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Investigation of Operability and Controllability Properties of a Pilot-Scale Distillation Column
Copyright © IFAC Dynamics and Control of Chemical Reactors (DYCORD+'95), Copenhagen, Denmark, 1995
INVESTIGATION OF OPERABILITY AND CONTROLLABILITY PROPERTIES OF A PILOT-SCALE DISTILLATION COLUMN Kurt E. Haggblom and Kari Lehtinen
Process Control Laboratory, Department of C1Jemical Engineering, Abo Akademi University, FIN-20500 Abo, Finland
Abstract: The results of an operability and controllability study of a pilot-scale distillation column separating a mixture of ethanol and water are presented. The relative gain for diagonal variable pairing in the LV -structure as well as the condition number are used as measures of controllability, Of particular interest are illconditioned operating regions characterized by strong directionality, which is typical of high-purity distillation. Operating constraints imposed by the column and the existing instrumentation are also taken into account. The investigation is carried out by means of a simulator, which has been calibrated with experimental data. Keywords: Distillation columns, process control, multi variable control systems, controllability, condition numbers, simulators.
1. INTRODUCTION
by the relative amount O.25A -1 . For a plant with A 25, which is not a particularly high relative gain for an ill-conditioned plant, this means that the gain matrix becomes singular if the gains are perturbed by more than 1% in unfavourable directions. Since there exists no controller with integral action that can stabilize two systems whose gain matrices have determinants of different signs (Garcia and Morari, 1985; Skogestad and Morari, 1987b), there is also no such multivariable controller that is certain to stabilize both the plant and the model if the relative uncertainties of the gains are uncorrelated and larger than O.25A -1 .
=
Distillation is typically an "ill-conditioned" process, especially if the product purities are high, Such a plant has a gain matrix with a high condition number, Another measure of ill-conditioning, which is particularly useful for 2 x 2 systems, is the relative gain (Bristol, 1966) - the higher the relative gain, the more ill-conditioned is the plant. Furthermore, an ill-conditioned plant is characterized by a strong directionality, which means that the (normed) plant gain depends strongly on the input direction , that is, on the interplay between the control variables. Although this is a property of a linear system, the behaviour resembles that of a nonlinear system.
If the errors are correlated in a certain way, larger errors can be tolerated. The design of the identification experiment so as to affect the distribution and correlation of errors in a favourabl e way is therefore important. It is, for example, necessary that the plant is sufficiently excited This may be in all relevant gain directions, achieved by identification in open loop (Koung and MacGregor, 1993) or closed loop (Andersen and Kiimmel, 1992) ,
Because of the strong directionality, good setpoint tracking in arbitrary directions cannot be obtained for an ill-conditioned plant using decentralized (multi-loop SISO) feedback control. where the controllers operate independently of each other, Good control performance requires a multi variable controller that takes the directionalitv into account, However, such a controller for an ill-conditioned plant can be very sensitive to model errors and disturbances in the control variables (Skogestad and Morari, 1987a; Skogestad et al., 1988),
In this paper, the results of an operability and controllability study of a pilot-scale distillation column are presented. The research was done in order to find an operating region, where identification and control problems typical of illconditioned plants such as high-purity distillation columns could be studied. Operating constraints imposed by the column and existing instrumen-
It can be shown that a gain matrix with the relative gain A for a given variable pairing may become singular if the individual gains are perturbed 339
The adjusted parameters are the tray efficiencies in the rectifying and stripping sections (assumed constant in each section) , the efficiency of the reboiler , the parameter associated with the energy balance, two parameters related with the heat removal in the condenser, and the specific enthalpy of condensation for the steam used to heat the reboiler.
tation were also taken into account. The investigation was carried out by means of a simulator, which was calibrated with experimental data from a few operating points. The relative gain and the condition number are used as measures of ill-conditioning and controllability. The directionality properties are clearly illustr~ted and verified experimentally.
Data from four open-loop experiments with different values for the feed flow rate , the feed composition, the reflux flow , and the steam flow to the reboiler , were used to calibrate the simulator. The product compositions, the temperature on tray 14, the reflux temperature , and the product flow rates were considered in the fit. As a result of the optimization, realistic values for the adjusted parameters were obtained. The behaviour of the simulator was also verified with experimental data not used in the optimization.
