Ind. Eng. Chem. Res. 2006, 45, 8973-8984
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Theory Analysis of Nonlinear Data Reconciliation and Application to a Coking Plant Minghui Hu* and Huihe Shao School of Electrionic, Information & Electrical Engineering, Shanghai Jiaotong UniVersity, Shanghai 200240, PRC
The estimation methodology of process variables usually consists of three parts: classification of process variables, gross error detection, and data reconciliation. In this paper, we proposed a modified M-estimator method for the covariance estimator which depends on the results from robust statistics to reduce the effect of the gross errors. We consider the Lagrange multipliers method and successive linearization method for nonlinear data reconciliation. Finally, the example of a coking plant is presented to illustrate the effectiveness of the revised M-estimator method and nonlinear data reconciliation methods. In this paper, the classifying, estimating, and adjusting of the process variables are based on a components balance and total flow rates balance. The comparative results of the introduced methods are given and demonstrate the successful application of the proposed method to reconcile actual plant data from a complex chemical process. 1. Introduction Chemical process data inherently contains some degree of error, and this error may be random or gross error due to sensor noise, disturbances, instrument degradation, and human errors. An additional difficulty is caused by the fact that not all variables can be measured because of cost considerations or technical infeasibility. Therefore, it is necessary to adjust the measured variables and estimate the values of the unmeasured variables so that they satisfy the balance constraints; this is known as the data reconciliation problem.1 In the course of the daily operation of a chemical plant, it is common practice to adjust and correct the measurements taken from the process, so that random or gross errors can be accounted and eliminated. The reliability of data is of great significance if they are to be used effectively in process monitoring for operational optimization, control, or identification.2 Measurements corrupted with undetected gross or random errors result in false control, optimization, or simple process identification. The operation is commonly represented by a nonlinear system of algebraic equations. It is made up of total flow rates and components balances and may include thermodynamic relationships and some physical behavior of the system. In this case, data reconciliation is based on the solution of a nonlinear constrained optimization problem. Real-time on-line optimization provides set points to the process’s distributed control system (DCS) and therefore maintains the process near its optimum operating conditions.3 The optimization requires an accurate process model and reconciled process data. The process model is a set of inequality and equality constraints and describes the fundamental relationship of process units, such as material and energy balances, rate equations, and equilibrium relations. Reconciled process data is used to specify the current status of the plant model and for estimation of the model parameters for plant-model matching.4-6 In a complex chemical plant, data reconciliation usually consists of three parts: (1) classification of process variables, (2) gross error detection, and (3) covariance estimation and variables rectification. Variable classification is the essential tool * To whom correspondence should be addressed. Tel.: +86 21 34204264. Fax: +86-21-54260762. E-mail:
[email protected].
for the design or revamp of monitoring systems. After fixing the degree of required knowledge of the process, that is to say, the subset of variables that must be known. The mathematical formulation of the variable classification problem is stated and some structural properties are discussed in terms of graphical techniques. An elegant classification strategy using projection matrices was proposed by Crowe et al.7 (in 1983) for linear systems and extended later (in 1989)7 to bilinear ones. A classification procedure8 is used in nonlinear systems, which is based on row and column permutation of the occurrence matrix corresponding to the Jacobian matrix of the linearized model. The presence of gross errors invalidates the statistical basis of data reconciliation procedures. The elimination of the less frequent gross errors is achieved by gross error detection. Therefore, simultaneous data reconciliation and gross error detection have emerged as a key part of real-time on-line optimization for the solution of optimal control problems. As is shown to all, the conventional method for data reconciliation is that of weighted least squares, in which the adjustments to the data are weighted by the inverse of the measurement noise covariance matrix so that the model constraints are satisfied. The main assumption of the conventional approach is that the errors follow a normal Gaussian distribution. When this assumption is satisfied, conventional approaches provide unbiased estimates of the plant states. The presence of gross errors violates the assumptions in the conventional approach and makes the results invalid. A solution of the nonlinear data reconciliation problem via successive linearization is described by Veverka and Madron.9 Barbosa et al.10 demonstrated that using nonlinear programming instead of successive linearization remarkably improved reconciliation results. By establishing an analogy between the generalized likelihood ratio test (GLR) and robust regression,11,12 the GLR method13 allows us to identify multiple gross errors of any type. It uses a serial compensation strategy. Chen et al.14 have developed a robust strategy (M-estimator method) for covariance estimation, which is insensitive to the presence of gross errors in the data set. Johnston and Kramer15 reported the feasibility and better performance of the robust estimators as the objective function in the data reconciliation problem especially when the data contain gross errors. Subsequently, different types of robust estimators and their performance in data reconciliation were
10.1021/ie060077h CCC: $33.50 © 2006 American Chemical Society Published on Web 11/09/2006
