Fluid temperature predictions of geothermal borefields using load estimations via state observers

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 0.1${}^{\circ }$C and 1${}^{\circ }$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.

Keywords:

• Geothermal modelling
• fluid temperature prediction
• load estimation
• state observers
• Kalman filter
• moving horizon estimation

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))
$\mathrm{\Delta }t$	=	Load aggregation resolution (s))
H	=	Borehole length (m)
α	=	Thermal diffusivity (m s${}^{-2}$)
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 (${}^{\circ }$C), or shanking space (m)
$\hat{x}$	=	Estimated state (W), (K) or (${}^{\circ }$C)
u	=	Model input (W)
y	=	Model output (K) or (${}^{\circ }$C)
p	=	Model parameter (kg s${}^{-1}$)
O	=	Observer
R	=	Thermal resistance (m K W${}^{-1}$) or measurement noise matrix
k	=	Thermal conductivity (W m${}^{-1}$ K${}^{-1}$)
n	=	Number of instances (-)
N	=	Number of time-steps (-)
T	=	Temperature (K) or (${}^{\circ }$C)
AAOE	=	Average absolute output error (K) or (${}^{\circ }$C)
AAD	=	Average absolute temperature difference (K) or (${}^{\circ }$C)
$\dot{m}$	=	Mass flow rate (kg s${}^{-1}$)
ρ	=	Density (kg m${}^{-3}$)
V	=	Volume (m${}^3$)
$c_p$	=	Specific heat capacity (J kg${}^{-1}$ K${}^{-1}$)
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)
$\dot{Q}$	=	Thermal power (W)
$\dot{q}$	=	Thermal power per unit length (W m${}^{-1}$)
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.

Additional information

Funding

This work was supported by Horizon 2020 Framework Programme [723649].