A new data reconciliation strategy based on mutual information for

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Process Systems Engineering

A new data reconciliation strategy based on mutual information for industrial processes Chunhua Yang, Sen Xie, Xiaofeng Yuan, Xiaoli Wang, and Yongfang Xie Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b02104 • Publication Date (Web): 31 Aug 2018 Downloaded from http://pubs.acs.org on September 2, 2018

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Industrial & Engineering Chemistry Research

Data reconciliation of mass balance layer The l Data reconciliation measured model of mass variables balance layer

Industrial process

The m-l measured variables

Reconciled results Estimated results

State Mutual transition information algorithm Data reconciliation model of heat balance layer

Data reconciliation of heat balance layer

Reconciled results Estimated results

ACS Paragon Plus Environment

Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

A new data reconciliation strategy based on mutual information for industrial processes Chunhua Yang, Sen Xie, Xiaofeng Yuan∗, Xiaoli Wang, Yongfang Xie School of Information Science and Engineering, Central South University, Changsha, 410083, Hunan, P. R. China

Abstract In modern industrial processes, data reconciliation plays a significant role in reducing measurement errors and adjusting process measurements to meet conservation laws and constraints of process models. Most traditional data reconciliation methods are based on the weighted least squares technique with diagonal weight matrix, in which the variable errors are assumed to be independent of each other. However, the measurement errors are often nonlinearly correlated due to reasons like kinetic relationships, external disturbances and feedback control, etc. To deal with this problem, a new data reconciliation strategy based on mutual information is proposed in this paper. In the new method, mutual information is first utilized to measure the nonlinear variable correlations. Then, a new objective function is designed for data reconciliation with a full weight matrix, whose elements are the mutual information coefficients between different variables. Finally, the effectiveness of the proposed data reconciliation method is demonstrated through a nonlinear numerical example and an industrial application.

Keywords:

Industrial processes; Data reconciliation; Mass and heat balance; Mutual information matrix; Measurement error

∗

Corresponding author. E-mail address: [email protected] (X. Yuan)


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1. Introduction Accurate and reliable process measurements often play significant roles for process behavior understanding and operation instruction in industrial processes. They are of great importance for process modeling1-5, performance monitoring6-11, real-time optimization12, and process control13-14. However, process measurements are often inevitably contaminated by measurement errors, which do not obey the conservation laws (like mass and energy conservation laws) and constraints of process models. In addition, there are some variables that are unmeasured due to technical and economic limitations. To better understand the process state, it is necessary to adjust the measured variable data and estimate the unmeasured data so that they can satisfy the conservation laws, which is known as data reconciliation. Data reconciliation is considered as a crucial data processing technique to reduce the effect of the measurement errors and enhance the quality of process measurements by using the redundancies in the measurements. Data reconciliation was firstly introduced by Kuehn and Davidson,15 which aims to minimize the difference between measured and reconciled values of process variables while keeping the constraints of process model. Over the past decades, numerous researches have been carried out on data reconciliation. Moreover, data reconciliation has been largely applied to various industrial processes, such as chemical processes16-18, mineral and metal processes19-20, and steam turbine systems21-22, etc. In early data reconciliation studies, researchers mainly focused on data reconciliation model with linear constraints. Mah et al.23 discussed the problem of linear data reconciliation with unmeasured variables, in which the relationship between linear algebra and graph theory was explained. Crowe et al.24 developed the projection matrix method which is used to deal with the linear data reconciliation problem. Then, iterative linearization method was



