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Active Fault Detection and Identification using Transient Data
Antonio Espuña, Moisès Graells and Luis Puigjaner (Editors), Proceedings of the 27th European Symposium on Computer Aided Process Engineering – ESCAPE 27 October 1st - 5th, 2017, Barcelona, Spain © 2017 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/B978-0-444-63965-3.50283-X
Active Fault Detection and Identification using Transient Data Kyle A. Palmera, George M. Bollasa a
Department of Chemical & Biolmolecular Engineering, University of Connecticut, 191 Auditorium Road, Unit 3222, Storrs, CT, 06269-3222, USA, [email protected]
Abstract Active Fault Detection and Isolation (FDI) methods can improve the detection of residuals between measured and expected outputs, isolate fault cause(s) and, thus, identify faults with higher confidence than conventional passive methods. This work focuses on quantifiably illustrating the value of transient experiments in active FDI, as compared to steady-state testing. Fault identification is treated as a set of constrained steady-state or dynamic optimization problems using D-Optimality as the objective of the test design. The identifiability of faults is examined using steady-state and transient measurements in two testbed systems with uncertainty in their parameters and boundaries. The inclusion of transient information during FDI decreases fault identification error. This is illustrated through application of the framework on a plate fin heat exchanger of an aircraft environmental control system (ECS). Keywords: active fault diagnosis, fault detection, optimization, dynamic systems
1. Introduction With modern engineered systems becoming more complex in their design and functionality, active FDI methods have gathered interest over the last decade (Niemann 2006). Traditional FDI techniques focus on the design of robust filters as well as thresholds for the sensors used for the detection of faults. Methods such as unknown input observers (Zarei and Poshtan 2010) are commonly used to optimize the calculation of residuals so that they are sensitive to faults and insensitive to uncertain or unknown disturbances. Another way to generate residuals sensitive to faults is by adjusting the admissible inputs to create an optimal test for active FDI. In this work, FDI test design is handled as a set of constrained optimization problems focused on maximizing information about the test. The system inputs are manipulated to improve the identifiability of system faults and uncertain conditions by maximizing information with respect to faults in the form of sensitivities of outputs. In a classical model-based design of experiments approach (Han et al. 2016a,b), steady-state or dynamic tests are designed to maximize the identifiability of fault(s), expressed in the system model as parameter(s). For a given number of system sensors the system uncertainty is also estimated (if identifiable). The test designed can be steady-state or transient, depending on the model structure and the accuracy of the system actuators. The FDI test optimization is applied for the identification of fouling in a heat exchanger with unknown moisture content its fluid streams, and uncertainty in the cold stream inlet temperature and mass flow rate.
2. Methods The FDI test designs presented in this work use physics-based models or semi-empirical correlations that are assumed to accurately represent the tested system. These models are
K. Palmer et al.
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often written as series of implicit equations, shown in Eq.(1), where f is the system governing equations, x the vector of state variables, u the controllable inputs, T the model parameters, t time, and yˆ are the estimated measurable outputs. f x t , x t , u t , T, t 0, yˆ
(1)
h x t , u t , T, t .
If the system is at steady-state, then t and x t are omitted. The parameter vector is split into three subvectors to represent model faults, T f , uncertain parameters, Tu , and design parameters, T p . The input vector, u, is split in a similar manner into certain system inputs, up, and uncertain inputs, uu. The target set of faults and uncertain parameters and variables that can impact the accuracy of fault detection are compiled into a vector, [ : [
ª¬T f , Tu º¼ >uu @.
The vector, [ , is assigned an anticipated set of values, [ , containing an estimate of the faults and mean value of the uncertain inputs and parameters. The test design vector, M , is used to determine the optimal FDI test within the design space, ) . Typically, M comprises the admissible input trajectories, up, number of tests, Ntest, initial system states, y0, sampling times, tsp, and overall timespan, W . In this work, M contains only the inputs, up, and the number of test, Ntest. Table 1 shows the optimal test design formulations for steady-state and dynamic tests, where the optimization objective is the minimization of the determinant of the inverse of the Fisher Information Matrix, H [ (D-optimality). Table 1. Mathematical formulation of the FDI steady-state and dynamic test design
Steady-state M
* StS
ªu , , u ¬ * p ,1
Dynamic * * p , Nttest
,N
M*Dyn
p
(2)
p
p
U p
L
s.t. f x t , x t , u p t , T p , [, t
0,
yˆ t h x t , u p t , T p , [, t ,
p
u d u p d u , x d x t d x . L p
* ª¬u*p t , N test t 1º¼ arg min det ª H [1 [, M º ¼ ¬ M)
f x, u , T , [ 0, yˆ h x, u , T , [ ,
arg min det ª H [1 ¬ M) s.t.
