208 Research Conference Proceedings - IGSHPA Research Conference 2018 IGSHPA Research Track Stockholm September 18-20, 2018 Willem Mazzotti (willem.mazzotti@energy.kth.se) is a PhD candidate at the Royal Institute of Technology (KTH) and a consultant at the Swedish firm Bengt Dahlg ren AB (willem.mazzotti@bengtdahlgren.se ). Husni Firmansyah is a recently graduated student from the Royal Institute of Technology (KTH). Mila n Stokuca is a consultant at Bengt D ahlgren AB. José Acuña is a PhD and researcher at the Royal Institute of Technology (KTH) as well as a manager in Bengt Dahlgren AB . Björn Palm is a professor at the Royal Institute of Technology (KTH). The Newton-Raphson Method Applied to the Time-Superposed ILS for Parameter Estimation in Thermal Response Tests Willem Mazzotti Husni Firmansyah José Acuña Milan Stokuca Björn Palm ABSTRACT Thermal Response Testing is now a well-known and widely-used method allowing the determination of the local thermal or geometrical properties of a Borehole Heat Exchanger (BHE), those properties being critical in the design of GSHP systems. The analysis of TRTs is an inverse problem that has commonly been solved using an approximation of the ILS solution. To do this, however, the heat rate during a TRT must be kept constant, or least be non time-correlated, during the test, which is a challenging constraint. Applying temporal superposition to the ILS model is a way to account for varying power, although it requires the use of an optimization algorithm to minimize the error between a parametrized model and experimental values. In this paper, the Newton-Raphson method is applied to the time-superposed ILS for parameter estimation in TRTs. The parameter estimation is limited to the effective thermal conductivity and the effective borehole resistance. Analytical expressions of the first and second derivatives of the objective function, chosen as the sum of quadratic differences, are proposed, allowing to readily inverse of the Hessian matrix and speed the convergence process. The method is tried for 9 different TRTs, 2 of which are reference datasets used for validation of the method (Beier et al., 2010). Differences between estimated and reference thermal conductivities are of 3.4% and 0.4% for the first and second reference TRTs, respectively. The method is shown to be stable and consistent: for each of the 9 TRTs, 11 realizations are performed with different initial values. Convergence is reached in all cases and all realizations lead to the same final values for a given TRT. The proposed convergence method is about 70% to 90% faster than the Nelder-Mead simplex and require about 8 times less iterations in average. The convergence speed varies between 0.3 to 13.6 s with an average of 3.7 s for all TRTs. INTRODUCTION Thermal Response Testing is a well-known in-situ method used to determine the local effective thermal conductivity, � ∗ , and effective borehole thermal resistance, � �∗ , both parameters being critical for the design of Ground- Source Heat Pump (GSHP) systems and/or Borehole Thermal Energy Storage (BTES). The Thermal Response Test (TRT) method for Borehole Heat Exchangers (BHEs) was introduced by Mogensen (1983) who discussed the experimental setup as well as the analysis procedure. The first TRT units were not specifically built to be transported and mobile units only appeared later (Eklöf & Gehlin, 1996). Today, most TRT rigs are built as mobile units (IEA ECES Annex 21, 2013). A more detailed description of the historical development of TRT units can be found in Spitler and Gehlin (2015). TRTs are now performed worldwide for both commercial and research purposes (IEA ECES Annex 21, 2013; Spitler & Gehlin, 2015; Witte, 2016; Zhang et al., 2014). Several developments of the TRT method have been undertaken since Mogensen’s initial proposal, including: IGSHPA Research Track Stockholm September 18-20, 2018 Willem Mazzotti (willem. mazzotti@energy.kth.se) is a PhD ca ndidate at the Royal Institute of Technology (KTH) and a consultant at the Swedish firm Bengt Dahlgren AB (willem.mazzotti@bengtdahlgren.se ). Husni Firmansyah is a recently graduated student from the Royal Institute of Technology (KTH). Milan Stokuca is a consultant at Bengt Dahlgren AB. José Acuña is a PhD and researcher at the Royal Institute of Technology (KTH) as well as a manager in Bengt Dahlgren AB. Björn Palm is a professor at the Royal Institute of Technology (KTH). The Newton-Raphson M th d Applied to the Time-Superposed ILS for Parameter Estimation in Thermal Response Tests Willem Mazzotti Husni Firmansyah José Acuña Milan Stokuca Björn Palm ABSTRACT Thermal Response Testing is now a well-known and widely-used method allowing the determination of the local thermal or geometrical properties of a Borehole Heat Exchanger (BHE), those properties being critical in the design of GSHP systems. The analysis of TRTs is an inverse problem that has commonly been solved using an approximation of the ILS solution. To do this, however, the heat rate during a TRT must be kept constant, or least be non time-correlated, during the test, which is a challenging constraint. Applying temporal superposition to the ILS model is a way to account for varying power, although it requires the use of an optimization algorithm to minimize the error between a parametrized model and experimental values. In this paper, the Newton-Raphson method is applied to the time-superposed ILS for parameter estimation in TRTs. The parameter estimation is limited to the effective thermal conductivity and the effective borehole resistance. Analytical expressions of the first and second derivatives of the objective function, chosen as the sum of quadratic differences, are proposed, allowing to readily