1. ARTICLE IN PRESS
Water Research 39 (2005) 3686–3696
www.elsevier.com/locate/watres
Reliability of parameter estimation in respirometric models
Nicola Checchi, Stefano Marsili-LibelliÃ
Department of Systems and Computers, University of Florence, Via Santa Marta, 3-50139 Firenze, Italy
Received 18 September 2004; received in revised form 23 June 2005; accepted 25 June 2005
Available online 3 August 2005
Abstract
When modelling a biochemical system, the fact that model parameters cannot be estimated exactly stimulates the
definition of tests for checking unreliable estimates and design better experiments. The method applied in this paper is a
further development from Marsili-Libelli et al. [2003. Confidence regions of estimated parameters for ecological
systems. Ecol. Model. 165, 127–146.] and is based on the confidence regions computed with the Fisher or the Hessian
matrix. It detects the influence of the curvature, representing the distortion of the model response due to its nonlinear
structure. If the test is passed then the estimation can be considered reliable, in the sense that the optimisation search
has reached a point on the error surface where the effect of nonlinearities is negligible. The test is used here for an
assessment of respirometric model calibration, i.e. checking the experimental design and estimation reliability, with an
application to real-life data in the ASM context. Only dissolved oxygen measurements have been considered, because
this is a very popular experimental set-up in wastewater modelling. The estimation of a two-step nitrification model
using batch respirometric data is considered, showing that the initial amount of ammonium-N and the number of data
play a crucial role in obtaining reliable estimates. From this basic application other results are derived, such as the
estimation of the combined yield factor and of the second step parameters, based on a modified kinetics and a specific
nitrite experiment. Finally, guidelines for designing reliable experiments are provided.
r 2005 Elsevier Ltd. All rights reserved.
Keywords: Respirometry; Parameter estimation; Activated sludge models; Nitrification
1. Introduction et al., 1998; Petersen, 2000; Petersen et al., 2002),
resulting in systematic estimation protocols (Petersen et
The structural and practical identifiability of Monod- al., 2003b; Sin, 2004; De Pauw, 2005). This evolution
based biochemical models has greatly progressed since was also stimulated by the introduction of the ASM
the first studies of Pohjanpalo (1978) and Holmberg models (Henze et al., 2000) whose parameter identifica-
(1982) and is still a viable research topic (Kesavan and tion is now considered such an important issue in
Law, 2005). Important contributions now exist both on wastewater treatment that the ASM3 model has
the principles of estimation (Dochain et al., 1995; replaced ASM1 not only for its better understanding
Vanrolleghem and Keesman, 1996; Petersen et al., of the biochemical mechanisms, but also for its
2000; Brun et al., 2002; Petersen et al., 2003a) and on improved identifiability (Gernaey et al., 2004). In this
its practical aspects (Vanrolleghem et al., 1995; Brouwer context, respirometry is a primary tool for model
identification and a great deal of research has been
ÃCorresponding author. Tel./fax: +39 055 47 96 264. devoted to providing uncertainty limits to parameter
E-mail address: marsili@dsi.unifi.it (S. Marsili-Libelli). estimates and designing better experiments, especially
0043-1354/$ - see front matter r 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.watres.2005.06.021

2. ARTICLE IN PRESS
N. Checchi, S. Marsili-Libelli / Water Research 39 (2005) 3686–3696 3687
Nomenclature rend endogenous respiration rate
(mg O2 LÀ1 minÀ1)
ASM activated sludge model SNH ð0Þ initial concentration of ammonium-N
C parameter covariance matrix, <np Ânp (mg N-NH4 LÀ1)
CJ approximate covariance matrix based on the SNO ð0Þ initial concentration of nitrite-N (mg N-
Fisher Information Matrix (FIM), <np Ânp NO2 LÀ1)
CH approximate covariance matrix based on the SNH ammonium-N concentration (mg N-
Hessian matrix, <np Ânp NH4 LÀ1)
E(p) error functional for parameter estimation SNO nitrite-N concentration (mg N-NO2 LÀ1)
y
F ap ;NÀnp
n F statistics at the 100(1Àa)% confidence SPj output sensitivity to parameter pj
level for np parameters and NÀnp degrees of s2 estimated variance of the residuals
freedom (mg LÀ1)2
a=2
J Fisher Information Matrix (FIM), <np Ânp tNÀnp two-tails Student’s t distribution for the
H Hessian (second derivative) matrix, <np Ânp given confidence level 100(1Àa)% and N-np
KNH ammonium oxidisers half velocity constant degrees of freedom
(mg N-NH4 LÀ1) X A1 ammonium oxidisers biomass concentration
KNO nitrite oxidisers half velocity constant (mg COD LÀ1)
(mg N-NO2 LÀ1) X A2 nitrite oxidisers biomass concentration
N number of experimental data (mg COD LÀ1)
NOD nitrogen oxygen demand (mg O2 LÀ1) Y A1 ammonium oxidisers biomass yield
np number of estimated parameters Y A2 nitrite oxidisers biomass yield
RNH composite parameter related to the first dpJ individual parameter confidence interval
oxidation stage (mg O2 LÀ1 minÀ1) estimated with the FIM approximation
RNO composite parameter related to the second dPH individual parameter confidence interval
oxidation stage (mg O2 LÀ1 minÀ1) estimated with the Hessian approximation
rNO synthesis oxygen uptake rate on nitrite-N mmaxA1 ammonium oxidisers maximum growth rate
(mg O2 LÀ1 minÀ1) (minÀ1)
rexp
o experimental respiration rate mmaxA2 nitrite oxidisers maximum growth rate
(mg O2 LÀ1 minÀ1) (minÀ1)
ro model respiration rate (mg O2 LÀ1 minÀ1) t time constant of nitrite-N modified respira-
tion model (min)
when the model structure is a priori specified as in ASM estimation of the second step parameters is critical and
models (Vanrolleghem et al., 1995; Vanrolleghem and assessing the influence of the number of data. Later, a
Keesman, 1996; Brouwer et al., 1998; Petersen, 2000; specific nitrite experiment is performed, leading to a
Petersen et al., 2000; Marsili-Libelli and Tabani, 2002; better estimation of those parameters. Eventually, an
Sin, 2004; De Pauw, 2005). estimation protocol is defined for the reliable estimation
Based on the assumption that any given model of the whole parameter set.