2. DISTILLATION COLUMN SIMULATOR The distillation column under study (Haggblom et al. , 1984; WaIler et al., 1988; WaIler, 1992) separates a mixture of ethanol and water. It has 15 bubble-cap trays and is 0.30 m in diameter. It is equipped with a thermosyphon-type reboiler and a total condenser, and it operates under atmospheric pressure. To facilitate the operability and controllability study, a simulator for the binary distillation column was constructed (Lehtinen , 1994) . The simulator was implemented in Simulink (Math Works, Inc.), using a "graphical" block-oriented programming technique.
The above parameters primarily affect the steadystate behaviour of the simulator. The dynamic behaviour can be adjusted by a number of other parameters (e.g., parameters affecting holdups), but this was not done in the present work.
2.1 Modelling 3. OPERABILITY AND CONTROLLABILITY STUDY
The simulator was essentially developed from first principles with support of some empirical relationships. Each tray is modelled by a total and a component material balance, an energy balance, Francis weir formula for tray hydraulics , and an empirical relationship for the vapour-liquid equilibrium. The Murphree tray efficiency is used to model the deviation from 100 % tray efficiency. The efficiency of the reboiler is modelled similarly.
An operability and controllability study of the distillation column was carried out by means of the calibrated simulator. First a feasable operating region was located, after which a more detailed investigation was made. The opet'ability and controllability properties are related both to open-loop and closed-loop specifications.
It is assumed that the specific enthalpies of liquid and vapour can be expressed as linear functions of their compositions. The dynamic energy balance can then be reduced to an algebraic relationship , where the contribution from the energy balance is contained in a single parameter, which expresses the relative difference between the partial specific enthalpies of vaporization of the two components, ethanol and water (Haggblom, 1991).
3.1 Operating Region The critical operating constraints of the distillation column are: reflux flow rateL:S 200 kg/h , steam flow rate (to reboiler) V :S 120 kg/h. The feed composition z can be varied as desired , whereas the feed flow rate F should not exceed 200 kg/h so as to enable experiments of sufficient duration without switching offeed tanks (and feed composition). The distillate flow rate D should be as large as possible to facilitate control of the reflux drum inventory when using the L l-Tstructure. Furthermore, the bottoms composition x (wt- % ethanol) should not be exceedingly low so as to avoid measurement problems and severe nonlinearity. Since ethanol-water forms an azeotropic mixture at 94 wt-% ethanol, the distillate composition y is always below this value, and does not give rise to any particular measurement problems.
The column is equipped with a total condenser producing a subcooled reflux. In the modelling of the condenser it has to be taken into account that neither the heat removal through condensation nor the temperature or the enthalpy of the reflux are constant when the coolant flow rate is constant. The column could not be adequately modelled by assuming any of these quantities to be constant. In accordance with a suggestion by Haggblom (1991) , the heat removal was therefore assumed to depend linearly on the flux of enthalpy from the column to the condenser.
Given the above constraints and requirements, it was desired to find an operating region where the column could be classified as "ill-conditioned". By means of simulations, it was found that a satisfactory compromise between the competing requirements was obtained with the feed flow rate F = 150 kg/h and the feed composition z 25 wt-% ethanol. This made possible a range of operating points satisfying D ::::: 40 kg/h, 85 kg/h :S L :S 160 kg/h , 70 kg/h :S V :S 105kg/h, 90.5% :S y :S 91.5%, 0.5% :S x :S 1.5%,5:S >':s 20 .
2.2 Calibration In the model , there are a number of uncertain parameters, which were adjusted to fit the behaviour of the simulator to the real column. Since Simulink runs under Matlab (Math Works, Inc.) , the parameters could be adjusted through numerical optimization, where the simulator was part of the objective function .
=
340
compositions, see Fig. 10. In the region where the relative gain is high, the condition number has a distinct minimum.
3.2 Opell-Loop Specificatiolls A more detailed simulation was performed for the range L 120 . .. 160 kg/h , V 85 ... 105 kg/h. Figures 1 and 2 show the distillate and bottoms composition surfaces vs. L and V. These look relatively well-behaved. although there is a rather sharp change of the slope along the line L ~ 2.2V - 70 kg/h. This is revealed more clearly by the gain plots in Figs. 3 - 6. In the region defined 2.2V -70 kg/h , ~ V ±1 kg/h, the gains by L of y change by a factor of 5 and the gains of x by two orders of a magnitude. The nonlinearity is thus severe in this region. This also means that the compositions and their gains are very sensitive to input uncertainty in the region .