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reported.16 In the present work, we have developed an approach which allows the estimation of the desired unmeasured variables,17 the identification and rectification of gross errors in the measurements,18,19 and the reconciliation of the process variables. Solution techniques for nonlinear data reconciliation, the Lagrange multipliers method, and the successive linearization method, are based on the solution of a nonlinear constrained optimization problem.20-22 Most importantly, the presented methods are employed on a coking plant. The topological character of the complex process is exploited for an easy classification of the measured and unmeasured variables independent of the nonlinearity of the balance equations. Rules have been developed; meanwhile, simple stochastic tests have been formulated to identify and rectify gross measurement errors.23-26 For clear interpretation of the proposed data reconciliation, this paper is organized as follows. Section 2 provides some analysis of the GLR method on gross error detection for the complex chemical plant. Section 3 provides the M-estimator method on the covariance estimator for the complex chemical plant. In section 4, the problem formulation and solution strategy of nonlinear data reconciliation are presented. In section 5, a simulation example is presented to illustrate the utility of the proposed method applied to the nonlinear data reconciliation problem of a coking plant. Some conclusions are addressed in the last section, 6. In the appendix, achievable classification of the process variables is discussed. 2. Gross Error Detection Using the GLR Method The presence of gross errors invalidates the statistical basis of data reconciliation procedures. It is also impossible, for example, to prepare an adequate process model on the basis of erroneous measurements or to assess production accounting correctly. Contrary to random errors, gross errors are usually due to one or more isolated factors that cause the displacement of the measurement in one direction. To avoid these shortcomings, we need to check for the presence of gross systematic errors in the measurement data. The measurement vector in the absence of gross errors can be written as
y ) x + , y ∈ Rn
(1)
With the following assumptions, (1) The expected value of , i.e., E(), ) 0. (2) The covariance matrix of is known and given by E(iiT) ) ψ. The following generalized likelihood ratio (GLR) method requires a model that describes the effect of each type of gross error. The measurement model with instrument bias is defined as
f(x,u) - aνi ) 0
(4)
where νi is a vector representing different constraints and a in eq 4 is the unknown magnitude of leak in a constraint. With either measurement bias or a process leak, the constraint residual is defined as
r ) f(x,u) + A + Aaδi or r ) f(x,u) + A - Aaνi
(5)
If no gross errors are present, under the null hypothesis, the expected values of r can be determined by the expected values of and the coefficient matrix of constraints, i.e.,
E(r) ) E(A) ) AE() ) 0
(6)
and the covariance matrix of r is the expected values of the squared differences between the individual constraint residual and its mean, i.e.,
cov(r) ) E[{r - E(r)}{r - E(r)}T] ) E[{A}{A}T] ) E[A(T)AT] ) AE[T]AT ) AΣ AT ) H (7) where H is the covariance matrix of constraint residual. The constraint residuals r follow a normal distribution with zero mean and covariance matrix H under the null hypothesis (no gross errors in measurements). Hence, the sum of squared ri values weighted by the variance will follow the χ-square distribution, if no gross errors are present in measurements. Mah proposed to test the null hypothesis H0, E(r) ) 0 which assumes that no gross errors are present, against the alternative hypothesis H1, E(r) ) aAδi or aνi which assumes that one gross error is present in either the measurement bias or process leak, by the likelihood ratio test. This test also estimates the unknown magnitude of the gross error if a gross error is indicated. The likelihood ratio test is given by
λ ) sup remum
P(r|H1)
(8)
P(r|H0)
where P(r|H1) and P(r|H0) are the probabilities of constraint residuals under the alternative and null hypotheses, respectively. The sup remum in eq 8 is computed over all possible values of the parameters (δi,νi, and a) present in the hypothesis. If constraint residuals r are normally distributed, then the distribution functions of P(r|H0) and P(r|H1) are written as follows: n
P(r|H0) )
P(ri|H0) ) ∏ i)1
1 (2π)n/2H1/2
[
exp -
]
rTH-1r 2
(9)
and n
y ) x + + aδi
(2)
where δi is an unit vector with one in position i and zero elsewhere and a is the unknown magnitude of a bias (gross error). A leak occurring in a process unit will not affect the measurement model in eq 2, but it affects the constraint equations with the leak. The nonlinear algebraic equations
f(x,u) ) 0, u ∈ Rm, f ∈ Rn can be rewritten as the following equation with a leak.
(3)
P(r|H1) )
P(ri|H1) ) ∏ i)1 1 (2π)n/2H1/2
[
]
(r - afi)TH-1(r - afi)
exp -
2
(10)
Substituting eqs 9 and 10 into eq 8 and taking a logarithm of eq 8 gives
T ) 2 ln λ ) sup remum [rTH -1r afi (r - afi)TH- 1(r - afi)] (11)
Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006 8975
where
fi ∈ (Aδi, i ) 1, 2, ..., n)
(12)
In eq 11, the possible outcome from either measurement error Aδi or process leak νi is combined and represented by fi as shown in eq 12. The computation of T proceeds as follows. For any given vector fi, the estimated gross error magnitude a is determined by maximizing eq 11. The solution of the maximization of eq 11 for given vector fi is the following: -1
-1
a ) (fiH fi) (fiH
-1
r)
(13)
Substituting eq 13 into eq 11 gives test statistic Ti for each case fi as follows:
a ) (fiH-1r)/(fiH-1fi)
In the work of Chen et al., the following simultaneous estimation strategy is adopted to construct the M-estimator: Assuming the covariance V is positive definite, the Cholesky decomposition of V is V ) CCT. Then, eqs 16 and 17 can be written as
E{w1(|C-1(x - u)|)C-1(x - u)} ) 0 E{w2(|C-1(x - u)|2)[C-1(x - u)][C-1(x - u)]T} ) I
(18) (19)
where |‚| denotes the norm of vectors or matrixes and I stands for the identity matrix. Let e ) |C-1(x - u)|. Then, the functions w1 and w2 will be determined by the following minimax problem:
φ(n,c) ) max {-c,min (n,c)}
(20)
This calculation is performed for every possible vector fi, and the test statistic is therefore obtained as follows:
with φi ) nui(n), i ) 1, 2, where φ(n,c) is known as the Huber’s psi function; c is a constant specified by the user to take into account the acceptable loss in efficiency of the Gaussian model in exchange for resistance to outliers.
T ) sup remum Ti
Taking φ1 ) φ(n,c), φ2 ) φ(n,c2)/β
(14)
(15)
Let f * be the vector that leads to the sup remum in eq 15. The test statistic T is compared with a prespecified threshold (critical values) C determined by the distribution function of T at the selected significant level R. If T exceeds C, then the measurement or constraint that corresponds to f* is identified as having a gross error or a leak, and its magnitude is estimated by eq 13 using f* for fi. For each case of fi, Ti has a central χ-square distribution with one degree of freedom under null hypothesis H0. 3. M-Estimator Method for the Covariance Estimator If outliers are present in the sampling data, the assumption about the error distribution will be violated. In such situations, the final estimation of the covariance of the measurements will not be a maximum likelihood estimator. In other words, this may lead to an incorrect estimation. For this reason, an alternative robust procedure is needed. Definition. An estimator is called robust if it is insensitive to mild departures from the underlying assumptions and is also only slightly more inefficient relative to conventional approaches when these assumptions are satisfied. The basic idea of an M-estimator is to assign weights to each vector, based on its own Mahalanobis distance, so that the amount of influence of a given point decreases as it becomes less and less characteristic. Let x1, ..., xn be samples of n dimension. Then, the Mestimator of the location (vector u) and the scatter (matrix V) are obtained by solving the following set of simultaneous equations:
1 i)n
∑w1(di)(xi - u) ) 0
n i)1 1
(17)
where w1 and w2 are arbitrary weights and di2 ) (xi - u)TV - u), i ) 1, ..., n, are the squared distances of the observation vectors from the current estimate of location u.
