Ventilation and Temperature Control for Energy-efficient and Healthy Buildings: A Differentiable PDE Approach

Abstract

In this paper, we introduce a novel framework for building learning and control, focusing on ventilation and thermal management to enhance energy efficiency. We validate the performance of the proposed framework in system model learning via two case studies: a synthetic study focusing on the joint learning of temperature and CO2 fields, and an application to a real-world dataset for CO2 field learning. For building control, we demonstrate that the proposed framework can optimize the control actions and significantly reduce the energy cost while maintaining a comfort and healthy indoor environment. When compared to existing traditional methods, an optimization-based method with ODE models and reinforcement learning, our approach can significantly reduce the energy consumption while guarantees all the safety-critical air quality and control constraints. Promising future research directions involve validating and improving the proposed PDE models through accurate estimation of airflow fields within indoor environments. Additionally, incorporating uncertainty modeling into the PDE framework for HVAC control presents an opportunity to enhance the efficiency and reliability of building HVAC system management.

Figures (8)

• Figure 1: Schematic of the proposed PDE-based learning and control framework for healthy and energy-efficient buildings. Left figure: we model the airflow, CO$_2$ and temperature dynamics with PDEs, including Navier-Stokes and convection-diffusion PDEs, with the unknown model parameters denoted by $\Theta$. The HVAC control and external disturbances are represented as $u_t$ and $d_t$, respectively. Right figure: we present the formulation of the learning system and control problem as PDE-constrained optimization problems. The goal is to learn unknown system parameters and determine optimal control actions. The results show that the proposed framework is able to model spatiotemporal dynamics with PDEs, leading to improved energy efficiency, ensured thermal comfort, and a healthy indoor environment.
• Figure 2: (a) The physic representation of the simulation testbed: a typical room with a ventilation system including air supply and air return vents on the ceiling. All the walls are insulated, and the right side features a glass window wall. (b) We define a 2D region and model it using 2D PDEs. The figure visualizes the airflow velocity, governed by the Navier-Stokes equations. All boundary conditions are highlighted in blue text.
• Figure 3: Algorithm for the PDE-based learning and control framework. For the "diffPDE" block, "diff" signifies "differentiable", denoting that it allows the computation of gradients. Left figure shows how to derive the gradients: we first solve the PDEs in a forward pass, then we solve the adjoint variables $\lambda_t, t = 0, ..., T$ in a backward pass with \ref{['eq:adjoint']}. The gradient of model learning loss w.r.t. system parameters, and the gradient of control loss w.r.t. control actions are computed using \ref{['eq:gradl']} and \ref{['eq:grad_control']}, using the adjoint variables. Right figure illustrates updates of the model parameters and control via the obtained gradients.
• Figure 4: Convergence results for joint temperature and CO$_2$ field experiment: the learning loss and the parameter estimation loss curves.
• Figure 5: Comparison of measured CO$_2$ concentrations from a real world dataset jin2015sensing (top) and estimated CO$_2$ concentrations (middle) based on the learned PDE model. The bottom figure represents the open status of the CO$_2$ pump in the real world experiment. An open CO$_2$ pump (indicated by a value of 1) corresponds to $g(z,t)=1$, while a closed CO$_2$ pump (indicated by a value of 0) corresponds to $g(z,t)=0$.