proposed to deal with nonlinear data reconciliation problem25. Then, nonlinear programming was further developed for nonlinear data reconciliation problems, which is more efficient than the iterative linearization technique26. Subsequently, Swartz used orthogonal triangle factorization method combining with projection matrix to solve data reconciliation problem27. Kelly studied a large-scale data reconciliation problem with quantity-quality bilinear constraints28. With the development of optimization and programming techniques, different optimization methods were adopted to handle data reconciliation problems, like improved genetic algorithm and particle swarm optimization algorithm29-30. Moreover, Vasebi et al.31 investigated data reconciliation for mineral and metallurgical process by Kalman filtering. To keep the effectiveness of data reconciliation, it is required that the process measurements are with redundant information32. For example, a generalized data reconciliation method was applied to improve data redundancy in thermal technology with characteristic constraints including the steam flow capacity, the equation of the pressure drop, the turbine internal efficiency, and heat flow33. Guo et al.34 introduced characteristic constraints which contain ellipse law and isentropic efficiency equation for estimation of exhaust steam enthalpy and steam wetness fraction for steam turbines. Moreover, the presence of gross errors in process measurements can largely affect the results of data reconciliation if they are not sufficiently detected or eliminated. To avoid distorted adjustment, robust data reconciliation methods have attracted the attention of many researchers. Many robust methods like the quasi-weighted least squares estimator, correntropy estimators have been developed in the past decades35-37. Alternatively, some researches were carried out to estimate the uncertainty of measurement noises by covariance matrix analysis38. Moreover, measurement uncertainties can be evaluated via process measured variables or process


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constraint residual39-40. Narasimhan and Shah considered the influence of the correlation of measurement errors41. Vasebi et al.20 analyzed uncertainty sources and selected proper uncertainty model for steady-state data reconciliation. Typically, most data reconciliation models are formulated as a weighted least squares problem, in which the objective function is the sum of least squares of standard residuals for measured variables. Hence, only the sensor errors are taken into account for data reconciliation. Moreover, the measurement errors are assumed to be independent between different variables. The weight of each variable in the weighted least squares is just the reciprocal of its variance. In this way, the weight matrix takes an inverse form of diagonal covariance matrix in the weighted least squares objective function. As variable measurements are often collected from different locations of the process plants, variable errors are naturally correlated due to reasons like dynamic process behaviors, feedback control and shared environmental disturbances. Hence, the weight matrix is usually not diagonal. Though measurement errors caused by model errors or external disturbances are considered in some literatures, the correlation of the measurement errors is rarely discussed in the previous researches. In general, only correlation coefficient is considered to construct data reconciliation model with a full weight matrix20. However, in actual industrial processes, the measurement variables have strong nonlinear correlations due to the complex process physicochemical relationship and the extensively used feedback control. In this way, the variable errors are also nonlinearly correlated to each other. To deal with the problem of data reconciliation containing the influence of the nonlinear relationship between measured variables, mutual information entropy is introduced to describe the nonlinear variable correlations. Mutual information not only shows whether there is relationship between two random variables, but also



reflects the strength of their relationship. Especially, mutual information, as a powerful a measure of interdependence of variables, is largely applied in process modeling and monitoring.42-48 Thus, a novel data reconciliation strategy is proposed based on mutual information theory in this paper. In the proposed strategy, mutual information is utilized to analyze the nonlinear correlations of process measured variables. The mutual information matrix of variables is used as the full weight matrix to construct the objective function in the data reconciliation model. Therefore, variable correlations are considered to build a more accurate data reconciliation approach. At last, the novel data reconciliation model based on mutual information is applied to a numerical example and an industrial processes. The remainder of this paper is organized as follows. In Section 2, the basic data reconciliation method is described. Then, Section 3 gives detailed description about the mutual information theory and the proposed mutual information based data reconciliation strategy. A numerical simulation is provided to validate the effectiveness of the proposed method in Section 4. In Section 5, a real industrial application on the evaporation process of sodium aluminate solution is adopted to verify its reconciliation performance. Finally, conclusions are made in Section 6.

