Maps of Process Dynamics

The visualization of the process data is an important factor in understanding process behavior. In this paper, both static and dynamic process information is mapped in a self-organizing map. This method is experimented in the modeling of the simulated Tennessee Eastman benchmark process. Introduction When the processes to be controlled are becoming more and more complex, understanding the process behavior is getting exceedingly important. When there are dozens of measurement variables, it is difficult to see their significance. Usually, many of the variables are mutually dependent and highly redundant, and the number of them can be reduced. There are mathematical tools for achieving data compression. However, the mathematically motivated models with their computationally derived parameters do not help in understanding the process structure. An intuitively appealing approach to representing the dependencies between the process variables in a visual form is by using the self-organizing map (SOM) as presented in [Kohonen, 1995]. There exist various reports on how process state information can be mapped in SOM. Typically, only the static variables are mapped, or the measurement vector is directly used as input to the self-organization algorithm. This way, the operating regimes can be visualized, but no intuition on the process behavior within these regimes is obtained. From the control point of view, the dynamics of the process is a much more important factor than the static variable values themselves: the dynamics may remain invariant for a wide range of process variables, but for some special variable combinations, the dynamics changes abruptly. It is these regions of special dynamic phenomena that should be modeled in more detail. In this paper, the dynamic nature of the processes is modeled in SOM. The adaptation of the map is based not only on the process measurements directly, but also on the identified process parameters. It is assumed that within each operating region the overall process model can be linearized, so that the process dynamics can be represented by a set of ARX model parameters. Using the identified parameters as input to the SOM algorithm results in a ‘mode map’. The obtained model library could be applied for adaptive control of the process. As an application, modeling of the Tennessee Eastman benchmark problem [Downs and Vogel, 1993] is discussed. This process is highly nonlinear, and its dynamic behavior in different operating points varies considerably. The experiments are based on a process simulator in the Matlab/Simulink environment. The Tennessee Eastman process The Tennessee Eastman (TE) process is a highly nonlinear, non-minimum phased, and open-loop unstable chemical process consisting of a reactor/separator/recycle arrangement. This process produces two products G and H from four reactants A, C, D and E. Also an inert B and byproduct F are present in the process. The simultaneous, irreversible and exothermic gas-liquid reactions are: A(g) + C(g) + D(g) → G(liq), Product 1, A(g) + C(g) + E(g) → H(liq), Product 2, A(g) + E(g) → F(liq), Byproduct, 3D(g) → 2F(liq), Byproduct. The process has 12 valves available for manipulation and 41 measurements available for monitoring or control. The detailed description of these variables, process disturbances and base case operating conditions, is given in [Downs and Vogel, 1993]. The process flowsheet is presented in Figure 1. Figure 1. The Tennessee Eastman process flowsheet The Tennessee Eastman process has been the subject of several studies [see for example Banerjee and Arkun, 1995, McAvoy and Ye, 1994, Ricker and Lee, 1995a and 1995b], but most of them have concentrated to the control of the process and the modeling of the process using input-output data has been mostly unexplored. The research work of [Sriniwas and Arkun, 1997] has made progress in this area by using input-output data of the process for the identification of an empirical model, further used to construct a Model Predictive Controller to control the process. In the study of [Banerjee and Arkun, 1995] the process control scheme was divided into two tiers. The first tier controlled the critical variables that affected the reactor stability, and the second tier controlled the variables, which primarily affected the compositions of streams entering and leaving the reactor. In the work of [Sriniwas and Arkun, 1997] the first tier with its PID loops was used to maintain the process stability while four manipulated variables of the second tier (the set points for the controllers of reactor pressure, reactor level, D feed flow and E feed flow) were excited with input sequences and the sampled outputs of four controlled variables (reactor pressure, reactor level, product flow rate, and mass ratio of G and H in the product) were collected for the process identification. The same procedure is used in this work. Identification of the process In the work of [Sriniwas and Arkun, 1997] one general model was used as a reference model over the whole operating region of the TE process. However, because the process is highly nonlinear, this one single model may be too a crude approximation of the process dynamics in different operating regimes.The purpose of this current work is to identify models for the different operating points of the process to reach a set of more appropriate models for it. The same