-1(x i
where β is chosen to make V an asymptotically unbiased estimator of the covariance matrix in a multivariate normal situation, the following algorithm can be devised to estimate the covariance matrix of the residuals Φ. Algorithm. Let y1, y2, ..., yr be the g-dimensional vector of data with yi ) [y1i, y2i, ..., ygi]T, A ) (aij)m×g is the matrix for the constraint equations. The algorithm14 for robust covariance estimation can be implemented as follows. (Step 1) Calculation of the residuals, ri, by means of eq 5. (Step 2) Calculation of the covariance matrix of residuals: • Variable initialization. Initialize the mean mj, scale bj, and covariance matrix Φ with / ) median(rji); j ) 1, 2, ..., m; i ) 1, 2, ..., r mj.0 i
(22)
bj ) median(|rji - mj|)/0.6745 i
(23)
Φ/0 ) diag(b12,b22,...,bm2)
(24)
• Iterative calculation. The solution of the M-estimator eqs 16 and 17 is obtained by means of an iterative procedure that, for iteration q, is made up of the following steps: (Step i) Estimation of the weight function u1, u2:
{
ωi < c 1 w1(ωi) ) c/ω i otherwise
(25)
w2(ωi2) ) (w1(ωi))2/β
(26)
with β ) G(c2/2,1.5) + 2c2(1 - F(c))
(27)
(16)
i)n
w2(di2)(xi - u)(xi - u)T ) V ∑ n - 1 i)1
(21)
where F represents the multivariate normal cumulative distribution function and G(a,f) is the gamma distribution with f degrees of freedom. In this work, c2 is taken to be the 90% point of the χ2 distribution. The Mahalanobis distances / / / di2 ) (ei - mq-1 )T(Φq-1 )-1(ei - mq-1 ), i ) 1, 2, ..., r (28)
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of the sample vectors from the current estimate of location m* are measured in the metric of Φ*, the current estimate of the covariance matrix. (Step ii) Estimation of the location m*: n
/ m/q ) mq-1 +{
n
/ w1(ωi)(ei - mq-1 )}/{∑wi(ωi)} ∑ i)1 i)1
(29)
problems that can be solved only using a nonlinear data reconciliation solution technique. 4.1. General Problem Formulation. As in the linear case, it is assumed that the random measurement errors follow a normal distribution with zero mean and a variance-covariance matrix ∑. The general nonlinear data reconciliation problem can be formulated as a least-squares minimization problem as follows:
(Step iii) Estimation of Φ*:
m/q )
1
Min(y - x)T x,u
n
w2(ωi2)(ei - m/q)(ei - m/q)T ∑ n i)1
(30)
The iteration process terminates when the maximum difference in the elements of Φ between two successive iterations is smaller than a prespecified threshold, in this work taken to be 10-6. (Step 3) Robust maximum likelihood estimation of Ψ: First, calculate the maximum likelihood estimator of vec(Ψ):
vec(Ψ) ) f ) (GTG)-1GT vec(Φ)
(31)
subject to f(x,u) ) 0, g(x,u) e 0
(32)
Remark. The selection of weight functions w1 and w2 is basically arbitrary. However, these functions have to satisfy certain conditions in order to ensure the existence and uniqueness of a solution. The weight functions w1 and w2 utilized in this approach have been verified as meeting all these conditions. Since w1 and w2 solve the minimax problem (20), they are also known as Huber type weights. It is worthwhile to mention that the breaking point of the M-estimator, the fraction of outliers that can be tolerated without the estimator potentially breaking down, is about 1/(g + 1), where g is the dimension of the data; the breaking point property is reasonably good for practical implementations. By assigning a weight to each of the observations, the individual observation’s contribution to covariance can be adjusted in order to avoid outliers governing the estimation of the covariance. The empirical convergence behavior of the iterative calculation procedure is good, but the iterative algorithm will maintain the positive definiteness of the covariance matrix. 4. Solution Techniques for Nonlinear Data Reconciliation The operation of a plant under steady-state conditions is commonly represented by a nonlinear system of algebraic equations. It is made up of total flow rates and component balances and may include some physical behavior of the system. If we are interested in only overall flow balance reconciliation of such processes, then linear data reconciliation techniques are adequately described. If we wish to take into consideration component balance relationships, then nonlinear data reconciliation techniques must be used. We have considered only equality constraints corresponding to the conservatio of material and flow rates and have not imposed even simple bounds on the variables. The reconciled estimates of variables can therefore become infeasible. Fox example, negative reconciled estimates for flows or components can be obtained. If we impose bounds on the estimates of variables or other feasibility constraints, then these give rise to inequality constraints in the data reconciliation,
(33) (34)
where f is an n × 1 vector of equality constraints, g is a q × 1 vector of inequality constraints, ∑ is an n × n variancecovariance matrix, u is an m × 1 vector of unmeasured variables, x is an n × 1 vector of measured variables, and y is an n × 1 vector of measured values of measurements of variables x. 4.2. Methods Using Lagrange Multipliers. The equality constrained nonlinear data reconciliation problem can be solved by using the classical method of Lagrange multipliers. The Lagrangian for the problem is given by
Then, the robust covariance Ψ can be obtained by reshaping vec(Ψ) as follows:
Ψ ) vec-1(vec(Ψ)) ) vec-1(f)
∑-1(y - x)
L(x,u,λ) ) (y - x)T
∑-1(y - x) + 2λTf(x,u)
(35)
The solution of the data reconciliation problem can be obtained by setting the partial derivatives of eq 35 with respect to the variables x, u, and λ to zero. The necessary conditions for an optimal solution point of a problem defined by eqs 33 and 34 are used in solving the resulting equations. The following equations are obtained:
∂L )∂x
∑-1(y - x) + JxTλ ) 0
(36)
∂L ) JuTλ ) 0 ∂u
(37)
∂L ) f(x,u) ) 0 ∂λ
(38)
where
Jx )
∂f ∂x
(39)
Ju )
∂f ∂u
(40)
We can use an iterative approach for solving the normal eqs 36-38 based on successive linearization. Let xˆk and uˆk represent the estimates of the variables obtained at the start of iteration k. A linear approximation can be obtained for the nonlinear constraint from the Taylor’s expansion of the function f(x,u) in eq 34 and retaining only the constant term and the first-order derivative term.