structure is necessarily a simplified approximation of a
complex reality, the paper applies a new estimation 1.1. Theoretical identifiability of a two-step nitrification
reliability test, derived from a previous method based on model
the approximate confidence regions (Marsili-Libelli
et al., 2003), to a respirometric model based on ASM A two-step nitrification model derived from the ASM
kinetics. Its aim is to determine under which conditions model suite (Henze et al., 2000) is considered, with two
routine respirograms yield reliable estimates, without zero-growth autotrophic populations of ammonium
performing complex experiments with inhibitors (e.g. oxidisers (XA1) and nitrite oxidisers (XA2). The zero-
Nowak et al., 1995; Surmacz-Gorska et al., 1996) or growth assumption is justified by the short duration of
combining respirometric and titrimetric measurements the experiments and the small amount of added
(Petersen, 2000; Ficara et al., 2002; Gernaey et al., 2002). substrate. A similar model was used by Petersen (2000)
After summarising the main results of Marsili-Libelli and Gernaey et al. (2001) for respirometric studies,
et al. (2003), the theory is taken a step further, whereas a more recent paper (Insel et al., 2003) includes
presenting a new reliability test which is applied to the hydrolysis. The endogenous respiration rend is directly
estimation of a two-step nitrification kinetics. Initially, estimated by averaging the endogenous data. The
the full two-step model is considered, showing that the identifiability of respirometric models without biomass

3. ARTICLE IN PRESS
3688 N. Checchi, S. Marsili-Libelli / Water Research 39 (2005) 3686–3696
growth has been definitively assessed (Vanrolleghem et 1.3. Approximations of confidence regions for the
al., 1995; Petersen, 2000; Brun et al., 2002; Petersen et estimated parameters
al., 2003a) and particularly by Brouwer et al. (1998),
who estimated a model similar to Eq. (3) save for the Confidence regions represent the set of parameter
yield coefficients which are supposed known. The values producing a model response within prescribed
following parameter combinations can be estimated statistical boundaries. Being related to the error func-
tional Eq. (2), any level EðpÞ4Eð^ Þ defines a region with
p
W1 ¼ ð3:43 À Y A1 ÞRNH ; W4 ¼ ð1:14 À Y A2 ÞRNO ;
a given degree of confidence. However, it is difficult to
W2 ¼ ð3:43 À Y A1 ÞS NH ð0Þ; W5 ¼ ð1:14 À Y A2 ÞSNO ð0Þ; specify statistically significant levels of the increment
W3 ¼ ð3:43 À Y A1 ÞK NH ; W6 ¼ ð1:14 À Y A2 ÞK NO ; DE ¼ EðpÞ À Eð^ Þ unless N is large, in which case DE
p
(1) has the required w2 asymptotic properties to apply the F
statistics (Seber and Wild, 1989). The numerical
where the two composite parameters RNH ¼ difficulty in estimating the exact confidence regions
^ ^
mmax A1 X A1 =Y A1 and RNO ¼ mmax A2 X A2 =Y A2 are intro- has been examined by Vanrolleghem and Keesman
duced for identifiability reasons (Dochain et al., 1995). (1996) and Dochain and Vanrolleghem (2001) who, on
The first-step parameters ðY A1 ; RNH ; K NH Þ are always the basis of a previous work by Lobry and Flandrois
identifiable if S NH ð0Þa0, whereas if S NO ð0Þ ¼ 0 only (1991), proposed a successive contraction method to
two of the three second-step parameters find the value of E(p) corresponding to the prescribed
ðY A2 ; RNO ; K NO Þ can be identified. In practice this F value. In this paper the approach of Marsili-Libelli
condition implies a large estimation uncertainty of the et al. (2003) is used, based on linear or quadratic
second step parameters, as will be shown later. The aim approximations, as suggested by Press et al. (1986) and
of this paper is to assess the reliability of the estimates as Seber and Wild (1989). Confidence regions in the np
a function of SNH ð0Þ, showing that this parameter plays parameter space can be expressed as a the quadratic
a crucial role in the model identifiability, and suggesting form
appropriate values for this quantity.
ðp À pÞC À1 ðp À pÞT pnp F ap ;NÀnp ,
^ ^ n (4)
1.2. Calibration of model parameters where the matrix C is the equivalent of the parameter
covariance matrix of the linear case (Ljung, 1999) and
A numerical optimisation search is used to minimise 100(1Àa)% is the required confidence level of the F
the sum of squared errors between respiration data rexp
o statistics with np parameters and NÀnp degrees of
and model responses ro at the sampling times freedom. This matrix can be approximated either by
i ¼ 1; 2; . . . ; N, extending the results of the linear theory or through a
X
N second-order expansion of the error functional Eq. (2).
EðpÞ ¼ ðrexp ðiÞ À ro ðiÞÞ2 ,
o (2) The first approach, used by Petersen (2000) and Dochain
i¼1 and Vanrolleghem (2001), approximates C with the
where p ¼ ½Y A1 RNH K NH Y A2 RNO K NO ŠT is the para- inverse of the Fisher Information Matrix (FIM) J
meter vector, N is the number of respirometric data (Ljung, 1999), defined as a quadratic form of the output
and the model output ro is computed from Eq. (3) sensitivity qy=qpjp
^
N exp
3:43 À Y A1 S NH 1 X qyðkÞ T qyðkÞ
ro ¼ ^
mmaxA1 X A1 C J ¼ J À1 ; where J ¼ . (5)
Y A1 K NH þ SNH s2 k¼1 qp qp
1:14 À Y A2 SNO
þ ^
mmaxA2 X A2 þ rend . ð3Þ The measurement error variance is estimated as
Y A2 K NO þ SNO
s2 ¼ Eð^ Þ=N À np . As an alternative, C can be approxi-
p
The minimisation of the error functional Eq. (2) with mated by a second-order expansion of the objective
the model constraint Eq. (3) is obtained with a combined error function in the neighbourhood of the minimum
search algorithm starting with a modified genetic Eð^ Þ (Press et al., 1986; Seber and Wild, 1989)
p
algorithm (Marsili-Libelli and Alba, 2000) to determine

4. the initial search region containing the global minimum, q2 EðpÞ

6. .
