Model Predictive Control for Dynamic Hydraulic Balancing in Building Radiator Heating Systems: Modeling, Design, and Experimental Validation

1. Introduction

• Existing radiator MPC often uses high-level inputs or cascaded controllers, without explicitly representing low-level controller dynamics in its prediction model.These approaches can leave a mismatch between MPC decisions and their implementation.Source: §1 · p. 2
• Radiators draw water from shared sources, so opening one valve can reduce flow and heating capacity in other zones and contribute to comfort violations.Prior work modeled hydraulic interactions or used simple control, while another MPC study handled balancing separately rather than optimizing valve positions within MPC.Source: §1 · p. 2
• The proposed MPC reduced comfort violations by at least 27% versus formulations that cannot control radiator valves or neglect radiator hydraulic interactions, while requiring the same or lower cumulative water supply temperature.Experiments were conducted in a real six-zone building with 10 radiators.Source: §1 · p. 3
• Accounting for hydraulic interactions reduced zone-temperature modeling RMSE by 11% compared with a model that neglects them.The comparison used real measurements.Source: §1 · p. 3
• The paper presents a physics-based thermal–hydraulic model and two grey-box parameter-estimation approaches using real measurements.The model represents radiator heat output as a function of water flow and temperature difference, including valves, pumps, and boilers.Source: §1 · p. 3

2. Building Thermal–Hydraulic Model

• The section focuses on the physical model formulation, with model-parameter identification and adjustment from data deferred to Section 3.Table 1 is titled “Thermal model parameters and variables.”Source: §2 · p. 3, Table 1 · p. 4

2.1. Building

• Each zone has a thermal state for its air temperature and for the mid-plane temperature of every enclosing wall, including the ceiling and floor.The air-temperature balance includes wall heat transfer, window exchange, infiltration, internal gains, radiator heating, and secondary solar gains.Source: §2.1 · p. 5
• Outdoor temperature, internal heat gains, and solar gains are treated as exogenous inputs, whereas radiator heat is the only indirectly controllable variable.Source: §2.1 · p. 5
• Direct solar gains transmitted through windows are redistributed among the walls surrounding the adjacent zone in proportion to wall area.The wall-temperature balance also represents heat transfer between a wall and adjacent zones, and solar radiation absorbed directly by the wall.Source: §2.1 · p. 5

2.2. Solar Irradiance

• Direct, diffuse, and ground-reflected solar radiation are modeled as incident on a wall, with the wall angle, incidence angle, and ground reflectance used in the calculation.The ground reflectance coefficient is typically approximately 0.4.Source: §2.2 · p. 6, Figure 2 · p. 6
• The incidence angle depends on the wall orientation and solar position, including location latitude, wall azimuth, and hour angle.The hour angle is computed using standard time, standard meridian, and local longitude.Source: §2.2 · pp. 6–7
• Solar irradiance is calculated assuming hourly integrated DNI, DHI, and GHI values and constant irradiance within each one-hour interval.The resulting solar gains include direct transmission and reflection toward surrounding walls, followed by secondary inward solar heat flux.Source: §2.2 · pp. 5–7, Figure 1 · p. 6

2.3.1. Radiator Modeling

• Because water-radiator heat output cannot be commanded arbitrarily, the model tracks coupled water and metal nodes and heat transfer between them.Heat output is represented as heat leaving the radiator metal, while water flow supplies heat to the water node.Source: §2.3 · p. 7, §2.3.1 · pp. 7–8
• The radiator exponent is typically about 1.3, and the nominal radiator-metal-to-zone-air temperature difference is often about 42 K.Source: §2.3.1 · p. 8
• The model assumes one heat source and parallel-installed radiators, giving all radiators the same supply temperature when pipe heat losses are neglected.The authors state that the model can be extended to multiple heat sources.Source: §2.3.1 · p. 8

2.3.2. Hydraulic Model

• The network model combines a pump head–flow curve with the common pressure drop across parallel radiators to derive system flows.Source: §2.3.2 · p. 9
• Each valve's local PI controller uses zone-temperature error and its integral to set valve position, with anti-windup correction when the valve saturates.Source: §2.3.2 · pp. 9–10
• Each radiator's flow depends on its own valve position and the positions of other valves connected to the same pump.Source: §2.3.2 · p. 10

2.4. State-Space Representation

• The combined building, radiator, and valve dynamics are represented by a vector field in a compact state-space formulation.Source: §2.4 · p. 11
• Solar irradiation values are stored only once for walls with identical orientation to avoid redundant entries.Source: §2.4 · p. 11
• The general hydronic-radiator model is formulated for a case-study building using the state-space representation.Source: §2.4 · p. 11

2.5. Case-Study Modeling

• The case study covers the Rensen Building’s first floor, comprising six zones and a 65-state system model.The model includes six zone-temperature states and 35 wall-temperature states; the full model also includes 24 radiator-related states.Source: §2.5 · p. 11, §2.5.1 · p. 11, §2.5.2 · p. 15

