19 Modeling the Impact of Passive Ventilation Systems on Multi-Zone Thermal Dynamics
19.1 Overview
19.2 Review of the paper
19.2.1 Summary
The paper proposes the Locally-interactive Bilinear Flow (LiBF) model—an extension of standard linear RC thermal network models where thermal conductances \(K_{ij}\) become functions of passive element opening factors \(\theta\). This introduces bilinearity: the system dynamics depend on the product of state variables (temperatures) and control inputs (opening factors).
Key contributions:
LiBF model formulation: Extends the standard multi-zone RC thermal model by replacing fixed thermal conductances \(K_{ij}\) with opening-factor-dependent functions \(K_{ij}(\theta_{ij})\). The “locally interactive” assumption means each passive element (window or door) only affects the thermal conductance between its two directly connected zones, keeping the model sparse and physically interpretable.
Two-step parameter estimation:
- Step 1: For each fixed opening configuration, use Non-Negative Least Squares (NNLS) to identify the thermal parameters (\(K_{ij}\), \(K_{i\infty}\), \(C_i\)).
- Step 2: Fit constrained 2nd-order polynomials \(K(\theta) = a_0 + a_1\theta + a_2\theta^2\) to the parameter estimates across configurations, with monotonicity constraints ensuring that conductance increases with opening factor.
Measurement efficiency: Only \(O(N + M)\) training configurations are needed to predict \(O(NM)\) unseen configurations, where \(N\) is the number of zones and \(M\) is the number of passive elements. This is a significant reduction in the experimental burden.
Validation: On a 3-zone EnergyPlus commercial building model with 3 passive elements (2 windows, 1 door), the LiBF model achieves RMSE of 0.21–0.53\(°\)C on 5-day validation data after training on 30 days of data.
19.2.2 What do we know already?
This paper connects directly and deeply to the thermal modeling content from Lecture 5 and Lecture 6. The table below maps each key concept from the paper to its origin in our course material:
| Paper Concept | Lecture Source | Connection |
|---|---|---|
| Thermal resistance \(R\) and conductance \(K = 1/R\) | Lecture 5: Fourier’s law, thermal resistance networks | The paper’s \(K_{ij}\) parameters are the same thermal conductances we derived from Fourier’s law for conduction through walls |
| Thermal capacitance \(C_i\) per zone | Lecture 5: \(C = \rho V c\), RC circuit analogy | Each zone’s lumped thermal mass stores energy; the paper estimates \(C_i\) for each zone just as we did for single-zone models |
| Energy balance at each node (Kirchhoff’s laws) | Lecture 6: KCL applied to thermal circuits | The LiBF governing equation is an energy balance at each zone node—heat flows in from adjacent zones, ambient, and HVAC, exactly as in our RC network analysis |
| Multi-node RC network \(\rightarrow\) coupled ODEs | Lecture 6: lumped parameter models, finite difference discretization | The paper extends the multi-zone coupled ODE system from Lecture 6 by making some conductances state-dependent |
| State-space form \(\dot{\mathbf{x}} = A\mathbf{x} + B\mathbf{u}\) | Lecture 6: state-space representation of thermal networks | When all opening factors are fixed, the LiBF model reduces exactly to the linear state-space form from Lecture 6 |
| Infiltration/ventilation as parallel thermal resistance | Lecture 6: infiltration heat transfer paths | The paper’s passive ventilation conductances \(K_{ij}(\theta)\) generalize the fixed infiltration resistance \(R_{inf} = 1/(\dot{m}c_p)\) from Lecture 6 to a controllable, opening-dependent path |
The key extension beyond Lectures 5–6: In our lecture material, all thermal conductances \(K_{ij}\) are fixed constants determined by material properties and geometry. The paper’s central innovation is making these conductances functions of controllable opening factors \(\theta_{ij}\), so:
\[C_i \dot{T}_i = \sum_{j \in \mathcal{N}(i)} K_{ij}(\theta_{ij})(T_j - T_i) + K_{i\infty}(\theta_{i\infty})(T_\infty - T_i) + Q_{HVAC,i}\]
This introduces bilinearity: the term \(K_{ij}(\theta_{ij}) \cdot (T_j - T_i)\) is a product of a function of one input (\(\theta\)) and a function of the state (\(T_j - T_i\)). The system is no longer linear in the classical sense, but it retains enough structure to be tractable for identification and control.
To fully understand this paper’s contributions, students may need background on:
Bilinear dynamical systems: Systems where the dynamics include products of state variables and inputs, i.e., \(\dot{x} = Ax + Bu + \sum_k N_k x \cdot u_k\). These are more general than linear systems but more structured than fully nonlinear ones, enabling specialized identification and control methods.
Non-Negative Least Squares (NNLS): A constrained least squares method that enforces \(x \geq 0\) on the solution. Physical thermal parameters (conductances, capacitances) must be non-negative, so NNLS naturally encodes this physical constraint during estimation.
Constrained polynomial fitting with monotonicity guarantees: Fitting polynomials \(K(\theta) = a_0 + a_1\theta + a_2\theta^2\) subject to the constraint that \(K\) is monotonically non-decreasing in \(\theta\) (opening a window more should not decrease thermal conductance). This requires constrained optimization during the curve fitting step.