t 1º ¼ º [, M ¼
* test
U
(3)
f x t0 , x t0 , u p t0 , T p , [, t0 0, ° ® °¯ yˆ t0 h x t0 , u p t0 , T p , [, t0 , L u p d u p t d uUp , t >0, W @ , y0
x L d x t d xU , t >0, W @.
3. Results and Discussion 3.1. Heat Exchanger Fouling Identification Particulate fouling on the ECS heat exchanger significantly reduces heat transfer efficiency and performance, leading to increased maintenance costs and peripheral component failures. Here, the method described in Section 2 is employed to improve the identifiability of fouling in the heat exchanger of Fig. 1, which is part of an ECS described elsewhere (Palmer et al. 2016). In brief, as shown in Fig. 1, the plate fin heat exchanger
Active Fault Detection and Identification using Transient Data
1689
Figure 1. Input-Output architecture of the ECS cross-flow plate fin heat exchanger.
has one admissible input, two measured outputs, one fault and three uncertain variables. The target of FDI is to quantify the extent of the fault (fouling) in lieu of the uncertainty in system inputs. The faults and uncertain inputs considered in this document are listed in Table 2, along with their anticipated values, lower and upper bounds. Fouling is expressed in the model as the thermal fouling resistance, Rf, although it also impacts the cross-sectional area available for flow (along with flow characteristics). Uncertainty exists in air moisture, cold stream flow rate and inlet temperature. The mass flow of the hot stream was assumed to be perfectly controlled by upstream ECS components (Palmer et al. 2016). The performance of the FDI tests designed was studied for two instances of the system model: one that is fault-free (Rf = 0) and one with faults of varying severity. The heat exchanger model was first experimentally validated with literature data of steady-state and dynamic heat transfer experiments (Palmer et al. 2016). The validated model was then used to calculate optimal tests for FDI at low (20 % blocked), medium (50 % blocked) and high (80 % blocked) levels of fouling. Table 2. Anticipated values of the faults and uncertain conditions for varying levels of fouling blockage in the heat exchange, along with their respective lower and upper bounds.
Uncertain Parameter
Lower Bound 0.00
20% 1.60
Blocked 50% 4.00
80% 6.40
Upper Bound 8.00
wH2O %
0.1
0.1
0.1
0.1
5.0
Tci °C
15
40
40
40
50
mc kg/s
0.75
1.00
1.00
1.00
1.25
R f u 103 m 2 K/W
Three types of tests are examined in this work: nominal tests that use standard operating conditions during the test, optimal tests with only steady-state outputs observed, and optimal tests with steady-state and dynamic responses observed. The ECS heat exchanger nominal state and input conditions were set according to specifications described in Palmer et al. 2016). The total duration of the FDI test design was kept at 300 s, with sampling frequency of 1 s-1. During this period the test was allowed to have up to one stepwise admissible input change, i.e. a maximum of two input settings could be used for FDI. In the steady-state analysis, the samples were taken at the same sampling frequency of 1 s-1, with varying noise during each test. In the dynamic analysis, the step change was