f(x,u) ) f(xˆ k,uˆ k) + Jkx(x - xˆk) + Jku(u - uˆ k)
(41)
where the Jacobian matrixes Jkx and Jku are as defined by eqs 39 and 40, with the superscript k indicating that they are evaluated at the estimates xˆ k, uˆ k. These linear equations can be decoupled by eliminating the vector x from eqs 37 and 38 using eq 36.
Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006 8977
From eq 36,
∑ (Jkx)Tλ
x)y-
(42)
Using eqs 41 and 42 in eqs 37 and 48 and rearranging, we obtain the following linear equations involving u and λ.
[∑
][ ] [
Jkx (Jkx)T Jku λ -f(xˆ k,uˆ k) + Jkxxˆ k + Jkuuˆ k - Jkxy ) k T (Ju) 0 0 u
]
(43)
Figure 1. Mass flow node diagram for a section of a coking plant. Table 1. Classification of Units
Equation 43 can be solved for obtaining the new estimates for u and λ. The estimates for λ are used in eq 42 to obtain the new estimates for x. This procedure is repeated using these new estimates as initial guesses for the next iteration. A disadvantage with these methods is that it requires more computational time. To reduce the size of the problem, the structure of the resulting matrix provides useful information for variable classification. 4.3. Method of Successive Linearization. A simpler way to handle the nonlinear data reconciliation is to successively solve a series of linear data reconciliation problems by linearization of the nonlinear constraints. We can obtain a linear data reconciliation problem for minimizing eq 33 subject to the linear equality constraints of eq 41. Their solutions for the estimates that have to be used at the next iteration are given by
unit type
units
splitter reactor separator mixer
1, 2, 4, 5 3, 6, 8, 9, 11 7 10
Table 2. Measurements for the Example measurement
stream
total flow rates components
1 2 3 4 5 6 7 10 12 14 15 16 1 2 4 5 7 8 9 10 12 13 15 16 17
Otherwise using these new estimates, repeat the procedure starting with step 2. 5. Simulation Example Study
uˆ k+1 ) uˆ k - [(Jku)TR-1Jku]-1(Jku)TR-1{f(xˆ k,uˆ k) + Jkx(y - xˆ k)] (44) xˆk + 1 ) y -
∑(Jkx)TR - 1{f(xˆk,uˆ k) + Jkx(y - xˆk) +
Jku(uˆ k+1 - uˆ k)] (45)
where
∑(Jkx)T
R ) Jkx
(46)
Equations 44 and 45 were derived by parameter estimation in nonlinear regression. An approach that can be used in general is based on the use of Crowe’s projection matrix technique to solve the linear data reconciliation problem in each iteration. The basic steps involved in this approach are as follows: (Step 1) Start with the measured values as initial estimates for the variables x and initial estimates for u which are provided by the user. (Step 2) Evaluate the Jacobian matrices of the nonlinear constraints with respect to variables x and u at the current estimates. (Step 3) Compute the projection matrix Pk such that it satisfies
PkJku ) 0 The projection matrix can be obtained using QR factorization of the matrix Jku. (Step 4) Compute new estimates for x using
xˆ ) y -
∑(PkJkx)T[PkJkx∑(PkJkx)T]Pk[Jkxy + f(xˆk,uˆk) - Jkxxˆk]
(Step 5) Compute the new estimates for u by utilizing the QR factorization of matrix Jku. (Step 6) Stop if the new estimates are not significantly different from those obtained in the preceding iteration.
To illustrate the application of the nonlinear data reconciliation theories for the coking plant, Let us consider an example which consists of a section of a coking plant. The process flowsheet and measured variables are presented in Figure 1 and Table 2, respectively. It consists of 11 units, 7 components, and 19 streams. This section of a coking plant includes CO transform, mixed gases refining, refrigeration, and compression. A simplified node diagram of the process is given in Figure 1. The mixed streams, 1 (CO, H2, CO2, N2, CH4, Ar, H2S), enter the separation columns (unit 1); they are separated and enter the cycle unit 2, the CO transform reactor unit 8, and unit 11. A portion of the stream from unit 2 is used as the recycle stream. The hydrogenated gaseous stream (units 3 and 6) enters the cold units 4 and 7, where it is passed through a number of heat exchangers and separators. The cooled cracked streams (unit 7 and 4) enter the mixed columns (unit 10), where part of the cooled cracked stream from unit 4 enters the splitter unit 5. The stream from the CO transform reactor (unit 8) enters the purify column (unit 9), and then, the depurative stream enters the mixed columns (unit 10). In the chemical process of the example, the different units may be classified as seen in Table 1. Streams 1-17 were analyzed for CO, H2, CO2, N2, CH4, Ar, and H2S. The measurements are assumed to have a relative standard deviation of 8%. Streams 18 and 19 are constant values that have not affected these measured variables and need not be reconciled. In this example, the concentrations of a stream are assumed to be either all measured or all unmeasured. All the theoretical results discussed in the previous sections of this work and in the Appendix for nonlinear systems are fully exploited for variable classification, gross error detection, and data reconciliation. The normalization constraints are taken into account by eliminating the measured concentration of one component and calculating this concentration by difference after reconciliation. There are 77 process constraints. Eleven equations belong to total mass balances, and 66 correspond to component mass balances. Enthalpy balances are not included.
