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Ind. Eng. Chem. Res. 1996, 35, 2288-2298
An Efficient Formulation for Batch Reactor Data Reconciliation Manuel Fillon, Michel Meyer, Herve´ Pingaud,* and Michel Enjalbert INPT/ENSIGC, 18 Chemin de la loge, 31078 Toulouse Cedex, France
An efficient formulation of the data reconciliation problem applied to batch reactor experiments is proposed and discussed in comparison with other works reported on this subject in the literature. Under the assumption that all components in the reaction are known experimentally and that their chemical compositions are defined, atomic balances are used as constraints in order to correct raw data values. The formulation is based on an a posteriori simultaneous treatment of all measurements over the whole time of experimentation. The main point of interest in this formulation is the preservation of the measurement identity in each sample. Even if component holdups are systematically involved in mass balances by pairs to estimate a difference in time, here we correct each component measurement and each sample independently. On this basis, it is shown how classical statistical tests can be used to discriminate gross errors. A case study built from simulated true and noisy data gives an overview of the performances of the method. The results state that the overall time formulation is more powerful than the classical one, which is essentially based on a sequential treatment by periods of time. Evaluation of the performance for three adapted statistical tests leads us to conclude that the generalized likelihood ratio (GLR) is not only the most appropriate but also the most efficient test. Introduction Measurement is the main source of information for acquiring knowledge about process behavior. However, measurements, as they are not free of random error, have to be used carefully and, if possible, have to be checked before use. Normally, if the measuring instrument is used correctly, the random error is small and can be neglected. But if a defective feature in the sensor has not been identified or if it is not used properly, a gross error might possibly occur. It is therefore not recommended to use experimental results without a fine analysis of their intrinsic quality. Methods which are able to diagnose such a situation are undoubtedly of great interest. The detection of gross error improves confidence in quality and avoids the propagation of erroneous information in subsequent data processing. This is especially true for kinetics estimation where nonlinear laws have to be designed directly from the results of experiments. When a gross error is systematically suspected on the same component for the same period in many experiments, an indication will be given on how to investigate the physicochemical analytical methods or some procedures of the experimentation itself to detect what is wrong. Data reconciliation is a statistical method where a measurement is interpreted as a random variable, the error of which is assumed to follow a probability law with known moments. The quality of measurements is controlled by means of a set of mathematical constraints. The redundancy that the raw measurement values provide when the constraints are quantified alongside them is exploited to compute an optimal estimation of all observable variables appearing in the constraints. The unsuitability of raw data with respect to the mathematical constraints could be achieved by statistical tests. In the field of chemical engineering, this method has been mainly studied in two directions: (1) problem formulation and analysis including two aspects, observability and redundancy, and (2) gross error detection methods. * Author to whom correspondence is addressed. 62.25.23.18. Email:
[email protected].
Fax:
S0888-5885(95)00152-7 CCC: $12.00
In the first direction, the problem is analyzed in order to find the relationship between available measurements and constraints which are mass and energy balances. The redundant equations are deduced from specific algorithms, after state variable observability has been established. Many algorithms have been proposed for the analysis of incompletely observed linear systems. The methods may rely on a matrix-based approach (Crowe et al. (1983), Crowe (1989), Darouach et al. (1986), Romagnoli and Stephanopoulos (1980)) or on the graph theory (Kretsovalis and Mah (1987), Meyer et al. (1993)). Apart from bilinear systems, which may be processed by the method developed by Meyer et al. (1993), there are no algorithms available for generalized nonlinear systems. The second point intensively investigated in chemical engineering literature is gross error detection. The importance of this aspect in data reconciliation was recognized very early on (Ripps, 1965), and since then, numerous tests have been proposed. Iordache et al. (1985) reviews the statistical tools available in 1985. Within 2 years, Madron (1985), Serth and Heenan (1986), and Rosenberg et al. (1987) had introduced detection algorithms with boundary conditions on the variables. This idea was resumed and generalized by Narasimhan and Harikumar in 1993. Statistical tests are always defined with a view of maximizing their ability to rightly detect gross error, i.e., to effectively detect an existing gross error, on the one hand, and, on the other hand, to avoid the wrongful questioning of an acceptable measurement. No test stands out for remarkable performance on both criteria. Nevertheless, the better tests perform well enough to justify their application. Reactor mass balance equations reflect the formation of products and the dissipation of chemical reactants. The quantification of the reaction dynamics thus requires knowledge of the chemical mechanisms and their effects. This is often the limiting factor when mass balances are chosen as constraints in the data reconcilation problem. The idea of using constraints which do not require such a type of knowledge was first introduced by Madron et al. (1977), who suggested using atomic balances as an adapted support for reactor data © 1996 American Chemical Society
Ind. Eng. Chem. Res., Vol. 35, No. 7, 1996 2289
analysis. This formulation was applied to batch reactors by Wang and Stephanopoulos (1983) and by Van der Heijden et al. (1994b). Such constraints are linear, and theoretical results are available for their analysis. Hence, the observability check and the redundant equation design are not discussed in this paper. The methods proposed by Van der Heijden et al. (1994a) can be applied, without any change, using the formulation introduced hereafter. The focus is on the mathematical formulation itself. The dynamic behavior of a perfectly mixed batch reactor is defined by the information contained in the samples taken. The constraints are based on the quantity of material transformed during each time interval. In previous applications, a correction was estimated for each interval independently of the other ones. The complete processing of the data treatment thus leads to the resolution of several local subproblems. Darouach and Zasadzinski (1991) and Rollings and Devanathan (1993) have introduced a more general formulation of the data reconciliation problem for dynamic linear systems. They both used an overall treatment of available measurements over the given period of observation. This idea is extrapolated here to the case of a batch reactor, with an a posteriori estimation of the errors that satisfy atomic balance constraints for the whole time of experiment. In a first part of this publication, the mathematical formulation of data reconciliation problems with linear constraints is recalled. Its application to the batch reactor follows. The computational requirements of the proposed formulation over time are emphasized. The linear system could be large, but its sparsity is exploited in order to provide an accurate solution. The second part is devoted to the detection of gross errors. After a review of the tests used in data reconciliation for steady-state continuous processes, three are selected as the most suited to our approach to the problem. In the last part, these techniques are compared in a case study using simulated data. Performance indices are discussed and conclusions are drawn. Data Reconciliation Linear Systems. If X*[n] is the vector of true variables, the m linear constraints could be written in matrix notation as follows:
AX* ) 0
(1)
where A[m,n] is the rectangular matrix of known coefficients. The vector of measurement errors is assumed to be a random variable vector which follows a centered Gaussian distribution:
≈ N(0,V) V is defined as the variance-covariance matrix of the measurement errors. As the errors are assumed to be independent, the matrix V is diagonal and positive. The measurement vector Xm is related to the true values and the errors by the following relationship,
Xm ) X* +
(2)
and, with respect to (2), also obeys a normal probability distribution:
Xm ≈ N(X*,V)
The likelihood function FV(Xm) of Xm is a scalar function that defines the probability to find the measurement result, Xm, X* being known:
Fv(Xm) ) 1 exp(-0.5 (Xm - X*)tV-1(Xm - X*)) (3) 0.5 (2π det V) The best estimation X h of the true values X* is chosen to be the one which maximizes FV. For consistency, this optimal value must comply with the model’s constraints (1). So the whole estimation is a constrained optimization problem:
max FV(X)
(Problem P1)
X
s.t.
