- Email: [email protected]
Experimental results and validation of a method to reconstruct forces on the ITER test blanket modules
Fusion Engineering and Design 96–97 (2015) 1021–1025
Contents lists available at ScienceDirect
Fusion Engineering and Design journal homepage: www.elsevier.com/locate/fusengdes
Experimental results and validation of a method to reconstruct forces on the ITER test blanket modules Christian Zeile ∗ , Ivan A. Maione Institute for Neutron Physics and Reactor Technology (INR), Karlsruhe Institute of Technology (KIT), Eggenstein-Leopoldshafen, Germany
h i g h l i g h t s • • • • •
An in operation force measurement system for the ITER EU HCPB TBM has been developed. The force reconstruction methods are based on strain measurements on the attachment system. An experimental setup and a corresponding mock-up have been built. A set of test cases representing ITER relevant excitations has been used for validation. The influence of modeling errors on the force reconstruction has been investigated.
a r t i c l e
i n f o
Article history: Received 2 September 2014 Received in revised form 26 March 2015 Accepted 8 April 2015 Available online 29 April 2015 Keywords: Force reconstruction Test blanket module
a b s t r a c t In order to reconstruct forces on the test blanket modules in ITER, two force reconstruction methods, the augmented Kalman filter and a model predictive controller, have been selected and developed to estimate the forces based on strain measurements on the attachment system. A dedicated experimental setup with a corresponding mock-up has been designed and built to validate these methods. A set of test cases has been defined to represent possible excitation of the system. It has been shown that the errors in the estimated forces mainly depend on the accuracy of the identified model used by the algorithms. Furthermore, it has been found that a minimum of 10 strain gauges is necessary to allow for a low error in the reconstructed forces. © 2015 Karlsruhe Institute of Technology. Published by Elsevier B.V. All rights reserved.
1. Introduction The blanket systems experience high mechanical and thermal loads in a fusion reactor. The high mechanical loads are mainly related to static and transient electromagnetic effects: as the blankets are made of a magnetic material, Maxwell forces are pulling the structure toward the plasma. In addition, eddy currents are induced in the conductive structure during electromagnetic transients (e.g. plasma disruptions or vertical displacement events (VDE), which interact with the magnetic fields to generate high Lorentz forces. Therefore, particularly with regard to the development of blanket system for future fusion reactors, e.g. DEMO, it is of high importance to develop the engineering models and codes to a high degree of confidence and to check the accuracy of theoretically calculated effects of the environmental conditions on the blanket systems. For this purpose, a force reconstruction method can be applied to the
test blanket modules (TBM) in ITER to reconstruct the forces acting on the structure during operation. Two force reconstruction methods, namely the augmented Kalman filter (AKF) and the model predictive controller (MPC), identified as suitable for the application to the TBM in ITER, have been developed and validated by means of a dedicated experimental setup, as reported in [1]. Both algorithms work with measurements of strain sensors placed on the attachment system, which connects the TBM box to the shield. For the validation of the force reconstruction methods, the attachment system concept based on a cylindrical element [2] has been taken as reference. Fig. 1 shows this concept together with the EU-Helium Cooled Pebble Bed TBM (HCPB-TBM). Nevertheless, the application of the methods is not limited to this concept.
2. Force reconstruction methods
∗ Corresponding author. E-mail address: [email protected] (C. Zeile).
The force reconstruction methods suitable for the application on the TBM in ITER have to able to reconstruct transient forces that are distributed over the structure. The force reconstruction
http://dx.doi.org/10.1016/j.fusengdes.2015.04.028 0920-3796/© 2015 Karlsruhe Institute of Technology. Published by Elsevier B.V. All rights reserved.