C H ð^ Þ ffi 2s2 H À1 ; where H ¼
p (6)
which is then refined with a modified simplex search qpqpT

7. p ^
(Marsili-Libelli, 1992). The calibration software was
developed in the Matlab 6.5 platform (The Mathworks, Substituting either CJ or CH or in place of C in Eq. (4)
Natick, MA, USA), using the stiff-Rosenbrock integra- two differing approximate confidence ellipsoids are
tion method with a relative accuracy of 10À6. The details obtained. Numerical algorithms for computing these
of model implementation are described in Marsili-Libelli approximations are described in Press et al. (1986),
and Tabani (2002). Marsili-Libelli et al. (2003) and De Pauw (2005).

8. ARTICLE IN PRESS
N. Checchi, S. Marsili-Libelli / Water Research 39 (2005) 3686–3696 3689
From Eqs. (5) or (6) the confidence interval of the (Bates and Watts, 1988). Consider the maximum and
individual parameter can be computed minimum curvature radii rmax ¼ ð1 À lmax ÞÀ1=2 and
a=2
pffiffiffiffiffiffiffiffiffiffiffiffi rmin ¼ ð1 À lmin ÞÀ1=2 , where lmax and lmin are the
dpi ¼ ÆtNÀnp Cði; iÞ, (7) maximum and minimum eigenvalues of the matrix B
a=2
where tNÀnp is the two-tails Student’s t distribution for related to the Hessian and FIM matrices by the
the given confidence level 100 (1Àa)% and NÀnp relationship (Seber and Wild, 1989)
degrees of freedom. This statistics is consistent with B ¼ I np À 1K T HK, (8)
2
the multivariate F distribution for np ¼ 1, since
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À1 T
a=2
tNÀnp ¼ F a where K ¼ R and R R is the upper-triangular
1;NÀnp . Substituting CJ or CH in place of C
decomposition of s2 J. Eq. (8) is the result of a
in Eq. (7) yields the approximate confidence bounds of coordinate transformation in the parameter space which
^
the estimated parameters pi . It is important to notice attempts at reducing the confidence ellipsoid to a sphere
that though dpi refers to a single parameter, it takes into (Seber and Wild, 1989). In the absence of curvature both
account the full np-dimensional confidence region rmax and rmin tend to 1, therefore the extent to which
through the matrix C. their ratio exceeds unity yields an indication of the
residual curvature in the neighbourhood of p. A ^
1.4. A parameter estimation reliability test normalized threshold value for the ratio rmax =rmin can
be obtained in the form
In addition to yielding confidence regions, the two rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
approximations Eqs. (5) and (6) provide a reliability test np
r¼1þ
¯ Fa , (9)
for the estimated parameters based on their inherent N À np np ;NÀnp
conceptual difference. The FIM approximation CJ is
where the square root represents the tolerance to keep
based on the sensitivities, whereas the Hessian approx-
into account the effective ellipsoids dimensions in the
imation CH depends on the shape of the error surface.
tangent plane (Bates and Watts, 1988; Seber and Wild,
For nonlinear systems CH includes the effect of the
1989) normalized to be insensitive to scaling errors.
curvature, reflecting the degree of nonlinearity induced
Further, Quinn et al. (2005) recently demonstrated that
by the model structure (Donaldson and Schnabel, 1987;
the F statistics can be used to determine the confidence
Bates and Watts, 1988; Seber and Wild, 1989; Marsili-
limits of a parameter function such as Eq. (8).
Libelli et al., 2003, Appendix B). Conversely CJ, being a
If rmax =rmin or the estimation can be considered
¯
linear approximation, does not contain this term. Since
reliable at the 100(1Àa)% level, implying that the
the curvature effect vanishes in the neighbourhood of
search terminated at a point on the error surface
the minimum of E(p), comparing the two confidence
where the effect of the curvature is negligible. This
regions yields a measure of the estimation reliability,
test is more practical than the mere visual compari-
because if the two regions diverge, this implies that the
son between ellipses because it does not involve
search terminated at a point where the effect of the
the subjectivity of the examiner and it takes into account
curvature is still significant and therefore this cannot be
the full np dimension of the parameter space rather
the real minimum of E(p). Conversely, if the two
than the 2-dimensional subspace of the projected
regions coincide the curvature effect is negligible and the
ellipsoids.
identification can be considered reliable. In this case
these regions coincide also with the exact confidence
region determined on the basis of the error surface 1.5. Optimal experiment design (OED) criteria
(Lobry and Flandrois, 1991). It should be reminded,
however, that the curvature, amplifying the estimation Other estimation accuracy indexes may be obtained as
errors, is an indicator of a failure to reach the minimum, scalar functions of the covariance matrix C. Each of
but is not influenced by model inadequacy and residual them emphasises differing aspects of the confidence
characteristics. Its vanishing merely indicates that ellipsoid, such as its volume or axes. Based on
the residuals are orthogonal to the response surface in these indicators, a theory of optimal experiments has
^
the neighbourhood of p, but does not attempt at been developed (Fedorov, 1972) with the aim of
characterizing them, e.g. whether they are gaussian minimising the estimates covariance. Table 1 illustrates
and uncorrelated. This test is therefore an assessment of the main criteria used in this and similar studies (see e.g.
the quality of the optimisation and not of the model Atkinson and Donev, 1992; Versyck et al., 1998;
structure. Petersen, 2000; De Pauw, 2005). Of all the criteria
Visual inspection of the agreement or divergence mentioned in Table 1, only D has the property of being
between the confidence regions involves a fair amount of scale invariant, whereas all the others are scale sensitive,
subjectivity. To obtain an objective test the curvature particularly mod E. The least sensitive of all appears to
radii are considered, which are related to the curvature be mod A.