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[]
The set of process constraints for this example is
[
A1 O1 O2 A2 O3 O4 B1 B2 B5 B3
where
[
[
]
]
fM fch θ )0 fU V
[ ]
[ ]
A O O A O B11 ) O1 B 1 B 2 , B22 ) B2 , B33 ) B 3 4 1 2 5 3 1 0 0 0 0 A1 ) 0 0 0 0 0 0
-1 1 0 0 0 0 0 0 0 0 0
-1 0 0 0 0 0 0 1 0 0 0
-1 0 0 0 0 0 0 0 0 0 1
0 -1 1 0 0 0 0 0 0 0 0
0 0 -1 1 0 0 0 0 0 0 0
[]
0 0 0 0 -1 0 0 0 0 1 0
f1 f2 f3 f4 f5 f fM ) f6 , fU ) 7 f10 f12 f14 f15 f16
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 -1 1 0 0 0 0
0 0 0 0 0 0 -1 0 0 1 0
0 0 0 0 0 0 0 -1 1 0 0
[] f8 f9 f13 f17 f11
[ ]
0 0 0 -1 0 A2 ) 0 0 0 0 1 0
0 0 0 -1 1 0 0 0 0 0 0
0 0 0 0 0 0 -1 0 0 0 0
0 0 0 0 0 0 0 0 0 -1 0
-I I O O O O O O O O O
O -I I O O O O O O O O
O O O O -I O O O O I O
O O O O O O O O O O -I
-I O O O O O O O O O I
0 -1 0 0 0 1 0 0 0 0 0
O O O O O -I I O O O O
O O O O O O O -I I O O
]
O O O -I I O O O O O O
O O O O O O -I O O O O
O O O O O O , B3 ) O O O -I O
-I O O O O O O I O O O
[ ] [ ]
1 0 0 I) 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
0 0 0 O) 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
O O O O O O O O -I I O
O O -I I O O O O O O O
O -I O O O I O O O O O
[ ]
o o o -d8 o B5 ) o o o o d8 o
o o o -d9 d9 o o o o o o
o o o o o o -d13 o o o o
[]
o o o o o o o o o -d17 o
0 0 0 o) 0 0 0
[ ]
I O O O O B1 ) O O O O O O
0 0 0 0 0 0 0 0 -1 1 0
[ ][ ]
O O O O O B2 ) O O O O I O
[[ ]] ‚‚‚ ‚‚‚ ‚‚‚ ‚‚‚ ‚‚‚ ‚‚‚ ‚‚‚ ‚‚‚ ‚‚‚
f1,6 f2,6 f4,6 f5,6 f7,6 f10,6 f12,6 f15,6 f16,6
θ8,1 θ θT ) θ9,1 13,1 θ17,1
‚‚‚ ‚‚‚ ‚‚‚ ‚‚‚
θ8,6 θ9,6 θ13,6 θ17,6
V3,1 V V ) V6,1 11,1 V14,1
‚‚‚ ‚‚‚ ‚‚‚ ‚‚‚
V3,6 V6,6 V11,6 V14,6
f1,1 f2,1 f4,1 f5,1 T fch ) f7,1 f10,1 f12,1 f15,1 f16,1
T
[ ]
o -d11 o o o d11 o o o o o
O O O O O O -I O O I O
Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006 8979 Table 3. Reconciliation Results for the Example of the GLR Method (units m3/h) GLR method measured components stream
CO
H2
CO2
N2
CH4
Ar
measd tot H2S flow rates
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
63205 11658 19057 32356 6819 6661 680 77 680 17138 4839 4727 4342 537 10466 10267 11560
46728 8624 14097 23936 5044 5022 1312 487 4311 40644 3580 3564 6 3405 22689 22465 27671
23774 4388 7173 12178 2567 0 0 0 0 28578 1821 0 0 0 15766 1290 1290
444 82 134 228 48 47 8 1 7 235 34 33 29 5 133 161 176
135 25 41 69 15 14 0 0 0 69 10 10 10 0 39 51 53
175 32 53 90 19 19 2 0 2 96 13 13 12 1 54 69 69
202 37 61 103 22 0 0 0 0 104 15 0 0 0 59 0 0
134623 24847 40615 68960 14533 11765 2000 565 5000 86864 10314 8349 4400 3949 49206 34303 40819
reconciled components CO
H2
CO2
63236 11579 18990 32667 6764.2 6716 670.39 77.2 670.39 16693 4812.3 4736.1 4288.6 525.86 10379 10305 11652
46479 8510.4 13958 24010 4971.7 4966.3 1344.5 512.4 4449.7 39984 3538.6 3526.3 6.29 3532.3 22570 22506 27959
23837 4364.6 7158.5 12314 2549.8 0.36 0 0 0 28287 1814.8 0 0 0 15770 1289.7 1289.7
In the Appendix, we discuss how to do variable classification. In classifying the measured and unmeasured process variables, the following procedure is applied: (1) A Q-R decomposition is performed on matrix B33, and QB1, QB2, RB1, RB2, ΠV matrices are obtained. (2) Matrix D is calculated, and the application of the orthogonal projection approach leads to QD1, QD2, RD1, RD2, ΠV matrices. (3) With the calculation of matrix Ga (using eq 18), all the necessary information for the measured variables classification is available. (4) As RD2 is an empty matrix, the subset fUnd-rf is also empty, so all the unmeasured total flow rates are determinable, as can be seen from eq 22. The subsets VrV, Vnb-rV, and matrices RIB and RIFI are then obtained:
[
]
V3,1 ‚‚‚ V3,6 VrV ) V11,1 ‚‚‚ V11,6 , Vnb-rV ) [V3,1 ‚‚‚ V3,6 ] V14,1 ‚‚‚ V14,6 In this example, matrix RIB is
[] [
V V ‚‚‚ V3,6 ‚‚‚ V11,6 Vd ) V11,1 , Vi ) V3,1 V 14,1 ‚‚‚ 14,6 6,1 ‚‚‚ V6,6
]
As all total flow rates are measured or determinable, the concentrations of streams 11 and 14 are determinable, while those of streams 3 and 6 are indeterminable. The operation of a plant under steady-state conditions is commonly represented by a nonlinear system of algebraic equations. It is made up of the total flow rates and component balances and some physical behavior of the system. In this case, data reconciliation is based on the solution of a nonlinear constrained optimization problem applying nonlinear data reconciliation theories to the coking plant which can be formulated as a Lagrange multipliers problem as follows:
L(xij,ul,λijl) ) (yij - xij)T
Ar
H2S
recd tot flow rates
457.83 138.71 182.59 201.02 134532.15 83.83 25.4 33.43 36.81 24633.47 137.49 41.66 54.83 60.37 40400.85 236.51 71.66 94.32 103.85 69497.34 48.97 14.84 19.53 21.5 14390.54 47.63 14.12 18.96 0 11763.37 7.97 0 0.17 0 2023.03 0.92 0 0.02 0 590.54 7.97 0 0.17 0 5128.23 236.51 71.66 94.32 103.85 85470.34 34.86 10.56 13.9 15.31 10240.33 34.86 10.56 13.9 0 8321.72 29.15 10.56 12.66 0 4347.26 5.71 0 1.24 0 4065.11 137.49 41.66 54.83 60.37 49013.35 153.16 48.21 56.26 0 34358.33 152.09 41.66 56.26 0 41150.71
Let i ) 1, 2, ..., 17 be the stream number, let j ) 1, 2, ...,7 be the component number, and let l ) 8, 9, 11, 13, and 17 be the unmeasured stream number. The necessary conditions for an optimal solution point of the problem defined by eqs 23 and 24 can be used in solving the resulting equations. The following equations are obtained:
∂L )∂xij
∑-1(yij - xij) + Jx Tλijl ) 0 ij
∂L ) JulTλijl ) 0 ∂ul ∂L ) f(xij,ul) ) 0 ∂λijl where
Jxij )
xˆ ij ) yij -
and RIFI is empty. As the first three rows of RIB are nonzero, the following Vd and Vi vectors result as follows:
]
CH4
∂f ∂f , Jul ) ∂xij ∂ul
combining these equations with eqs 31-36, using the method of successive linearization, we can compute new estimates for xij using
-I O RIB ) O O