1022
C. Zeile, I.A. Maione / Fusion Engineering and Design 96–97 (2015) 1021–1025
Fig. 1. TBM box with attachment system [1].
methods can be grouped into three categories: deterministic methods, stochastic methods and methods based on artificial intelligence [3]. Stochastic methods use a regression model to estimate forces from measured parameters. However, the main field of application is not the reconstruction of transient forces. In addition, the regression model is obtained by a minimization on sample data. For a high reliability of the force reconstruction it has to be assured that the sample data represents well the real conditions. The same issue applies for the self-learning process of methods based on artificial intelligence, which is also based on sample data. This problem does not occur for deterministic methods as the model is based on a model identification and not on sample data. Therefore, two deterministic methods, the augmented Kalman filter (AKF) [4], an unknown input observer, and the model predictive controller (MPC), an optimization algorithm, have been selected. Both algorithms are based on a reduced order modal model of the system and consider a regularization parameter to ease the solution of the inverse problem. In addition, the iterative algorithms are able to compensate modeling errors by continuously updating the states of the system. Due to the predictor–corrector nature of the AKF, a certain time delay in the reconstructed forces can occur, if a high amount of regularization is specified. Although this time delay can be detected and reduced, the MPC does not face this problem. On the other hand, the AKF has a significantly shorter computation time. As the sensors have to be placed in such a way to detect all considered mode shapes with a signal as high as possible, a method based on a genetic algorithm has been developed to optimize the placement of the sensors considering the total number of sensors. Both force reconstruction methods and the integrated models are discussed in detail in [1]. 3. Experimental setup and results The experimental study of the force reconstruction methods is based on a mock-up in combination with a set of test cases to represent the TBM as well as the possible excitations of the system. In order to excite the system, a dedicated test device has been built that is able to apply punctual forces by two solenoids with a force up to 2000 N. In addition, the mock-up is equipped with 16 strain gauges. 3.1. Mock-up and test cases The mock-up in Fig. 2 has been designed in such a way that it represents well the modal behavior of the TBM. This means that mock-up and the TBM can both be described by a modal model containing the first six modes with identical modes shapes, which have been selected by the effective modal mass criterion. This assures
Fig. 2. Mock-up with strain gauges.
that the mock-up reacts to excitations in the same way as the TBM. In combination with a pipe as cylindrical element representing the attachment system, it can also be assured that the knowledge gained concerning the number and placement of the sensors can be transferred to the real TBM. The difference in the eigenfrequencies of the numerical model and real system in Table 1 also points out the necessity to conduct an experimental modal analysis for the application of the force reconstruction on the real TBM. The experimental setup used to excite the structure allows for the application of punctual forces as described in detail in [1]. As the forces acting on the TBM due to electromagnetic effects are distributed over the TBM box, the mock-up has been design in such a way that the punctually applied forces cause the same reactions of the mock-up as distributed forces. This has been confirmed by numerical modal analyses comparing the eigenfrequencies and eigenvectors of a model where the top plate is represented by a rigid body and a model with a flexible top plate. A set of test cases has been defined in order to represent a complete set of possible excitations of the system. A test case consists of one or two force application points on the top plate and a time history of the force at each force application point. The time histories are based on the duration of plasma disruptions and VDE in the range of tens of ms. In addition, rise and fall times of the forces from 2 to 10 ms have been considered to represent typical rise and fall times as expected during a plasma disruption in ITER. The in total 14 test cases cover the excitations of different combinations of the modes considered in the model. Therefore, although the excitation pattern of the TBM during a plasma disruption in ITER is only
Table 1 Eigenfrequencies of considered modes. Mode no.
TBM (Hz)
Mock-up (numerical) (Hz)
Mock-up (experimental) (Hz)
1 2 3 4 5 6
65 91 112 260 286 417
42 52 80 279 298 360
40 48 82 262 267 336
C. Zeile, I.A. Maione / Fusion Engineering and Design 96–97 (2015) 1021–1025
Fig. 3. Example of reconstructed force component in x-direction.