type of approach was used in the earlier work of the authors [Hyötyniemi and Ylöstalo, 1997]. In Fig. 2 typical responses for the step changes in the pressure are illustrated in different operating points. It seems that an oscillatory behavior emerges when the pressure rises. The input-output data for the process identification was obtained by making step changes to the four different input variables in different operating points. These input variables were the reference values of the reactor pressure (u1), the reactor level (u2), the D-feed (u3) and the E-feed (u4) controllers. The simulation step size was 1 second and the data was collected once in a minute. The structures of the four process models were chosen according to the work done by [Sriniwas and Arkun, 1997]. The models for the reactor pressure and the reactor level were SISO models and for the production rate and the product G/H ratio MISO models were used. In fact, as can be seen in Fig.2 (at time 111.5 h) there is an interaction between the pressure and some other process variable: it turns out that the setpoint of the reactor level has been changed at that point. Nevertheless, the simplified model structures of [Sriniwas and Arkun, 1997] were thought to be sufficient. All the models were of ARX type with delay time 1 minute. The following models were used in the identification: Figure 2. a) Signals u1 and y1 when the operating point was p0 = 2650 kPa, and b) u1 and y1 when p0 = 2800 kPa y1(t) = -a11 y1(t-1) a12 y1(t-2) + b11 u1(t-1) + b12 u1(t-2) + b13 u1(t-3) y2(t) = -a21 y2(t-1) a22 y2(t-2) + b21 u2(t-1) + b22 u2(t-2) + b23 u2(t-3) y3(t) = -a31 y3(t-1) a32 y3(t-2) + k= ∑ 1 3 b31k u1(t-k) + k= ∑ 1 3 b32k u2(t-k) + k= ∑ 1 3 b33k u3(t-k) + k= ∑ 1 3 b34k u4(t-k) y4(t) = -a41 y4(t-1) a42 y4(t-2) + k= ∑ 1 5 b41k u1(t-k) + k= ∑ 1 5 b42k u2(t-k) + k= ∑ 1 5 b43k u3(t-k) + k= ∑ 1 5 b44k u4(t-k) where y1(t) is the reactor pressure, y2(t) is the reactor level, y3(t) is the product flow rate and y4(t) is the mass ratio of G/H in the product. The sampling interval was 1 minute. The parameters a and b for the four models were identified in different operating points by using a standard least squares algorithm. Constructing the maps The self-organizing maps were used for the classification and organizing of the data, which consisted of the parameters of the four process models in the different operating points, and the numerical values describing the corresponding operating points, e.g. the reactor pressure, the reactor level, the D-feed into the reactor, and the E-feed into the reactor. The input vectors for the self-organizing maps included 50 parameters altogether, with four operating point values and 46 model parameters. All parameters in the input vectors were first scaled so that the standard deviations of the 50 different parameters were equal. To visualize the difference between the 'mode maps' and the traditional maps where the process measurements are directly modeled, three different maps were constructed: in the first map, the model parameters and the operating point values were weighted equally, while in the other maps, either the model parameters or the process measurements alone were used. The parameters of the input vector for the second map were weighted so that the weight of the 46 model parameters become tenfold and the parameters of the input vector for the third map were weighted so that the weight of the four operating point parameters become tenfold. The first map is a mixture of a mode map and a traditional map, whereas the other maps are morelike examples of the two alternative basic approaches. Two-dimensional maps with three by three neurons were used for modeling the process. The first map (Fig. 3) was initialized with random weight values. Figure 3. The first map, using the operating point values as well as model parameters as training data. The map is visualized by projecting it so that the corresponding reactor pressure and level are shown The second and the third map (Figs. 4 and 5) were initialized with the weight matrix obtained as a result from the training of the first map. The maps were trained for 5000 cycles. The long training time was needed for the map to converge. All the calculations were done in Matlab using the Neural Network Toolbox [Matlab, 1994]. All the maps cover best the area near the base case values, since that is the optimal operating regime of the process and most of the data were concentrated there. Even if the set of operating points that were experimented and identified consisted of a rather narrow band of values, so that the dynamics did not vary remarkably, it seems that the maps clearly differ from each other. Using the model parameters in the map construction seems to be a good choice. The mode map extends wider over the variation range than the other two maps. This results from the fact that the mode map emphasizes the differences in the process dynamics. For example, when the pressure rises, the process dynamics becomes more oscillatory rather abruptly. Abrupt changes in the process dynamics and therefore also in the model parameters are reflected in the final map that is stretched towards the dynamically 'more interesting' areas. The map of models could be further used in the adaptive control of the TE process. In the earlier work of [Hyötyniemi and Ylöstalo, 1997] this type of control scheme was used for the control of a pH process.