=
=
=
Since the scaling of the variables can be related to control specifications in terms of allowed deviations in these variables, the condition number can be taken as a measure of how difficult the specifications are to meet. The relative gain. on the other hand , is more related with the sensitivity to modelling errors and (input) uncertainty. that is, with robustness issues. The relative gain and the condition number thus complement each other.
=
3.3 Closed-Loop Specifications
Figure 7 shows the relative gain ,\ vs. L and V. In the region defined above , the relative gain has a sharp maximum. At lower values of V (and L) , ,\ changes from 5 to 10 , and at higher values of V , from 5 to more than 20 , when V is perturbed by 1-2 kg/h . The variation of ,\ is due to the nonlinearities, but their effect on ,\ is reduced by the fact that the nonlinearities of the gains are strongly correlated. The controllability implied by the relative gain is thus very sensitive to modelling errors and input uncertainty in this region.
The simulations with open-loop specifications cover a large range of product compositions. In practice, the operating region is much more limited. Therefore, a set of simulations in the composition range y 90 .. . 92 wt-% , x 0.05 . .. 2 wt% was performed.
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Figure 11 shows contour lines of L and V vs. y and x. The L and V lines are nearly parallell in the whole region , even in the nonlinear part , where the contour lines are curved. This means that the gain matrix is nearly singular.
At high reflux rates and low boilup rates the compositions approach their upper limits (the azeotropic composition 94 % for y and the feed composition 25 % for x) , which naturally results in an increase of the relative gain.
The figure clearly illustrates the gain directionality of the plant. To move from operating point A (y 91 %, x 1 %) to operating point B (y = 91.1 %, x = 1.5 %) , the required changes in the inputs are ~L = 0.54 kg/h, ~ V = -0.14 kg/h . A change to operating point C (y ~1.1 %, x = 0.5 %), on the other hand, requires the input changes ~L 15.8 kg/h, ~ V 7.64 kg/h, which are more than 30 times larger than the previous changes in L and V, although the composition changes are equally large in both cases.
=
=
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Figure 8 shows the condition number vs. Land V. It is striking that the condition number has a minimum in the region where the relative gain has a maximum, and vice versa. Since the condition number denotes the ratio between the maximum and minimum plant gains, this means that the variability of the normed plant gain is, in fact, smaller in the region where the relative gain is large. This might seem to be in conflict with the previous results and interpretations.
=
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The previous example is not even extreme. A change to operating point D (y = 91.5 %, x 0.9 %) , which is orthogonal to the change to point B, requires the input changes ~L = 38.5 kg/h , ~ V = 17.5 kg/h, which are almost two orders of a magnitude larger than the required changes between point A and B. The change from point A to point B is thus in a high-gain direction , whereas the change from point A to point D is in a lowgain direction. The ratio between these gains is (approximately) equal to the condition number.
However, the condition number should not be directly related with the relative gain, but with the gains themselves. In the region where the bottoms composition gains are close to zero, the condition number is high since a plant gain close to zero can be obtained by choosing the inputs such that ~y ~ O. The fact that the gains for x are close to zero means that it is very difficult to affect x by the inputs L and V. One can also make the interpretation that it is difficult to reduce variations in y to the same level as in x.
Note that the principal output directions corresponding to the high- and low-gain directions change significantly when the bottoms composition moves towards zero. The principal input directions. on the other hand , are not much affected , since the angle between the contour lines for L and V remain essentially constant. It can, in fact, be proved that the principal input directions are aligned with changes in the rates of external (product) flows (Haggblom, 1995). These directions can therefore easily be determined from the flow gains.
It is well-known that the condition number is scaling dependent. If y is scaled by the factor 0.2 and x is scaled by 5, the result shown in Fig. 9 is obtained. Now the condition number is highest in the region where the distillate composition gains are small (note that the scaling has increased the gains related with x and decreased the gains related with y). According to the condition number, the plant is then most difficult to control in this region.