AX ) 0
If the matrix V is known, which assumes a prerequisite knowledge of the accuracy of the measurement method, problem P1 is equivalent to problem P2:
min (X - Xm)tV-1(X - Xm)
(Problem P2)
X
s.t.
AX ) 0
This problem P2 has an analytical solution X h because the constraints are linear in X and the objective function is quadratic in X. The optimal solution is calculated using expression (4).
X h ) (I - VAt(AVAt)-1A)Xm
(4)
It can be easily shown that this estimation has good statistical properties. It is a nonbiased and minimal variance estimate. Batch Reactor Constraints. A fundamental assumption embedded in the data reconciliation concept is that the model is exact. So the consistency of measurements can be approached confidently only if the process model is truly representative of its process dynamics. Unfortunately, in the case of a reactor, the mass balances take into account the effects of one or many reactions for which the mechanisms, the stoichiometry, or the kinetics are not necessarily precisely known. That is why component mass balances do not constitute the best choice for constraint formulation. Keeping atomic elements in the reactor is a better alternative as it does not necessitate so much theoretical knowledge. Here it is sufficient to have the definitions of the chemical composition of pure substances involved in the reaction. Let Xj[nc] be the vector of molar holdups of the different measured components at time and E[ne‚nc] the matrix of the chemical compositions, the calculation of Xja[ne], the vector of atomic molar holdups at the given time tj, is obtained simply from the following product:
Xja ) EXj
(5)
In most cases, E is determined a priori and kept constant over time. But chemical reactions are continuously changing the amount of molar component holdups during time, when the kinetic effects are sufficiently slow. That is why Xj is a function of time. However, there is no possible variation in the amount of each atomic element into a closed system if the nuclear reactions are excluded. As a consequence, this quantity is independent both of time and of reaction dynamics.
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Ind. Eng. Chem. Res., Vol. 35, No. 7, 1996
The principle is an efficient means of checking the consistency of the raw measurements without worrying about reaction mechanisms. It can be written in the following form:
{holdup of element k at time tj} {holdup of element k initially present} ) 0, k ) 1, ne; j ) 1, ns This first kind of formulation refers to the initial conditions of the experiment, i.e., to the initial composition of the material in the batch reactor. Frequently, this mixture is defined as a multicomponent one prepared by measuring given quantities of pure chemical substances, in which case, as an error is inherent in the composition and should be considered as a random factor in the reconciliation phase. A second formulation is introduced that seems to be more generic:
there are only ns intervals and ns estimations of ∆Xj differences while there are ns + 1 values of Xj (including j ) 0). This approach incurs a loss of identity for the sample variables. To preserve primary measurements, a new formulation, based on an a posteriori simultaneous processing of all the samples, is proposed. The whole set of constraints (7) is considered in only one optimization problem P2. In order to simplify the presentation, new notations are introduced. Let Xm be the vector of all measured samples over the entire time of experiment. Xm is built by a concatenation of the Xj,m (j ) 0, ns) in a chronological order. Let A be the elemental balance matrix corresponding to system (7). A can be broken down into an interval basis to allow the use of matrix E. Then, it is written in a block-bidiagonal form:
[
{holdup of element k at time tj+1} {holdup of element k at time tj} ) 0, k ) 1, ne; j ) 0, ns - 1 The mathematical translation, in matrix notation, of this formulation leads to
∆Xa ) Xj+1a - Xja ) 0,
j ) 0, ns - 1
(6)
and, following (5), to
E(Xj+1 - Xj) ) 0,
j ) 0, ns - 1
[][
[]
]
[
E(V0 + V1)Et -EV1Et 0 0
0 0 E ... ...
... ... ... ... ...
(8)
j
... ... ... ... -E
0 0 0 E
]
(9)
(10)
where matrix V is a block-diagonal matrix based on the variance matrices Vj (j ) 0, ns). It is easy to show that the left-hand side operator AVAt is a block-tridiagonal matrix see (Chart 1). Sparsity is used to minimize computational errors by using an appropriate solver. Finally, the estimate is taken from:
X h ) Xm - VAtλ
Formulation of the Estimation Problem. Previous papers relating to this subject of batch reacting systems have used atomic balance constraints (7). Numerical processing was done in a sequential manner. Samples were considered two by two, and a subproblem of type P2 was solved for each time interval between two successive samples. The optimal estimation of the material transformed for each component on a time interval is obtained for each P2j. Consequently, the correction for an intermediary sample is necessarily involved in both forward and backward estimations. The correction of the sample cannot be obtained because Chart 1
0 E -E ... 0
(AVAt)λ ) AXm
xCH4 xH2O
1 0 1 0 1 ) 0 1 1 0 2 x CO 4 2 0 2 0 x H2 xCO2
E -E 0 ... 0
The solution is still calculated using expression (4). However, due to the relatively big size of the A matrix, some numerical refinements should be used to properly compute the estimate X h . Expression (4) is broken into two parts. First of all, a vector of Lagrangian factors λ is found by solving the linear system (10),
(7)
Let us take an example where five pure chemical components (CH4, H2O, CO, H2, and CO2) are involved in a set of chemical reactions. Three elements (carbon, hydrogen, and oxygen) will be quantified if all the components are measured. For each time tj, the canonical form of (5) is defined as:
xaC xaH xaO
-E 0 A) 0 ... 0
(11)
Example Data Collection. In order to compare the two formulations discussed above, we have simulated a set of true data X*, representing the oxidation of propane in the liquid phase of a batch reactor (Bulygin et al. (1972)). The integration of mass balances is performed with the help of kinetics laws assumed to be perfectly true. This example involves 10 components (nc ) 10), and 13 samples have been taken (ns ) 13). Table 1 shows the results of this simulated experiment. Atomic balances were expressed using the three elements involved: carbon, hydrogen, and oxygen. Addition of Gaussian Noise. Next, these “true” measurements are disturbed, in order to obtain a vector Xm of virtual measurements (Table 2). The Gaussian noise is obtained by a pseudorandom numbers generator. A relative error arbitrarily chosen to be 5% is
-EV1Et 0 0 t t E(V1 + V2)E -EV2E -EV2Et ... -EVjEt E(Vj + Vj+1)Et -EVj+1Et 0 -EVj+1Et 0 ...
