Discussion of “Assessing Parameter Identifiability of Activated Sludge Model Number 1” by Pedro Afonso and Maria da Conceição Cunha

Discussion of “Assessing Parameter Identifiability of Activated Sludge Model Number 1” by Pedro Afonso and Maria da Conceição Cunha

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  Discussion of ‘‘Assessing ParameterIdentifiability of Activated Sludge ModelNumber 1’’ by Pedro Afonsoand Maria da Conceic¸a ˜ o Cunha August 2002, Vol. 128, No. 8, pp. 748–754. DOI: 10.1061/   ASCE  0733-9372  2002  128:8  748  Dirk J. W. De Pauw 1 ; Gu¨rkan Sin 2 ; Gu¨clu¨ Insel 3 ; StijnW. H. Van Hulle 4 ; Veronique Vandenberghe 5 ; and PeterA. Vanrolleghem 6 1 PhD Candidate, Biomath—Dept. of Applied Mathematics, Biometricsand Process Control, Ghent Univ., Ghent, Belgium. 2 PhD Candidate, Biomath—Dept. of Applied Mathematics, Biometricsand Process Control, Ghent Univ., Ghent, Belgium. 3 PhD Candidate, Biomath—Dept. of Applied Mathematics, Biometricsand Process Control, Ghent Univ., Ghent, Belgium. 4 PhD Candidate, Biomath—Dept. of Applied Mathematics, Biometricsand Process Control, Ghent Univ., Ghent, Belgium. 5 PhD Candidate, Biomath—Dept. of Applied Mathematics, Biometricsand Process Control, Ghent Univ., Ghent, Belgium. 6 Prof., Biomath—Dept. of Applied Mathematics, Biometrics and ProcessControl, Ghent Univ., Ghent, Belgium. Parameter identification, as discussed by the authors, is a veryimportant task within modeling, and particularly, model calibra-tion. We would like to congratulate the authors for their contribu-tion to this field. However, in order to increase the understandingof this rather complicated modeling task we would also like tobring forward some points of discussion.As a first point we would like to stress that a clear distinctionshould be made between structural and practical identifiability.The structural identifiability as a first step addresses whetherthere is any chance of obtaining a unique value for the param-eters, given the structure of the model and the measurements to beperformed. Structural identifiability can be performed in the ab-sence of any prior information on the value of the parameters andeven before collecting any data on the system to be studied. Manytechniques exist to assess the structural identifiability   Walter andPronzato 1997  , including the technique used by the authors, the‘‘numerical local approach,’’ in which noise-free data is used toassess the identifiability of the parameters by trying to estimatepotential identifiable parameters to a generated set of noiselessdata. However, it should be noted that this technique can only beused to assess the local structural identifiability of a parameterand not its global identifiability, as presented by the authors.The practical identifiability of parameters, on the other hand,depends not only on the model structure, but also on the experi-mental conditions together with the quality and quantity of themeasurements. It gives an assessment of the accuracy with whichparameters can be estimated. Most methods for the evaluation of practical identifiability are based on the parameter estimation co-variance matrix or its inverse, the Fisher Information Matrix   seeDochain and Vanrolleghem 2001  . Note that, if parameters arepractically identifiable, they are also locally structurally identifi-able, and that is the essence of the numerical local approach.However, it means that the accuracy of these parameter estimatesmust be evaluated too, something not done by the authors.The authors investigated the structural identifiability of ASM1using the numerical local approach. A number of parameter esti-mation algorithms were used, including gradient- andnongradient-based methods. From the results the authors reportedthat the algorithms did not converge for some situations. To us, itis not very clear what the authors meant by nonconvergence. Wasit because the parameters were correlated and the parameter esti-mation algorithm stopped at a local minimum or was it caused bynumerical problems of the algorithms. It would also be very in-teresting to know if different sets of initial parameter values wereused and what the values of the final estimates and their varianceswere. This is a very important and well-known check of the iden-tifiability of parameters as expressed by the uniqueness of theparameters.In the introduction of the identification algorithms, the authorsmentioned that numerical problems were to be expected with thegradient-based method. Dochain and Vanrolleghem   2001   alsolist a number of authors who have found poor convergence for theLevenberg-Marquardt algorithm. In contrast to that statement, itwas found that the Levenberg-Marquardt   