known to a certain degree, this pattern is still a combination of the excitation patterns of the test cases. 3.2. Experimental results The experimental setup was used to apply the defined set of test cases to the mock-up. As the force reconstruction algorithms can only estimate a limited number of independent inputs, the model has been defined in such a way that the estimated input vector [Fx , Fy , Fz , Mx , My , Mz ] corresponds to the resultant forces and moments of the applied forces in relation to the center of the top plate. The regularization parameter has been chosen individually for each test case by means of the L-curve method [5]. For the AKF, a method to reduce the time delay by comparing the time shift between recorded strain measurements and reconstructed strain measurements have been applied. All 16 strain sensors have been used as input for the force reconstruction methods. Figs. 3 and 4 show the result of a test case where a single force is applied at the top plate as indicated in the figure. The two graphs only show the components of the estimated input vector for which the input is non-zero. The force in x-direction Fx as well as the moment around the z-axis Mz are well estimated. The reconstruction of the impact force of about 2 ms in Fig. 5 shows the already mentioned small time delay in the reconstruction by the AKF, which cannot be completely removed. As a force reconstruction algorithm in combination with a structure has the task of a force transducer, the attempt has been made to define an accuracy of the force reconstruction methods in analogy to a standard force transducer. Nevertheless, it should be noted that a force transducer is a direct measurement method and the accuracy definitions in the corresponding guidelines do not consider
1023
Fig. 5. Example of time delay for AKF reconstruction.
transient forces in an appropriate way. Table 2 lists the relative linearity error based on the conservative definition of the upper limit of the measurement range as maximum force of the corresponding test case. The given values represent the lower limit of the accuracy for each force component considering all test cases. 4. Influence of errors in the model As the force reconstruction methods use a modal model of the system, a model identification by an experimental modal analysis is necessary in order to apply the methods to experimentally obtained strain data. In contrast, if the force reconstruction methods are applied to test cases that are simulated with a FEA software, the modal model can be directly extracted by a numerically solved modal analysis. While this model corresponds exactly to the model used in the simulation in terms of a reduced-order model, the modal model obtained by an experimental modal analysis represents the real system only to a certain degree. Hence, in order to investigate the influence of modeling errors on the force reconstruction, errors are intentionally introduced in the numerical modal model and the methods are applied to simulated strain data of the defined test cases. For this purpose, three parameters have been identified in previous test to have the strongest influence on the error in the reconstructed forces: the errors in the eigenfrequencies and eigenvectors of the modal model used by the algorithms and the number of strain sensors considered for estimation of the input forces. As the strain data is simulated by the FEA software ANSYS, the perfectly matching modal model is calculated and the errors in the eigenfrequencies and eigenvectors are intentionally inserted into the model. In order to evaluate the influence of the modeling errors and the number of sensors on the reconstructed forces, a performance criterion has to be defined. In literature related to force reconstruction, there exists no consistent definition of such a criterion. Relative, absolute or performance measures based on a single point in time are used. As the force reconstruction of electro-magnetic forces deals with the reconstruction of transient forces, not only the accuracy of the estimated magnitude of the force is relevant, but also the accuracy of the reconstructed time history. For this reason, a performance criterion based on a single point in time is not suitable. Furthermore, a relative performance criterion cannot be defined as Table 2 Relative linearity error.
Fig. 4. Example of reconstructed moment around z-axis.
Fx
Fy
Fz
Mx
My
Mz
12%
10%
10%
25%
28%
9%
1024
C. Zeile, I.A. Maione / Fusion Engineering and Design 96–97 (2015) 1021–1025
Fig. 6. Schematic time history of excitation forces used for the evaluation of RSMEF and RSMEM .