[
N2
∑-1(yij - xij) + 2λijlTf(xij,ul)
∑ (PkJ xk )TPkJ xk ∑(PkJ xk )T × ij
ij
ij
Pk[J xkijyij + f(xˆij(k),uˆ l(k)) - J xkijxˆ ij(k)] where Pk denotes the projection matrix such that it satisfies PkJ ku ) 0. The projection matrix can be obtained using QR factorization of the matrix J ku. Let xˆ ij(k) and uˆ l(k) represent the estimates of the variables obtained at the start of iteration k. The process constraints for the flotation circuit are the material balances. The results were obtained using the successive linearization data reconciliation algorithm described above. The results are shown in Tables 3-5. We choose the total mass flow of stream 1 to be unity, so that the remaining unknown total mass flow rates will be calculated relative to this basis. We note that stream 1 belongs to category 1. Since there are no chemical reactions occurring, this is purely a separation process. So, the reconciled components ratio of streams 1-5 must be same. In Table 3, we notice that reconciliation results for this example with only the treatment of the total flow rates balance. For multiple gross error cases, the GLR method can adjust the measurement or constraint that is declared as containing gross
201.64 36.95 60.57 104.13 21.59 0 0 0 0 104.13 15.36 0 0 0 60.57 0 0 183.54 33.63 55.13 94.78 19.65 19.65 0.16 0.02 0.16 94.78 13.98 13.98 12.77 1.21 55.13 55.13 56.52 468.06 85.76 140.6 241.7 50.12 50.12 7.85 0.9 7.85 241.7 35.64 35.64 30.04 5.6 140.6 140.6 154.95 62948 11683 18933 32332 6849.7 6849.7 658.82 71.59 658.59 16427 4833.6 4833.6 4321.3 512.35 10383 10383 11626 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
45905 8417 14072 23416 4895.2 4895.1 1372.3 514.19 4372 39321 3521.9 3521.8 5.2 3515.3 22622 22622 28024
24082 4395 7200.4 12488 2572.9 61.68 0 0 0 28393 1820.4 0 64.19 0 15750 1281.8 1281.5
515.25 80.56 139.66 294.03 49.13 49.66 6.66 8.71 6.55 294.07 31.72 31.643 27.11 3.27 140.03 139.81 152.68
143.05 23.25 44.4 74.87 14.02 13.59 0 0 0 75.11 10.31 10.945 10.77 0 44.42 44.2 43.71
189.14 31.17 58.99 98.65 18.98 19.95 0.59 0 2.85 98.62 13.6 12.942 11.84 0.45 57.2 58.24 61.9
202.9 37.63 60.08 104.63 21.7 99.24 0 0 0 105.54 15.11 0 0 0 60.56 0 0
133985.34 24667.61 40508.53 68808.18 14421.63 11988.92 2038.37 594.49 5039.99 84714.34 10246.64 8410.93 4440.41 4031.37 49057.21 34529.05 41189.79
63336.46 11604.64 19025.14 32706.68 6781.59 6781.59 656.08 75.3 656.08 16619.11 4823.05 4823.05 4312.1 510.94 10380.85 10380.85 11623.17
46304.84 8484.07 13909.15 23911.62 4957.97 4957.97 1353.12 509.42 4438.5 39999.19 3526.1 3526.1 6.47 3519.62 22553.45 22553.45 27935.62
23902.09 4379.39 7179.76 12342.94 2559.26 0 0 0 0 28430.5 1820.14 0 0 0 15824.06 1283.09 1283.09
139.4 25.54 41.87 71.99 14.93 14.93 0 0 0 71.99 10.62 10.62 10.62 0 41.87 41.87 41.87
H2S Ar CH4 N2 CO2 N2 CO2 H2 CO stream
reconciled components
CH4
Ar
H2S
recd tot flow rates
CO
H2
reconciled components
successive linearization method Lagrange multipliers method
Table 4. Reconciliation Results for the Example of the Lagrange Multipliers and Successive Linearization Methods (units m3/h)
134536.03 24649.98 40412.22 69473.84 14405.11 11824.26 2017.21 585.64 5102.59 85561.4 10244.89 8409.39 4372 4037.37 49056.53 34454.99 41095.22
Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006 recd tot flow rates
8980
Table 5. Results of the Example for Different Weighting Matrices R (units m3/h) reconciled flow rates stream
R)Q
R ) 1000Q
R ) 1000000Q
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
133990 24668 40508 68809 14422 11827 2040.2 586.9 5039.9 84714 10246 8410.3 4376.4 4033.9 49058 34529 41190
134374.4 24699.2 40521 69154.21 14423.9 11843.8 2050.7 593.4 5050.4 85252.7 10275.3 8444.6 4385.6 4058.9 48977.2 34564.1 41267.1
134496.4 24690.5 40596.3 69209.6 14450.9 11879.2 2030.7 648.7 5030.4 85411.1 10239.5 8450.1 4452.1 3998.1 48979.9 34460.5 41138
error. If a gross error is identified, the estimated magnitude of the error is used to adjust the measurement or constraint associated with the detected gross error. And then, the GLR test is repeated again until no gross error is detected. The advantage of the GLR test is that it can identify the gross error source as an instrument error or process leak. However, its applications are still restricted to linear process constraints or approximate linear ones. The linearization of nonlinear constraints brings in great errors in the approximation of nonlinear constraints and distribution when the process is highly nonlinear and gross errors are large. Also, the implementation of GLR for searching gross errors is not efficient. It is not applicable for complicated and highly nonlinear processes of on-line optimization. In Table 4, the reconciliation results for this example treatment of gross errors, component balances, and total flow rate balances with the Lagrange multipliers and successive linearization methods are listed, respectively. The results of the reconciled components and total flow rates are balanced within the confidence level (R ) 0.1). The Lagrange multipliers method determines the minimum of the variance weighted closure residuals of the mass balance equations through direct analytical differentiation of the successive linearization method. In general, the Lagrange multipliers method tends much more to shift the reconciled values in a biased direction; that is, it would maintain a positive bias for one variable and a negative for another and not give a normal distribution of errors. In doing so, it does give a higher degree of dynamic predictability of the adjustments, but it does this at the expense of adjustment accuracy. The successive linearization method has the advantage of relative simplicity and fast calculation. In addition, it can