]
Ind. Eng. Chem. Res., Vol. 35, No. 7, 1996 2291 Table 1. “True” Simulated Values (X*) (mol) sample
C3H8
O2
(CH3)2CO
H2O
C3H7OH
CH3CHO
CH3OH
CH3COOH
CO2
HCOOH
0 1 2 3 4 5 6 7 8 9 10 11 12 13
359.52 330.80 307.25 268.52 241.39 215.56 204.38 188.57 159.24 131.07 104.67 78.43 49.95 22.93
578.43 530.40 494.97 426.65 390.22 344.87 310.80 273.09 240.25 200.00 153.94 104.88 69.14 27.67
0.00 8.14 12.50 18.29 23.29 28.38 29.79 30.07 40.06 41.86 47.75 54.15 57.18 64.21
0.00 30.54 52.42 98.33 121.79 152.60 172.49 201.48 223.89 248.10 280.37 310.58 331.30 362.03
0.00 8.57 17.19 25.56 31.74 39.90 45.94 46.52 53.92 60.97 66.69 74.40 77.14 78.63
0.00 5.73 10.89 31.75 45.92 53.33 41.97 56.07 70.90 90.62 99.61 102.58 129.95 152.50
10.00 12.86 9.97 11.00 12.35 8.07 10.41 10.15 11.18 14.60 15.45 20.01 27.75 26.22
0.00 1.61 9.03 12.65 19.42 24.88 31.55 32.58 32.70 35.69 42.01 44.74 45.48 45.91
0.00 13.25 17.30 33.89 37.75 46.61 63.69 77.21 78.71 85.22 97.65 113.90 116.62 120.30
0.00 5.26 10.64 17.78 18.50 25.93 27.09 28.39 31.80 34.46 34.91 39.05 40.54 47.71
Table 2. Values with Standard Deviation (Xm) (mol) sample
C3H8
O2
(CH3)2CO
H2O
C3H7OH
CH3CHO
CH3OH
CH3COOH
CO2
HCOOH
0 1 2 3 4 5 6 7 8 9 10 11 12 13
351.85 326.77 307.11 267.80 239.72 215.43 200.51 190.45 158.49 130.52 103.01 75.69 49.81 22.31
575.89 534.90 486.50 426.70 392.75 340.92 306.23 266.40 238.51 209.88 151.16 103.85 68.92 27.76
0.00 7.85 12.59 19.00 22.66 28.27 30.24 30.66 39.45 42.56 47.82 55.68 56.26 64.76
0.00 31.65 52.82 91.86 119.92 151.93 174.14 194.23 227.62 242.55 275.19 314.02 332.66 368.11
0.00 8.29 17.15 24.67 31.11 40.07 45.56 45.85 53.38 58.68 64.41 74.65 76.80 77.65
0.00 5.65 11.27 33.23 44.85 52.39 40.14 55.34 70.37 90.38 101.31 103.89 132.64 155.21
9.86 12.54 10.00 10.76 12.43 8.56 10.20 10.09 11.51 14.37 15.42 19.75 27.35 25.99
0.00 1.57 8.98 12.81 19.55 25.45 31.58 32.35 31.65 37.34 41.74 44.14 44.92 45.87
0.00 12.91 17.85 33.29 36.46 45.29 65.15 76.20 78.15 86.36 95.84 114.63 114.60 119.10
0.00 5.48 10.36 17.55 18.55 26.51 28.12 28.26 31.45 33.93 35.97 38.07 40.31 48.12
Table 3. Corrected Values for the Simultaneous Formulation (X) (mol) sample
C3H8
O2
(CH3)2CO
H2O
C3H7OH
CH3CHO
CH3OH
CH3COOH
CO2
HCOOH
0 1 2 3 4 5 6 7 8 9 10 11 12 13
357.88 329.76 305.23 267.22 241.77 213.97 202.81 188.24 158.94 132.11 103.56 75.64 49.50 22.27
576.60 528.91 492.29 426.81 391.70 343.95 306.25 274.16 239.87 201.71 152.52 103.25 69.22 27.73
0.00 7.92 12.57 18.76 22.71 28.31 30.29 30.57 39.65 42.27 47.93 55.56 56.12 64.48
0.01 30.48 53.20 97.41 119.02 150.71 174.30 199.64 223.02 246.28 280.14 311.73 330.50 361.24
0.00 8.29 17.15 24.66 31.14 40.04 45.68 45.83 53.46 58.68 64.75 74.45 76.24 77.07
0.00 5.68 11.26 32.74 44.97 52.49 40.19 55.21 70.83 89.28 101.73 103.50 132.30 153.97
9.89 12.42 10.01 10.81 12.42 8.56 10.20 10.10 11.51 14.38 15.44 19.74 27.32 25.94
0.00 1.58 8.97 12.74 19.57 25.49 31.61 32.36 31.76 37.02 41.86 44.01 44.94 45.72
0.00 13.16 17.80 32.32 36.57 45.50 65.21 76.03 79.15 83.66 96.11 113.79 115.76 118.74
0.00 5.49 10.36 17.48 18.55 26.54 28.13 28.31 31.52 33.70 36.05 37.899 40.37 47.99
applied to the above values for all components. Variances are estimated on the basis of a 5% risk. Results of Estimation. We have applied both approaches to this set of noisy data. The results are summarized in Tables 3 (overall formulation) and 4 (sequential formulation). Since the sequential formulation only provides estimations of ∆Xj (see Table 4), comparison of the two approaches, in terms of estimation quality, can only be done using this variable (the quantity of material transformed between two successive samples). From the true values X*, we calculate the quantities of material transformed between two successive samples: ∆X* ) Xj+1* - Xj* (these values are reported in Table 5). In such a way, from the overall corrected values X h , we compute ∆X h (Table 6). In order to provide a more synthetic evaluation of both formulations, two criteria are introduced. The first one is classical: the variance of the estimation. The calculation of this variance is done by a method described in Appendix I. The other one is average error reduction (AER). It indicates the percentage of correction carried out with respect to the simulated noise and is evaluated in Table 7. The closer AER is to 100%, the better the method performs. The average values of the variance
for each component are recorded in Table 8.
AERi )
Ei1 - Ei2
ns-1
Ei1 )
∑ j)0
(11)
Ei1 ns-1
i |∆Xm,j - ∆X*ij|
Ei2 )
|∆X h ij - ∆X*ij| ∑ j)0
(12)
Treatment by means of the simultaneous formulation is preferred for the components of greatest magnitude (C3H8, O2, H2O). On the basis of the AER, the sequential method appears to give better results for three substances: CH3H7OH, CH3OH, and HCOOH. But a study of the variances in Table 8 proves that CH3H7OH, CH3OH, and HCOOH have relatively small errors. For these components, the two methods of calculation do not lead to significant differences in the reconciled values. Considering the results of variance estimation, the simultaneous formulation always gives a better value than the sequential one, even if the benefit is not very high.