2.5.1. Building-Related System Parameters

• Building parameters are obtained from construction drawings, with real measurements used later to adjust parameter estimates to actual system behavior.The model accounts for the wall materials and window properties described for the building.Source: §2.5.1 · pp. 11, 13, §2.5.2 · p. 15

2.5.2. Radiator-Related System Parameters

• A single boiler and pump supply the radiators; the model uses one supply-temperature input and four valve-setpoint inputs for zones 2–5.The pump operates at its lowest speed, and its parameters are fitted from the corresponding pump curve; some radiator-branch flow coefficients are challenging to determine accurately.Source: §2.5.2 · pp. 14–15

3.1. Monotonic Two-Block Optimization Method

• The two-block method jointly estimates the initial state, process noise, and system parameters, with nonincreasing objective cost across iterations under its stated solver condition.It alternates between state/noise and parameter updates; the nonincrease relies on local solvers not returning a worse objective than their warm starts.Source: §3.1 · pp. 17–18

3.2. Heuristic Iterative EKF–Identification Method

• The heuristic method alternates EKF state estimation with parameter identification, yielding errors almost the same as the two-block method at substantially lower computation time.Unlike the two-block method, it does not guarantee nonincreasing cost because its objective changes as the EKF state estimate is updated.Source: §3.2 · pp. 19–20, §3.2.2 · p. 22

3.2.1. Warm-Up Phase

• During the four-day warm-up, the EKF filters measured zone temperatures and produces an end-of-period state estimate for initializing identification.The EKF prediction and update steps are repeated over the warm-up data.Source: §3.2.1 · p. 20

3.2.2. Identification Phase

• Identification separates building and radiator parameters and weights residuals toward samples that excite each group; heating influence decays after valve closure to reflect stored radiator heat.Samples with open valves are informative for radiator parameters, while closed-valve samples are most informative for building parameters; setting the minimum weight near zero improves computational efficiency by reducing less-informative samples.Source: §3.2.2 · p. 21

3.3. Effect of Hydraulic Interactions on Grey-Box Modeling Accuracy

• Accounting for hydraulic interactions reduced overall RMSE from 1.03 [°C] to 0.92 [°C], approximately 11%, in the six-zone case-study building.The paper notes that the effect may be more important in larger buildings.Source: §3.3 · p. 24
• Table 4 compares open-loop RMSE with and without hydraulic balancing, but the supplied passage gives no values or winner.The comparison concerns grey-box modeling accuracy.Source: Table 4 · p. 25
• The model is used in the next section’s MPC framework to regulate radiator heat output and pursue the case-study building’s objectives.Source: §3.3 · p. 25

4. Model Predictive Control

• At each sampling instant, MPC uses an EKF state estimate and exogenous-input forecasts to predict system behavior and optimize future control inputs.The nonlinear model is linearized along operating points over the prediction horizon to form a linear time-varying affine approximation.Source: §4 · pp. 25–26
• The MPC penalizes comfort-range slack and high boiler supply temperatures, while constraining targeted-zone temperatures within time-varying comfort bounds.The comfort bounds may vary with occupancy.Source: §4 · p. 27

4.1. Numerical Case Study

• The numerical benchmarks isolate valve-control and hydraulic-interaction effects: one fixes all valves fully open, while the other controls valve positions but neglects radiator interactions.Source: §4.1 · p. 28
• 34.31 °C total comfort violation was lower than Benchmark MPC 1’s 70.90 °C and Benchmark MPC 2’s 47.22 °C, reductions of 51.6% and 27.3%, respectively.The proposed framework’s realized cost was approximately 92% higher for Benchmark MPC 1 and 33.5% higher for Benchmark MPC 2.Source: §4.1 · p. 29, Table 5 · p. 30

4.2. Experimental Validation

• 4.8 °C total comfort violation versus 18.2 °C for the benchmark—a 73.6% reduction—while using a lower cumulative boiler supply temperature in the Rensen-building experiment.The benchmark kept all valves fully open and could control only boiler supply temperature.Source: §4.2 · p. 30
• Real-building controller comparisons were conducted on consecutive days to reduce, but not eliminate, differences in exogenous inputs such as outdoor temperature.Source: §4.2 · p. 31

• The two grey-box parameter-identification approaches produced approximately the same results, while the heuristic approach was computationally efficient and attractive for buildings with multiple zones.One approach offers theoretical guarantees; the other is heuristic.Source: §5 · p. 31
• Accounting for hydraulic interactions reduced zone-temperature RMSE by 11%, while the proposed MPC reduced total comfort violations by at least 27% with lower or nearly identical cumulative boiler supply temperature.The cumulative supply temperature is used as a proxy for energy use.Source: §5 · pp. 32–33
• Future work will account for pipe heat losses and model inaccuracies and use longer experiments to assess long-term effects on energy savings and thermal comfort.Source: §5 · p. 33

• ChatGPT was used to represent radiator metal and water in Figure 3 and add graphical details to Figure 1; both figures are explanatory only.Source: Declaration of generative AI and… · p. 34