A moving horizon estimator for aquifer thermal energy storages

TL;DR

This work addresses the challenge of integrating ATES into building MPC by requiring accurate estimation of spatial ground temperatures, which are difficult to measure directly. It introduces a moving horizon estimator (MHE) that exploits a hybrid, piecewise surrogate ATES model, reformulated as a quadratic program to avoid costly mixed-integer optimization. The authors demonstrate that the MHE yields tightly bounded state estimates that respect physical constraints and outperforms UKF and LTV-KF in accuracy and reliability, while also revealing how far ground states influence MPC decisions. Overall, the approach enables more robust, energy-efficient operation of ATES-enabled buildings and provides practical guidance on spatial-domain considerations for MPC in groundwater-heat systems.

Abstract

Aquifer thermal energy storages (ATES) represent groundwater saturated aquifers that store thermal energy in the form of heated or cooled groundwater. Combining two ATES, one can harness excess thermal energy from summer (heat) and winter (cold) to support the building's heating, ventilation, and air conditioning (HVAC) technology. In general, a dynamic operation of ATES throughout the year is beneficial to avoid using fossil fuel-based HVAC technology and maximize the ``green use'' of ATES. Model predictive control (MPC) with an appropriate system model may become a crucial control approach for ATES systems. Consequently, the MPC model should reflect spatial temperature profiles around ATES' boreholes to predict extracted groundwater temperatures accurately. However, meaningful predictions require the estimation of the current state of the system, as measurements are usually only at the borehole of the ATES. In control, this is often realized by model-based observers. Still, observing the state of an ATES system is non-trivial, since the model is typically hybrid. We show how to exploit the specific structure of the hybrid ATES model and design an easy-to-solve moving horizon estimator based on a quadratic program.

Figures (4)

• Figure 1: Schematic illustration of a building with an ATES system in heating mode comprising one warm (red) and cold (blue) aquifer, an HX, two APUs, and two groundwater pumps. Components illustrated by dotted lines are not considered by the ATES surrogate model. $T_{\mathrm{w}}$ and $T_{\mathrm{c}}$ refer to the temperature of the warm and cold storage.
• Figure 2: Infinity norm of the difference of the OCP's solution $\boldsymbol{u}_{N}$ over the perturbation radius $\hat{r}$. Light blue lines indicate the results of single tests (20), whereas the dark blue line represents the mean. The black and gray vertical lines denote the bounds of the spatial domain and the mesh's cell boundaries. The green vertical line indicates the maximal temperature movement according to \ref{['eq:max-penetration-depth']}.
• Figure 3: Illustration of the MHE model error (mean, 95% confidence interval $2\sigma$, max, and min) in comparison to the surrogate ATES model. The vertical solid line corresponds to the center of each partition ($\mathbb{U}_{i}$), whereas the vertical dotted lines represent the bounds of each partition.
• Figure 4: Illustration of the mean estimation error (solid line) for all $33$ states and its 95% confidence interval (dashed line) for the MHE in blue, UKF in red, and LTV-KF in green. The MHE starts at $t_{40}$, which corresponds to the MHE's time horizon $M$.