9. ARTICLE IN PRESS
3690 N. Checchi, S. Marsili-Libelli / Water Research 39 (2005) 3686–3696
Table 1
Optimal experiment design criteria based on FIM
Criteria Method Effect
A minðtrðJ À1 ÞÞ Minimization of the arithmetic mean of parameter errors.
mod A maxðtrðJÞÞ Same as A, but insensitive to FIM ill-conditioning
D maxðdetðJÞÞ Minimizes the volume of the confidence ellipsoid
E maxðlmin ðJÞÞ Minimizes the length of the largest axis of the confidence ellipsoid
mod E minðlmax =lmin Þ Minimizes the ill-condition number
ro (mgO2 L-1 min-1)
2. Calibration of the respirometric model 1
SNH (0) = 2.5 mg L-1
0.8 model
The purpose of this section is to investigate the data
influence of the initial amount of substrate SNH(0) on 0.6
the identifiability of the model parameters 0.4
p ¼ ½Y A1 RNH K NH Y A2 RNO K NO ŠT , using three data sets 0.2
obtained with the respirometer described in Marsili- 0 5 10 15 20 25 30
Libelli and Tabani (2002). The validity of the identified time (min)
parameters is assessed through several tests: sensitivity,
curvature radii and OED criteria. ro (mgO2 L-1 min-1) 1.2
1 SNH (0) = 5 mg L-1
model
2.1. Assessment of estimated parameters 0.8 data
0.6
Though the estimation results, shown in Fig. 1, 0.4
appear visually acceptable in all cases, the inspection 0.2
of the parameter values and confidence intervals of 0 5 10 15 20 25 30 35 40 45
Table 2 reveals that under certain experimental condi- time (min)
tions the results may be seriously flawed. As a
ro (mgO2 L-1 min-1)
preliminary test, parameter values should always check 1.2
1 SNH (0) = 10 mg L-1
for physical constraints; in this sense the negative values model
YA2 for low and medium SNH(0) indicates an unreliable 0.8 data
estimation, though its wide confidence interval encom- 0.6
passes reasonable values. On the other hand, imposing 0.4
positivity constraints to the simplex algorithm may 0.2
disrupt the search and produce awkward numerical 0 10 20 30 40 50 60 70 80 90
results, as demonstrated by the results of Alfonso and da time (min)
-˜
Conceic ao Cunha (2002), later criticized by De Pauw Fig. 1. Fitting the model Eq. (3) to the batch respirometric
et al. (2004). For this reason it was preferred to use an data.
unconstrained method, but rather use the confidence
regions approach to design reliable experiments. The
most serious flaws appearing in Table 2 are the negative whereas the full data set failed. This results is in
values of the yield coefficients for low and medium agreement with De Pauw (2005), who found that
initial SNH(0) and the abnormally large values RNO and under-sampling may results in an improved accuracy
KNO for SNH(0) ¼ 5 mg LÀ1. Further, since the duration and reduced parameter correlation.
of the three experiments differs, the number of available
data is not the same and this may affect the estimation 2.2. Sensitivity analysis
accuracy. To obtain a fair comparison among the three
experiments, the data sets for SNH(0) ¼ 5 and 10 mg LÀ1 The sensitivity trajectories for the three initial
were under-sampled selecting every other data, in order substrate conditions SNH(0) are shown in Fig. 2. They
to have approximately the same experimental basis for were computed with an incremental method similar to
all of them. The results were rather counterintuitive: not that described by De Pauw and Vanrolleghem (2003)
only the estimation did not get worse, but rather it and De Pauw (2005). The sensitivity to YA1 and YA2 is
improved considerably for SNH(0) ¼ 5 mg LÀ1, particu- very small for all initial SNH(0). This is in agreement
larly for the yield coefficients and the critical second-step with the results of Table 2 and suggest that the best way
parameters. The radii test Eq. (9) was also passed, to identify the yield coefficients is to estimate their sum

10. ARTICLE IN PRESS
N. Checchi, S. Marsili-Libelli / Water Research 39 (2005) 3686–3696 3691
Table 2
Estimated parameters for the respirometric experiments, confidence intervals and reliability indicators
SNH(0) No. of data Y A1 RNH K NH Y A2 RNO K NO Y A ¼ Y A1 þ Y A2 rend rmax =rmin r
¯
2.5 18 ^
p 2.5288 0.1909 0.2124 -2.2039 2.4849 1.8595 0.3249 0.2725 36.4446 2.3900
dpJ 75.9618 70.0110 70.0498 75.7584 713.9352 712.1508 70.2115
dpH 75.2857 70.0077 70.0280 75.1212 71.7371 71.3545 70.1721
5.0 15 ^
p 0.1973 0.2063 0.2006 -0.0258 0.2036 0.1371 0.1714 0.2710 1.1565 2.6387
dpj 70.7820 70.0078 70.0351 70.7577 70.0513 70.2092 70.0492
dpH 70.7023 70.0072 70.0352 70.6809 70.0473 70.1893 70.0478
5.0 29 ^
p 0.7340 0.2032 0.2088 -0.4982 13.4362 24.443 0.2358 0.2710 202.6126 1.9416
dpj 70.7867 70.0037 70.0273 70.7582 7584.9534 71081.8783 70.0451
dpH 70.5556 70.0022 70.0207 70.5379 73.8382 76.3424 70.0393
10.0 21 ^
p 0.0657 0.2061 0.2440 0.1557 0.1777 0.0819 0.2214 0.2657 1.6819 1.8830
dpj 70.0998 70.0015 70.0214 70.0986 70.0097 70.0513 70.0232
dpH 70.0946 70.0014 70.0216 70.0939 70.0069 70.0340 70.0232
10.0 42 ^
p 0.0388 0.2058 0.2416 0.1936 0.1824 0.1103 0.2324 0.2657 1.3155 1.6205
dpj 70.0870 70.0013 70.0196 70.0875 70.0079 70.0467 70.0205
dpH 70.0820 70.0013 70.0208 70.0829 70.0066 70.0389 70.0204
Y A ¼ Y A1 þ Y A2 and perform a separate experiment amount SNH(0) and the consumed oxygen expressed as
for calibrating YA2, as will be shown later. For NOD, where YA is the yield factor of the two steps
SNH ð0Þ ¼ 2:5 mg LÀ1 , the proportional sensitivities of combined and a fourth experiment with SNH ð0Þ ¼
the second step parameters RNO and KNO denote poor 1 mg LÀ1 was added for improved accuracy. The
identifiability. This effect decreases as SNH(0) increases, resulting value Y A ¼ 0:245 is very close to 0.24, which
but still underlines the critical identifiability of the is generally indicated as the typical yield factor for
second step parameters. It should also be noticed that autotrophs (Orhon and Artan, 1994; Henze et al., 2000;
the shape of the RNO trajectory is largely influenced by Petersen, 2000; Sin, 2004).