2. Fundamentals of data reconciliation method In industrial processes, process measurements often contain errors due to different kinds of reasons. Data reconciliation tries to adjust process measurements and parameters so that the conservation laws and process constraints can be fulfilled. The steady-state data reconciliation problem is generally formulated as an objective function represented by weight least squares with equation and inequation constraints, which can be expressed as


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% ) = min( X − X % )W(X − X % )T = min( X − X % ) ∑ −1 ( X − X % )T min f ( X, X

  n  = ∑   x1n − x%1n j =1    n

m

= ∑∑

(x

i j

xn2 − x%n2

− x% ij )

σ 12 0  0 σ 22 L xnm − x%nm   L L  0  0

-1

L 0  L 0  x1n − x%1n L L   L σ m2 

xn2 − x%n2

  T L xnm − x%nm   (1)   

2

σ i2

j =1 i =1

Subject to

(

)

% ,U = 0 H X x% ≤ x% ≤ x%Ui i = 1, 2,K , m i L

i

(2)

uLq ≤ u q ≤ uUq q = 1, 2,K , M − m

% = [x% 1 , x% 2 ,K, x% m ] represent the measured and the reconciled sample where X = [ x1 , x 2 ,K , x m ] and X

sets of measured variables, respectively; x i = [ x1i , x2i ,K , xni ]T and x% i = [ x%1i , x% 2i ,K , x%ni ]T represent the sets of measured values and reconciled values of the ith measured variable, respectively; xij and x% ij represent the jth sampled value and jth reconciled value of the ith measured variable,

respectively; σ i represents the standard deviation of the ith variable; ∑ represents the variable variance matrix; W is the diagonal weight matrix; H represents the set of steady state constraint equations; x%Li and uLq represent the lower bounds of the ith reconciled measured variable and the qth unmeasured variable, respectively; x%Ui and uUq are the corresponding upper bounds of the two variables, respectively; m is the number of measured variables; M-m is the number of unmeasured variables. Practically, the data reconciliation problem can be handled as an optimization problem. The aforementioned data reconciliation problems are mostly constructed based on the assumption that measurement errors follow normal distributions with a zero mean and a known standard deviation. In this way, it allows that the measured data can be reconciled and the unknown parameters can be estimated. The objective function defines the sum of the weighted



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squares for each measurement deviation. Eq. (2) represents the set of equation and inequation constraints. In most industrial processes, equation constraints are generally defined by mass and heat balance equations.

3. Data reconciliation strategy based on mutual information As can be seen, the objective function in the basic data reconciliation model is a weighted least squares of data errors for all measured variables. Moreover, the variable errors are assumed to be independent of each other. However, in actual industrial processes, the variable errors are interrelated since they are caused by not only sensor errors but also other factors. Thus, the correlations of measured variables should be taken into account in the data reconciliation modeling. In this part, a novel data reconciliation method is described in detail, in which mutual information is utilized to measure the complex nonlinear correlations of variables.

3.1. Mutual information In information theory, mutual information is a nonlinear method to measure the variable relevances. Mutual information indicates the degree of association between two random variables, it represents the amount of uncertainty reduction of information z due to introduction of information y, where z and y are two random variable. In general, entropy is a measurement of uncertainty of variable. Denote the sets of observed data for variable z and y as z = [z1 , z2 ,K , zn ] and y = [ y1 , y2 ,K , yn ] . The entropy information H (z) is defined as: n

H (z ) = −∑ p(zi ) log p( zi )

(3)

i =1

For z and y, the two-dimensional joint entropy is defined as: n

m

H (z, y ) = −∑∑ p (z i , y j ) log p( zi , y j ) i =1 j =1


(4)

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Moreover, the conditional entropy probability descripts the amount of uncertainty in one variables when the other one is introduced, which can be expressed as: n

m

H ( y z ) = −∑∑ p ( zi , y j ) log p( y j z i )

(5)

i =1 j =1

Additionally, the relationship between joint entropy and conditional entropy is H ( z, y ) = H ( z ) + H ( y z ) = H ( y ) + H ( z y )

(6)

Thus, mutual information shared in both variables is defined as: n

m

I (z; y ) = H (z) − H (z y ) = H (y) − H (y z ) = ∑∑ p(zi , y j ) log i =1 j =1

p( zi , y j ) p( zi ) p( y j )

(7)

where p ( z , y ) is the joint probability distribution function of z and y; p( z ) and p ( y ) are the marginal probability distribution functions of z and y, respectively.