K. Palmer et al.
1690
implemented at t = 100 s. The exit temperature sensors were used to collect observable data used for FDI. It was assumed that each sensor contains white measurement noise at a standard deviation of 0.5 qC . 3.2. Case Study I: Identification of heat exchanger fouling with uncertainty in one input The objective in this case study was to determine the optimal test design for fouling identification in a heat exchanger with uncertain moisture content in its inlet streams, then to compare the results obtained with optimal steady-state or transient information. The heat exchanger FDI test was executed in the virtual system and fouling was estimated through the use of two different fault identification approaches. The first approach was a maximum likelihood estimation (MLE), as shown in Eq.(4): N
ξ*
N
sp y 2 0.5 1 ª º arg max « 2SV ik2 exp §¨ 2V ik2 yi x tk , u*p tk , θ p , ξ, tk yˆi x tk , u*p tk , θ p , ξ, tk ·¸ ». ξ; © ¹¼ k 1 i 1 ¬
(4)
The second approach performed state and parameter estimation with a nonlinear observer based on an Extended Kalman Filter (EKF) (Wenzel et al. 2006). Eqs.(2) and (3) of Table 1 were used to determine the optimal FDI test designs for identifying faults and uncertain inputs, assuming that all other parameters are certain. Fig. 2 shows the determinant of the Fisher Information Matrix as a function of the hot mass flow rate with two input settings using steady-state (left) and dynamic (right) outputs. The virtual heat exchanger was initialized at a nominal steady-state and simulated for 300 s. The hot mass flow rate was then set to the values calculated by Eqs.(2) and (3) for steady-state and dynamic FDI, respectively. In the steady-state tests, the data was collected when the system was no longer in a transient state, and in the dynamic tests the data was collected continuously during the test period. The optimal steady-state test for FDI was calculated with one test having the hot mass flow rate set to 0.43-0.47 kg/s (depending on the anticipated level of fouling) followed by a test with the hot mass flow rate at 1.0 kg/s (upper bound). The dynamic optimal test
Figure 2. Determinant of the Fisher Information Matrix for the heat exchanger FDI test over a range of admissible hot mass flow rates, with two inputs settings in the test design at only steadyy * * state (left, M StS =[ mh =[0.43 kg/s, 1.0 kg/s], Ntest = 2]) or with transient response (right, M Dyn =[ mh =[0.1 kg/s, 1.0 kg/s], Ntest = 1]). The uncertain parameters are the thermal fouling resistance and the moisture content, with anticipated values at the high fouling level (80 % blockage).
Active Fault Detection and Identification using Transient Data
1691
had the hot mass flow rate set to the lower bound (0.1 kg/s) followed by a step change to its upper bound (1.0 kg/s). This test design generates the greatest dynamic response, within the system constraints reported in Table 2. For this comparison, the virtual system fouling resistance was set to a value corresponding to 50 % blockage and the moisture content was set to its upper bound. It is shown in Table 3 that the results from the EKF observer were consistently less accurate than those of the MLE. Overall, fault identifiability was feasible when the system had only one uncertain input and test designs with two input settings were used. Measurement noise and the relative lack of sensitivity of exit stream temperatures with respect to air moisture content led to inaccurate fouling severity estimates at nominal conditions or when the EKF observer was used. Table 3. Estimates of the thermal fouling resistance and moisture content obtained with MLE and EKF at nominal, optimal steady-state and dynamic FDI test designs.
Uncertain Parameter R f u 103 m 2 K/W
wH2O %
True Values 4.00
Initial Guess 6.40
Nominal MLE EKF 4.62 5.66
Optimal StS MLE EKF 4.23 5.33
Optimal Dyn MLE EKF 3.96 4.84
5.00
0.10
3.52
4.89
5.00
0.06
0.09
0.96
3.3. Case Study II: Identification of heat exchanger fouling with multiple uncertain parameter and inputs Fouling identification was also assessed with the cold mass flow rate and cold inlet temperature as uncertain variables in order to make the case study more realistic. All of the parameters and inputs of Table 2 were considered to be unknown or uncertain. The anticipated values and lower and upper bounds of the uncertain inputs were the same as in the previous case study. The optimal steady-state test design was calculated using Eq.(2) of Table 1 and the hot mass flow rates were set to 1.0 and 0.29 kg/s for the first and second tests. The optimal dynamic test design was calculated from Eq.(3) to be 0.1 kg/s for the first step and 1.0 kg/s for the second step. The resulting test designs were used to estimate fouling using MLE and EKF, as shown in Fig. 3. Fig. 3 and Table 4 show a significant improvement in the fault and uncertain variable estimation with the dynamic optimal test. The insensitive uncertain parameters and variables could not be estimated in this case study from steady-state tests. The same low
Figure 3. The steady-state and dynamic optimal test designs (a) and the estimated thermal fouling resistance using MLE and EKF with the steady-state (b) and dynamic (c) optimal test design.