SNH(0), exhibiting a large peak at the beginning of the
second step for SNH(0) ¼ 10 mg LÀ1. This indicates that 2.5. Respiration on nitrite
the transition between the first and the second step is a
critical event for parameter estimation and it will be The estimation of the second step has always posed
investigated further with a specific nitrite experiment. special problems and in the past the estimation of both
steps was achieved by introducing selective inhibitors
2.3. Evaluation of design criteria (Nowak et al., 1995; Surmacz-Gorska et al., 1996) such
as Allylthiourea (ATU) or sodium chloride. However,
The OED indicators of Table 1 were evaluated for the the use of inhibitors, in addition to requiring a sample of
calibrated model of Fig. 1 and the results of Table 3 were fresh biomass for each experiment, is not advisable for
obtained. In agreement with the confidence intervals of their lack of selectivity. On another front, Ficara et al.
Table 2, the uncertainty affecting the estimates of RNO (2002) characterize the two steps by measuring pH or
and KNO is not monotonic and this reflects into the OED performing two separate experiments, if only DO
criteria of Table 3, save for mod A, which is not affected measurements are considered. To investigate further
by FIM ill-conditioning (Petersen, 2000) and shows a the dynamics of the second step, a nitrite respirogram
monotonic increase with SNH(0). In fact it was found that with SNO ð0Þ ¼ 2:7 mg LÀ1 was performed. According to
if RNO and KNO are not estimated all the OED indexes Eq. (3) the pertinent model should be
exhibit a monotonic behaviour. Further, all the OED
SNO
criteria were only slightly affected by data decimation. rNO ¼ ð1:14 À Y A2 ÞRNO þ rend . (10)
K NO þ SNO
2.4. Estimation of the yield factors This, however, does not take into account the gradual
response of the nitrite oxidisers observed by Vanrolle-
Fig. 3 shows the fitted regression line nitrogen oxygen ghem et al. (1998) and Vanrolleghem et al. (2004). To
demand (NOD) ¼ (4.57ÀYA) SNH(0) between the initial account for this biochemical fact, an exponential term is

11. ARTICLE IN PRESS
3692 N. Checchi, S. Marsili-Libelli / Water Research 39 (2005) 3686–3696
SNH (0) = 2.5 mg L-1 SNH (0) = 5 mg L-1
1°step 2°step 1°step 2°step
4 4 0.05
0.15
KNH
2
RNO KNO
2 0.1
KNH 0 0
RNO
YA1 0.05 KNO -2 YA1
0
-4 -0.05
0
-2 -6
-0.05
-8 -0.1
-4
-0.1 -10
RNH YA2
RNH YA2
-12 -0.15
-6 -0.15
-14
-8 -0.2 -16 -0.2
0 10 20 30 0 10 20 30 10 20 40 60 10 20 40 60
time (min) time (min) time (min) time (min)
SNH (0) = 10 mg L-1
1°step 2°step
5 2
kNH KNO
0 0
YA1 YA2
-2
-5
-4
-10
-6
-15
-8
RNH RNO
-20 -10
-25 -12
0 50 100 0 50 100
time (min) time (min)
Fig. 2. Sensitivity trajectories of the respirometric model Eq. (3) for the three values of SNH(0) over the two steps. The solid lines refer
to Y’s, the dotted lines to R’s and the dashed lines to K’s.
introduced to model the initial time lag t. From the identification viewpoint, the initial rising
data are incompatible with the Monod term, which
SNO cannot fit the OUR initial increase. It is therefore fair to
rNO ¼ ð1 À eÀt=t Þð1:14 À Y A2 ÞRNO þ rend .
K NO þ SNO omit them and calibrate the model Eq. (10) only with the
(11) data which it can explain. The technique of eliminating

12. ARTICLE IN PRESS
N. Checchi, S. Marsili-Libelli / Water Research 39 (2005) 3686–3696 3693
Table 3
Optimal experiment design criteria evaluated for the three respirograms. The FIM results are in italic, the Hessian in bold
SNH(0) A mod A D E mod E
2.5 8.6480  101 1.9961  106 7.1748e  1012 1.3325  10À2 1.4888  108
1.2432 Â 101 1.9859 Â 106 7.9761 Â 1014 8.5083 Â 10À2 2.3194 Â 107
5 3.5347 Â 105 4.7588 Â 106 7.4034 Â 109 2.8290 Â 10À6 1.6758 Â 1012
1.2983 Â 101 4.6728 Â 106 3.5911 Â 1014 9.7351 Â 10À2 4.7818 Â 107
10 4.4101 Â 10À3 1.5971 Â 107 2.9120 Â 1027 2.6742 Â 102 5.3342 Â 104
3.8515 Â 10À3 1.6782 Â 107 4.9547 Â 1027 3.0053 Â 102 4.9163 Â 104
50 High High
NOD= 4.325SNH (0)
sensitivity to sensitivity to
YA = 4.57−4.325=0.245 τ KNO , RNO
40
R2 = 0.9992
NOD (mgO2 L-1)
rNO
30
0
20
YA2
-0.5
10
-1
0 0 5 10 15 20 25
0 2 4 6 8 10 12 1
SNH(0) (mg NH4+− N L-1 )
RNO
0
Fig. 3. Computation of the combined yield factor YA from
respirograms. For improved accuracy a fourth experiment with -1
0 5 10 15 20 25
SNH ð0Þ ¼ 1 mg LÀ1 was considered.