3.2. Mutual information based data reconciliation modeling In actual industrial process, the sensor errors involved in measurement errors are often independent of each other. However, variable errors caused by model errors or external disturbances may be relevant. Thus, the data reconciliation model in the form of diagonal weight matrix cannot meet the accuracy requirement in industrial processes. Particularly, nonlinear correlation of measured variables should be considered in data reconciliation modeling. Here, mutual information is used to describe the nonlinear relationship of measured variables. Hence, a data reconciliation model is constructed based on mutual information to improve the accuracy of data reconciliation results. m n×m 1 2 Denote the sample dataset of measured variables as X = [x , x ,K, x ] ∈ R , where n and m

are the total number of samples and dimension of variables, respectively. In addition, T

xi =  x1i ,..., x ij ,..., xni  is all the sample data of the ith measured variable. The objective of data

reconciliation model based on mutual information can be expressed as follows



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% )W(X − X % )T min f = min( X − X x% , u

(8)

x% , u

where

 x%11  1 % =  x%2 X L  1  x%n

x%12 L x%1m   x%22 L x%2m  L L L  x%n2 L x%nm 

(9)

% represents the reconciled dataset of measured variables. W represents the mutual where X

information matrix which can be described as Eq. (10).  w11 w W =  21 L   wm1

w12 w22 L wm 2

L w1m  L w2 m  L L   L wmm 

(10)

where wrs represents the mutual information between the rth measured variable and the sth measured variable. Thus, the formulation of mutual information based data reconciliation model comes down to: % ) W(X − X % )T min f = min( X − X x% , u

  w11  w  = ∑   x1n − x%1n xn2 − x%n2 L xnm − x%nm   21 L j =1     wm1  % ,U = 0 s.t. H X n

(

w12 w22 L wm 2

w1m  L w2 m  1  x − x%1n L L  n  L wmm  L

)

xn2 − x%n2

   L xnm − x%nm      T

x% ≤ x% ≤ x%Ui i = 1, 2,K , m i L

i

uLq ≤ u q ≤ uUq q = 1, 2,K , M − m

(11) From the comparison of the proposed and the basic weighted least squares data reconciliation methods, the objective function of the basic weighted least squares data reconciliation model is the least squares of normalized residuals for measured variables. The form of the weight matrix is a diagonal one with the reciprocal of variable variance as its diagonal elements, and the measurement errors are assumed to be uncorrelated. While for the correlation coefficient based


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method,20 the model errors is considered, and the diagonal weight matrix is replaced by the linear correlation coefficient matrix of measured variables. However, in actual industrial process, the measurement errors are often nonlinearly correlated due to complex kinetic relationships, external disturbances and feedback control. In this case, data reconciliation model based on mutual information can properly describe the nonlinear relationship of measured variables. Thus, the uncertainties of nonlinear correlation are considered fully in the proposed data reconciliation strategy, which can improve the performance of data reconciliation.

4. Nonlinear numerical case In this section, a nonlinear numerical example is carried out to present the performance of the proposed data reconciliation method based on mutual information. Moreover, the proposed data reconciliation method based on a full weight matrix is compared with data reconciliation methods based on a diagonal weight matrix and a correlation coefficient matrix, respectively. To deal with the nonlinear optimization problem of the data reconciliation approaches, the state transition algorithm (STA) is used in this paper. STA is a global stochastic optimization algorithm with very efficient convergence performance. More details about STA can be found in references49-50. The nonlinear example is applied to demonstrate the effectiveness of the proposed data reconciliation method by Tjoa and Biegler51 and Alighardashi et al.52. This nonlinear system contains five measured variables, three unmeasured variables, and six nonlinear constraint equations. The setting of the nonlinear simulation can be described as Eq. (12). 0.5 x12 − 0.7 x2 + x3 u1 + x22 u1u2 + 2 x3u32 − 255.8 = 0   x1 − 2 x2 + 3x1 x3 − 2 x2 u1 − x2 u2 u3 + 111.2 = 0  x u − x + 3x + x u − x u − 33.57 = 0  3 1 1 2 1 2 3 3  2  x4 − x1 − x3 + u2 + 3u3 = 0 x − 2x u u = 0 3 2 3  5  2 x1 + x2 x3u1 + u2 − u3 − 126.6 = 0


(12)


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where x = [ x1 , x2 ,..., x5 ] is the measured variable vector; u = [u1 , u2 , u3 ] is the unmeasured variable

vector.