K. Palmer et al.
1692
sensitivity led to poor estimates from the EKF observer. The optimal dynamic test with an MLE observer was the only scenario where FDI could lead to accurate estimates of the fault severity and the system uncertainty. Table 4. Estimates of the thermal fouling resistance, moisture content, cold mass flow rate and inlet temperature obtained with MLE and EKF at nominal, optimal steady-state and dynamic FDI test designs.
Uncertain Parameter
True Values 4.00
Initial Guess 6.40
Nominal MLE EKF 6.19 5.74
Optimal StS MLE EKF 4.39 5.36
Optimal Dyn MLE EKF 4.08 4.96
wH2O %
5.00
0.10
0.39
0.11
4.34
1.17
5.00
4.65
Tci °C
40
40
39.4
39.6
39.9
40.3
40.2
40.6
mc kg/s
1.00
1.00
0.99
0.99
0.99
1.01
1.00
0.99
R f u 103 m 2 K/W
4. Conclusions The proposed active FDI methodology was tested with a plate fin heat exchanger model in order to identify fouling at optimal input conditions for steady-state and transient tests. Therefore, the importance of dynamic information in FDI test designs was assessed. The results obtained from nominal and optimal FDI through MLE and EKF were compared. It was found that the FDI tests that used dynamic information to identify faults were consistently more accurate than those with only steady-state measurements, and that FDI with MLE could correctly and consistently identify anticipated system faults.
5. Acknowledgements This project was sponsored by the UTC Institute for Advanced Systems Engineering (UTC-IASE) of the University of Connecticut and the United Technologies Corporation. Any opinions expressed herein are those of the authors and do not represent those of the sponsor.
References L. Han, Z. Zhou, G.M. Bollas, 2016a, Han L, Zhou Z, Bollas GM. Model-based analysis of chemical-looping combustion experiments. Part I: Structural identifiability of kinetic models for NiO reduction, AIChE Journal, 62, 7, 2419–2431. L. Han, Z. Zhou, G.M. Bollas, 2016b, Han L, Zhou Z, Bollas GM. Model-based analysis of chemical-looping combustion experiments. Part II: Optimal design of CH 4 -NiO reduction experiments, AIChE Journal, 62, 7, 2432–2446. H. Niemann, 2006, A Setup for Active Fault Diagnosis, IEEE Transactions on Automatic Control, 51, 9, 1572–1578. K.A. Palmer, W.T. Hale, K.D. Such, B.R. Shea, G.M. Bollas, 2016, Optimal design of tests for heat exchanger fouling identification, Applied Thermal Engineering, 95, 382–393. T.A. Wenzel, K.J. Burnham, M.V. Blundell , R.A. Williams, 2006, Dual extended Kalman filter for vehicle state and parameter estimation, Vehicle System Dynamics: International Journal of Vehicle Mechanics and Mobility, 44, 2, 153–171. J. Zarei, J. Poshtan, 2010, Design of nonlinear unknown input observer for process fault detection, Industrial and Engineering Chemistry Research, 2010, 49, 11443–11452.
Active Fault Detection and Identification using Transient Data Kyle A. Palmera, George M. Bollasa a
Department of Chemical & Biolmolecular Engineering, University of Connecticut, 191 Auditorium Road, Unit 3222, Storrs, CT, 06269-3222, USA, [email protected]
Abstract Active Fault Detection and Isolation (FDI) methods can improve the detection of residuals between measured and expected outputs, isolate fault cause(s) and, thus, identify faults with higher confidence than conventional passive methods. This work focuses on quantifiably illustrating the value of transient experiments in active FDI, as compared to steady-state testing. Fault identification is treated as a set of constrained steady-state or dynamic optimization problems using D-Optimality as the objective of the test design. The identifiability of faults is examined using steady-state and transient measurements in two testbed systems with uncertainty in their parameters and boundaries. The inclusion of transient information during FDI decreases fault identification error. This is illustrated through application of the framework on a plate fin heat exchanger of an aircraft environmental control system (ECS). Keywords: active fault diagnosis, fault detection, optimization, dynamic systems
1. Introduction With modern engineered systems becoming more complex in their design and functionality, active FDI methods have gathered interest over the last decade (Niemann 2006). Traditional FDI techniques focus on the design of robust filters as well as thresholds for the sensors used for the detection of faults. Methods such as unknown input observers (Zarei and Poshtan 2010) are commonly used to optimize the calculation of residuals so that they are sensitive to faults and insensitive to uncertain or unknown disturbances. Another way to generate residuals sensitive to faults is by adjusting the admissible inputs to create an optimal test for active FDI. In this work, FDI test design is handled as a set of constrained optimization problems focused on maximizing information about the test. The system inputs are manipulated to improve the identifiability of system faults and uncertain conditions by maximizing information with respect to faults in the form of sensitivities of outputs. In a classical model-based design of experiments approach (Han et al. 2016a,b), steady-state or dynamic tests are designed to maximize the identifiability of fault(s), expressed in the system model as parameter(s). For a given number of system sensors the system uncertainty is also estimated (if identifiable). The test designed can be steady-state or transient, depending on the model structure and the accuracy of the system actuators. The FDI test optimization is applied for the identification of fouling in a heat exchanger with unknown moisture content its fluid streams, and uncertainty in the cold stream inlet temperature and mass flow rate.