Figure i showed that the relative gain is high only in a narrow region in terms of L and V . Relatively small perturbations in these variables will drop the relative gain to around 5. However , Fig. 12 shows that a large relative gain cannot be
By another scaling (y scaled by 0.6 , x scaled by 1.6) , large condition numbers are obtained both for high distillate compositions and low bottoms 341
95~ 901~ l8S
80
75
lOO 105
105
Fig. 1. Distillate composition versus reflux and steam flow .
Fig. 2. Bottoms composition versus reflux and steam flow.
120
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Fig. 3. Gain between distillate composition and reflux.
Fig. 4. Gain between distillate composition and steam flow.
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105
Fig. 5. Gain between bottoms composition and reflux.
Fig. 6. Gain between bottoms composition and steam flow .
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342
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Fig. 8. Condition number versus reflux and steam flow.
Fig. 7. Relative gain versus reflux and steam flow .
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Fig. 9. Condition number versus reflux and steam flow ; scaling 0.2y , 5x.
Fig. 10. Condition number versus reflux and steam flow; scaling 0.6y , 1.6x.
Fig. 11. Contour plot of reflux and steam flow versus product compositions (L= - , V= --).
Fig. 12. Contour plot of relative gam versus product compositions. 343
with robustness issues, and the condition number , which is more related with directionality and control specifications, can be conflicting measures of controllability. This means that they complement each other rather than substitute each other .
~: :tJ. . .- .- -,:.~ . 9100
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100
150
200
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ACKNOWLEDGEMENT Financial support from Tekes and the Academy of Finland is gratefully acknowledged.
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Fig. 13. Experimental illustration of directionality.
avoided if high product purities are desired. This implies that it is in practice difficult to maintain high product purities due to the ever present input uncertainty. The distillation column is, in other words, inherently different to control at high product purities. 4. EXPERIMENTAL VERIFICATION The simulator was calibrated using data from four steady states, only one of which had the feed flow rate F = 150 kg/h and the feed composition z 25 wt-%. Therefore, a number of experimental verifications were made.
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One such experiment, which also illustrates the directionality properties, are shown in Fig. 13. The initial steady steady is y 91.35 %, x ::::::: 1 %, L 140 kg/h , V 96 kg/h, which is in good agreement with Fig. 11 . At t 50 min, a sequence of three step changes satisfying tlL/ tl V = 2 was initiated , and at t = 140 min, this was continued with a sequence of step changes with tlL/ tl V = -0 .5. These ratios were determined from flow gains, which were known from previous experiments. As can be seen, the first sequence of step changes results in a low-gain response . while the second sequence gives a high-gain response. The difference in gains is an order of a magnitude.
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5. CONCLUSION The results of an operability and controllability study of a pilot-scale distillation column separating a mixture of ethanol and water has been presented. In particular, ill-conditioned operating points characterized by strong directionality as well as sensitivity to model errors and input uncertainty were studied. An interesting result of the study is that the relative gain. which is related 344
REFERENCES Andersen, H.W. and M. Kiimmel (1992). Evaluating estimation of gain directionality. (Part 1 and 2.) J. Proc. Cont. 2 , 59- 66 , 67- 86. Bristol, E.H. (1966). On a new measure of interactions for multi variable process control. IEEE Trans. Autom. Control AC-ll , 133- 134. Garcia, C.E. and M. Morari (1985) . Internal model control. 2. Design procedure for multivariable systems. Ind. Eng. Chem . Process Des. Dev. 24, 474-484. Haggblom, K.E. (1991). Modeling of flow dynamics for control of distillation columns. Proc. American Control Conference, pp. 785790 , Boston, USA. Haggblom, K.E. (1995). Static directionality of 2 x 2 systems: A control-relevant property with application to distillation. Submitted to the European Control Conference '95, September 5-8, Roma, Italy. Haggblom, K.E. , K.E. Wikman, S.E. Gustafsson and K.V. Waller (1984) . Abo Akademi 's pilotplant distillation column and its control system (in Swedish). Proc. Finnisll Automation Days '84, pp. 11/427-441 , Espoo , Finland. Koung , C.-W. and J.F. MacGregor (1993) . Design of identification experiments for robust control. A geometric approach for bivariate processes. Ind. Eng. Chem. Res. 32 , 16581666. Lehtinen, K. (1994). Investigation of the behaviour of a distillation column at different operating points (in Swedish). M.Sc. Thesis, Process Control Laboratory, Abo Akademi University, Abo , Finland. Skogestad, S. and M. Morari (1987a). Implications of large RGA elements on control performance. Ind. Eng. Chem. Res. 26, 2323-2330. Skogestad, S. and M. Morari (1987b). Design of resilient processing plants - IX. Effect of model uncertainty on dynamic resilience. Chem. Eng. Sci. 42, 1765-1780. Skogestad, S., M. Morari and J.G. Doyle (1988) . Robust control of ill-conditioned plants: highpurity distillation. IEEE Trans. Autom. Control AC-33. 1092-1105. Waller. K.V. (1992). Experimental comparison of control structures. In: Practical Distillation Control (W.L. Luyben (Ed.)) , pp. 313- 330. Van Nostrand Reinhold , New York. Waller , K.V., D.H. Finnerman, P.M. Sandelin K.E. Haggblom and S.E . Gustafsson (1988 ): An experimental comparison of four control structures for two-point control of distillation . hld. Eng. Cllem . Res. 27, 624-630.