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Ind. Eng. Chem. Res., Vol. 35, No. 7, 1996
Table 4. Corrected Values ∆Xs for the Sequential Formulation (mol) sample
C3H8
O2
(CH3)2CO
H2O
C3H7OH
CH3CHO
CH3OH
CH3COOH
CO2
HCOOH
0 1 2 3 4 5 6 7 8 9 10 11 12
-28.13 -24.50 -38.05 -25.41 -27.79 -11.14 -14.47 -29.20 -26.86 -28.52 -27.83 -26.32 -27.42
-47.71 -36.64 -65.55 -34.89 -47.71 -37.57 -31.78 -33.99 -38.52 -49.67 -49.21 -34.07 -41.48
7.94 4.65 6.17 3.93 5.59 1.99 0.27 9.06 2.61 5.62 7.61 0.58 8.43
30.52 22.79 44.24 21.44 31.67 23.49 25.11 23.15 23.54 34.25 31.57 18.77 30.76
8.29 8.85 7.52 6.47 8.90 5.62 0.15 7.64 5.15 5.97 9.66 1.84 0.86
5.69 5.58 21.56 12.26 7.52 -12.24 14.99 15.56 18.48 12.42 1.73 29.01 21.90
2.49 -2.47 0.82 1.62 -3.87 1.64 -0.10 1.41 2.87 1.06 4.30 7.59 -1.38
1.59 7.38 3.75 6.85 5.91 6.11 0.72 -0.60 5.29 4.87 2.15 0.93 0.77
13.17 7.11 14.53 4.12 8.92 19.61 10.69 3.00 4.71 12.77 17.70 1.88 2.82
5.49 4.68 7.11 1.07 7.98 1.59 0.17 3.19 2.20 2.38 1.93 2.37 7.61
Table 5. True Values ∆X* (mol) sample
C3H8
O2
(CH3)2CO
H2O
C3H7OH
CH3CHO
CH3OH
CH3COOH
CO2
HCOOH
0 1 2 3 4 5 6 7 8 9 10 11 12
-28.72 -23.55 -38.73 -27.13 -25.83 -11.19 -15.80 -29.33 -28.18 -26.40 -26.24 -28.48 -27.01
-48.03 -35.43 -68.32 -36.44 -45.35 -34.07 -37.71 -32.85 -40.25 -46.06 -49.05 -35.74 -41.47
8.14 4.36 5.79 5.00 5.09 1.41 0.27 9.99 1.80 5.89 6.40 3.02 7.03
30.54 21.87 45.92 23.46 30.82 19.88 29.00 22.40 24.21 32.27 30.21 20.72 30.74
8.57 8.62 8.37 6.19 8.16 6.04 0.58 7.39 7.05 5.72 7.72 2.74 1.49
5.73 5.16 20.86 14.17 7.41 -11.36 14.11 14.83 19.71 8.99 2.97 27.37 22.64
2.86 -2.88 1.03 1.35 -4.28 2.34 -0.25 1.03 3.42 0.85 4.56 7.74 -1.53
1.61 7.42 3.62 6.77 5.46 6.67 1.03 0.12 2.99 6.32 2.73 0.74 0.43
13.25 4.06 16.58 3.87 8.86 17.08 13.53 1.49 6.51 12.44 16.25 2.71 3.69
5.26 5.38 7.14 0.72 7.43 1.16 1.30 3.41 2.65 0.45 4.14 1.49 7.17
Table 6. Estimated Values ∆X (mol) sample
C3H8
O2
(CH3)2CO
H2O
C3H7OH
CH3CHO
CH3OH
CH3COOH
CO2
HCOOH
0 1 2 3 4 5 6 7 8 9 10 11 12
-28.13 -24.53 -38.01 -25.45 -27.80 -11.16 -14.57 -29.30 -26.83 -28.55 -27.92 -26.14 -27.23
-47.69 -36.61 -65.49 -35.10 -47.75 -37.70 -32.10 -34.29 -38.15 -49.20 -49.26 -34.03 -41.49
7.92 4.65 6.19 3.95 5.59 1.99 0.28 9.08 2.63 5.65 7.63 0.56 8.36
30.47 22.72 44.21 21.61 31.69 23.59 25.35 23.38 23.25 33.86 31.59 18.76 30.75
8.29 8.86 7.52 6.48 8.90 5.64 0.15 7.63 5.22 6.07 9.70 1.78 0.83
5.68 5.58 21.47 12.23 7.52 -12.30 15.02 15.62 18.45 12.45 1.77 28.81 21.67
2.53 -2.41 0.80 1.61 -3.86 1.64 -0.10 1.41 2.87 1.06 4.30 7.58 -1.38
1.58 7.40 3.76 6.83 5.92 6.13 0.74 -0.60 5.26 4.84 2.15 0.93 0.78
13.16 4.64 14.51 4.26 8.93 19.71 10.83 3.12 4.51 12.45 17.68 1.98 2.98
5.49 4.87 7.12 1.08 7.98 1.59 0.18 3.21 2.18 2.36 1.93 2.38 7.62
Table 7. Comparison of AER (%) sequential method overall method
C3H8
O2
(CH3)2CO
H 2O
C3H7OH
CH3CHO
CH3OH
CH3COOH
CO2
HCOOH
46.90 47.05
64.14 64.14
18.49 18.79
68.07 68.29
-1.95 -3.67
-31.99 -33.20
-3.48 -4.95
8.82 9.26
9.86 23.16
-11.41 -9.13
Table 8. Comparison of Variances for the Two Formulations C 3H 8
O2
raw data 55.4119 139.0027 sequential method 3.5607 9.2848 overall method 3.5062 9.1337
(CH3)2CO
H 2O
C3H7OH
1.7090 1.5249 1.5220
55.8792 6.1556 6.0661
3.1402 2.5172 2.5050
CO2
HCOOH
6.8411 3.5395 3.4877
1.0342 1.0070 1.0065
CH3CHO CH3OH CH3COOH raw data sequential method overall method
7.1649 5.3928 5.3557
0.2797 0.2772 0.2769
1.2004 1.1460 1.1455
Gross Error Detection The method of estimation described in the previous paragraph assumes that the random errors follow a Gaussian distribution. When this is not the case, the presence of gross errors strongly disturbs the estimation result. The basis of all gross error detection methods is to define a sensitive statistical variable so that hypotheses can be checked. After a brief review of the
main variables of interest, a study is done in order to determine which works best. Statistical Variables and Their Properties. Normalized Residual Vector. From expression (2) and the normal distribution hypothesis, it is easily shown that the residual vector of constraints R follows a normal law:
R ) AXm ) AX* + A ) A
(13)
R ≈ N(0,Vr) where matrix Vr is equal to the matrix AVAt. The components of vector R are normalized to give the variables Rn(i). These normalized residuals are calculated in the following way:
Rn(i) ) R(i)/Vr(i,i)1/2
(14)
Ind. Eng. Chem. Res., Vol. 35, No. 7, 1996 2293
They follow a Gaussian distribution law.