The

true

values

of

measured

variables

are

given

as

xtrue = [4.5124 5.5819 1.9260 1.4560 4.8545] , and the standard deviations of measured variables

are σ = [0.5 0.6 0.2 0.2 0.5] . 100 samples are generated for data reconciliation. Totally, three different methods are investigated to construct the data reconciliation models. Method 1 is the case that the covariance matrix in data reconciliation model is in the form of diagonal matrix, which is just the traditional weighted least squares method. For method 2, the correlation coefficients of measured variables are used in the covariance matrix for data reconciliation model. Method 3 is just the proposed approach, in which the mutual information matrix is adopted to measure variable correlations for data reconciliation. For different data reconciliation models, the comparison results of the reconciled data and measured data of the five measured variables are shown in Fig. 1. It can be seen that the fluctuation range of reconciled data with method 3 is narrower than that of the reconciled data with method 1 and method 2. The reconciled data based on method 3 fluctuates little around true values. Since there are nonlinear correlations between the measured variables, data reconciliation model based on mutual information can reflect the real changes of measured variables indirectly.


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a

6 4 2 0

10

20

30

40

50

60

70

80

90

100

60

70

80

90

100

60

70

80

90

100

60

70

80

90

100

60

70

80

90

100

Sample points

b

10 5 0 0

c

10

20

30

40

50

Sample points 3 2 1 0

d

10

20

30

40

50


e

10

20

30

40

50


10

20

30

40

50

Sample points Measured data

Reconciled data with method 1



Fig. 1. Comparisons of reconciled data and measured data for five variables

In addition, the relative error between the reconciled data, the measured data and the real data are calculated for different methods. The distribution of relative error is illustrated in Fig. 2. From Fig. 2, although the reconciled data is closer to the real data than the measured data, the reconciled data with method 3 is more accurate than that with method 1 and method 2. Furthermore, Fig. 3 shows the standard deviation of measured data and reconciled data of five variables. It shows that the standard deviation of the reconciled data obtained by method 3 is largely reduced compared with those of the measured data, the reconciled data obtained by method 1 and method 2. That is, the reconciled data obtained by method 3 fluctuates in a narrow range and is closer to the real data than other results. Based on these simulation results, it is concluded that method 3 can improve the accuracy of data reconciliation by taking nonlinear correlation between measured data into account.


Relative error Frequency Relative error

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Frequency



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e

2

1

0 -0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Relative error Relative error

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Frequency

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0.5

0

-0.5 0

10

20

30

40

50

60

70

80

90

100

Sample points Measured data Measured data

Reconciled data with method 1 Reconciled data with method 1



Fig. 2. The distribution of the relative error which relatives to the real data 0.7 Measured data Reconciled data with method 1 Reconciled data with method 2 Reconciled data with method 3

0.6

0.5

0.4

0.3

0.2

0.1

0 1

2

3

4

5

Measured variables

Fig. 3. The standard deviation of measured data and reconciled data for five variables with different methods.

Besides, to evaluate the performance of the proposed data reconciliation method based on mutual information, two assessment indices are calculated, which are the mean relative error (MRE) and the root mean square error (RMSE). The MRE is the average relative error between the reconciled or measured values and the true values. The RMSE is the arithmetical square root of the mean value of the square of the difference between the reconciled or measured value and the true value. The reconciliation results for 5 variables with the three methods are illustrated in Table 1. From Table 1, it is seen that the accuracy of data reconciliation model based on mutual information is higher than those of data reconciliation models with diagonal matrix and correlation coefficient matrix. Taking variable 1 as an example, the MRE of the reconciled data with method 3 are reduced by 16.93%, 33.3% and 51.25% compared with those with method 2, method 1 and the measured data, respectively. Moreover, the RMSE of the reconciled data with method 3 are



reduced by 14.88%, 31.48% and 54.23% compared with those with method 2, method 1 and the measured data, respectively. From this numerical example, the mutual information based data reconciliation method can improve the accuracy of measured data effectively. Table 1. Assessment indexes for the data reconciliation method. Measured variables

