Li Bo, Chen Bingzhen, Wang Jian, Cong Songbo
(Department of Chemical Engineering, Tsinghua University, Beijing, 100084, China)

1. STEADY-STATE ONLINE DATA RECONCILIATION
1.1 Basic model 
With the assumptions of steady state process, credible measurements, the random measurement errors are normally distributed with zero means and known variance/covariance, the most common approach formulates data reconciliation as the following constrained least squares problem:
(1)
(2)
    In this model, the weighted sum of errors is to be minimized, subject to the constraints encapsulated in the term F(X,U). These constraints arise because the mass balances, energy balances and any other performance equations must be satisfied.
1.1.1 Linear case
When the constraints are linear or almost linear, the model can be written as follows:
(1)
                (3)
    The solution is obtained by means of Lagragian multipliers, and is given by:
(4)
       (5)
1.1.2 Nonlinear case
When the constraints are nonlinear, the model can be written as follows by linearising the constraints:
(1)
                (6)
    where F_x and F_u are the Jacobin matrices of the constraint functions to the measured and unmeasured variables respectively.
One can solve the nonlinear problem directly using successive quadratic programming algorithm. For the purpose of performing data reconciliation online, successive linearization programming method given by Kneepper & Gorman (Knepper & Gorman,1980)is adopted, which calculates the reconciled values as follows: (7)
                (8)
    where X₀ is the vector of measured values，X_i is the vector of reconciled values of measured variables at the ith iteration, X_i+1 is the vector of reconciled values of measured variables at the i+1th iteration，U_i is the vector of estimated values of unmeasured variables at the ith iteration, U_i+1 is the vector of estimated values of unmeasured variables at the i+1th iterations. D=(F_xQF_x^T)^-1。
1.2 Data classifying
Chemical process variables can be classified into four types. The measured variable can be defined as redundant or nonredundant according to its spatial redundancy, the unmeasured variable can be defined as observable or unobservable according to its observability. The existences of nonredundant variables or unobservable variables will produce a singular matrix during the procedure of data reconciliation, which will result in that the procedure can not be completed correctly.
    The most commonly used method for data classifying is of projection matrix method presented by Crowe (Crowe et. al., 1983) which classify variables into measured and unmeasured ones. The measured variables are reconciled and the unmeasured variables are then estimated to complete the data reconciliation work. However, if some unmeasured variables are unobservable, the problem of calculating the inverse of a singular matrix still exists in the procedure of estimating the unmeasured variables.
    In order to perform data reconciliation online, it is necessary to develop a new method to classify the process data thoroughly. For this purpose, we developed a two level transformation method based on projection matrix(Wang Jian et. al., 1999).
1.2.1 Data classifying of unmeasured variables
In equation (3), matrix B is divided into two parts B1^* and B2^*. B1^* consists of the columns linearly independent with the largest number in matrix B, B2^* consists of the other columns of matrix B.
B=[B1^*|B2^*]
    We define P₂ as the projection matrix of B2^*. In the matrix P₁B, the columns having all elements zero are corresponding to the observable unmeasured variables while the other columns are corresponding to the unobservable ones. The matrix B can be divided into two parts B1 and B2 corresponding to the columns having all elements zero and other columns of P₁B respectively. The unmeasured variables then can be divided into U₁ including the observable ones and U₂ including the unobservable ones.
1.2.2Data classifying of measured variables
P₁ is defined as the projection matrix of P₁B, multiplying equation (3) by P₁P₁ will remove the term of unmeasured variables, giving:
(9)
    In matrix P₁P₂A, the columns having all elements zero are corresponding to the nonredundant measured variables while the other columns are corresponding to the redundant ones.
    The matrix A then can be divided into two parts A1 and A2 corresponding to the redundant variables and the nonredundant variables respectively. The measured variables are then divided into two sets, the redundant ones constitute X₁ while the non-redundant ones constitute X₂.
1.2.3 Two level transformation method
We define the following symbols:

     
      