Rn(i) ≈ N(0,1) Normalized Corrective Terms. Let j be the estimated corrective term:
j ) Xm - X h
(15)
From (4), the following relationship is established:
j ) VAt(AVAt)-1AXm ) VAt(AVAt)-1A
(16)
The moments of j are deduced from this relationship:
E(j) ) VAt(AVAt)-1AE() ) 0
(17)
E(j jt) ) W ) VAt(AVAt)-1AV
(18)
The vector of estimated corrective terms follows a normal centered distribution, of variance W. The normalization of these terms produces a generalized statistical variable called the normalized corrective term. Each component of vector En is defined by (19) and follows a reduced centered normal distribution.
En(i) ) j(i)/Wii1/2
(19)
with
En(i) ≈ N(0,1) The variables in expressions (14) and (19) are constructed from the diagonals of the variance-covariance Vr and W. Residual Criterion. Let us consider the objective function of the optimization problem P2:
h ) jtV-1j
(20)
Some relationship deduced from (16) links j to R:
j ) VAt(AVAt)-1R
(21)
h ) RtVr-1R
(22)
thus
Since the elements of j and R obey centered normal distributions, and h is defined as the sum of random variable quadratic means, h follows a χ2 distribution with n degrees of freedom.
h ≈ χ2n, with n ) (ns - 1) × rank(E)
(23)
Detection Algorithms. The gross error location procedures may be classified into two main families: (a) a priori testing and (b) a posteriori testing. The former set contains all the detection methods that do not require the solution to problem P2, whereas the techniques grouped in the second set need X h. A Priori Tests. The normal distribution of the residuals can be tested using the objective function h, which in the absence of gross errors actually follows a χ2 distribution. This test was evolved by Reilly and Carpani (1963) and applied by Romagnoli and Stephanopoulos (1980). Wang and Stephanopoulos (1983) used it to determine erroneous measurements in a sequential formulation for a batch biological reactor. If the value of h is greater than the χ-value calculated for a given risk and for a degree of freedom corresponding to the number of constraints, the gross error hypothesis is
valid. This approach gives only an overall indication, without any information to localize the suspect sample. The value of h must be recalculated for each set of measurements where one sample has been rejected. If the test is passed with the rejection of only one sample, the gross error is localized. When this is not the case, the procedure is applied again for all combinations of two samples, and so on. This method is efficient as long as there are only one or two biased measurements. Once the number of abnormal measurements increases, the algorithm leads to combinatorial complexity. Let us take the example introduced above: 140 measurements are divided into 14 samples (13 samples + initial conditions). Suppose that there is at least one gross error in 4 different samples. The localization of these gross errors, therefore, implies 3472 calculations of h. Having regard to such an important computation cost, we have not applied this test in what follows. A local approach can be used that relies on the normalized residual vector Rn. Nogita (1972) was the first to suggest testing the null hypothesis H0: “The normalized residuals Rn(i) follow a reduced normal centered law.” This method then results in the following discrimination function: If Rn(i) > rc(R), then measurement i is suspect. The value rc(R) is obtained from a Gaussian distribution with a error I type of risk R probability. In our formulation, two samples are associated in the balance residual Rn(i). The rejection of a Rn component does not lead to the elimination of measurements in one sample but to the suspicion of measurements in two successive samples without affording the means to go beyond this doubt. Narasimhan and Mah (1987) have proposed a statistical test which gives more precise information about the nature of the gross error. The sensitive variable is the generalized likelihood ratio, and the test is known under its abbreviation, GLR. This method (Appendix II) is able to localize biased measurement (sample and chemical component) and to estimate the amplitude of the biases. This avoids the need to eliminate a whole sample as previously concluded. Moreover, the loss of information can be minimized by reconstructing the measurement. A Posteriori Test. The normalized corrective terms En are the basis for a whole set of algorithms. Serth and Heeman (1986), and then Rosenberg et al. (1987), published comparative studies on many of these algorithms when applied to steady-state processes. In their respective opinions, the most effective tests are the “Modified Iterative Measurement Test” (MIMT) and the “Dynamic Measurement Test” (DMT). Unfortunately, both are difficult to apply in the batch reactor case, as upper and lower boundary limits are needed for each measurement. It is very difficult to estimate these limits for each sample period because there is often a wide range within the variables. The “Iterative Measurement Test” (IMT) algorithm has been used. This test may be represented in three steps: Let F be the set of rejected samples and T be the set of still valid samples. Step 0: k ) 0, F ) φ, T ) {ns samples, initial conditions} Step 1: calculation of En by resolving P2 for the sample of T Step 2: calculation of the measurement number jmax such that En(jmax) ) max En(j), j ) 0, (ns)nc - k*nc Step 3: if En(jmax) < ec(R), where R is the risk, then stop. Else, remove sample i which contains the measurement jmax: F ) F + {i}, T ) T - {i}, k ) k + 1 and return to step 1
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ec(R) is the critical value recommended by Serth and Heeman (1986). The RC (residual criterion) test relies on the contribution of sample i to the objective function h. This value is calculated using the following formula:
hi ) jitVi-1ji
(24)
where ji represents the error estimations for measurements of sample i. The function hi follows a χ2 distribution with rank (E) degrees of freedom, generally equal to ne. This property may be used as a discrimination function. We can now construct a decision algorithm which will lead to the set of suspected measurements: Step 1: calculation of ji and hi, i ) 0, ns Step 2: calculate himax ) max hi Step 3: if H < χ2ne(R), then stop. Else, remove sample imax and return to Step 1. Synthesis. Three tests have been retained. One a priori test, the GLR, and two a posteriori tests, the IMT and the RC. Of these, only the GLR allows both the detection of erroneous measurements and the estimation of the bias amplitude. The others only localize samples where abnormal measurements are assumed to be present.