Variable 1

Variable 2

Variable 3

Variable 4

Variable 5

MRE

RMSE

Measured data

0.086

0.485


0.063

0.324


0.050

0.261


0.042

0.222

Measured data

0.095

0.658


0.040

0.274


0.034

0.222


0.028

0.182

Measured data

0.077

0.185


0.058

0.174


0.041

0.115


0.031

0.082

Measured data

0.171

0.315


0.139

0.230


0.105

0.181


0.067

0.119

Measured data

0.079

0.495


0.062

0.344


0.050

0.287


0.042

0.244

5. Industrial application In this section, the proposed data reconciliation method based on mutual information is applied to a real evaporation process of sodium aluminate solution. Firstly, Section 5.1 gives a description of the evaporation process of sodium aluminate solution. Then, the data reconciliation model based on mutual information is constructed for the evaporation process in Section 5.2. Finally, the results are discussed in Section 5.3.

5.1. Process description The evaporation process of sodium aluminate solution is the major process in Bayer alumina


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production. The aim of the evaporation process is to evaporate an excess amount of water which comes from red mud washing and heating steam condensate, so that the concentrated sodium aluminate solution can meet the production requirement of the digestion process. A flow chart of the sodium solution evaporation process is shown in Fig. 4. There are six evaporators, four flash evaporators, five pre-heaters and parts of condensed water tanks. The sodium aluminate solution is concentrated by using high-temperature and high-pressure steam through a series of reactors. The liquid material is fed into the sixth and the fifth evaporators simultaneously, and flows backward to other four evaporators successively. Then, the liquid material further condensed by the fourth flash evaporator is sent to the next plant. The steam is supplied as heat source in the whole process. In generally, the evaporation process is a typical chemical process containing a lot of units and equipments, and there are many nonlinear measured variables for process modeling, optimization, control and monitoring.

Fig. 4. Flow chart of sodium aluminate solution evaporation in the alumina production process.

The evaporation process of sodium aluminate solution contains many process variables, such as the flow rate of the live steam, the flow rate of the feed, the temperature of the live steam and the temperature of the liquid material, are measured. In addition, other variables like the flow rate



of the liquid material and the flow rate of the secondary steam are unmeasured due to the limited analyzers and harsh measuring conditions. Detailed descriptions of measured and unmeasured variables are shown in Tables 2 and 3. There are totally 30 constraint equations and 103 variables, which include 34 measured variables and 69 unmeasured variables in the evaporation process. In fact, the number of unmeasured variables is larger than the constraint equations, resulting in the problem of data insufficient redundancy.53 Therefore, it is necessary to build the layered data reconciliation models based on mutual information to improve the accuracy of measurements. Table 2. Detailed description of measured variables. Measured variables

F01 , F02 , F4s , V0 C 0 , C 4s ′ , Tms , T0 , TV 0 , Tn , TVn , Twn s (n=1,2,…,6, m=1,2,3,4) TVm

Description Flow rate of feed of the sixth evaporator and the fifth evaporator, flow rate of outlet liquid material of the fourth flash evaporator, flow rate of live steam. Concentration of feed and concentration of outlet liquid material of the fourth flash evaporator. Temperature of feed and live steam, temperature of outlet liquid material, outlet secondary steam and outlet condensed water of the 1st to 6th evaporators, temperature of outlet liquid material of the 1st to 4th flash evaporators.

Table 3. Detailed description of unmeasured variables. Unmeasured variables

Description

Fn , C n , ρ n , cpn , Vn , H n

Flow rate, concentration, density, specific heat capacity of outlet liquid

(n=1,2,…,6)

material, flow rate and thermal enthalpy of secondary steam of the 1st to 6th evaporators. Flow rate, density, specific heat capacity of outlet liquid material, flow

Fms , C ks , ρ ms , cpms , V ms , H ms

rate and thermal enthalpy of secondary steam of the 1st to 4th flash

(m=1,2,3,4, k=1,2,3)

evaporators. Concentration of outlet liquid material of the 1st to 3rd flash evaporators.