the estimated input vector may contain zero components. Hence, an absolute performance criterion based on the root-mean-square error (RMSE) was defined as an intuitive performance criterion with the same unit as the reconstructed force component to evaluate the influence modeling errors on the force reconstruction. As the estimated input vector contains forces and moments, two different RMSE have been defined:
n k=1
RMSEF =
2
2
(Fˆx,k − Fx,k ) + (Fˆy,k − Fy,k ) + (Fˆz,k − Fz,k )
2
(1)
n
RMSEM
=
n k=1
2
2
ˆ x,k − Mx,k ) + (M ˆ y,k − My,k ) + (M ˆ z,k − Mz,k ) (M n
2
(2)
with time step k, the length of the considered period n, the reconstructed force components Fˆx,k , Fˆy,k , Fˆz,k , the reference force components Fx,k , Fy,k , Fz,k , the reconstructed moment components ˆ y,k , M ˆ z,k and the reference moment components Mx,k , My,k , ˆ x,k , M M Mz,k . Although the RSME scales with the magnitude of the input forces, the influence of the modeling errors or numbers of sensors can be easily assessed by means of the relative change of the RSME. Therefore, the average RSMEF and RSMEM of all test cases with a maximum force magnitude of 1000 N during the constant time period of 20 ms according to the schematic time history in Fig. 6 has been calculated. Figs. 7 and 8 show the two RSME depending on the force reconstruction algorithm, the number of sensors (NumS) and the error in the eigenvectors of the model. The average RSMEF is for both algorithms and all sensor configurations at a low level of about 50 N for an error in the eigenvectors of 0% compared 2400 N for an error in the eigenvectors of 30%. Starting from an error in the eigenvectors of 10%, the influence of the number of sensors
Fig. 7. Averaged RMSEF in function of eigenvectors error and sensor number (eigenfrequencies error 0%).
Fig. 8. Averaged RMSEM in function of eigenvectors error and sensor number (eigenfrequencies error 0%).
becomes more significant. An error in the eigenfrequencies of 5%, which is not shown in the figure, increases the RSMEF by about 20% each. The same observations can be made for the RSMEM . From this assessment it can be seen that the difference in the RSMEF and RSMEM between MPC and AKF is small compared to the increase of both RSME due to modeling errors. Therefore, it can be concluded that the RSME mainly depends on the accuracy of the model as both algorithms use the same model. Nevertheless, the RSME of the MPC is in general 20% lower, as the AKF always shows a minor time shift in the reconstructed forces. Regarding the number of sensors, 6 sensors are only sufficient for a model with an error in the eigenfrequencies and eigenvectors close to 0%. As this requirement is very unlikely to be met by an experimental modal analysis, 10 sensors is the minimum that should be used in the application of the system. Nevertheless, in order to be able to compensate for the failure of single sensors, an even higher number should be considered.
5. Conclusion Two methods, the AKF and MPC, have been selected and developed to reconstruct forces on the TBM during operation to check the accuracy of theoretically calculated effects on the blanket system and therefore validate the applied engineering models and codes. A dedicated experimental setup and corresponding mock-up has been designed and built to validate the force reconstruction methods. The application of the defined test cases to a mock-up by a testing device has demonstrated the overall feasibility of the proposed methods. The capability of the force reconstruction methods to reconstruct the force time histories has been illustrated by the results of a sample test case. The obtained results show the possibility to reconstruct loads with a lower limit of the relative linearity error of about 10% for the force components and about 20% for the moment components. Nevertheless, it has to be noted that this error strongly depends on the accuracy of the model identified by the experimental modal analysis. As a standard method has been used for the experimental modal analysis in this study, a lower error is very likely to be achieved with more sophisticated methods, e.g. polyreference curve fitting methods. Although the application of the force reconstruction methods have been demonstrated on a concept of a cylindrical attachment system, the methods can be easily transferred to other attachment system concepts, as the methods are mainly based on the design and boundary conditions of the TBM in ITER.
C. Zeile, I.A. Maione / Fusion Engineering and Design 96–97 (2015) 1021–1025
References [1] C. Zeile, I.A. Maione, Design and setup of a testing device to investigate a reduced sized attachment system mock up for the ITER EU HCPB-TBM under different mechanical loading conditions, Fusion Eng. Des. 89 (2014) 1284–1288. [2] C. Zeile, H. Neuberger, A revised design approach of the attachment system for the ITER EU-HCPB-TBM based on a central cylindrical connection element, Fusion Eng. Des. 87 (2012) 859–863.
