Control configurations in distillation columns: A

Control configurations in distillation columns: A comparative study Max Thone, Max Potters, and Simone Baldi

Abstract— Model-plant mismatch in model predictive controlled industrial processes is a phenomenon that may lead to relevant performance losses. Research in finding a suitable procedure to effectively detect and revert model plant mismatch often uses distillation columns as benchmark testing models. However, it was found that sub optimal distillation columns and control configurations were used in these studies. This paper aims to provide answers for the issues as just discussed. At first a benchmark distillation column is designed using the work of Sigurd Skogestad. Secondly, two common control configurations are compared, in which it is shown that in high purity regions the a more accurate model can be identified in the (L/D)(V /B)configuration. Further, in the case of disturbances on the feed rate and feed composition channels of the distillation column, it is shown that the (L/D)(V /B)-configuration out performs the LV -configuration in terms of disturbance rejection and handling model-plant mismatch.

I. I NTRODUCTION Model predictive control (MPC) is a control technique frequently used in the process industry. The two main advantages of model predictive controllers is that they are able to predict the impact of disturbances, and can accommodate for constraints on inputs and outputs of the dynamic systems they control. The design of model predictive controllers and their implementation in example industrial processes, most notably distillation columns, have been investigated thoroughly in literature, see for example [1] and [2]. In [3] and [4], non-linear MPC methods are investigated using a distillation column as benchmark. A disadvantage of MPC is the fact that it relies on a mathematical model of the plant that is controlled. As such, it may happen that the plant changes over time, while the model in the model-predictive controller stays the same. This results in the phenomenon of model-plant mismatch (MPM), which can lead to reduced closed-loop control performance. Indeed, in [5], it is argued that MPM may lead to increased variance of the controlled output of the MPC controlled system. In [6] it is argued that in an industrial setting, increased variance in the output often leads to reduced ’constraint pushing’ which may lead to economic losses. The problem of detecting MPM and reverting the subsequent loss in performance has already been discussed extensively in literature, see for example [5] and [7]. However, one major gap has been found with respect to this literature. Most often a distillation column model is used as benchmark in these studies. However, the so-called LV configuration is always used in these benchmark models, *The authors are with the Delft Center for Systems and Control, Delft University of Technology, Delft 2628CD, The Netherlands [email protected].

while it is discussed in [8] that this is almost never the most optimal control configuration, especially when high purity outputs are desired. Based on this issue in literature, the following is discussed in this work. In the first part a new benchmark distillation column is designed, where design equations derived in [8] and [9] are used to ensure proper design parameters are chosen. The goal of this part is to encourage the use of a standard benchmark distillation column while researching for new techniques that performance monitoring in model predictive controlled systems. In the second part of this work, linear models of both the distillation column control configurations are identifed using prediction-error identification (PEI). Subsequently, model predictive controllers are designed for both these configurations, and the closed-loop performance of both configurations is compared. II. D ESIGN OF A BENCHMARK DISTILLATION COLUMN In current research very disparate distillation columns are used as benchmark models. Often, the distillation columns that are used are also not optimally designed. The main purpose of this section is thus to provide guidelines on designing a suitable benchmark distillation column for research in process control. The distillation theory and design guidelines are based on [9] and [8]. The Matlab code that is used to build a benchmark column is available online 1 . As distillation theory is well known, this will be described only superficially. For more in depth distillation theory, the author refers to theory described in [8] and [9]. A. The binary distillation column Consider the diagram of a binary distillation column in Fig. 1. A feed F with light component mole fraction zF and liquid fraction q enters the system at stage NF . The main goal of the column is to separate the incoming binary mixture feed F in to a distillate product D with a high purity of the light component xL , and a bottoms product B with a high purity of the heavy component xH = 1 − xL . This again means we want a high purity

1 http://www.nt.ntnu.no/users/skoge/book/matlab

m/cola/cola.html

of the light product xD in the top, and a low purity of the light product xB in the bottom product.

in the column can be adjusted, or the the reflux and boil up flows LT and VB can be adjusted. Based on this, design equations were derived in [8] to set guidelines for developing a benchmark distillation column. 1) Design equation for number of stages: In [8] an expression for the minimum amount of stages required is derived by considering the theoretical setting in which there is infinite energy (i.e. infinite reflux LT and infinite boil-up VB ) available, also called Fenske’s formula.