Ql (l=1,2,…,10)

Heat loss of the 1st to 6th evaporators and the 1st to 4th flash evaporators.

5.2. Layered data reconciliation model based on mutual information In the evaporation process, a layered data reconciliation model based on mutual information is built due to insufficient redundancy of measured data. First, the data reconciliation model based on mutual information is built for mass balance layer and the measured variables with regard to mass balance are reconciled, like the flow rate of the feed and the outlet liquid material, the


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concentration of the feed and the outlet liquid material. Then, the reconciled results are used to reconcile measured variables in heat balance layer, like the temperature of the liquid material, the temperature of the steam and the flow rate of the live steam. In addition, the following assumptions are invoked to keep mechanism model simple yet effective. (1) The distribution of the steam and the liquid material in the devices are balanced. (2) The temperature, the pressure of the steam and the liquid material are uniformly distributed in the evaporator heating tube. (3) The effect of scale formation can be neglected. Hence, according to the mass balance and the heat balance of the evaporation process, the layered data reconciliation models based on mutual information are defined as follows. Mass balance layer: n

{

min f1 = ∑  x1n − x%1n j =1

  n   = ∑   F01, n − F%01, n j =1     F01, n − F%01, n s.t.

(

xn2 − x%n2 L xnl − x%nl  W  x1n − x%1n

F02, n − F%02, n

F02, n − F%02, n

C0, n − C% 0, n

C0, n − C% 0, n

C4,s n − C% 4,s n

C4,s n − C% 4,s n

xn2 − x%n2 L xnl − x%nl 

 w11 w  21 F4s, n − F%4,s n   w31   w41  w51 F4,s n − F%4,s n 

T

T

}

w12

w13

w14

w22

w23

w24

w32

w33

w34

w42

w43

w44

w52

w53

w54

} + λ r(F ) + λ r(C ) 1

w15  w25  w35   w45  w55 

2

)

% ,U = 0 H mass X

x% ≤ x% ≤ x%Ui i = 1, 2,K , l i L

i

u ≤u ≤u q L

q

q U

(13)

q = 1, 2,K , L

where f1 is the objective function for data reconciliation model with mass balance; l is the number of measured variables included in mass balance; n is the number of samples; x ij and x% ij represent the jth sampled data and jth reconciled data for the ith measured variable, respectively; r ( F ) n

9

n

9

represents ∑∑ Fji − F% ji , and r (C ) represents ∑∑ C ij − C% ij ; λ is a weight that is determined by j =1 i =1

2

2

j =1 i =1

the empirical rule; H mass represents the set of constraint equations for mass balance layer; L is the



Page 20 of 29

number of unmeasured variables for mass balance layer. Heat balance layer: n

{

xn2 − x%n2 L xnm −l − x%nm − l  W  x1n − x%1n

L eTi ,n

eTV 1,n

min f 2 = ∑  x1n − x%1n j =1

n

{

= ∑  eT1,n j =1  w1,1 w  2,1 L   w29,1  eT1,n s.t.

L eTVi ,n

xn2 − x%n2 L xnm −l − x%nm −l 

eTw′1,n

L eTwm′ ,n

eV0 ,n

eT0,n

eTV 0 ,n 

eTw′1,n

L eTwm′ ,n

eV0,n

eT0 ,n

eTV 0,n 

w1,2 L w1,29  w2,2 L w2,29  L L L   w29,2 L w29,29  L eTi ,n

(

eTV 1,n

L eTVi ,n

T

T

}

}

)

% ,U = 0 H heat X x% ≤ x% ≤ x%Ui i = 1, 2,K , m − l i L

i

u ≤u ≤u q L

q

q U

(14)

q = 1, 2,K , M − L − m

where f2 is the objective function for data reconciliation model with heat balance; e represents measurement error; m-l is the number of measured variables included in heat balance; H heat represents the set of constraint equations for heat balance layer; M − L − m is the number of unmeasured variables for heat balance layer.