1
KNO
0
the initial respiration data has been frequently used in -1
the literature (Vanrolleghem et al., 1995; Vanrolleghem 0 5 10 15 20 25
and Coen, 1995; Vanrolleghem and Keesman, 1996;
0
-
Ossenbruggen et al., 1996; Ubay C okgor et al., 1998;
¨
Mathieu and Etienne, 2000) in estimating a Monod -0.5
τ
kinetics. Conversely, Eq. (11), including both start-up -1
and nitrification, can explain all the experimental data 0 5 10 15 20 25
and the introduction of the exponential term in Eq. (11) time (min)
is fully supported by experimental evidence and bio-
Fig. 4. Sensitivity analysis of the nitrite respirogram. The
chemical theory (Vanrolleghem et al., 1998; Petersen,
dotted line represents the reference rNO trajectory.
2000; Baeza et al., 2002; Vanrolleghem et al., 2004).
Further, this additional term does not pose special
estimation problems given the sensitivity separation with correlation matrix R
the other parameters YA2, RNO or KNO. In fact the
trajectories of Fig. 4 indicate two high-sensitivity zones: Y A2 RNO K NO t
at the beginning of the experiment, both the time lag t 2 3
1:0000 À0:1859 À0:2803 À0:4453 Y A2
and the yield factor YA2 are the most sensitive
R ¼ 6 À0:1859
6 1:0000 0:8912 0:3143 77 RNO
parameters, whereas in the last part of the respirogram 6 7
4 À0:2803 0:8912 1:0000 0:2863 5 K NO
the two most critical parameters are RNO and KNO. The
introduction of the time lag allows the preservation À0:4453 0:3143 0:2863 1:000 t
of the biological meaning of YA2, RNO or KNO. The (12)
relative fit of the two models is shown in Fig. 5 and
the estimated parameters of the two models are that apart from the couple (RNO, KNO), all other
compared in Table 4. It is evident from the parameter correlations are moderate.

13. ARTICLE IN PRESS
3694 N. Checchi, S. Marsili-Libelli / Water Research 39 (2005) 3686–3696
The primary aim of this experiment is the direct Y A1 þ Y A2 with the area method of Section 2.4 and then
estimation of YA2, and in this sense model Eq. (11) is perform a separate nitrite respirogram.
definitely superior to Eq. (10), in which the large
estimated value for YA2 can be explained observing that
the oxygen consumption is given by NOD ¼ (1.14ÀYA2) 3. Conclusion
SNO(0). If the respirogram area decreases, as a
consequence of discarding the initial four data, NOD This paper has advanced the results of a previous
decreases and therefore the right-hand-side must de- study to obtain an estimation reliability test, which was
crease. Since SNO(0) is constant, this can only be applied to a respirometric model to assess its estimates
accomplished by increasing YA2, which leads to the and design efficient experiments. A test (Eq. (9)) is
observed error. This value is clearly unfeasible, being presented, based on the computation of the approximate
larger than the sum of YA1 and YA2 of Table 2 for confidence regions, from which a discriminating thresh-
SNH ð0Þ ¼ 10 mg LÀ1 whereas its confidence interval is old is derived: if the ratio of the curvature radii is below
unrealistically narrow, indicating a great confidence in a a given value, then the estimation can be considered
wrong estimate. On the other hand, the parameter reliable in the sense that the estimation procedure ended
values of Eq. (11) are in good agreement with those of in a well-behaved region of the error functional,
Table 2 for SNH ð0Þ ¼ 10 mg LÀ1 and are consistent with corresponding to acceptable parameters. This test is
the literature values (Orhon and Artan, 1994; Petersen, based on the effect of curvature on parameter estimation
2000). They confirm, however, the large inherent and detects possible obstacles preventing the successful
uncertainty, about 36%, in the estimation of YA2, in termination of the search, but is not influenced by model
line with the results found by Chandran and Smets inadequacy and residual characteristics, therefore it
(2001) by an indirect method based on electron balance. should not be directly used to discriminate among
Respirometric experiments based on titrimetric and off- model structures. In the paper this test has been used to
gas analysis (Gapes et al., 2003) report a similar range assess the role played by the initial amount of
(0.02–0.07). As expected, both models satisfy the ammonium-N SNH(0) in identifying a two-step respiro-
curvature criterion Eq. (9), confirming that this test is metric model and the effect of decimating the data,
capable of detecting only error in the optimisation and which may produce more accurate estimates. Increasing
should not be used to discriminate among model the amount of SNH(0) produces a general decrease in the
structures. From these experiments it appears that the uncertainty of the estimated parameters. However, it is
best way to estimate YA2 is to compute the sum Y A ¼ necessary to take into account other factors which limit
this quantity: (1) the model does not consider biomass
growth (as in Nowak et al., 1995; Brun et al., 2002) and
therefore the experiment must have a rather short
Model Eq. (11)
Model Eq. (10) duration; (2) the nitrification process is very sensitive
Data not used in calibrating Model Eq. (10) to several factors: not only pH and temperature, but also
Data used in calibrating both models Eqs. (10) and (11) a high ammonium-N concentration as an inhibiting
0.45 agent. This conflicts with the requirement of a high
rNO (mg L-1 min-1)
SNO (O)= 2.7 mg L1 SNH(0) for good identifiability, though a high SNH(0)
0.4
produces more data, which increase the estimation
0.35 accuracy. Therefore one of the aims of the paper has
been that of determining the best compromise for
0.3
producing a well-identifiable two-step respirogram with-
rend
0.25 out sacrificing either the accuracy of the first step
0 5 10 15 20 25 30 with an insufficient SNH(0) or inhibiting the process
time (min) with a too large SNH(0). It was also shown that a
Fig. 5. Calibration of the second step with nitrite data. The nitrite experiment could be used to estimate the second
initial transient data (hollow circles) were not used in the step alone, adding an exponential term to take into
calibration of model Eq. (10). account the gradual nitrite uptake by the biomass. The
Table 4
Estimated parameters of the nitrite respiration models
Y A2 RNO K NO rend t rmax =rmin r
¯
Model Eq. (10) 0.361570.0155 0.23670.007 0.14570.0138 0.263 — 1.1330 1.9891
Model Eq. (11) 0.063770.0233 0.166570.066 0.092170.0428 0.263 1.069470.3425 1.2358 1.9430

14. ARTICLE IN PRESS
N. Checchi, S. Marsili-Libelli / Water Research 39 (2005) 3686–3696 3695
estimation of the yield factors, given their strong Gapes, D., Pratt, S., Yuan, Z., Keller, J., 2003. Online
correlation, is best accomplished by computing the titrimetric and off-gas analysis for examining nitrification
global yield Y A ¼ Y A1 þ Y A2 obtained from the oxygen processes in wastewater treatment. Water Res. 37,
consumption and then perform a separate nitrite 2678–2690.