2. Methods The FDI test designs presented in this work use physics-based models or semi-empirical correlations that are assumed to accurately represent the tested system. These models are
K. Palmer et al.
1688
often written as series of implicit equations, shown in Eq.(1), where f is the system governing equations, x the vector of state variables, u the controllable inputs, T the model parameters, t time, and yˆ are the estimated measurable outputs. f x t , x t , u t , T, t 0, yˆ
(1)
h x t , u t , T, t .
If the system is at steady-state, then t and x t are omitted. The parameter vector is split into three subvectors to represent model faults, T f , uncertain parameters, Tu , and design parameters, T p . The input vector, u, is split in a similar manner into certain system inputs, up, and uncertain inputs, uu. The target set of faults and uncertain parameters and variables that can impact the accuracy of fault detection are compiled into a vector, [ : [
ª¬T f , Tu º¼ >uu @.
The vector, [ , is assigned an anticipated set of values, [ , containing an estimate of the faults and mean value of the uncertain inputs and parameters. The test design vector, M , is used to determine the optimal FDI test within the design space, ) . Typically, M comprises the admissible input trajectories, up, number of tests, Ntest, initial system states, y0, sampling times, tsp, and overall timespan, W . In this work, M contains only the inputs, up, and the number of test, Ntest. Table 1 shows the optimal test design formulations for steady-state and dynamic tests, where the optimization objective is the minimization of the determinant of the inverse of the Fisher Information Matrix, H [ (D-optimality). Table 1. Mathematical formulation of the FDI steady-state and dynamic test design
Steady-state M
* StS
ªu , , u ¬ * p ,1
Dynamic * * p , Nttest
,N
M*Dyn
p
(2)
p
p
U p
L
s.t. f x t , x t , u p t , T p , [, t
0,
yˆ t h x t , u p t , T p , [, t ,
p
u d u p d u , x d x t d x . L p
* ª¬u*p t , N test t 1º¼ arg min det ª H [1 [, M º ¼ ¬ M)
f x, u , T , [ 0, yˆ h x, u , T , [ ,
arg min det ª H [1 ¬ M) s.t.
t 1º ¼ º [, M ¼
* test
U
(3)
f x t0 , x t0 , u p t0 , T p , [, t0 0, ° ® °¯ yˆ t0 h x t0 , u p t0 , T p , [, t0 , L u p d u p t d uUp , t >0, W @ , y0
x L d x t d xU , t >0, W @.
3. Results and Discussion 3.1. Heat Exchanger Fouling Identification Particulate fouling on the ECS heat exchanger significantly reduces heat transfer efficiency and performance, leading to increased maintenance costs and peripheral component failures. Here, the method described in Section 2 is employed to improve the identifiability of fouling in the heat exchanger of Fig. 1, which is part of an ECS described elsewhere (Palmer et al. 2016). In brief, as shown in Fig. 1, the plate fin heat exchanger
Active Fault Detection and Identification using Transient Data
1689
Figure 1. Input-Output architecture of the ECS cross-flow plate fin heat exchanger.
