•..;..ht 0 IFAC Artificiallntclligcnce in Real·Time Control, Copy&__ . Kuala Lumpur, Malayma, 1997
A HYBRID MODEL FOR CHEMICAL PROCESS MODELING
Low Soon Tiong and Arshad Ahmad Department ojChemical Engineering, Universiti Teknologi Malaysia, Locked Bag 791, 80990 Johor Bahru Negeri Johor Darul Ta 'zim, Malaysia
[email protected]
the development of first principle models (FP M) using fundamental laws of science. Since it is founded on actual physical and chemical laws, the outcome is a robust model with the ability to represent the process in wide operating conditions as prescribed during the derivation of the models. However, the development of an accurate FPM requires a complete understanding of the process. Since most chemical processes are complicated, the development of such models are normally time consuming and costly. Simplifying assumptions are often inevitable due to mathematical limitations as well as lack of understanding on some physical phenomena.
ABSTRACT This paper proposes a hybrid approach to chemical process modeling. The scheme combines empirical and theoretical models in various different configurations. Performance evaluations were carried out on two case studies, i.e., a laboratory scaled liquid level apparatus and a simulation of the Tennessee Eastman Plant. Results obtained from these studies have shown that hybrid models are more superior compared to the individual theoretical or black box models. Among all structures considered, the structure with FPM preceding SIM in series was most promising. These investigations have proved the potentials of hybrid strategy in providing solution to complex modeling problem. COpyright © 1998lFAC
1
On the contrary, data based mode ling does not necessitates complete understanding of the process. What is required is sufficiently rich data consisting of the inputoutput relationships between variables of interest to fit a selected model structure. System identification models (SIM) such as the Autoregressive with exogeneous input (ARX) model fall into this category. If the data is sufficiently rich and the structure is adequate, the model can be an accurate representation of the process. However, the domain of validity of the SIMs is limited to the domain of operation as defmed by the data. If the process is non linear, any extrapolations beyond the calibration data may produce unwanted results.
Introduction
In recent years, dynamic process models have found widespread applications in process engineering. Among others, process models have been applied for process monitoring, estimation, control, and fault diagnosis. It has been noted that the success of these application are highly dependent on the quality of developed process model. For example, in the application of modelbased control strategy, the development of a sufficiently accurate dynam ic model of the process is a prerequisite for success. If the model is accurate, the dynamics of the process can be predicted, thus enabling the determination of effective control action. In fact, since controller performance is directly dependent on how well the model describes the salient process characteristics, without an adequate model of the plant to be controlled, the synthesis of a control algorithm is not possible.
In this paper, a hybrid approach consisting of both the FPM and SIM is proposed. It is expected that when these mode ling approaches are incorporated within a single framework, a better process model will be produced. Furthermore, by formulating suitable structure, hybrid models can be designed with flexibility to allow updating of critical process parameters to be carried out either online or offline. This will accommodate any changes in operating condition.
In general, process models can be developed using either knowledgebased or databased methodologies. Knowledgebased approach refers to
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To evaluate the model performance, two systems are considered. The first is a laboratory scaled liquid level system, installed at the Department of Chemical Engineering, Universiti Teknologi Malaysia. This set up is a simple tank with a computeraided data acquisition facilities . All the proposed modeling will be evaluated on this apparatus. The second case study involves a published versi01l of the Tennessee Eastman Plant [I] .