     
then Equation (3) can be rewritten as:
(10)
    Equation (10) is the constraint after the first level transformation. In this equation, the unobservable unmeasured variables have been eliminated, the nonredundant measured variables have been removed from the measured data set and merged into the constant term C₂, so that the observable unmeasured variables can be estimated using this equation.
    The equation (9) can be rewritten as:
(11)
    Equation (11) is the constraint after the second level transformation. In this equation, only the redundant measured variables are considered to be reconciled.
1.2.4 Solution for linear case after data classifying
With the two level transformed constraints using projection matrix, we can solve the data reconciliation problem as follows:
    First to calculate the reconciled value of the redundant measured variables, the model is given by:
(12)
         (11)
    where Q is the variance/covariance matrix of redundant measured variables.
    The reconciled value of the redundant variable is given by:
(13)
    From equation (10), the observable unmeasured variable can be estimated as follows:
        (14)
There is no more work to do with the nonredundant measured variables and the unobservable unmeasured variables.
1.2.5 Solution for nonlinear case after data classifying
For nonlinear case, after data classifying, using the successive linearization programming method, the redundant variables are reconciled by:
              (15)
the observable unmeasured variables are then estimated by:
(16)
    where A₁ 、A₂ 、B₂ is the transformed matrixes of F_x and F_y. F_x and F_y are the Jacobin matrixes of the constraint function to measured and unmeasured variables respectively.
1.3 Steady State Test
The methods based on statistical technique are commonly used in steady state test. For chemical engineering processes, Narasimhan et. al. ( 1986 and 1987) provided two steady state test methods based on statistical theory, they are Composite Statistical Test method (CST, Narasimhan et. al., 1986) and Mathematical Theory of Evidence method(MTE, Narasimhan et. al., 1987). All these two methods need the specific assumption that a time period is to be selected within which the process variables are assumed to be in steady state.
    In fact, process variables often change disorderly, which will make it very difficult for us to determine the location of a series of successive time periods satisfying the specific assumption. Therefore these two methods are greatly limited in industrial cases.
    On the purpose of performing steady state test online, this paper developed a new method based on statistic test to find a steady state in complicated industrial case(Cong Songbo, 1998). This method uses a low-pass dynamic filter to reduce the variation magnitude of dynamically changed process variables, while steady process variables will remain unaffected by the filter. Then the differences between the filtered and unfiltered data are detected. A steady state is concluded if no significant differences exist. It should be emphasized that no specific assumption mentioned above is needed. Furthermore, the time period can be selected arbitrarily, one can move the time period step by step along time axis as a moving window, specifying whether it goes into a period of steady state.
1.3.1 Algorithm description
The basic assumptions are :
    a. Measurements contain only random errors varying at a relatively high frequency and are normally distributed with zero means.
    b. Successive measurements are mutually independent.
    c. The variance/covariance matrix of measurement errors are constant and can be estimated by process data. In fact, from assumption b, the covariance matrix is diagonal. It’s element can be used as an index representing how much change can be tolerated for a steady state.
    According to these assumptions, the measurement model is given by:
j=1,2,…,N
j=1,2,…,N
    where is the jth measurement of variable i, is the jth true value of variable i and is the random error of variable i at jth measurement.
    The variables pass through a low-pass filter are

where F is the function of a low-pass filter such as one order and two order linear filter. In this paper, a one order low-pass filter is introduced with its transfer function as follows

where T  is the time constant of the low-pass filter.
An offset for each variable is then available:

For a low-pass filter with its gain equal to 1, it is obvious that when the process variable is in steady state, will be equal to and will be a normal distributed noise with zero mean. This characteristic will be very useful for steady state detection. There are two methods according to the statistic quantity selected.
1.3.2 Mean-value test method
In mean-value test method, the statistic quantity is:
, i =1,…,p
In statistical theory, S_i is a statistical variable with normal distribution N(0,1). If the mean value of variable S_i is zero, a steady state can be then concluded. The hypothesis to be inspected is:
H₀ : E(S_i)=0, for all i=1,2,…,p
H₁ : E(S_i) ¹0, for at least one i
The null hypothesis is rejected if S_i >S_i(a ), where S_i(a ) is the test criterion at a level of significance a . For multivariable case, because the variables are mutually independent, we can transform it to a series of single-variable statistical test problem through following formulations,

where, , i=1,2,…,p
1.3.3 Variance test method
In this method, a steady state is concluded if the process variable does not exceed a pre-specified value (or its normal variance) at a level of significance  . The variance of a process variable is used to form the statistic quantity:
i =1,…,p
In statistical theory, C_i is a statistical variable with c² distribution. The c² test is then taken for the following hypothesis,
H₀ : C_i  < C(a ), for all i=1,2,…,p
H₁ : C_i > C(a ), for at least one of i
A series of single-variable test can be carried out for multivariable case through the following formulations:

where, , i=1,2,…,p
1.3.4 Parameter tuning
Both mean-value method and variance method have their own characteristics. The first method is suitable for cases where all the process variables possess nearly the same time constants.
    If the process variables, with significantly different time constants, are included in a mean-value statistic, and the time period for detection is selected according to the longer time constant, then the variation of process variable with small time constant will be taken as a process noise. The variation at different direction from its mean value, will be canceled mutually to some extent, leading to an incorrect result. The second method will be more effective than the first one in these cases. If we don’t pay much attention to the variation with a relatively high frequency, the first method will be a good choice.
    The parameter T of the low pass filter is a key factor affecting the result. If T is set too small , the difference between filtered and unfiltered data will not be significant enough to be detected. On the other hand, if T is set too large, the algorithms will not be sensitive to the change of process state. Generally speaking, T is determined according to the time constants of process variables, and is often assigned with a value of 1-2 hour for most chemical processes.
    The variance/covariance matrix is unknown in both of the two methods and need to be estimated with industrial experience. In fact, the criterion C(a ) can be used as an index representing the variation range within which the process data will be taken as in steady state. This physical meaning provides an approach estimating the variance/covariance matrix. Generally, the variance can be assigned to 2-5 percent of full measurement span of the corresponding variable.
    The length of time period is another parameter to be chosen. It is recommended that the length of time period should be a bit longer than the setting time for all process variables concerned. For most industrial chemical processes, the length of time period should be assigned to 1-4 hours.
1.4 Gross Error Handling   
The bottleneck of data reconciliation is of gross error handling (Crowe et. al., 1996; Narasimhan et. al., 1987; Cheng Lifei, 1998; Tjoa et. al., 1991; Albuquerque et. al., 1996). The sources of gross errors include malfunction instruments, decrease of measurement precision, process leaks, non-steady state operation, mismatching model, and so on(Crowe et. al., 1996).
    Many methods based on statistical technique have been developed to detect and eliminate gross errors. All these methods are based on spatial redundancy of the data reconciliation problem. Among them, global test, constraint test and measurement test or modified measurement test are most commonly used (Crowe et. al., 1996; SIMSCI, 1996).
    In many industrial cases, as shown in Fig. 1, gross error often occurs as the result of concurrence of measurement bias and decrease of measurement precision. It should be noted that the gross error exists not only at one measurement but at all the measurements until the instrument is adjusted, we can call this long-time existing gross error as Integral Gross Error (IGE). When we perform data reconciliation in a real plant, the common condition is that many instruments especially the flow-rate instruments have IGE, the spatial redundancy is small so that the methods mentioned above are not capable of handling the gross errors reasonably. The shortcoming of the existing methods will result in that process of data reconciliation can not be performed online. In this paper, a better new method is developed to preprocess the raw data before data reconciliation. During the data preprocessing, the sequential measurements is used to estimate the magnitudes of IGE and compensate them with a correcting coefficient.

Fig. 1 Integral Gross Error in Industrial Case

1. Collect process data from DCS
2. Steady state test, if the process is not under steady state then go to 1; else do 3
3. Data classifying, calculate the spatial redundancy of that problem. If the spatial redundancy is negative, data reconciliation is not feasible, then give some advice of adding instrument, improve the precision of instruments, and so on, go to 1; else do 4
4. Data preprocessing to handle the IGEs
5. Solve the problem of data reconciliation, gross error detection and identification, the results satisfying the constraints then can be achieved
6. Generate the result report
7. Go to 1

2. APPLICATION TO A CRUDE OIL DISTILLATION UNIT 
The technique of steady-state online data reconciliation is implemented in a crude oil distillation unit whose flowchart is shown in Fig. 2. This unit consists of 13 facilities, 37 streams. There are 74 variables to be reconciled, 67 variables including 31 flowrates and 36 temperatures are measured , 7 variables including 6 flowrates and 1 temperature are unmeasured. For nonlinear case with mass balances and energy balances, the model of data reconciliation contains 26 equality constraints.

Fig. 2 Flowchart of A Crude Oil Distillation Unit

Note: MV Measured Value RV Reconciled Value VT Variable Type
PVR Percentage Variance Reduction MR Measured Redundant Variable
UO Unmeasured Observable Variable ME Measured Variable having Gross Errors
The streams with (A) in their description enter or leave from the atmospheric column
The streams with (V) in their description enter or leave from the vacuum column.

3. CONCLUSION
The technique of steady-state online data reconciliation is studied in detail in this paper. With the purpose of online performance, two level transformation method is developed to classify process variables thoroughly, new algorithm including mean-value test method and variance test method is presented to avoid the specific assumption which is necessary to the MTE and CST method, we also handle Integral Gross Errors through data preprocessing using correcting coefficient method which can overcome the limitation of the spatial redundancy to the existing methods. An industrial application to a crude oil distillation unit is also discussed. The reconciled result is validated by comparing with the result of tank scaling, the variance reduction after data reconciliation is also shown. We can conclude from the result that more precise data satisfying the constraints can be achieved through data reconciliation and can be applied for works such as administrative management, online process optimization, and so on.

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