Table 9. Comparison of ETI (%) no. of errors
GLR
IMT
RC
3 6 9 12
0.88 1.04 0.72 0.81
0.31 0.35 0.28 0.35
0.32 0.25 0.31 0.28
Table 10. Comparison of OP (%) no. of errors
GLR
IMT
RC
3 6 9 12
55.3 59.5 63.5 65.5
51.5 55.2 57.4 61.4
49.1 52.2 59.3 59.0
Table 11. OP (%) for Runs with Maximum One Gross Error Per Sample no. of errors
GLR
IMT
RC
3 6 9
61.1 56.3 57.7
57.9 52.0 53.3
54.0 48.1 49.1
samples (error type I):
ETI )
no. of samples wrongly identified no. of samples simulated
(29)
Application Addition of Gross Errors to Measurements. Comparative studies based on simulated data have been done for the three tests. From the set of “true” data, the measurement vector is constructed in the following way:
Xm ) X* + E + δ
(25)
where E is the vector of Gaussian noises as previously introduced, and δ represents the vector of gross errors. The location of a gross error is defined by its sample and component. The amplitude of δ is determined using a uniform pseudo random number generator. It should be noted that for component i (i ) 1, nc)
δi ) (X* + E)(Ri - 1) with Ri ∈ [0.5; 1 - err(i)]
(26)
∪ [1 + err(i); 1.5]
where err(i) is the relative error of the component i measurement. The number of gross errors is fixed for each study and is allowed to take integer values between 3 and 12. Performance Indices. Three indices of performance were defined to reflect the behavior of the tests. The first represents the capacity of the tests to detect the samples which contain simulated gross errors. This is the overall power (OP): OP ) no. of samples with at least one gross error correctly identified no. of samples in which at least one gross error has been simulated
(27) The second is more precise and specific to the GLR test. It represents the capacity to detect measurements where gross errors have been simulated:
OP* ) no. of erroneous measurements correctly identified no. of erroneous measurements simulated (28) The last one takes into account abusive detections of
The first two indices should be close to 100% if the test performs well, and the last one should be close to zero. Results and Significances. In each case study (3 biases, 6 biases, 9 biases, and 12 biases), we have produced 1000 sets of simulated data, and the three tests have been applied to this database. The results presented here are the average responses. Table 9 shows small values of ETI (the maximum being 1%), whatever the number of biases. We can therefore suppose that these techniques are not sensitive to type I error. For the overall power, the calculation reported in Table 10 leads to the following observations: (1) The efficiency of the tests is not very sensitive to the number of biases. There is even an improvement in the power of the tests when the number of gross errors is increased. This tendency should obviously not be interpreted in such a way. When increasing the number of gross errors, we may simulate more gross errors in the same sample. This increase simply reflects the fact that the more gross errors a sample contains, the easier detection is. When a single gross error per sample is simulated (Table 11), OP remains virtually constant. (2) There exists a hierarchy in the effectiveness of the tests: GLR, IMT, RC. (3) The values of the OP are not greater than 65%. Component by component analysis in the 9 biases set of data (Table 12) reveals a great probability of gross error detection for the major components (C3H8, O2, H2O, and CO2). But for the other oness(CH3)2CO, C3H7OH, CH3OH, CH3COOH, and HCOOHsthe tests are quite inefficient. It is obvious that a large error in these latter components only produces a very small variation in the residual balance, which in most cases cannot be compared with the order of magnitude of the noise added to major components. The power of the GLR to identify erroneous measurement(s) is given by OP*. In Table 13, with 9 biases imposed, the detection of biased measurements is carried out properly, once more, for the major components: CH3H8, O2, H2O, and CO2. In the opposite case, it is not recommended to use the results of the GLR. The identification of the component(s) supported by the GLR is a way of limiting the loss of measurements. It
Ind. Eng. Chem. Res., Vol. 35, No. 7, 1996 2295 Table 12. Comparison of OP (%) for Each Component constituent
C3H8
O2
(CH3)2CO
H 2O
C3H7OH
CH3CHO
CH3OH
CH3COOH
CO2
HCOOH
GLR IMT RC
89.7 87.3 85.4
89.3 87.6 86.6
65.2 60.6 59.6
90 88.3 86.9
61.6 57 54.7
69.2 64.7 61.8
39 33.9 30.2
51.1 45 42.3
81.8 79 78.1
43.5 38.4 35.4
Table 13. Comparison of OP* (%) for Each Component constituent
C3H8
O2
(CH3)2CO
H 2O
C3H7OH
CH3CHO
CH3OH
CH3COOH
CO2
HCOOH
3 6 9 12
79.5 78.4 73.2 69.1
76.2 79.4 74.2 70.7
31.1 26.7 23.9 24.3
71.9 68.6 69.3 63.2
32.7 31.1 29.4 24.8
28.4 21.3 20 18.5
4.6 4.6 6 6.6
16.8 14.2 16.7 12.2
62.8 54 53.2 46.4
7.2 10.5 8.1 8.3
Table 14. Effectiveness of the Reconstruction (%) AIGER AGER
C3H8
O2
(CH3)2CO
H2O
C3H7OH
CH3CHO
CH3OH
CH3COOH
CO2
HCOOH
80.3 75.6
77.7 76.2
29.3 -11.5
82.2 79.6
40.1 21.5
30.1 3.6
-4.4 -75.7
18.1 -23.2
66.8 58.3
7.8 -42.0
is important to review the quality of the biases estimated by the GLR in order to reconstruct the suspected measurements. Two situations will be considered: (1) the evaluation only considers the correct diagnostics of the GLR (ideal case); (2) the evaluation considers all the biases detected by the GLR (real case). To illustrate these two situations, we have computed 1000 sets of data. For each run nine gross errors were dispatched in the data. The following criteria are computed from the GLR test results: First Criterion. In the ideal case and for each run, the sum of the simulated gross errors for component i, called E3i, is calculated. Then the reduction of errors which can be obtained by the reconstruction of the biased measurements (E4i) is assessed: nsge
E3i )
Figure 1. C3H8 gross error estimation in the ideal case.
nsge
∑ |δik| k)1
E4i )
∑ |δh ik - δik|
(30)
k)1
The average value of the simulated gross error, called E3i,t, is the sum of E3i over the runs divided by the number of runs. In the same manner, the average reduction of errors (E4i,t) is also computed. Then, the average identified gross error reduction (AIGER) is defined as the relative difference between E3i,t and E4i,t:
AIGERi )
(E3i,t - E4i,t) E3i,t
(31)
Second Criterion. In the real case, we have to take into account the incorrect detections of the GLR. So, for each run and each component i, to E3i (the sum of all simulated biases), we add the sum of the simulated errors erroneously detected by the GLR as gross errors. The error reduction obtained by the reconstruction of the measurements which are detected as erroneous leads to the E6i index. nde
i
i
E5 ) E3 +
∑ |k | k)1 i
nde
i
i
E6 ) E4 +
|jki - ki| ∑ k)1
(32)
By analogy with (31), we introduce the average gross error reduction (AGER):
AGERi )
(E5i,t - E6i,t) E5i,t
(33)
In the ideal case (AIGER), reconstitution allows the errors to be reduced for the quasi totality of the
Figure 2. C3H8 gross error estimation in the real case.
components (apart from CH3OH, which is a low order of magnitude component holdup). The estimation of the biases is thus correct if the gross errors are correctly identified. Gross error detection is, however, correctly carried out only for major components (Table 12). In the real case, there is a correlation between the effectiveness of the reconstruction (AIGER) and the pertinence of gross error detection in the components. In the case of the following components, (CH3)2CO, CH3CHO, CH3OH, CH3COOH, and HCOOH, the use of bias estimation visibly disturbs reconstructed measurements which are more erroneous than the raw measurements themselves (Table 14). Figures 1 and 2 show a representation of the results for the major component C3H8 considering the whole set of runs. In the ideal case (cf. Figure 1), simulated errors (E3) are reported on the abscissa and estimated errors are ordinates. The estimation is correct; all the points are located between the first bisectrix and the abscissa axis. Furthermore, these corrections are very effective, with most of the points being close to the axis of the abscissa. The real case is shown in Figure 2. The
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Figure 3. C3H7OH gross error estimation in the ideal case.