be modified to choose a step size that minimizes a penalty function. The step size is chosen by the method of interval halving. Table 5 shows the results of the reconciliation for different values of the weighting matrix R. As expected, larger corrections are performed to the measured values in order to satisfy the constraints. It is clear that using this general procedure we have additional degrees of freedom in terms of assigning weights to the satisfaction of constraints. Furthermore, this general formulation clarifies the contributions to the error in the final estimate from the processing of both the constraints and the measurements. To compare the results of data reconciliation for different methods, we define the percent ratio of errors pi as follows: pi ) i/xˆ . The results of the reconciliation for the full data are depicted in Figures 2-6. It can be observed in Figure 2 that
Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006 8981
Figure 2. Reconciliation results for the example with the GLR method.
Figure 3. Reconciliation results for the example with the Lagrange multipliers method.
some reconciled values are wrong and go over the confidence level. If a gross error is identified, the estimated magnitude of the error is used to adjust the measurement or constraint associated with the detected gross error. And then, the GLR test is repeated again until no gross error is detected. It is necessary to reconcile measurements of components and total flow rates from the streams of a chemical process at the same time. This usually involves the minimization of a least-squares objective function which is subjected to a set of material balance constraints. Figure 3 should be compared with Figure 4, which show the example using the Lagrange multipliers and successive linearization methods, respectively. Figure 3 shows the results of the reconciled values for the Lagrange multipliers method within a deviation of 10%. The advantage of the Lagrange multipliers method is that it determines the minimum of the variance weighted closure residuals of the mass balance equations through direct analysis. In Figure 4, we can observe that the successive linearization method has the advantage of relative simplicity and fast calculation. The M-estimator method is insensitive to gross errors, which is its most important feature. The drawback can be eliminated by using the M-estimator method. The performance of the M-estimator method is better than the GLR method when gross errors are present in the data set as shown in Figure 4. Figure 5 shows the results of the reconciled total flow rates for different values of the weighting matrix R. It can be observed
Figure 4. Reconciliation results for the example with the successive linearization and M-estimator methods.
Figure 5. Results of the example for different weighting matrices R.
Figure 6. Results of this example for different weighting matrices R.
that the 8th stream has gross errors. As expected, larger corrections are performed to the measured values in order to satisfy the constraints. By eliminating gross errors, the results of the reconciled values are depicted in Figure 6. This example illustrates the use of the techniques described in the previous sections in an actual on-line framework, using industrial hardware. Furthermore, the usefulness of data reconciliation prior to process modeling and optimization is clearly demonstrated.
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6. Conclusions In this paper, we have stated the data reconciliation problem and explored some available techniques for solving it. This study of a nonlinear process using available industrial data illustrates the problematic nature of applying theoretical tools to operating chemical plants. Data reconciliation is an important step in realtime on-line optimization of a plant. It adjusts the process data to satisfy the constraints of the system model and provides estimates for unmeasured variables and process parameters, which are used in the consecutive economic optimization step. In this study, the focus has been on the simultaneous data reconciliation and covariance estimation strategies to improve this initial step in on-line optimization. The example study discussed in this paper clearly shows the importance of the techniques developed for data reconciliation, when used in a practical environment. Using total flow rate and component balances within a section of a coking plant, fully exploited the theoretical results for complex nonlinear systems, namely, variable classification, gross error detection, and data reconciliation. For gross error detection, the GLR method has been investigated along with weighted least squares for nonlinear models. For the covariance estimator, M-estimator methods are designed so that they are insensitive to gross error. The study shows that the M-estimator approaches for the simultaneous data reconciliation and covariance estimate of chemical processes can provide similar or better results compared with the GLR method for gross error detection. Solution techniques for nonlinear data reconciliation, the Lagrange multipliers method, and the successive linearization method are listed, respectively. The results of the reconciled total flow rates for different values of the weighting matrix R that shows larger corrections are performed to the measured values in order to satisfy the constraints. In conclusion, we have demonstrated the application of data reconciliation in an actual plant and successfully estimated the assumed constant value. This work has also explicitly demonstrated the need for complete process knowledge in order for the results of the reconciliation to be obviously interpretable.