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[3] T. Uhl, The inverse identification problem and its technical application, Arch. Appl. Mech. 77 (2007) 325–337. [4] E. Lourens, E. Reynders, G. De Roeck, G. Degrande, G. Lombaert, An augmented Kalman filter for force identification in structural dynamics, Mech. Syst. Signal Process. 27 (2012) 446–460. [5] P.C. Hansen, Analysis of discrete ill-posed problems by means of the L-curve, SIAM Rev. 34 (1992) 561–580.
Contents lists available at ScienceDirect
Fusion Engineering and Design journal homepage: www.elsevier.com/locate/fusengdes
Experimental results and validation of a method to reconstruct forces on the ITER test blanket modules Christian Zeile ∗ , Ivan A. Maione Institute for Neutron Physics and Reactor Technology (INR), Karlsruhe Institute of Technology (KIT), Eggenstein-Leopoldshafen, Germany
h i g h l i g h t s • • • • •
An in operation force measurement system for the ITER EU HCPB TBM has been developed. The force reconstruction methods are based on strain measurements on the attachment system. An experimental setup and a corresponding mock-up have been built. A set of test cases representing ITER relevant excitations has been used for validation. The influence of modeling errors on the force reconstruction has been investigated.
a r t i c l e
i n f o
Article history: Received 2 September 2014 Received in revised form 26 March 2015 Accepted 8 April 2015 Available online 29 April 2015 Keywords: Force reconstruction Test blanket module
a b s t r a c t In order to reconstruct forces on the test blanket modules in ITER, two force reconstruction methods, the augmented Kalman filter and a model predictive controller, have been selected and developed to estimate the forces based on strain measurements on the attachment system. A dedicated experimental setup with a corresponding mock-up has been designed and built to validate these methods. A set of test cases has been defined to represent possible excitation of the system. It has been shown that the errors in the estimated forces mainly depend on the accuracy of the identified model used by the algorithms. Furthermore, it has been found that a minimum of 10 strain gauges is necessary to allow for a low error in the reconstructed forces. © 2015 Karlsruhe Institute of Technology. Published by Elsevier B.V. All rights reserved.
1. Introduction The blanket systems experience high mechanical and thermal loads in a fusion reactor. The high mechanical loads are mainly related to static and transient electromagnetic effects: as the blankets are made of a magnetic material, Maxwell forces are pulling the structure toward the plasma. In addition, eddy currents are induced in the conductive structure during electromagnetic transients (e.g. plasma disruptions or vertical displacement events (VDE), which interact with the magnetic fields to generate high Lorentz forces. Therefore, particularly with regard to the development of blanket system for future fusion reactors, e.g. DEMO, it is of high importance to develop the engineering models and codes to a high degree of confidence and to check the accuracy of theoretically calculated effects of the environmental conditions on the blanket systems. For this purpose, a force reconstruction method can be applied to the
test blanket modules (TBM) in ITER to reconstruct the forces acting on the structure during operation. Two force reconstruction methods, namely the augmented Kalman filter (AKF) and the model predictive controller (MPC), identified as suitable for the application to the TBM in ITER, have been developed and validated by means of a dedicated experimental setup, as reported in [1]. Both algorithms work with measurements of strain sensors placed on the attachment system, which connects the TBM box to the shield. For the validation of the force reconstruction methods, the attachment system concept based on a cylindrical element [2] has been taken as reference. Fig. 1 shows this concept together with the EU-Helium Cooled Pebble Bed TBM (HCPB-TBM). Nevertheless, the application of the methods is not limited to this concept.
2. Force reconstruction methods
∗ Corresponding author. E-mail address: [email protected] (C. Zeile).
The force reconstruction methods suitable for the application on the TBM in ITER have to able to reconstruct transient forces that are distributed over the structure. The force reconstruction
http://dx.doi.org/10.1016/j.fusengdes.2015.04.028 0920-3796/© 2015 Karlsruhe Institute of Technology. Published by Elsevier B.V. All rights reserved.