respirogram for YA2 using the modified Eq. (11) which Gernaey, K., Petersen, B., Ottoy, J.P., Vanrolleghem, P.A.,
2001. Activated sludge monitoring with combined respir-
includes an exponential term for taking into account the
ometric–titrimetric measurements. Water Res. 35 (5),
initial start-up data.
1280–1294.
Gernaey, K., Petersen, B., Dochain, D., Vanrolleghem, P.A.,
2002. Modelling aerobic carbon source degradation pro-
Reference cesses using titrimetric data and combined respirometric-
titrimetric data: structural and practical identifiability.
-˜
Alfonso, P., da Conceicao Cunha, M., 2002. Assessing Biotechnol. Bioeng. 79, 754–769.
parameters identifiability of activated sludge model number Gernaey, K.V., van Loosdrecht, M.C.M., Henze, M., Lind, M.,
1. J. Environ. Eng. ASCE 128, 748–754. Jørgensen, S.B., 2004. Activated sludge wastewater treat-
Atkinson, A.C., Donev, A.N., 1992. Optimum Experimental ment plant modelling and simulation: state of the art.
Designs. Clarendon Press, Oxford. Environ. Model. Soft. 19, 763–783.
Baeza, J.A., D. Gabriel, D., Lafuente, J., 2002. In-line fast Henze, M., Gujer, W., Mino, T., van Loosdrecht, M.C.M.,
OUR (oxygen uptake rate) measurements for monitoring 2000. Activated sludge models ASM1, ASM2, ASM2d, and
and control of WWTP. Water Sci. Technol. 45, 19–28. ASM3. IWA Scientific and Technical Report no. 9.
Bates, D., Watts, D., 1988. Nonlinear Regression Analysis and Holmberg, A., 1982. On the practical identifiability of microbial
its Applications. Wiley, New York. models incorporating Michaelis–Menten-type nonlineari-
Brouwer, H., Klapwijk, A., Keesman, K.J., 1998. Identification ties. Math. Biosci. 62, 23–43.
of activated sludge characteristics using respirometric batch- Insel, G., Orhon, D., Vanrolleghem, P.A., 2003. Identification
experiments. Water Res. 32, 1240–1254. and modelling of aerobic hydrolysis mechanisms—applica-
Brun, R., Kuhni, M., Siegrist, H., Gujer, W., Reichert, P., 2002. tion of optimal experimental design. J. Chem. Technol.
Practical identifiability of ASM2d parameters—systematic Biotechnol. 78, 437–445.
selection and tuning of parameter subsets. Water Res. 36, Kesavan, P., Law, V.J., 2005. Practical identifiability of
4113–4127. parameters in Monod kinetics and statistical analysis of
Chandran, K., Smets, B., 2001. Estimating biomass yield residuals. Biochem. Eng. J. 24, 95–104.
coefficients for autotrophic ammonia and nitrite oxidation Ljung, L., 1999. System identification—theory for the user.
from batch respirograms. Water Res. 35, 3153–3156. Prentice-Hall, Englewood Cliffs, NJ.
De Pauw, D.J.W., 2005. Optimal experimental design for Lobry, J.R., Flandrois, J.P., 1991. Comparison of estimates of
calibration of bioprocess models: a validated software Monod’s growth model parameters from the same data set.
toolbox. Ph.D. Thesis in Applied Biological Sciences, Binary 3, 20–23.
BIOMATH, University of Gent. Downloadable from Marsili-Libelli, S., 1992. Parameter estimation of ecological
http://biomath.ugent.be/publications/download/. models. Ecol. Model. 62, 233–258.
De Pauw, D.J.W.,Vanrolleghem, P.A., 2003. Practical aspects Marsili-Libelli, S., Alba, P., 2000. Adaptive mutation in genetic
of sensitivity analysis for dynamic models. In: Proceedings algorithms. Soft Computing 4 (2), 76–80.
of the IMACS Fourth MATHMOD Conference, Vienna, Marsili-Libelli, S., Tabani, F., 2002. Accuracy analysis of a
Austria. respirometer for activated sludge dynamic modelling. Water
De Pauw, D.J.W., Sin, G., Insel, G., Van Hulle, S.W.H., Res. 36, 1181–1192.