has one admissible input, two measured outputs, one fault and three uncertain variables. The target of FDI is to quantify the extent of the fault (fouling) in lieu of the uncertainty in system inputs. The faults and uncertain inputs considered in this document are listed in Table 2, along with their anticipated values, lower and upper bounds. Fouling is expressed in the model as the thermal fouling resistance, Rf, although it also impacts the cross-sectional area available for flow (along with flow characteristics). Uncertainty exists in air moisture, cold stream flow rate and inlet temperature. The mass flow of the hot stream was assumed to be perfectly controlled by upstream ECS components (Palmer et al. 2016). The performance of the FDI tests designed was studied for two instances of the system model: one that is fault-free (Rf = 0) and one with faults of varying severity. The heat exchanger model was first experimentally validated with literature data of steady-state and dynamic heat transfer experiments (Palmer et al. 2016). The validated model was then used to calculate optimal tests for FDI at low (20 % blocked), medium (50 % blocked) and high (80 % blocked) levels of fouling. Table 2. Anticipated values of the faults and uncertain conditions for varying levels of fouling blockage in the heat exchange, along with their respective lower and upper bounds.
Uncertain Parameter
Lower Bound 0.00
20% 1.60
Blocked 50% 4.00
80% 6.40
Upper Bound 8.00
wH2O %
0.1
0.1
0.1
0.1
5.0
Tci °C
15
40
40
40
50
mc kg/s
0.75
1.00
1.00
1.00
1.25
R f u 103 m 2 K/W
Three types of tests are examined in this work: nominal tests that use standard operating conditions during the test, optimal tests with only steady-state outputs observed, and optimal tests with steady-state and dynamic responses observed. The ECS heat exchanger nominal state and input conditions were set according to specifications described in Palmer et al. 2016). The total duration of the FDI test design was kept at 300 s, with sampling frequency of 1 s-1. During this period the test was allowed to have up to one stepwise admissible input change, i.e. a maximum of two input settings could be used for FDI. In the steady-state analysis, the samples were taken at the same sampling frequency of 1 s-1, with varying noise during each test. In the dynamic analysis, the step change was
K. Palmer et al.
1690
implemented at t = 100 s. The exit temperature sensors were used to collect observable data used for FDI. It was assumed that each sensor contains white measurement noise at a standard deviation of 0.5 qC . 3.2. Case Study I: Identification of heat exchanger fouling with uncertainty in one input The objective in this case study was to determine the optimal test design for fouling identification in a heat exchanger with uncertain moisture content in its inlet streams, then to compare the results obtained with optimal steady-state or transient information. The heat exchanger FDI test was executed in the virtual system and fouling was estimated through the use of two different fault identification approaches. The first approach was a maximum likelihood estimation (MLE), as shown in Eq.(4): N
ξ*
N
sp y 2 0.5 1 ª º arg max « 2SV ik2 exp §¨ 2V ik2 yi x tk , u*p tk , θ p , ξ, tk yˆi x tk , u*p tk , θ p , ξ, tk ·¸ ». ξ; © ¹¼ k 1 i 1 ¬
(4)
The second approach performed state and parameter estimation with a nonlinear observer based on an Extended Kalman Filter (EKF) (Wenzel et al. 2006). Eqs.(2) and (3) of Table 1 were used to determine the optimal FDI test designs for identifying faults and uncertain inputs, assuming that all other parameters are certain. Fig. 2 shows the determinant of the Fisher Information Matrix as a function of the hot mass flow rate with two input settings using steady-state (left) and dynamic (right) outputs. The virtual heat exchanger was initialized at a nominal steady-state and simulated for 300 s. The hot mass flow rate was then set to the values calculated by Eqs.(2) and (3) for steady-state and dynamic FDI, respectively. In the steady-state tests, the data was collected when the system was no longer in a transient state, and in the dynamic tests the data was collected continuously during the test period. The optimal steady-state test for FDI was calculated with one test having the hot mass flow rate set to 0.43-0.47 kg/s (depending on the anticipated level of fouling) followed by a test with the hot mass flow rate at 1.0 kg/s (upper bound). The dynamic optimal test
Figure 2. Determinant of the Fisher Information Matrix for the heat exchanger FDI test over a range of admissible hot mass flow rates, with two inputs settings in the test design at only steadyy * * state (left, M StS =[ mh =[0.43 kg/s, 1.0 kg/s], Ntest = 2]) or with transient response (right, M Dyn =[ mh =[0.1 kg/s, 1.0 kg/s], Ntest = 1]). The uncertain parameters are the thermal fouling resistance and the moisture content, with anticipated values at the high fouling level (80 % blockage).