1022
C. Zeile, I.A. Maione / Fusion Engineering and Design 96–97 (2015) 1021–1025
Fig. 1. TBM box with attachment system [1].
methods can be grouped into three categories: deterministic methods, stochastic methods and methods based on artificial intelligence [3]. Stochastic methods use a regression model to estimate forces from measured parameters. However, the main field of application is not the reconstruction of transient forces. In addition, the regression model is obtained by a minimization on sample data. For a high reliability of the force reconstruction it has to be assured that the sample data represents well the real conditions. The same issue applies for the self-learning process of methods based on artificial intelligence, which is also based on sample data. This problem does not occur for deterministic methods as the model is based on a model identification and not on sample data. Therefore, two deterministic methods, the augmented Kalman filter (AKF) [4], an unknown input observer, and the model predictive controller (MPC), an optimization algorithm, have been selected. Both algorithms are based on a reduced order modal model of the system and consider a regularization parameter to ease the solution of the inverse problem. In addition, the iterative algorithms are able to compensate modeling errors by continuously updating the states of the system. Due to the predictor–corrector nature of the AKF, a certain time delay in the reconstructed forces can occur, if a high amount of regularization is specified. Although this time delay can be detected and reduced, the MPC does not face this problem. On the other hand, the AKF has a significantly shorter computation time. As the sensors have to be placed in such a way to detect all considered mode shapes with a signal as high as possible, a method based on a genetic algorithm has been developed to optimize the placement of the sensors considering the total number of sensors. Both force reconstruction methods and the integrated models are discussed in detail in [1]. 3. Experimental setup and results The experimental study of the force reconstruction methods is based on a mock-up in combination with a set of test cases to represent the TBM as well as the possible excitations of the system. In order to excite the system, a dedicated test device has been built that is able to apply punctual forces by two solenoids with a force up to 2000 N. In addition, the mock-up is equipped with 16 strain gauges. 3.1. Mock-up and test cases The mock-up in Fig. 2 has been designed in such a way that it represents well the modal behavior of the TBM. This means that mock-up and the TBM can both be described by a modal model containing the first six modes with identical modes shapes, which have been selected by the effective modal mass criterion. This assures
Fig. 2. Mock-up with strain gauges.
that the mock-up reacts to excitations in the same way as the TBM. In combination with a pipe as cylindrical element representing the attachment system, it can also be assured that the knowledge gained concerning the number and placement of the sensors can be transferred to the real TBM. The difference in the eigenfrequencies of the numerical model and real system in Table 1 also points out the necessity to conduct an experimental modal analysis for the application of the force reconstruction on the real TBM. The experimental setup used to excite the structure allows for the application of punctual forces as described in detail in [1]. As the forces acting on the TBM due to electromagnetic effects are distributed over the TBM box, the mock-up has been design in such a way that the punctually applied forces cause the same reactions of the mock-up as distributed forces. This has been confirmed by numerical modal analyses comparing the eigenfrequencies and eigenvectors of a model where the top plate is represented by a rigid body and a model with a flexible top plate. A set of test cases has been defined in order to represent a complete set of possible excitations of the system. A test case consists of one or two force application points on the top plate and a time history of the force at each force application point. The time histories are based on the duration of plasma disruptions and VDE in the range of tens of ms. In addition, rise and fall times of the forces from 2 to 10 ms have been considered to represent typical rise and fall times as expected during a plasma disruption in ITER. The in total 14 test cases cover the excitations of different combinations of the modes considered in the model. Therefore, although the excitation pattern of the TBM during a plasma disruption in ITER is only
Table 1 Eigenfrequencies of considered modes. Mode no.
