ABSTRACT
Fluid temperature predictions of geothermal borefields usually involve temporal superposition of its characteristic g-function, using load aggregation schemes to reduce computational times. Assuming that the ground has linear properties, it can be modelled as a linear state-space system where the states are the aggregated loads. However, the application and accuracy of these models is compromised when the borefield is already operating and its load history is not registered or there are gaps in the data. This paper assesses the performance of state observers to estimate the borefield load history to obtain accurate fluid predictions. Results show that both Time-Varying Kalman Filter (TVKF) and Moving Horizon Estimator (MHE) provide predictions with average and maximum errors below 0.1C and 1
C, respectively. MHE outperforms TVKF in terms of n-step ahead output predictions and load history profile estimates at the expense of about five times more computational time.
Acknowledgments
The authors would like to acknowledge the funding of their research work by the EU within the H2020-EE-2016-RIA-IA programme for the project ‘Model Predictive Control and Innovative System Integration of GEOTABS;-) in Hybrid Low Grade Thermal Energy Systems - Hybrid MPC GEOTABS’ [grant number 723649 - MPC-; GT]. Furthermore, the InnoEnergy PhD School Programme and the European Institute of Technology (EIT) are acknowledged for supporting the international mobility from which this work was emerged.
This work emerged from the IBPSA Project 1, an international project conducted under the umbrella of the International Building Performance Simulation Association (IBPSA). Project 1 will develop and demonstrate a BIM/GIS and Modelica Framework for building and community energy system design and operation.
Disclosure statement
No potential conflict of interest was reported by the authors.
Nomenclature
t | = | Time (s)) |
= | Load aggregation resolution (s)) | |
H | = | Borehole length (m) |
α | = | Thermal diffusivity (m s |
τ | = | Moving-backwards time (s) |
κ | = | Weighting factor (-) |
L | = | Kalman gain |
Γ | = | Arrival cost |
S | = | Process noise matrix |
P | = | Error covariance matrix |
I | = | Identity matrix (-) |
r | = | Radius (m) |
x | = | Model state (W), (K) or ( |
= | Estimated state (W), (K) or ( | |
u | = | Model input (W) |
y | = | Model output (K) or ( |
p | = | Model parameter (kg s |
O | = | Observer |
R | = | Thermal resistance (m K W |
k | = | Thermal conductivity (W m |
n | = | Number of instances (-) |
N | = | Number of time-steps (-) |
T | = | Temperature (K) or ( |
AAOE | = | Average absolute output error (K) or ( |
AAD | = | Average absolute temperature difference (K) or ( |
= | Mass flow rate (kg s | |
ρ | = | Density (kg m |
V | = | Volume (m |
= | Specific heat capacity (J kg | |
C | = | Output matrix or Thermal capacity (J/K) |
g | = | g-Function (-) |
d | = | Buried depth (m) or model disturbance |
v | = | Model measurement noise |
w | = | Model process noise |
z | = | Axial direction |
e | = | Thickness (m) |
Q | = | Thermal energy (J) |
= | Thermal power (W) | |
= | Thermal power per unit length (W m | |
A | = | State matrix |
B | = | Input matrix |
GSHP | = | Ground source heat pump |
TES | = | Thermal energy storage |
BMS | = | Building Management System |
TV KF | = | Time-varying Kalman Filter |
MHE | = | Moving Horizon Estimator or Moving Horizon Estimation |
TABS | = | Thermally activated building system |
COP | = | Coefficient of performance |
ILS | = | Infinite line source |
FLS | = | Finite line source |
CHS | = | Cylindrical heat source |
RC | = | Resistance–capacitance |
SSM | = | State-space model |
TRT | = | Thermal response test |
KPI | = | Key performance indicator |
KPI | = | Key performance indicator |
b | = | Borehole or borehole wall |
f | = | Fluid |
g | = | Grout |
v | = | Vertical discretization |
fm | = | Mean/average fluid |
fm | = | Measured |
s | = | Soil |
x | = | State |
k | = | Time-step |
c | = | Cell |
d | = | Difference |
Q | = | Aggregated loads |
T | = | Temperatures |
agg | = | Aggregated |
acc | = | Accumulated |
out | = | Outlet |
fg | = | Fluid-to-grout |
gg | = | Grout-to-grout |
gb | = | Grout-to-wall |
e | = | Error |
sim | = | Simulation |
Notes
1 We represent the vectors with upright boldface notation while the matrices use non-bold italic notation.
2 In the literature, the process noise matrix is commonly referred as Q. We use the notation S to avoid confusion with the heat flow rates.