INVESTIGATION OF OPERABILITY AND CONTROLLABILITY PROPERTIES OF A PILOT-SCALE DISTILLATION COLUMN Kurt E. Haggblom and Kari Lehtinen
Process Control Laboratory, Department of C1Jemical Engineering, Abo Akademi University, FIN-20500 Abo, Finland
Abstract: The results of an operability and controllability study of a pilot-scale distillation column separating a mixture of ethanol and water are presented. The relative gain for diagonal variable pairing in the LV -structure as well as the condition number are used as measures of controllability, Of particular interest are illconditioned operating regions characterized by strong directionality, which is typical of high-purity distillation. Operating constraints imposed by the column and the existing instrumentation are also taken into account. The investigation is carried out by means of a simulator, which has been calibrated with experimental data. Keywords: Distillation columns, process control, multi variable control systems, controllability, condition numbers, simulators.
1. INTRODUCTION
by the relative amount O.25A -1 . For a plant with A 25, which is not a particularly high relative gain for an ill-conditioned plant, this means that the gain matrix becomes singular if the gains are perturbed by more than 1% in unfavourable directions. Since there exists no controller with integral action that can stabilize two systems whose gain matrices have determinants of different signs (Garcia and Morari, 1985; Skogestad and Morari, 1987b), there is also no such multivariable controller that is certain to stabilize both the plant and the model if the relative uncertainties of the gains are uncorrelated and larger than O.25A -1 .
=
Distillation is typically an "ill-conditioned" process, especially if the product purities are high, Such a plant has a gain matrix with a high condition number, Another measure of ill-conditioning, which is particularly useful for 2 x 2 systems, is the relative gain (Bristol, 1966) - the higher the relative gain, the more ill-conditioned is the plant. Furthermore, an ill-conditioned plant is characterized by a strong directionality, which means that the (normed) plant gain depends strongly on the input direction , that is, on the interplay between the control variables. Although this is a property of a linear system, the behaviour resembles that of a nonlinear system.
If the errors are correlated in a certain way, larger errors can be tolerated. The design of the identification experiment so as to affect the distribution and correlation of errors in a favourabl e way is therefore important. It is, for example, necessary that the plant is sufficiently excited This may be in all relevant gain directions, achieved by identification in open loop (Koung and MacGregor, 1993) or closed loop (Andersen and Kiimmel, 1992) ,
Because of the strong directionality, good setpoint tracking in arbitrary directions cannot be obtained for an ill-conditioned plant using decentralized (multi-loop SISO) feedback control. where the controllers operate independently of each other, Good control performance requires a multi variable controller that takes the directionalitv into account, However, such a controller for an ill-conditioned plant can be very sensitive to model errors and disturbances in the control variables (Skogestad and Morari, 1987a; Skogestad et al., 1988),
In this paper, the results of an operability and controllability study of a pilot-scale distillation column are presented. The research was done in order to find an operating region, where identification and control problems typical of illconditioned plants such as high-purity distillation columns could be studied. Operating constraints imposed by the column and existing instrumen-
It can be shown that a gain matrix with the relative gain A for a given variable pairing may become singular if the individual gains are perturbed 339
The adjusted parameters are the tray efficiencies in the rectifying and stripping sections (assumed constant in each section) , the efficiency of the reboiler , the parameter associated with the energy balance, two parameters related with the heat removal in the condenser, and the specific enthalpy of condensation for the steam used to heat the reboiler.
tation were also taken into account. The investigation was carried out by means of a simulator, which was calibrated with experimental data from a few operating points. The relative gain and the condition number are used as measures of ill-conditioning and controllability. The directionality properties are clearly illustr~ted and verified experimentally.