TBM (Hz)
Mock-up (numerical) (Hz)
Mock-up (experimental) (Hz)
1 2 3 4 5 6
65 91 112 260 286 417
42 52 80 279 298 360
40 48 82 262 267 336
C. Zeile, I.A. Maione / Fusion Engineering and Design 96–97 (2015) 1021–1025
Fig. 3. Example of reconstructed force component in x-direction.
known to a certain degree, this pattern is still a combination of the excitation patterns of the test cases. 3.2. Experimental results The experimental setup was used to apply the defined set of test cases to the mock-up. As the force reconstruction algorithms can only estimate a limited number of independent inputs, the model has been defined in such a way that the estimated input vector [Fx , Fy , Fz , Mx , My , Mz ] corresponds to the resultant forces and moments of the applied forces in relation to the center of the top plate. The regularization parameter has been chosen individually for each test case by means of the L-curve method [5]. For the AKF, a method to reduce the time delay by comparing the time shift between recorded strain measurements and reconstructed strain measurements have been applied. All 16 strain sensors have been used as input for the force reconstruction methods. Figs. 3 and 4 show the result of a test case where a single force is applied at the top plate as indicated in the figure. The two graphs only show the components of the estimated input vector for which the input is non-zero. The force in x-direction Fx as well as the moment around the z-axis Mz are well estimated. The reconstruction of the impact force of about 2 ms in Fig. 5 shows the already mentioned small time delay in the reconstruction by the AKF, which cannot be completely removed. As a force reconstruction algorithm in combination with a structure has the task of a force transducer, the attempt has been made to define an accuracy of the force reconstruction methods in analogy to a standard force transducer. Nevertheless, it should be noted that a force transducer is a direct measurement method and the accuracy definitions in the corresponding guidelines do not consider
1023
Fig. 5. Example of time delay for AKF reconstruction.
transient forces in an appropriate way. Table 2 lists the relative linearity error based on the conservative definition of the upper limit of the measurement range as maximum force of the corresponding test case. The given values represent the lower limit of the accuracy for each force component considering all test cases. 4. Influence of errors in the model As the force reconstruction methods use a modal model of the system, a model identification by an experimental modal analysis is necessary in order to apply the methods to experimentally obtained strain data. In contrast, if the force reconstruction methods are applied to test cases that are simulated with a FEA software, the modal model can be directly extracted by a numerically solved modal analysis. While this model corresponds exactly to the model used in the simulation in terms of a reduced-order model, the modal model obtained by an experimental modal analysis represents the real system only to a certain degree. Hence, in order to investigate the influence of modeling errors on the force reconstruction, errors are intentionally introduced in the numerical modal model and the methods are applied to simulated strain data of the defined test cases. For this purpose, three parameters have been identified in previous test to have the strongest influence on the error in the reconstructed forces: the errors in the eigenfrequencies and eigenvectors of the modal model used by the algorithms and the number of strain sensors considered for estimation of the input forces. As the strain data is simulated by the FEA software ANSYS, the perfectly matching modal model is calculated and the errors in the eigenfrequencies and eigenvectors are intentionally inserted into the model. In order to evaluate the influence of the modeling errors and the number of sensors on the reconstructed forces, a performance criterion has to be defined. In literature related to force reconstruction, there exists no consistent definition of such a criterion. Relative, absolute or performance measures based on a single point in time are used. As the force reconstruction of electro-magnetic forces deals with the reconstruction of transient forces, not only the accuracy of the estimated magnitude of the force is relevant, but also the accuracy of the reconstructed time history. For this reason, a performance criterion based on a single point in time is not suitable. Furthermore, a relative performance criterion cannot be defined as Table 2 Relative linearity error.
Fig. 4. Example of reconstructed moment around z-axis.