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2
Modeling Strategies
2.1
First Principle Modeling
Majority of the process in chemical industries can be approximated by lumped parameter models. This indicates that the system can be modeled by sets of firstorder ordinary differentialalgebraicequations. In addition, deadtime on process variables is included. Mathematically, the representation ofa typical FPM is as follows: Dynamic modeling equation,
dx
;;=f(x,u(tO),p,l) ThermodYnamic relationships equilibrium equation,
(1)
such
as
To develop the ARX model, data collected from the laboratory experiment are used to identify unknown parameters through the optimization of selected cost functions. The purpose of this optimization procedure is to reduce the error between the predicted and the actual values of the output. A typical cost function for this optimization is the sum of squared error:
E (2)
(3)
where x == state variables, u == manipulated variables, p parameters, I == load disturbances (measured or unmeasured), y == output or measured variables, 8 = deadtime between manipulated and state variables, and et> = deadtime between state and measured variables. This type of formulation may also be used for distributed parameter systems that have been "lumped" by discretizing the partial differential g~uations and forming o~dinary differential equations . The above equatlOns are normally solved numerically.
=:
2.3
SYstem Identification Modeling
B(Z)U(/)Zd
~~pnd y.,,)'
(8)

Hybrid Modeling
A few hybrid paradigms combining the FPM and ARX are proposed. Schematic diagrams of these strategies are given in the following figures. In figure I, ARX model is placed in series with FPM. In this case, ARX transforms the input, u(t}, into intermediate variable, x(t}. The intermediate X(I} then serves as the input to the FPM. In order to develop the ARX, x(t} must be determined by inverting the FPM. The ARX parameters can then be estimated by mapping input u(t) and intermediate input X(I}.
The autoregressive with exogenous inputs or simply the ARX model is a class of linear system identification model that is commonly used in process modeling and control application. This model can be represented by the following equation : A(Z)Y(/) =
=
Here, E is the sum of squared error (SS E), ypnJ is the predicted value of the output obtained from the model, YDCt is the actual output and N is the number of data used in the identification procedure. Due to the fact that the SSE is optimized, the procedure is often called least square. ARX model was identified using a built in tools provided by MATLAB software.
Stateoutput relationship,
2.2
(7)
=b1u(tI)+ .. .bnbu(tn) Here, no and nb are the number of output and the input sequences respectively. In practice, the delay term d is expressed as mUltiple of the sampling interval. These parameters can be determined using suitable parameter estimation techniques.
algebraic
gl (x,u,p ) = 0
and
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FPM~
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Figure I : Hybrid Structure A (HA)
where d is the time delay. A and B are the polynomials as shown below :
Another series arrangement is shown in Figure 2. In this case, the FPM is placed preceding
164
the ARX model. As such, the identification of an ARX model between x(t} and y(t} is relatively easier compared to the previous scheme. This is becau:e the intermediate output x(t} that can be conventently generated from the FPM serves as the input to ARX model.
volumetric now rate of the inlet and outlet streams respectively q""
1
R
qUIll
Figure 2 : Hybrid Structure B (HB) Figure 5. A liquid level storage tank system diagram Figure 3 shows another hybrid structure. In this case, the outputs of the ARX and FPM are combined to determine the total model output. The FPM serves as the main model while the ARX model is used to accommodate any discrepancy between the FPM and the actual process. As such, the ARX model is identified on the residual between the actual plant data and the FPM.
Choosing the level h as the system output and feed flow rate q as the input, the mathematical representation of a cylindrical tank system is as follows : A
FPM
+ ARX
(9)
dh =..!..(qin _~) (10) dt A R The term R, is a measure of valve resistance. Equation (10) serves as the FPM for the purpose of this study. The valve resistance R is determined experimentally. The ARX model is fitted using data collected from the experiment performed in the laboratory.
Figure 3: Hybrid Structure C (HC) Finally, the hybrid structure shown in Figure 4 utilizes ARX model as parametric estimator to the FPM. In this case, the ARX model was used to estimates physical properties in the model such as the valve resistant, R for the case of liquid level system.
3.1
Results and Discussion
As expected, the model structure influenced the quality of model prediction. For the purpose of this study, model having the smallest value of Sum of Squared Error (SSE) was considered most accurate. Two sets of data, each consisting of 320 pairs of qill and h were generated. Due to the presence of measurement noise, the experimental data were filtered using a first order exponential filter. For each type of hybrid structure, the parameters of ARX was identified by using one set of data. Prediction accuracy of all models were tested using the other set of data. The results are summarized in Table I below.