Data from four open-loop experiments with different values for the feed flow rate , the feed composition, the reflux flow , and the steam flow to the reboiler , were used to calibrate the simulator. The product compositions, the temperature on tray 14, the reflux temperature , and the product flow rates were considered in the fit. As a result of the optimization, realistic values for the adjusted parameters were obtained. The behaviour of the simulator was also verified with experimental data not used in the optimization.
2. DISTILLATION COLUMN SIMULATOR The distillation column under study (Haggblom et al. , 1984; WaIler et al., 1988; WaIler, 1992) separates a mixture of ethanol and water. It has 15 bubble-cap trays and is 0.30 m in diameter. It is equipped with a thermosyphon-type reboiler and a total condenser, and it operates under atmospheric pressure. To facilitate the operability and controllability study, a simulator for the binary distillation column was constructed (Lehtinen , 1994) . The simulator was implemented in Simulink (Math Works, Inc.), using a "graphical" block-oriented programming technique.
The above parameters primarily affect the steadystate behaviour of the simulator. The dynamic behaviour can be adjusted by a number of other parameters (e.g., parameters affecting holdups), but this was not done in the present work.
2.1 Modelling 3. OPERABILITY AND CONTROLLABILITY STUDY
The simulator was essentially developed from first principles with support of some empirical relationships. Each tray is modelled by a total and a component material balance, an energy balance, Francis weir formula for tray hydraulics , and an empirical relationship for the vapour-liquid equilibrium. The Murphree tray efficiency is used to model the deviation from 100 % tray efficiency. The efficiency of the reboiler is modelled similarly.
An operability and controllability study of the distillation column was carried out by means of the calibrated simulator. First a feasable operating region was located, after which a more detailed investigation was made. The opet'ability and controllability properties are related both to open-loop and closed-loop specifications.
It is assumed that the specific enthalpies of liquid and vapour can be expressed as linear functions of their compositions. The dynamic energy balance can then be reduced to an algebraic relationship , where the contribution from the energy balance is contained in a single parameter, which expresses the relative difference between the partial specific enthalpies of vaporization of the two components, ethanol and water (Haggblom, 1991).
3.1 Operating Region The critical operating constraints of the distillation column are: reflux flow rateL:S 200 kg/h , steam flow rate (to reboiler) V :S 120 kg/h. The feed composition z can be varied as desired , whereas the feed flow rate F should not exceed 200 kg/h so as to enable experiments of sufficient duration without switching offeed tanks (and feed composition). The distillate flow rate D should be as large as possible to facilitate control of the reflux drum inventory when using the L l-Tstructure. Furthermore, the bottoms composition x (wt- % ethanol) should not be exceedingly low so as to avoid measurement problems and severe nonlinearity. Since ethanol-water forms an azeotropic mixture at 94 wt-% ethanol, the distillate composition y is always below this value, and does not give rise to any particular measurement problems.
The column is equipped with a total condenser producing a subcooled reflux. In the modelling of the condenser it has to be taken into account that neither the heat removal through condensation nor the temperature or the enthalpy of the reflux are constant when the coolant flow rate is constant. The column could not be adequately modelled by assuming any of these quantities to be constant. In accordance with a suggestion by Haggblom (1991) , the heat removal was therefore assumed to depend linearly on the flux of enthalpy from the column to the condenser.
Given the above constraints and requirements, it was desired to find an operating region where the column could be classified as "ill-conditioned". By means of simulations, it was found that a satisfactory compromise between the competing requirements was obtained with the feed flow rate F = 150 kg/h and the feed composition z 25 wt-% ethanol. This made possible a range of operating points satisfying D ::::: 40 kg/h, 85 kg/h :S L :S 160 kg/h , 70 kg/h :S V :S 105kg/h, 90.5% :S y :S 91.5%, 0.5% :S x :S 1.5%,5:S >':s 20 .