Fx
Fy
Fz
Mx
My
Mz
12%
10%
10%
25%
28%
9%
1024
C. Zeile, I.A. Maione / Fusion Engineering and Design 96–97 (2015) 1021–1025
Fig. 6. Schematic time history of excitation forces used for the evaluation of RSMEF and RSMEM .
the estimated input vector may contain zero components. Hence, an absolute performance criterion based on the root-mean-square error (RMSE) was defined as an intuitive performance criterion with the same unit as the reconstructed force component to evaluate the influence modeling errors on the force reconstruction. As the estimated input vector contains forces and moments, two different RMSE have been defined:
n k=1
RMSEF =
2
2
(Fˆx,k − Fx,k ) + (Fˆy,k − Fy,k ) + (Fˆz,k − Fz,k )
2
(1)
n
RMSEM
=
n k=1
2
2
ˆ x,k − Mx,k ) + (M ˆ y,k − My,k ) + (M ˆ z,k − Mz,k ) (M n
2
(2)
with time step k, the length of the considered period n, the reconstructed force components Fˆx,k , Fˆy,k , Fˆz,k , the reference force components Fx,k , Fy,k , Fz,k , the reconstructed moment components ˆ y,k , M ˆ z,k and the reference moment components Mx,k , My,k , ˆ x,k , M M Mz,k . Although the RSME scales with the magnitude of the input forces, the influence of the modeling errors or numbers of sensors can be easily assessed by means of the relative change of the RSME. Therefore, the average RSMEF and RSMEM of all test cases with a maximum force magnitude of 1000 N during the constant time period of 20 ms according to the schematic time history in Fig. 6 has been calculated. Figs. 7 and 8 show the two RSME depending on the force reconstruction algorithm, the number of sensors (NumS) and the error in the eigenvectors of the model. The average RSMEF is for both algorithms and all sensor configurations at a low level of about 50 N for an error in the eigenvectors of 0% compared 2400 N for an error in the eigenvectors of 30%. Starting from an error in the eigenvectors of 10%, the influence of the number of sensors
Fig. 7. Averaged RMSEF in function of eigenvectors error and sensor number (eigenfrequencies error 0%).
Fig. 8. Averaged RMSEM in function of eigenvectors error and sensor number (eigenfrequencies error 0%).
becomes more significant. An error in the eigenfrequencies of 5%, which is not shown in the figure, increases the RSMEF by about 20% each. The same observations can be made for the RSMEM . From this assessment it can be seen that the difference in the RSMEF and RSMEM between MPC and AKF is small compared to the increase of both RSME due to modeling errors. Therefore, it can be concluded that the RSME mainly depends on the accuracy of the model as both algorithms use the same model. Nevertheless, the RSME of the MPC is in general 20% lower, as the AKF always shows a minor time shift in the reconstructed forces. Regarding the number of sensors, 6 sensors are only sufficient for a model with an error in the eigenfrequencies and eigenvectors close to 0%. As this requirement is very unlikely to be met by an experimental modal analysis, 10 sensors is the minimum that should be used in the application of the system. Nevertheless, in order to be able to compensate for the failure of single sensors, an even higher number should be considered.
5. Conclusion Two methods, the AKF and MPC, have been selected and developed to reconstruct forces on the TBM during operation to check the accuracy of theoretically calculated effects on the blanket system and therefore validate the applied engineering models and codes. A dedicated experimental setup and corresponding mock-up has been designed and built to validate the force reconstruction methods. The application of the defined test cases to a mock-up by a testing device has demonstrated the overall feasibility of the proposed methods. The capability of the force reconstruction methods to reconstruct the force time histories has been illustrated by the results of a sample test case. The obtained results show the possibility to reconstruct loads with a lower limit of the relative linearity error of about 10% for the force components and about 20% for the moment components. Nevertheless, it has to be noted that this error strongly depends on the accuracy of the model identified by the experimental modal analysis. As a standard method has been used for the experimental modal analysis in this study, a lower error is very likely to be achieved with more sophisticated methods, e.g. polyreference curve fitting methods. Although the application of the force reconstruction methods have been demonstrated on a concept of a cylindrical attachment system, the methods can be easily transferred to other attachment system concepts, as the methods are mainly based on the design and boundary conditions of the TBM in ITER.
C. Zeile, I.A. Maione / Fusion Engineering and Design 96–97 (2015) 1021–1025
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