Vandenberghe, V., Vanrolleghem, P.A., 2004. Discussion of Marsili-Libelli, S., Guerrizio, S., Checchi, N., 2003. Confidence
Assessing parameter identifiability of activated sludge regions of estimated parameters for ecological systems.
model number 1 by P. Afonso and M. da Conceicao ˜ Ecol. Model. 165, 127–146.
Cunha. J. Environ. Eng. ASCE, 110–112. Mathieu, S., Etienne, P., 2000. Estimation of wastewater
Dochain, D., Vanrolleghem, P.A., 2001. Dynamical Modelling biodegradable COD fractions by combining respirometric
and Estimation in Wastewater Treatment Processes. IWA experiments in various S0/X0 ratios. Water Res. 34,
Publishing, London. 1233–1246.
Dochain, D., Vanrolleghem, P.A., Van Daele, M., 1995. Nowak, O., Svardal, K., Schweighofer, F., 1995. The dynamic
Structural identifiability of biokinetic models of activated behaviour of nitrifying activated sludge system influenced
sludge respiration. Water Res. 29, 2571–2578. by inhibiting wastewater compounds. Water Sci. Technol.
Donaldson, J., Schnabel, R., 1987. Computational experience 31 (2), 115–124.
with confidence regions and confidence intervals for non- Orhon, D., Artan, N., 1994. Modelling of Activated Sludge
linear least squares. Technometrics 29, 67–82. Systems. Technomics Publ. Co. Inc., Basel.
Fedorov, V.V., 1972. Theory of Optimal Experiments. Aca- Ossenbruggen, P.J., Spanjers, H., Klapwik, A., 1996. Assess-
demic Press, New York. ment of a two-step nitrification model for activated sludge.
Ficara, E., Musumeci, A., Rozzi, A., 2002. Comparison and Water Res. 30, 939–953.
combination of titrimetric and respirometric techniques to Petersen B., 2000. Calibration, identifiability and optimal
estimate nitrification kinetics parameters. Water SA 26 (2), experimental design of activated sludge models. Ph.D.
217–224. Thesis in Applied Biological Sciences, BIOMATH,

15. ARTICLE IN PRESS
3696 N. Checchi, S. Marsili-Libelli / Water Research 39 (2005) 3686–3696
University of Gent. Downloadable from http://biomath.u- BIOMATH, University of Gent. Downloadable from
gent.be/publications/download/. http://biomath.ugent.be/publications/download/.
Petersen, B., Gernaey, K., Vanrolleghem, P.A., 2000. Improved Surmacz-Gorska, J., Gernaey, K., Demuynck, C., Vanrolle-
theoretical identifiability of model parameters by combined ghem, P.A., Verstraete, W., 1996. Nitrification monitoring
respirometric–titrimetric measurements. A generalization of in activated sludge by oxygen uptake rate (OUR) measure-
results. In: Proceedings of the IMACS Third Symposium ments. Water Res. 30 (5), 1228–1236.
on Mathematical Modelling, Vienna, Austria, vol.2, -
Ubay C okgor, E., Sozen, S., Orhon, D., Henze, M., 1998.
¨ ¨
pp. 639–642. Respirometric analysis of activated sludge behaviour—I.
Petersen, B., Gernaey, K., Henze, M., Vanrolleghem, P.A., Assessment of the readily biodegradable substrate. Water
2002. Evaluation of an ASM1 model calibration procedure Res. 32, 461–475.
on a municipal-industrial wastewater treatment plant. Vanrolleghem, P.A., Coen, F., 1995. Optimal Design of in-
J. Hydroinformatics 4, 15–38. sensor-experiments for on-line modelling of nitrogen
Petersen, B., Gernaey, K., Devisscher, M., Dochain, D., removal processes. Water Sci. Technol. 31, 149–160.
Vanrolleghem, P.A., 2003a. A simplified method to assess Vanrolleghem, P.A., Keesman, K.J., 1996. Identification of
structurally identifiable parameters in Monod-based acti- biodegradation models under model and data uncertainty.
vated sludge models. Water Res. 38, 2893–2904. Water Sci. Technol. 33, 91–105.
Petersen, B., Gernaey, K., Henze, M., Vanrolleghem, P.A., Vanrolleghem, P.A., Van Daele, M., Dochain, D., 1995.
2003b. Calibration of activated sludge models: a critical Practical identifiability of a biokinetic model of activated
review of experimental designs. In: Agathos, S.N., Reineke, sludge respiration. Water Res. 29, 2561–2570.
W. (Eds.), Biotechnology for the Environment: Wastewater Vanrolleghem, P.A., Gernaey, K. V., Petersen, B., De Clercq,
Treatment and Modelling, Waste Gas Handling. Kluwer B., Coen, F. and Ottoy, J.-P., 1998. Limitations of short-
Academic Publ., Dordrecht (NL), pp. 101–186. term experiments designed for identification of activated
Pohjanpalo, H., 1978. System identifiability based on the power sludge biodegradation models by fast dynamic phenomena.
series expansion of the solution. Math. Biosci. 41, 21–33. In: Proceedings of the Seventh IFAC Conference on
Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T., Computer Applications in Biotechnology CAB7, Osaka,
1986. Numerical Recipes: The Art of Scientific Computing. Japan.
Cambridge University Press, Cambridge, Mass. Vanrolleghem, P.A., Sin, G., Gernaey, K.V., 2004. Transient
Quinn, S.L., Harris, T.J., Bacon, D.W., 2005. Accounting for response of aerobic and anoxic activated sludge activities to
uncertainty in control-relevant statistics. J. Proc. Contr. 15, sudden substrate concentration changes. Biotechnol.
675–690. Bioeng. 86, 277–290.
Seber, G.A.F., Wild, C.J., 1989. Nonlinear Regression. Wiley, Versyck, K.J., Claes, J.E., Van Impe, J.F., 1998. Optimal
New York. experimental design for practical identification of unstruc-
Sin, G., 2004. Systematic calibration of activated sludge tured growth models. Math. Comput. Simulation 46,
models. Ph.D. Thesis in Applied Biological Sciences, 621–629.