2.2 Calibration In the model , there are a number of uncertain parameters, which were adjusted to fit the behaviour of the simulator to the real column. Since Simulink runs under Matlab (Math Works, Inc.) , the parameters could be adjusted through numerical optimization, where the simulator was part of the objective function .
=
340
compositions, see Fig. 10. In the region where the relative gain is high, the condition number has a distinct minimum.
3.2 Opell-Loop Specificatiolls A more detailed simulation was performed for the range L 120 . .. 160 kg/h , V 85 ... 105 kg/h. Figures 1 and 2 show the distillate and bottoms composition surfaces vs. L and V. These look relatively well-behaved. although there is a rather sharp change of the slope along the line L ~ 2.2V - 70 kg/h. This is revealed more clearly by the gain plots in Figs. 3 - 6. In the region defined 2.2V -70 kg/h , ~ V ±1 kg/h, the gains by L of y change by a factor of 5 and the gains of x by two orders of a magnitude. The nonlinearity is thus severe in this region. This also means that the compositions and their gains are very sensitive to input uncertainty in the region .
=
=
=
Since the scaling of the variables can be related to control specifications in terms of allowed deviations in these variables, the condition number can be taken as a measure of how difficult the specifications are to meet. The relative gain. on the other hand , is more related with the sensitivity to modelling errors and (input) uncertainty. that is, with robustness issues. The relative gain and the condition number thus complement each other.
=
3.3 Closed-Loop Specifications
Figure 7 shows the relative gain ,\ vs. L and V. In the region defined above , the relative gain has a sharp maximum. At lower values of V (and L) , ,\ changes from 5 to 10 , and at higher values of V , from 5 to more than 20 , when V is perturbed by 1-2 kg/h . The variation of ,\ is due to the nonlinearities, but their effect on ,\ is reduced by the fact that the nonlinearities of the gains are strongly correlated. The controllability implied by the relative gain is thus very sensitive to modelling errors and input uncertainty in this region.
The simulations with open-loop specifications cover a large range of product compositions. In practice, the operating region is much more limited. Therefore, a set of simulations in the composition range y 90 .. . 92 wt-% , x 0.05 . .. 2 wt% was performed.
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=
Figure 11 shows contour lines of L and V vs. y and x. The L and V lines are nearly parallell in the whole region , even in the nonlinear part , where the contour lines are curved. This means that the gain matrix is nearly singular.
At high reflux rates and low boilup rates the compositions approach their upper limits (the azeotropic composition 94 % for y and the feed composition 25 % for x) , which naturally results in an increase of the relative gain.
The figure clearly illustrates the gain directionality of the plant. To move from operating point A (y 91 %, x 1 %) to operating point B (y = 91.1 %, x = 1.5 %) , the required changes in the inputs are ~L = 0.54 kg/h, ~ V = -0.14 kg/h . A change to operating point C (y ~1.1 %, x = 0.5 %), on the other hand, requires the input changes ~L 15.8 kg/h, ~ V 7.64 kg/h, which are more than 30 times larger than the previous changes in L and V, although the composition changes are equally large in both cases.
=
=
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Figure 8 shows the condition number vs. Land V. It is striking that the condition number has a minimum in the region where the relative gain has a maximum, and vice versa. Since the condition number denotes the ratio between the maximum and minimum plant gains, this means that the variability of the normed plant gain is, in fact, smaller in the region where the relative gain is large. This might seem to be in conflict with the previous results and interpretations.
=
=
The previous example is not even extreme. A change to operating point D (y = 91.5 %, x 0.9 %) , which is orthogonal to the change to point B, requires the input changes ~L = 38.5 kg/h , ~ V = 17.5 kg/h, which are almost two orders of a magnitude larger than the required changes between point A and B. The change from point A to point B is thus in a high-gain direction , whereas the change from point A to point D is in a lowgain direction. The ratio between these gains is (approximately) equal to the condition number.
However, the condition number should not be directly related with the relative gain, but with the gains themselves. In the region where the bottoms composition gains are close to zero, the condition number is high since a plant gain close to zero can be obtained by choosing the inputs such that ~y ~ O. The fact that the gains for x are close to zero means that it is very difficult to affect x by the inputs L and V. One can also make the interpretation that it is difficult to reduce variations in y to the same level as in x.
