play an essential role in managing the stability of the power system in a smart grid. Traditionally, buildings assumed a passive rule in the day-to-day operation of the electric grid. Utilities control the supply of energy to match the demand for buildings. However, the dynamics of the power distribution grid are rapidly changing. A critical aspect of managing the balance between supply and demand in the distribution grid is estimating future demand through load forecasting. Loading forecasting requires accurate, adaptive, and simple models to account for the dynamic behavior of the electrical demand. In residential buildings, heat pumps are a significant source of energy consumption. This document describes a novel gray-box model (RL Model) to forecast a two-stage air-source heat pump's steady-state and transient current consumption in heating and cooling seasons. The RL Model is based on the solution of a first-order differential equation describing the characteristics of a resistor-inductor equivalent circuit and the steady-state power consumption of a heat pump. The steady-state power consumption of the heat pump is a strong function of the outdoor temperature. Key parameters of the RL Model were estimated using a learning algorithm. The performance of the RL Model is validated using measurement data from the Net-Zero Energy Residential Test Facility located on the National Institute of Standards and Technology (NIST) campus in Gaithersburg, MD. The predicted output of the RL Model, current consumption, is used to estimate the heat pump's real and reactive power consumption for different operating stages and seasons. A list of key parameters, like the time constant and temperature-dependent coefficients of the RL Model, are tabulated. Knowledge of the time constant provides critical information for analyzing the aggregated effect of controlling many heat pumps, as flexible loads, on grid stability and provision of ancillary services. The average root mean squared error between the RL Model's predicted current output and the measured current consumption is 0.06.
Technical Note (NIST TN) - 2249
Report Number


Data-driven model, energy use forecasting, gray-box model, heat pump modeling, heat pump real and reactive power, heat pump time constant, load forecasting, load modeling, parameter fitting, parameter optimization, RL model, temperature dependent heat pump model.


Omar, F. (2023), A Gray-Box Model of a Two-Stage Heat Pump for Electrical Load Forecasting in a Single-Family Residence, Technical Note (NIST TN), National Institute of Standards and Technology, Gaithersburg, MD, [online],, (Accessed July 19, 2024)


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Created March 10, 2023