gradient-based   methodperformed best. Important to mention is that Walter and Pronzato  1997   suggested not to use the Levenberg-Marquardt for localstructural identifiability analysis because of the regularizationprocedure used in the algorithm.The authors conclude that a large number of parameters   up to15   are structurally identifiable from oxygen measurements alone,including the yield coefficient. However, from the   many   struc-tural identifiability analyses that have been performed with ASM-type models using only oxygen measurements   e.g., Dochainet al. 1995; Petersen et al. 2001  , it was concluded that only com-binations of parameters with the yield coefficient ( Y   H  ), maxi-mum growth rate (  max ), substrate half-saturation coefficient( K  SH ), and initial biomass concentration (  X   H  ) could be structur-ally identified. The claim made in the paper that up to 15 param-eters can be identified uniquely using only oxygen measurementsis therefore very questionable. The fact that 15 parameters get avalue in a numerical parameter estimation procedure does notimply structural identifiability. Parameters of a model that is notstructurally identifiable can indeed get values, but these are notunique, and that is the property to evaluate. Rather than going intoa theoretical discussion, the point is illustrated in Fig. 1 and Fig.2 for the aerobic oxidation of COD. The simplified COD massbalance only considering the aerobic heterotrophic COD oxida-tion process   neglecting the dynamics   kinetics   of the process   isgiven by Eq.   1  :  S  S     X   H    S  O ,  H   (1)where   S  S   ( S  S  ,in  S  S  ). And   X   H   Y   H  *  S  S   (2)  S  O ,  H    1  Y   H   *  S  S   (3)where    X   H   biomass production   mg COD  ;   S  O ,  H   oxygenconsumption due to heterotrophic growth   mg COD  ; and   S  S   readily biodegradable substrate oxidized   mg COD  . DISCUSSIONS AND CLOSURES 110  / JOURNAL OF ENVIRONMENTAL ENGINEERING © ASCE / JANUARY 2004  It can clearly be seen from Eq.   1   that the COD mass balancehas two unknowns—substrate removed and biomass produced—whereas only the oxygen consumption is known. This means thatEq.   2   and Eq.   3   can be solved by any combination of (1  Y   H  ) *  S  S    Dochain et al. 1995   see Table 1 for some ex-amples  . This simple example demonstrates why only values forcombinations of parameters rather than values for the single pa-rameters can be estimated based on oxygen measurements alone  see Dochain et al.   1995   for further explanation  . Moreover, forthe full ASM1 model as studied by the authors we also need toconsider the autotrophic organisms, in addition to the het-erotrophic bacteria. Both organism types contribute to the oxygenconsumption:  S  O ,total   S  O ,  H    S  O ,  A   1  Y   H   *  S  S    4.57  Y   A  *  S  NH (4)Given only oxygen measurements, it is clear from Eq.   4   thatneither  Y   H   nor  Y   A  can uniquely be identified. This brings us to theconclusion that it is impossible to uniquely identify 15 parametersgiven only oxygen measurements, as claimed by the authors.Thus, it can be concluded that the identifiability of each parameterof the ASM1 model depends largely on the considered measuredvariables   Petersen et al. 2001  . It is clear that more information  e.g., NO 3  data, effluent COD, biomass production ...   about thesystem is required to uniquely identify the large number of pa-rameters considered in ASM1.As stated before, the practical identifiability is closely relatedto the quality and quantity of the data. This aspect is not suffi-ciently stressed by the authors. Calibration and identifiabilitystudies can only be carried out if sufficient and information-richdata is available.In order to catch all the dynamics of the treatment plant, it isadvisable to take as duration of the measurements several HRTs  hydraulic retention times   as opposed to the 0.2 days of measure-ments used by the authors   Petersen et al. 2002  . It is also pre-ferred to have time-varying conditions in the WWTP, in contrastto the presented case where all variables stay more or less con-stant. We also question the choice of the sampling interval   being7.5 min  , because the suggested O 2   DO, dissolved oxygen   mea-surements can be obtained using a much smaller interval   on theorder of seconds  . Besides O 2  and NO 3  measurements, the au-thors also propose to measure the readily biodegradable substrate S  S   in the aeration tanks. To our knowledge, no measurement tech-nique exists in order to measure this quantity on a full-scale plantin the mixed liquor. Therefore it should preferably not be used ina model calibration case study.The authors present a