Figure 5. Detection of systematic sensor failure.
validate the first nine samples and reject the others. This is a first part where the right detection of the failure has been achieved. The algorithm described above was applied given that water is a major component. Figure 5 proves that the reconstruction is very efficient. Conclusion Figure 4. C3H7OH gross error estimation in the real case.
conclusions are the same, since the component is present in a large quantity. Let us now consider the case of a minor component C3H7OH. Figure 3 gives the quality of the reconstruction in the ideal case. We notice some useful corrections; very few points are located above the first bisectrix, but they are not very effective as the points are not grouped close to the x-axis. In the real case (Figure 4), a large number of points are located above the first bisectrix. Measurement reconstruction for minor components is to be avoided. An Appropriate Use of GLR Results. In order to systematically use the GLR results with a good confidence, we have chosen to apply the following algorithm: Let Ek be the set of samples definitively rejected at stage k, Fk be the set of samples detected as being erroneous at stage k, and Tk be the set of samples still valid/representative at stage k. (1) k ) 0, Ek ) φ, Fk ) φ, and Tk ) {all samples} (2) k ) k + 1
perform the GLR, RC, and IMT method Ek ) Ek-1 + (Fk ∩ Fk-1) Fk ) Fk - (Fk ∩ Fk-1) (3) The elements of Fk which do not fulfill the following conditions are put into Ek: (a) the component rejected by the GLR corresponds to a major component; (b) the addition to the suspected value of the calculated bias gives a physically admissible measurement.
Tk ) T0 - Ek (4) For the remaining components in Fk, we reconstruct the measurement. If Fk * φ, then return to Step 2, else stop. At first, the set of measurements is normally disturbed; no gross error is added. Then, biases on a sensor (measurement of H2O) were simulated starting from the 9th sample (Figure 5, circles). The different tests
In this paper, an overall formulation for data reconciliation on batch reactors has been proposed. All the measurements over the total sampling period are simultaneously treated in order to obtain a reconciled measurements set. This method includes gross error detection or detection of a systematic sensor failure. Three tests have been used, and their efficiency has been proved to be pretty good for all major components. The amplitude of the gross error which can be estimated with the help of the GLR method needs to be used with care. A specific algorithm, combining information taken from the GLR, IMT, and RC tests, has been proposed in order to avoid the costly rejection of the whole set of measurements in a sample. In the near future, attempts will be made to deal with minor components. Notation A [ne(ns, (ns + 1)nc] ) overall matrix of elemental balances b ) magnitude of gross error E [ne, nc] ) matrix of elemental balances En [(ns + 1)nc] ) vector of normalized estimated values of FV(X) ) likelihood function of X I [(ns + 1)nc, (ns + 1)nc] ) overall identity matrix ne ) number of elemental balances nc ) number of components nde ) number of detected errors ns ) number of samples (initial conditions excluded) nsge ) number of simulated gross errors R [ne‚ns] ) overall vector of constraint residuals rc(R) ) upper R quantile of gaussian distribution Rn [ne‚ns] ) vector of normalized constraint residuals rank(A) ) rank of matrix A V [(ns + 1)nc, (ns + 1)nc] ) covariance matrix of measurement errors Vr [ne‚ns, ne‚ns] ) overall covariance matrix of constraint residuals Xj [nc] ) vector of molar holdup at time tj X* [(ns + 1)nc] ) vector of true measurements Xm [(ns + 1)nc] ) vector of measurements Xm,j [nc] ) vector of the jth holdup measurements Xim,j ) measurement of ith component holdup at time tj
Ind. Eng. Chem. Res., Vol. 35, No. 7, 1996 2297 Xa [nc] ) vector of atomic element holdups
an optimization problem P2
Greek Letters
{
min (X - Xm)tV-1(X - Xm)
E ) vector of standard deviation Eki ) kth error simulated on component i Ejki ) estimation of ki δ ) vector of gross errors δki ) kth gross error on component i δh ki ) estimation of δki
The solution of which is
Other Symbols
and therefore
P2
X
AX ) 0
X h ) (I - VAt(AVAt)-1A)Xm
At ) transpose of A A-1 ) inverse of A X h ) estimated value of X E(x) ) expected value of x χ2n ) chi-square distribution with n degrees of freedom ∆Xs ) estimation of ∆X with a sequential approach P(a/b) ) conditional probability of a, with b given
∆X h ) FnsX h ) Fns(I - VAt(AVAt)-1A)Xm So, it can be shown that
Var(∆X h) ) Fns(I - VAt(AVAt)-1A)V(I - VAt(AVAt)-1A)t(Fns)t
Appendix I
Appendix II
Calculation of the variance covariance matrix for the sequential and the overall method. Let Ir[nc‚ns, nc‚ns] be the identity matrix, Ic[nc, nc] be the identity matrix, Fj[nc‚j, nc‚(j + 1)] is the following block bidiagonal matrix:
In order to establish the theoretical basis for the GLR test, let us summarize the measurement model: (a) In the absence of bias, Xm ) X* + . (b) The vector Jj,i is the (i*j)th component of the canonic base Rmeas; it gives the ith component of sample j as being suspect. (c) In the case of a bias amplitude b (scalar) for the component i of the jth sample:
[
-Ic Ic 0 ... ... ... 0 -Ic Ic 0 ... ... 0 Fj ) 0 0 0 ... ... 0 -Ic Ic
]
Xm ) X* + E + bJj,i
D [ne‚ns,nc‚ns] is a block diagonal matrix:
[
E 0 ... ... 0 D ) 0 E 0 ... 0 0 0 ... 0 E
]
thus
E(R) ) bAJj,i The null hypothesis H0 is expressed as follows: “There is no error in the process: E(R) ) 0.” The opposite proposition H1 is as follows: “There exists at least one anomalous measurement: E(R) ) bAJj,i.” For these conditions, the likelihood ratio Rv is defined as:
Sequential Formulation. By definition
∆Xm,j ) (Xm,j+1 - Xm,j) ) F1[Xm,j, Xm,j+1]t and, by assumption, we have
Xm,j ≈ N(Xj*,Vj)
Rv ) P(R/H1)/P(R/H0)
So, it is obvious that
∆Xm,j ≈ N(∆Xj*,Wj) with Wj ) Vj + Vj+1 In order to obtain the estimation of ∆Xm, we have to solve ns optimization problems P1,j
P1,j
{
where P(R/H) is the probability of obtaining R when H is known. Rv is the relationship between two exponentials as the probabilities follow a Gaussian distribution. A new variable is therefore introduced:
t ) 2 ln(Rv) )
min (∆Xj - ∆Xm,j)tWj-1(∆Xj - ∆Xm,j)
RtVr-1R - (R - bAJj,i)tVr-1(R - bAJj,i) (24)
∆Xj
E∆Xj ) 0
The t variable, by definition, follows a χ2 law with 1 degree of freedom. If T is the greatest value of t,
with ns solutions of the following form:
∆Xj ) (Ic - WjEt(EWjEt)-1E)∆Xm,j
T ) max t
The second-order moment is obtained from this expression:
Var(∆Xj) ) (Ic - WjEt(EWjEt)-1E)(Vj + Vj+1)(Ic t
(25)
bJj,i
t -1
WjE (EWjE ) E)
t
Simultaneous Formulation. By assumption
Xm ≈ N(X*,V) In order to obtain the estimation of Xm, we have to solve
The value of b which maximizes t for the measurement (j, i) is obtained from expression (24) as
b h ) (RtVr-1AJj,i)(Jk,itAtVr-1AJj,i)-1
(26)
The value of T is obtained through the evaluation of t for each measurement (j, i). Narasimhan showed that, if T > χ21(R), then H1 is valid. There exists at least one error in the measurement i for sample k. The number of iterative processes (evaluation of T) is equal to the number of biases detected.