EnergyPlus simulation methodology and EMS scripting: EnergyPlus is the reference building energy simulation tool used to generate training and validation data. The Energy Management System (EMS) scripting interface allows custom control logic (e.g., setting window opening factors on a schedule) within the simulation.
Stack effect and wind-driven ventilation physics: The physical mechanisms behind natural ventilation—buoyancy-driven flow (stack effect) due to indoor-outdoor temperature differences and wind-induced pressure differences across openings. These are the underlying phenomena that the opening-factor-dependent conductances \(K(\theta)\) are capturing in a lumped fashion.
Graph-based representations of multi-zone buildings: The paper represents the building as a graph where nodes are zones and edges are thermal connections (walls, windows, doors). This representation enables systematic enumeration of heat flow paths and is the basis for the “locally interactive” assumption.
19.3 Methods
19.3.1 The LiBF Model
The governing equation for zone \(i\) in the LiBF model is:
\[C_i \dot{T}_i(t) = \sum_{j \in \mathcal{N}(i)} K_{ij}(\theta_{ij}) \big[T_j(t) - T_i(t)\big] + K_{i\infty}(\theta_{i\infty}) \big[T_\infty(t) - T_i(t)\big] + Q_{HVAC,i}(t)\]
where:
- \(C_i\) is the thermal capacitance of zone \(i\)
- \(K_{ij}(\theta_{ij})\) is the thermal conductance between zones \(i\) and \(j\), which depends on the opening factor \(\theta_{ij} \in [0, 1]\) of any passive element (window or door) connecting them
- \(K_{i\infty}(\theta_{i\infty})\) is the conductance between zone \(i\) and the ambient, similarly dependent on any exterior window’s opening factor
- \(T_\infty(t)\) is the ambient temperature
- \(Q_{HVAC,i}(t)\) is the HVAC heat input to zone \(i\)
Connection to Lecture 6: When all opening factors \(\theta\) are held constant, every \(K_{ij}(\theta)\) becomes a fixed scalar, and the equation above reduces exactly to the standard linear multi-zone RC model:
\[C_i \dot{T}_i = \sum_{j} K_{ij}(T_j - T_i) + K_{i\infty}(T_\infty - T_i) + Q_{HVAC,i}\]
which can be written in state-space form \(\dot{\mathbf{T}} = A\mathbf{T} + B\mathbf{u}\), as covered in Lecture 6.
The “locally interactive” assumption: Each passive element affects only the conductance between its two directly connected zones. A window between zones 1 and 2 changes \(K_{12}(\theta_{12})\) but does not affect \(K_{13}\) or \(K_{23}\). This keeps the number of parameters linear in the number of passive elements rather than combinatorial.
19.3.2 Two-Step Parameter Estimation
Step 1: NNLS for fixed configurations
For each training configuration (a specific set of fixed opening factors \(\boldsymbol{\theta}^{(k)}\)), the continuous-time ODE is discretized and the parameters (\(K_{ij}\), \(K_{i\infty}\), \(C_i\)) are estimated using Non-Negative Least Squares:
\[\min_{\mathbf{p} \geq 0} \| \Phi \mathbf{p} - \mathbf{b} \|_2^2\]
where \(\mathbf{p}\) is the vector of thermal parameters and \(\Phi\) is the regression matrix constructed from temperature measurements. The non-negativity constraint \(\mathbf{p} \geq 0\) ensures physical plausibility (no negative conductances or capacitances).
Step 2: Constrained polynomial fit
Once parameters are estimated for multiple fixed configurations, the conductance \(K_{ij}\) as a function of \(\theta_{ij}\) is fit with a constrained 2nd-order polynomial:
\[K_{ij}(\theta) = a_0 + a_1 \theta + a_2 \theta^2\]
subject to the monotonicity constraint:
\[\frac{dK_{ij}}{d\theta} = a_1 + 2a_2 \theta \geq 0, \quad \forall \theta \in [0, 1]\]
This ensures that increasing the opening factor never decreases the thermal conductance—a physically sensible requirement.
Measurement efficiency: The key insight is that training configurations can be chosen to vary one passive element at a time (plus a few joint configurations). This requires only \(O(N + M)\) experiments to predict the behavior of all \(O(NM)\) possible configurations.
19.3.3 Experimental Validation
Setup:
- 3-zone EnergyPlus model of a commercial building
- 3 passive elements: 2 exterior windows (zones 1 and 3) and 1 interior door (between zones 1 and 2)
- Opening factors discretized to 6 levels: \(\theta \in \{0, 0.2, 0.4, 0.6, 0.8, 1.0\}\)
- 30-day training period, 5-day validation period
Results:
- RMSE across zones: 0.21–0.53\(°\)C on validation data
- The model successfully predicts temperature dynamics for opening factor combinations not seen during training
- Sensitivity analysis shows that exterior windows (DOF/WOF) have a larger impact on zone temperatures than interior doors, consistent with the larger temperature differential between indoor and ambient vs. between adjacent zones