inverse of the Hessian matrix and speed the convergence process. The method is tried for 9 different RTs, 2 of which are reference datasets used for validation of the method (Beier et al., 2010). Differences between estimated and reference thermal conductivities are of 3.4% and 0.4% for the first and second reference TRTs, respectively. The method is shown to be stable and consistent: for each of the 9 TRTs, 11 realizations are performed with different initial values. Convergence is reached in all cases and all realizations lead to the same final values for a given TRT. The proposed convergence method is about 70% to 90% faster than the Nelder-Mead simplex and require about 8 times less iterations in average. The convergence speed varies between 0.3 to 13.6 s with an average of 3.7 s for all TRTs. INTRODUCTION Thermal Response Testing is a well-known in-situ method used to determine t e local effective thermal conductivity, � ∗ , and effective borehole thermal resistance, � �∗ , both parameters being critical for the design of Ground- Source Heat Pum (GSHP) systems and/or Borehole Thermal Energy Storage (BTES). The Thermal Response Test (TRT) method for Borehole Heat Exchangers (BHEs) was introduced by Mogensen (1983) wh discussed the experimental setup as well as the analysis procedure. The first TRT units were ot specifically built to be transported a mobile units only appeared later (Eklöf & Gehlin, 1996). T day, most TRT rigs are built as mobile units (IEA ECES Annex 21, 2013). A more detailed description of the historical development of TRT units can be found in Spitler an Gehli (2015). TRTs are now performed worldwide for both commercial and research purposes (IEA ECES Annex 21, 2013; Spitler & Gehlin, 2015; Witte, 2016; Zhang et al., 2014). Several developments of the TRT method have been undertaken since Mogensen’s initial proposal, including: IGSHPA Res arch Track Stockholm S ptember 18-20, 2018 Willem Mazzotti (willem. mazzotti@energy.kth.se) is a PhD ca ndidate at the Royal Institute of Technology (KTH) and a consultant at the Swedish firm Bengt Dahlg ren AB (willem.mazzotti@bengtdahlgren.se ). Husni Firmansyah is a recently graduated student from the Royal Institute of Technology (KT H). Milan Stokuca is a consultant at Bengt D ahlgren AB. José Acuña is a PhD and researcher at the Royal Institute of Technology (KTH) as well as a manag er in Bengt Dahlgren AB. Björn Palm is a professor at the Royal Institute of Technology (KTH). The Ne ton-Raphson ethod Applied to the Time-Superposed ILS for Para eter Esti ation in Thermal Response Tests Willem Mazzotti Husni Firmansyah José Acuña M lan Stokuca Björn Palm ABSTRACT Thermal Response Testing is now a well-known and widely-used method allowing the determination of the local thermal or geometrical properties of a Borehole H at Exchanger (BHE), those properties being critical in the design of GSHP systems. The analysis of TRTs is an inv rse problem hat has comm nly b en solved using an approximation of the ILS solution. To do this, h wever, the heat rate during a TRT must be kept constant, r least be non ti e-c rrelated, during the test, which is a challenging constraint. Applying temp ral superposition o the ILS model is a way to ac ount for varying power, although it r quires the use of an optimization al orithm to minimize the error between a arametrized model an experimental values. In this paper, the Newton-Raphson method is applied to the time-superposed ILS for parameter estimation in TRTs. The parameter estimation is limited to t e effective t rmal c nductivity and the effective borehol resistanc . Analytical ex ressions of he first a d second derivatives of the objective funct on, chosen as the sum of quadratic d ffere ces, are proposed, allowing o readily inverse of the Hessian matr x and speed the converg nce process. The method is tried for 9 different TRTs, 2 of which are reference datasets used for validation of the method (Beier et al., 2010). Differences between estimat d and reference thermal co ductivities are of 3.4% and 0.4% for th first and second reference TRTs, respectiv ly. The method is shown to be stable and consiste t: for a h of the 9 TRTs, 11 realizations are performed with differe t initial valu s. Convergence is reached in all cases and all realiza ions lea t the same final values for a given TRT. The proposed convergence method is about 70% to 90% faster than the Nelder-Mead simplex and require about 8 times less iterations in average. The convergence speed a i s between 0.3 to 13.6 s wi h an aver g of 3.7 s for al TRTs. INTRODUCTION Thermal Response Testing is a well-known i -situ method used to determine the local effective thermal conductivity, � ∗ , and effectiv borehole thermal resista ce, � �∗ , both parameters b ing critical for the d sign of Ground- S urce Heat Pump (GSHP) systems and/or Borehole Thermal Energy Storage (BTES). The Thermal Response Test (TRT) m thod for Borehole H at Exchangers (BHEs) was introduced by Mogensen (1983) who di cussed the experimen al setup as well as the nalysis procedure. The first TRT units were not specifically built to be tran por ed and mobile uni s only appeared later (Eklöf & Gehlin, 1996). oday, most TRT rigs are built as m bile units (IEA ECES Annex 21, 2013). A more de ailed description of the historical devel pment of TRT units c n be found in Spitler and Gehli (2015). TRTs are now perform d worldwide for bo h commercial and research purposes (IEA ECES Ann x 21, 2013; Spitler & Gehlin, 2015; Witte, 2016; Zhang et al., 2014). Several developments of the TRT method have be n undertaken since Mogensen’s initial proposal, including: 2018 Research Conference DOI: 10.22488/okstate.18.000039

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