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From this discriminate function T, the detection algorithm is composed of four steps: Step 0: F ) φ, B ) φ Step 1: calculation of R Step 2: calculation of T (step with the highest cost) Step 3: if T < χ21, then go to step 4; else,
R)R-b h Jj,i F ) FUJj,i Step 4:
B ) BUbh and go to step 2 F contains the set of suspicious measurements B contains the set of estimated biases Literature Cited Bulygin, M. G.; Blyumberg, E. A.; Al’tshuler, L. A. The Liquid Oxidation Of Propane. Int. Chem. Eng. 1972, 12, 50. Crowe, C. M.; Garcia Campos, Y. A.; Hiymak, A. Reconciliation of Process Flow Rates By Matrix Projection. AIChE J. 1983, 29, 881. Crowe, C. M. Observability and Redundancy Of Process Data For Steady State Reconciliation. Chem. Eng. Sci. 1989, 44, 2909. Darouach, M.; Ragot, J.; Fayolle, I.; Maquin, D. Validation Des Mesures Pai Equilibrages Des Bilans Matieres. Int. J. Miner. Process 1986, 17, 273. Darouach, M.; Zasadzinski, M. Data Reconciliation In Generalized Linear Dynamic Systems. AIChE J. 1991, 37, 193. Iordache, C.; Mah, R. S. H.; Tamhane, A. C. Performance Studies Of Measurement Test For Detection Of Gross Errors In Process Data. AIChE J. 1985, 31, 1187. Kretsovalis, A.; Mah, R. S. H. Observability And Redundancy Classification In Multicomponent Process Network. AIChE J. 1987, 33, 70. Madron, F.; Veverka, V.; Vanceck, V. Statistical Analysis Of Material Balance Of A Chemical Reactor. AIChE J. 1977, 23, 482. Madron, F. A. New Approach To The Identification Of Gross Errors In Chemical Engineering Measurements. Chem. Eng. Sci. 1985, 40, 1855.
Meyer, M.; Kochret, B.; Enjalbert, M. Data Reconciliation On Multicomponent Networks Process. Comput. Chem. Eng. 1993, 17, 807. Narasimhan, S.; Mah, R. S. H. Generalized Likelihood Ratio Method For Gross Error Identification. AIChE J. 1987, 33, 1514. Narasimhan, S.; Ilarikumar, P. A Method To Incorporate Bounds in Data Reconciliation And Gross Error Detection. I. The Bounded Data Reconciliation Problem. Comput. Chem. Eng. J. 1993, 17, 1115. Nogita, S. Statistical Test And Adjustment Of Process Data. Ind. Eng. Chem. Process Des. Dev. 1972, 11, 197. Reilly, P. M.; Carpani, R. E. Application Of Statistical Theory Of Adjustment To Material Balances. Abstract of Papers; 13th Canadian Chemical Engineering Conference, Montreal, 1963. Ripps, D. L. Adjustment Of Experimental Data. Chem. Eng. Prog. Symp. Ser. 1965, 61 (No. 55), 8. Rollings, D. K.; Devanathan, S. Unbiased Estimation In Dynamic Data Reconciliation. AIChE J. 1993, 39, 1330. Romagnoli, J. A.; Stephanopoulos, G. On The Rectification Of Measurement Errors For Complex Chemical Plants. Chem. Eng. Sci. 1980, 35, 1067. Rosenberg, J.; Mah, R. S. H.; Iordache, C. Evaluation Of Schemes For Detecting And Identifying Gross Errors In Process Data. Ind. Eng. Chem. Res. 1987, 26, 555. Serth, R. W.; Heenan, W. A. Gross Error Detection And Data Reconciliation In Steam Metering Systems. AIChE J. 1986, 32, 733-742. Van der Heijden, R. T. J. M.; Heijnen, J. J.; Hellinga, C.; Romein, B.; Luyben, K. Ch. A. M. Linear Constraint Relations In Biochemical Reaction Systems: I. Classification Of The Calculability And The Balanceability Of The Conversion Rates. Biotechnol. Bioeng. 1994a, 43, 3. Van der Heijden, R. T. I. M.; Romein, R.; Heijnen, J. J.; Hellinga, C.; Luyben, K. Ch. A. M. Linear Constraint Relations In Biochemical Reaction Systems: III. Sequential Application Of Data Reconciliation For Sensitive Detection Of Systematic Errors. Biotechnol. Bioeng. 1994b, 44, 781. Wang, M. S.; Stephanopoulos, G. Application Of Macroscopic Balances To The Identification Of Gross Measurement Errors. Biotechnol. Bioeng. 1983, 15, 2177.
Received for review March 1, 1995 Revised manuscript received November 10, 1995 Accepted April 9, 1996X IE950152G Abstract published in Advance ACS Abstracts, June 1, 1996. X