Active Fault Detection and Identification using Transient Data
1691
had the hot mass flow rate set to the lower bound (0.1 kg/s) followed by a step change to its upper bound (1.0 kg/s). This test design generates the greatest dynamic response, within the system constraints reported in Table 2. For this comparison, the virtual system fouling resistance was set to a value corresponding to 50 % blockage and the moisture content was set to its upper bound. It is shown in Table 3 that the results from the EKF observer were consistently less accurate than those of the MLE. Overall, fault identifiability was feasible when the system had only one uncertain input and test designs with two input settings were used. Measurement noise and the relative lack of sensitivity of exit stream temperatures with respect to air moisture content led to inaccurate fouling severity estimates at nominal conditions or when the EKF observer was used. Table 3. Estimates of the thermal fouling resistance and moisture content obtained with MLE and EKF at nominal, optimal steady-state and dynamic FDI test designs.
Uncertain Parameter R f u 103 m 2 K/W
wH2O %
True Values 4.00
Initial Guess 6.40
Nominal MLE EKF 4.62 5.66
Optimal StS MLE EKF 4.23 5.33
Optimal Dyn MLE EKF 3.96 4.84
5.00
0.10
3.52
4.89
5.00
0.06
0.09
0.96
3.3. Case Study II: Identification of heat exchanger fouling with multiple uncertain parameter and inputs Fouling identification was also assessed with the cold mass flow rate and cold inlet temperature as uncertain variables in order to make the case study more realistic. All of the parameters and inputs of Table 2 were considered to be unknown or uncertain. The anticipated values and lower and upper bounds of the uncertain inputs were the same as in the previous case study. The optimal steady-state test design was calculated using Eq.(2) of Table 1 and the hot mass flow rates were set to 1.0 and 0.29 kg/s for the first and second tests. The optimal dynamic test design was calculated from Eq.(3) to be 0.1 kg/s for the first step and 1.0 kg/s for the second step. The resulting test designs were used to estimate fouling using MLE and EKF, as shown in Fig. 3. Fig. 3 and Table 4 show a significant improvement in the fault and uncertain variable estimation with the dynamic optimal test. The insensitive uncertain parameters and variables could not be estimated in this case study from steady-state tests. The same low
Figure 3. The steady-state and dynamic optimal test designs (a) and the estimated thermal fouling resistance using MLE and EKF with the steady-state (b) and dynamic (c) optimal test design.
K. Palmer et al.
1692
sensitivity led to poor estimates from the EKF observer. The optimal dynamic test with an MLE observer was the only scenario where FDI could lead to accurate estimates of the fault severity and the system uncertainty. Table 4. Estimates of the thermal fouling resistance, moisture content, cold mass flow rate and inlet temperature obtained with MLE and EKF at nominal, optimal steady-state and dynamic FDI test designs.
Uncertain Parameter
True Values 4.00
Initial Guess 6.40
Nominal MLE EKF 6.19 5.74
Optimal StS MLE EKF 4.39 5.36
Optimal Dyn MLE EKF 4.08 4.96
wH2O %
5.00
0.10
0.39
0.11
4.34
1.17
5.00
4.65
Tci °C
40
40
39.4
39.6
39.9
40.3
40.2
40.6
mc kg/s
1.00
1.00
0.99
0.99
0.99
1.01
1.00
0.99
R f u 103 m 2 K/W
4. Conclusions The proposed active FDI methodology was tested with a plate fin heat exchanger model in order to identify fouling at optimal input conditions for steady-state and transient tests. Therefore, the importance of dynamic information in FDI test designs was assessed. The results obtained from nominal and optimal FDI through MLE and EKF were compared. It was found that the FDI tests that used dynamic information to identify faults were consistently more accurate than those with only steady-state measurements, and that FDI with MLE could correctly and consistently identify anticipated system faults.
5. Acknowledgements This project was sponsored by the UTC Institute for Advanced Systems Engineering (UTC-IASE) of the University of Connecticut and the United Technologies Corporation. Any opinions expressed herein are those of the authors and do not represent those of the sponsor.
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