Note that the principal output directions corresponding to the high- and low-gain directions change significantly when the bottoms composition moves towards zero. The principal input directions. on the other hand , are not much affected , since the angle between the contour lines for L and V remain essentially constant. It can, in fact, be proved that the principal input directions are aligned with changes in the rates of external (product) flows (Haggblom, 1995). These directions can therefore easily be determined from the flow gains.
It is well-known that the condition number is scaling dependent. If y is scaled by the factor 0.2 and x is scaled by 5, the result shown in Fig. 9 is obtained. Now the condition number is highest in the region where the distillate composition gains are small (note that the scaling has increased the gains related with x and decreased the gains related with y). According to the condition number, the plant is then most difficult to control in this region.
Figure i showed that the relative gain is high only in a narrow region in terms of L and V . Relatively small perturbations in these variables will drop the relative gain to around 5. However , Fig. 12 shows that a large relative gain cannot be
By another scaling (y scaled by 0.6 , x scaled by 1.6) , large condition numbers are obtained both for high distillate compositions and low bottoms 341
95~ 901~ l8S
80
75
lOO 105
105
Fig. 1. Distillate composition versus reflux and steam flow .
Fig. 2. Bottoms composition versus reflux and steam flow.
120
85
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rc
0 .3
0 .2
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0.1
o 160
105
105
Fig. 3. Gain between distillate composition and reflux.
Fig. 4. Gain between distillate composition and steam flow.
105
105
Fig. 5. Gain between bottoms composition and reflux.
Fig. 6. Gain between bottoms composition and steam flow .
120
85
342
2.5 2.0
.8
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1.5
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8 0 .5 0 160 105
105
Fig. 8. Condition number versus reflux and steam flow.
Fig. 7. Relative gain versus reflux and steam flow .
2.5
2 .5
2 .0
2 .0
a;
a;
.D
§ c: c:
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15
§
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15
~ 1.0
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8
8 0.5
0 .5
0 160
0 160 105
105
Fig. 9. Condition number versus reflux and steam flow ; scaling 0.2y , 5x.
Fig. 10. Condition number versus reflux and steam flow; scaling 0.6y , 1.6x.
Fig. 11. Contour plot of reflux and steam flow versus product compositions (L= - , V= --).
Fig. 12. Contour plot of relative gam versus product compositions. 343
with robustness issues, and the condition number , which is more related with directionality and control specifications, can be conflicting measures of controllability. This means that they complement each other rather than substitute each other .
~: :tJ. . .- .- -,:.~ . 9100
f:l
::;;zo
50
100
150
200
250
ACKNOWLEDGEMENT Financial support from Tekes and the Academy of Finland is gratefully acknowledged.
00o:----:5O~---==--:1~00:-------:1::-50::----2,...00=:....--2:-!5O
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50
100
150
200
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50
100
150
200
1 250
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t(min)
Fig. 13. Experimental illustration of directionality.
avoided if high product purities are desired. This implies that it is in practice difficult to maintain high product purities due to the ever present input uncertainty. The distillation column is, in other words, inherently different to control at high product purities. 4. EXPERIMENTAL VERIFICATION The simulator was calibrated using data from four steady states, only one of which had the feed flow rate F = 150 kg/h and the feed composition z 25 wt-%. Therefore, a number of experimental verifications were made.
=
One such experiment, which also illustrates the directionality properties, are shown in Fig. 13. The initial steady steady is y 91.35 %, x ::::::: 1 %, L 140 kg/h , V 96 kg/h, which is in good agreement with Fig. 11 . At t 50 min, a sequence of three step changes satisfying tlL/ tl V = 2 was initiated , and at t = 140 min, this was continued with a sequence of step changes with tlL/ tl V = -0 .5. These ratios were determined from flow gains, which were known from previous experiments. As can be seen, the first sequence of step changes results in a low-gain response . while the second sequence gives a high-gain response. The difference in gains is an order of a magnitude.
=
=
= =
5. CONCLUSION The results of an operability and controllability study of a pilot-scale distillation column separating a mixture of ethanol and water has been presented. In particular, ill-conditioned operating points characterized by strong directionality as well as sensitivity to model errors and input uncertainty were studied. An interesting result of the study is that the relative gain. which is related 344
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