procedure for evaluation of the practicalidentifiability of the ASM1 model parameters. This procedure isbased on nonlinear regression analysis, which uses a sum-of-squared-errors objective function to minimize the difference be-tween model output and measurements. Further, the Fisher infor-mation matrix, which contains the second derivatives of theobjective function, is used to assess the precision of the parameterestimates. The variance of the observations is incorporated in thecalculation of the Fisher information matrix by using the objec-tive function value, the number of measurement points, and thenumber of estimated parameters. This approach holds only if allmeasurements have identical measurement error and no correla-tion between them exists. In our opinion this assumption does nothold   as the authors themselves accept by using a relative mea-surement noise   and a weighted sum-of-squared-errors objectivefunction that incorporates the measurement error for every mea-sured variable should be used instead. Also, the measurementerror covariance matrix should be used in the calculations of theFisher information matrix   Dochain and Vanrolleghem 2001  . Fig. 2.  Concept of yield coefficient based on COD unit Fig. 1.  COD mass balance around the aeration tank considering only the aerobic oxidation of COD by heterotrophic biomass. The oxygentransfer coefficient ( K   L a ), saturation oxygen concentration ( S  O ,sat ) and influent characteristics ( Q in ,  S  S  ,in  and  S  O ,in  etc....   are assumed to beknown a priori, while only  S  O  is measured. Table 1.  Infinite Number of Solutions to COD Mass Balance withOnly   S  O  Known. Examples Given for  S  O  10 mgCOD/l and  S  S  ,in  35 mgCOD/l  S  O mgCOD/l (1  Y   H  )  S  S  mgCOD/l Y   H  mgCOD/mgCOD   X   H  mgCOD/l S  S  mgCOD/l10 0.33 30.30 0.67 20.3 4.710 0.50 20.00 0.50 10.0 15.010 0.40 25.00 0.60 15.0 10.0— — — — — — JOURNAL OF ENVIRONMENTAL ENGINEERING © ASCE / JANUARY 2004 /   111  In order to assess practical identifiability, the authors addednoise to the data and ran the identification procedure. However, inorder to avoid meaningless estimation results, boundaries wereestablished on the parameters. From the results of the identifica-tion study it can be seen that some parameter estimates are at theirbound, meaning that the ‘‘true’’ value is probably outside theparameter bounds. This typically points to model inadequacy andthis should have been discussed.In order to rule out badly identifiable parameters, a sequentialselection procedure was used in which the least accurate param-eter was eliminated from the set. In our opinion it would be better,as in Weijers and Vanrolleghem   1997  , to investigate all possibleparameter subsets because removing one parameter could influ-ence the identifiability of the remaining parameters.Finally, the iterative identifiability process was stopped whenthe estimates for the reduced set of parameters were ‘‘close’’ tothe real values. It must be clear that, in practically any actualcalibration study, this can never be used as a stopping criterion,for the real values of the parameters are never known. Otherauthors have presented different stopping criteria in similar stud-ies   Brun et al. 2002; Weijers and Vanrolleghem 1997  , and theseshould have been discussed. References Brun, R., Kuhni, M., Siegrist, H., Gujer, W., and Reichert, P.   2002  .‘‘Practical identifiability of ASM2d parameters—Systematic selectionand tuning of parameter subsets.’’  Water Res.,  36  16  , 4113–4127.Dochain, D., and Vanrolleghem, P. A.   2001  .  Dynamic modelling and estimation in wastewater treatment processes , IWA Publishing,London.Dochain, D., Vanrolleghem, P. A., and Van Daele, M.   1995  . ‘‘Structuralidentifiability of biokinetic models of activated sludge respiration.’’ Water Res.,  29, 2571–2578.Petersen, B., Gernaey, K., and Vanrolleghem, P. A.   2001  . ‘‘Practicalidentifiability of model parameters by combined respirometric-titrimetric measurements.’’  Water Sci. Technol.,  43  7  , 347–356.Petersen, B., Gernaey, K., and Henze, M., and Vanrolleghem, P. A.  2002  . ‘‘Evaluation of an ASMI model calibration procedure on amunicipal-industrial wastewater treatment plant.’’  J. Hydroinformat-ics,  4, 15–38.Walter, E., and Pronzato, L.   1997  .  Identification of parametric models from experimental data , Springer, Paris.Weijers, S. R., and Vanrolleghem, P. A.   1997  . ‘‘A procedure for select-ing best identifiable parameters in calibrating activated sludge modelno. 1 to full scale plant data.’’  Water Sci. Technol.,  36  5  , 69–79. 112  / JOURNAL OF ENVIRONMENTAL ENGINEERING © ASCE / JANUARY 2004
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