Table A.1. Categories of Component and Total Flow Rates category
component
total flow rates
1 2 3 4
measured measured unmeasured unmeasured
measured unmeasured measured unmeasured
strategies may be divided into two major groups. One group applies the concepts of graph theory to achieve the categorization; the other makes use of matrix ordering techniques and computations. In this paper, the main feature of matrix ordering techniques application is discussed. Given the topology of the process and a measurement set, these strategies generate the system of model equations for the plant first. Different kinds of rearrangements and calculations, involving matrices and nonlinear equations, are then performed to classify process variables. Chemical process data of multicomponents can be divided into four categories depending on the combination of composition and total flow rates, as indicated in Table A.1. The total flow rates and component flow rate material balances are as follows:
(A1 X E)X1 + (A2 X EC2) vec(G2) + (A3G3 X E) vec(C3) + (A4 X E)X4 ) 0 (A.1) where A1, A2, A3, and A4 are compatible matrices of dimension m × nj (m is number of process model units and nj is number of process model streams of category j); X1 is vector of component flow rates of category 1; X4 is vector of component flow rates of category 4; C2 is component matrices of category 2 defined by eq A.2; C3 is component matrices of category 3; s is a number of stream components
Acknowledgment This work is supported by National Natural Science Foundation of China (60504033).
C2 )
d21 d22 l d2s
(A.2)
l
dm21 dm22 l dm2s
Appendix: Classification of the Process Variables The mathematical formulation of the variable classification problem is stated and some structural properties are discussed in terms of graphical techniques. Different strategies are available for carrying out process-variable classification. Both graph-oriented approaches and matrix-based techniques are briefly analyzed in the context of their usefulness for performing variable categorization. For reasons of cost, convenience, or technical feasibility, not every variable is measured, and some of them can be estimated using other measurements through balance calculations. Unmeasured variable estimation depends on the structure of the process flowsheet and on the instrument placement. Typically, there is an incomplete set of instruments; thus, unmeasured variables are divided into determinable (or estimable) and indeterminable (or inestimable) ones. Measured variables are classified into redundant (or adjustable) and nonredundant (or nonadjustable) ones. During the past three decades, several strategies have been formulated for performing process variable classification. These
[] d11 d12 l d1s
G2 is a diagonal matrix (m2 × m2) of total flow rates of category 2; G3 is a diagonal matrix (m3 × m3) of total flow rates of category 3; E is a matrix ((s + 1) × s) defined by eq A.4
G2 )
[
g21 g22
l
[ ]
g2m2
]
1 2 l l E) 1 s 1 1 ‚‚‚ 1 s + 1
(A.3)
1
1
(A.4)
Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006 8983
1. Classification of the Unmeasured Variables. In balance eq A.1, unmeasured data vec(G2), vec(C3), and x4 are linear, and their coefficient matrix is defined by eq A.5.
T ) [A2 X EC2|A3G3 X E|A4 X E]
(A.5)
The matrix T is divided into two submatrices T1 and T2. T1 is defined by max linear independent columns.
T ) [T1|T2]
(A.6)
The zero columns of matrix PT correspond to indeterminable unmeasured variables and others are determinable unmeasured variables. The projection matrix P2 of matrix T2 is defined by the following:
P2T2 ) 0
(A.7)
The columns of P2 span the null space of T2, and thus, the indeterminable unmeasured variables are eliminated. 2. Classification of the Measured Variables. 2.1. Classification of the Measured Total Flow Rates. First, using the projection matrix P4 of matrix A4 X E left, multiply eq A.1, and thus, the variables X4 of category 4 are eliminated.
P4(A1 X E)X1 + P4(A2 X EC2) vec(G2) + P4(A3G3 X E) vec(C3) ) 0 (A.8) Second, using the projection matrix P2 of matrix P4(A2 X EC2) left, multiply eq A.8, and thus, the unmeasured total flow rates G2 of category 2 are eliminated.
P2P4(A1 X E)X1 + P2P4(A3G3 X E)C3 ) 0
(A.9)
The zero columns of matrix P2P4(A1 X E) and P2P4(A3G3 X E) correspond to nonadjustable variables of total flow rates and others are adjustable variables. 2.2. Classification of the Measured Component. Using the projection matrix P3 of matrix P2P4(A3G3 X E) left, multiply eq A.9
P3P2P4(A1 X E)X1 ) 0
(A.10)
The zero columns of their coefficient matrix correspond to nonadjustable variables of stream components and others are adjustable variables.
Nomenclature x ) vector of measured variables y ) measurement vector ) measurement random errors n ) number of measured variables ψ ) covariance matrix of δi ) unit vector a ) magnitude of bias f ) nonlinear algebraic equations u ) vector of unmeasured variables m ) number of process model functions Vi ) a vector of different constraints A ) a general matrix r ) constraint residual H ) covariance matrix of constraint residual λ ) likelihood ratio
wi ) arbitrary weights φ(n,c) ) Huber’s psi function β ) asymptotically unbiased estimator of the covariance matrix Φ ) covariance matrix of the residuals G(a,f) ) gamma distribution with f degrees of freedom c2 ) the 90% point of the χ2 distribution di2 ) Mahalanobis distances vec(Ψ) ) maximum likelihood estimator g ) q × 1 vector of inequality constraints ∑ ) n × n variance-covariance matrix P ) projection matrix
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(22) Jiang, T.; Chen, B.; He, X.; Stuart, P. Application of steady-state detection method basied on wavelet transform. Comput. Chem. Eng. 2003, 27, 569-578. (23) Kelly, J. D. Techniques for solving industrial nonlinear data reconciliation problems. Comput. Chem. Eng. 2004, 28, 2837-2843. (24) Cochinwala, M.; Kurien, V.; Lalk, G.; Shasha, D. Efficient data reconciliation. Inf. Sci. 2001, 137, 1-15. (25) Bagajewicz, M. J. On the definition of software accuracy in redundant measurement systems. AIChE J. 2005, 51, 1201-1206.
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ReceiVed for reView January 17, 2006 ReVised manuscript receiVed June 12, 2006 Accepted September 25, 2006 IE060077H