ARX
R FPM Figure 4: Hybrid structure D (HD)
3
qout
Here, A is the cross sectional area of the tank. This equation is valid for a constant density system. In this experimental set up, the outlet flow rate is flXed by a Jllanual valve which was set open at a constant value. Thus, knowing the valve characteristics, the term qOl" in equation (9) can be expressed in term of the height h. Assuming that the valve is linearly related to the liquid level, the following relationship is obtained:
y(t)
u~
dh ;;;= qin 
Case Study 1 : Liquid Level System
Table I: SSE for Liquid level System.
The schematic diagram of the liquid storage tank is shown in Figure 5. Here, qin and qool are the
165
dh
=
Results tabulated in Table I suggest that the HB and HC were the best (see also Figure 7). Theoretically, both perfonnance HA and HB should be same, if the system is linear. However, in this case some degree of non linearity exist in the characteristic of the valve. Furthern10re, modeling errors are encountered in the computation of the intermediate variables.
dl
DATA steadystate h hi.)
h'" R*0.15 R*0.3
Y 143.0 96.9 94.3
4
Based on the result displayed in Table 2 we can conclude that although inaccurate ARX model was used, the performance of the hybrid structure HB and HC were still acceptable. Both models performed better than the individual ARX model. This is due to the contribution of the FPM built in within HA and HB. If this approach is taken, the training stage of the ARX model in the hybrid structure could be owing to the reduced amount of data used in fitting the model.
Dynamic information from equation (10) was ignored by letting dh/dt = O. Thus, the output of liquid level is just a function of Rand qin as shown in equation (12).
h •
=
A
R
HB 97 .96 107.90 120.88 97.18 95.23
He 98.58 94.53 101.32 93 .77 88.05
Case study 11: Tennessee Eastman Plant
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The Tennessee Eastrnan (TE) process has five major unit operations : a reactor, a product condenser, a vaporliquid separator, a recycle compressor and a product stripper. The flowsheet of the TE plant provided by Downs and Vogel [I] is reproduced here as Figure 6. In this plant, gaseous feedstocks react to form liquid products. The gas phase reactions are catalyzed by a nonvolatile catalyst dissolved in the liquid phase. The reactor has an internal cooling bundle for removing the heat of reaction. The products leave the reactor as vapors along with the unreacted feeds while the catalyst remains in the reactor. In this process, two products were generated from four reactants. Also present are an inert material and a byproduct making a total of eight components : A, B, e, D, E, F, G, and H. The reactions are as follows:
The second investigation involved inaccuracies of the FPM. In this case, sufficient data was provided to the ARX and errors were introduced to the FPM by modifying the term h in equation (l0) according to the following schemes: •
h)
03· R
Table 3 : SSE for models with inaccurate FPM
Table 2: SSE for Situation I. X* 9.2 9.5 9.1
A
The results are tabulated in Table:; below . The first row of Table 3 displays the performance of the steady state model of the plant. Other rows represent the prediction of the models under modeling errors involving the dynamic FPM. In all cases, both HB and HC have again produced acceptable performance. Significant improvement in prediction accuracy when compared to the original FPM. Comparing between the two hybrid structure, in this case, HC is found to be better than HB .
To further investigate the performance of hybrid model, HB and HC was developed under two adverse situation: I . Inaccuracies are imposed on the ARX by developing models using 50 data pairs collected from restricted operating region (data X*). 2. Inaccuracies are imposed on the FPM. The results of both situation I and 2 are shown in Table 2 and 3, respectively.