5.3. Results and analysis To evaluate the proposed data reconciliation method based on mutual information, 100 samples are collected from actual evaporation process (the sampling frequency is 2 min per sample) to construct the three data reconciliation models. For the flow rate of the feed of the sixth evaporator and the fifth evaporator, the temperature of the secondary steam of the sixth evaporator, the temperature, concentration and flow rate of the outlet liquid material of the fourth flash evaporator, the temperature and flow rate of the live steam, Fig. 5 shows their standard deviations of measured data and reconciled data with different data reconciliation models. From Fig. 5, the standard deviation of reconciled data with method 3 is less than that of method 1 and method 2. In


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detail, for the flow rate of the feed of the sixth evaporator, the standard deviation of reconciled data calculated by the method 3, is reduced by 5.23%, 5.57% and 3.44% when compared with those by method 2, method 1 and measured data, respectively. Similar conclusions can be obtained for other measured variables. This is because the proposed method takes the nonlinear variable correlations into data reconciliation modeling. In addition, Fig. 6 provides the relative standard deviations of reconciled data for different method, which is defined in Eq. (15).

relative standard deviation =

where

σ reconciled − σ measured σ measured

(15)

σ reconciled is the standard deviation of the reconciled data; σ measured is the standard

deviation of the measured data. Obviously, the relative standard deviations with method 3 are mostly less than zero for different variables, which implies the reconciled standard deviations are smaller than the measured standard deviations. However, the relative standard deviations obtained by the method 1 and method 2 are sometimes greater than zero, which means that the standard deviation of the reconciled data of some variables is larger than the standard deviation of the measured data. This indicates that reconciled data with the proposed method (method 3) is closer to real data than the other two methods. Furthermore, Fig. 7-9 also provide the reconciled data and the measured data for the flow rate of the feed of the sixth evaporator, the flow rate of the feed of the fifth evaporator and the flow rate of the outlet liquid material of the fourth flash evaporator with the three different methods. The fluctuation range of reconciled data with method 3 is narrower than that with method 1 and method 2, which implies that the reconciled data with method 3 is more accurate than the other two methods.



Fig. 5. Comparisons of standard deviations of measured data and reconciled data for different data reconciliation

models.

Fig. 6. The relative standard deviations of reconciled data with different methods 310 Measured data Reconciled data(method 1) Reconciled data(method 2) Reconciled data(method 3)

300

290

280

270

260

250 10

20

30

40

50

60

70

80

90

100

Sample points

Fig. 7. Measured data and reconciled data for the flow rate of the feed of the sixth evaporator.


Page 22 of 29

the fifth evaporator(m 3/h) the fourth flash evaporator(m 3/h)

Fig. 8. Measured data and reconciled data for the flow rate of the feed of the fifth evaporator.

The flow rate of the outlet liquid material of

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


The flow rate of the feed of

Page 23 of 29

Fig. 9. Measured data and reconciled data for the flow rate of the outlet liquid material of the fourth flash

evaporator

6. Conclusions In this paper, to reduce the influence of the contaminated process measurements and consider the nonlinear correlation of measured variables, a novel data reconciliation method based on mutual information is proposed for calibration of process measurements and estimation of unmeasured variables. In the proposed method, mutual information is first utilized to measure the nonlinear variable correlations. Then, a new reconciliation objective function is designed with a full weight matrix, whose elements are the mutual information between different variables. A nonlinear numerical example and an industrial application are performed to validate the effectiveness of the proposed data reconciliation method. The accurate process measurements with



the new data reconciliation method is beneficial for more effective real-time process modeling, optimization and control.

Acknowledgements This study is financially supported in part by the National Natural Science Foundation of China (61703440, 61621062), in part by the innovation-driven plan in Central South University (2016zzts054, 2018CX011), in part by the Natural Science Foundation of Hunan Province of China (2018JJ3687), and in part by Fundamental Research Funds for the Central Universities (222201717006).

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A new data reconciliation strategy based on mutual information for

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