Data ARX HB He
~. qtn
3D (sI
~ ~
~
G(liq), Product I,
( 15)
H(liq), Product 2,
( 16)
F(liql
• 2F(liql
Byproduct,
(17)
Byproduct, (18)
(13)
The reactions are all irreversible and exothermic. As in nornlal cases, the temperature dependence of the reaction rates follow the Arrhenius expression. It is important to indicate that the
Linear discrepancy was introduced by reduces R and substitute by 0. 15* Rand 0 3· R,
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Tennessee Eastlllan process is openloop unstable . As such, proponionalintegral (PI) feedback controllers proposed by McA voy and Ye [3] are used to maintain process constraints within their prescribed limits. For the purpose of this study, the modeling equations described by Downs and Vogel [I] were simulated to represent the actual plant. For this case study, FPM is represented by the steady state model where the dynamics of the process is not provided . Energy balance equations were also ignored. The measurement of flowrate, pressure and temperature were utilized as the input to the steady state FPM.
4.1
Results and Discussion
Four sets of data were generated at different operating regions and conditions. For each case, perturbations were imposed to the selected process inputs by subjecting the variables to a random noise in the viscinity of the steady state condition. These include the Downs and Vogel base case, and "optimal" operation at three different GIH production ratios: 50/50, 10/90, and 90110 (mass basis). The base case data was used to identify parameters in ARX models. Model order and delay were determined using Akaike Criterion. The followings are the prediction results for the different type of model structures. The SSE tabulated in the squared error in predicting one of the product composition (i.e., product G). Table 4: SSE for TE Plant
DATA
FPM
ARX
HB
HC
Base case Mode I (50/50) Mode 2 (10/90) Mode 3 (90110)
159.34
3.2ge4
8.51e5
0.0135
130.53
0.0384
0.0144
1.0652
171.13
0.8187
0.1349
30.67
325 .16
1.7031
0.3124
40.74
.
.
The results displayed in Table 4 indicate that improvement can be provided by hybrid modehng strategy (see also Figure 8). The performance of a steady state FPM is dependent of the na.ture of the process dynamics. This is confirmed in thiS study. The slowest mode (mode 3) exhibited the poorest performance of the FPM . The ARX model perform best in the base case since the model itself was generated from data that is ::.oenerated from that . operatmg region. The structure HB was found most accurate.
slgnlfi~ant
5
Discussion and Conclusion
The .res.ults of this investigation displayed the supeno~lty of the hybrid lllodeling strategy Our evaluations based on two case studies also indicated that both HB and HC have the potentials to be exploited for future modeling endeavor. Preliminary findings indicated that HB was the best In this study, it has also been proved that . despite the inaccuracies in either the FPM or the ARX, the performance of the resulting hybrid model is still adequate . This shows the strong complementary effect of the other component of the hybrid model. FPM extrapolate better outside to the domain of the data used to identify an ARX model. Due to this fact the hybrid models performed better than the ARX a; displayed in T~ble 2. Similarly, owing to the ability of the A.RX m c~pt~ring information from process data, the maccuracles 111 the FPM as shown in Table 3 was improved by the ARX . Similar trends were also observed in the simulation studies involving the Tennessee Eastman Plant. . The results also exposed the ability of hybrid modelmg strategy in providing solution to modeling problem for process plant. [n most cases, steady state plant models are available since the design stage of the plant. To provide dynamic information as well as to ~lim~ate . errors due process changes, system IdentificatIOn models can be employed usin o the hybrid paradigm as proposed in this study. F~rther evaluattons especially in utilizing nonlinear S[Ms IS needed to further strengthened the findings. As a conclusion, the hybrid structure propose.d in this study is proven promising as an alternative scheme for complex modeling problem.
References [1] D?wns J. J. and Vogel E. F. (1993), A PlantWide Industrial Process Control Problem Computers chem Engng, 17, No . 3, pp 245255.' [2] Bequette B. W . (1991), Nonlinear Control of Chemical Processes : A Review. Ind. Eng. Chem . Res.,30, No. 7,pp. 13911413.
(3] McAvoy T. 1. and Nan Ye (1994), Base Control for The Tennessee Eastman Problem , Computers chem. Engng, 18, pp 383413 .
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