Nmin =

Fig. 1: A binary continuous distillation column [9]. As is explained in [8], the purities xD and xB generally show very non-linear behaviour, especially in the high purity regions. To deal with this problem logarithmic purities are considered instead of the actual purities so that we have xD , 1 − xD B. Control configurations XD = ln

XB = ln

xB . 1 − xB

(1)

From a control perspective, in a binary distillation column LT VB , D and B can all be used as control inputs to control the logarithmic purities XL , XD and the levels in the condenser and reboiler [8]. The mathematical equations that govern how these entities are related are elaborated in [9]. Depending on which control inputs are used to control which outputs, different control configurations for the binary distillation column can be defined. In this work two control configurations are compared and evaluated. • In the LV -configuration, D and B are used for level control, so that the reflux flow LT and boil-up flow VB may be used for control of the top and bottom qualities. Mathematically the following relation is considered     XD LT = fDist (2) XB VB •

In the double ratio (L/D)(V /B)-configuration, (LT /D) and (VB /B) are used for the control of qualities, while D and B are used for level control in the condenser and the reboiler. Mathematically we have the following      XD D 0 (LT /D) = fDist (3) XB 0 B (VB /B) Note that the control inputs (LT /D) and (VB /B) must be multiplied by D and B, respectively before they enter the distillation column model

C. Distillation column design To achieve high purities in the distillate and bottom flows D and B, two things can be done in the design phase of a distillation column [8]. Either the the number of stages NT

ln S . ln α

(4)

where S is the separation index and α the relative volatility, which are both elaborated in [8]. A common rule of thumb is to select the actual amount of stages for the distillation column as N = 2Nmin . 2) Design equations for reflux and boil-up flows: Similarly, when an infinite amount of stages NT is assumed, expressions are derived for the minimum amount of reflux LT and boil-up flows VB required to achieve the desired purities xL and xH [8]

LT,min =

1 F, α −1

VB,min =

1 F + D. α −1

(5)

3) Design equation for feed stage: Finally, the location for the feed stage may be determined via

ln C=

h

1−y f xf

ih

ln α

xb 1−xd

i ,

NF =

[N + 1 −C] . 2

(6)

which gives an optimal feed stage location of NF = 30, halfway in the column. Intuitively this is logical, since the top and bottom products both have the same desired purity, and the feed is assumed to have equal amounts of light and heavy product. 4) Determining exact steady state values for LT and VB : It should be clear that equation (5) can only be used for initial guesses as to what LT and VB should be in order to obtain the desired purities xD and xB . A way to obtain exact steady state values for LT and VB is to run the distillation column model with the found values for NT and NF in closed-loop with simple PI controllers to steady state, where desired values for xD and xB are supplied as reference signals to the PI controllers. The parameters resulting from our design are listed in Table I.

parameter F z q D B LT VB xD xB N NF α τl Mi

value 10 [kmol/min] 0.5 1 5 [kmol/min] 5 [kmol/min] 33.28 [kmol/min] 38.28 [kmol/min] 0.99 0.01 60 30 1.36 0.063 [min] 5 [kmol]

note Feed rate Fraction of light component in feed Liquid fraction in feed Distillate product flow rate Bottom product flow rate Reflux flow boil-up flow light fraction in distillate product light fraction in bottom product Number of theoretical stages. Location of feed stage Relative volatility Liquid flow dynamics time constant Nominal liquid hold up on each stage

TABLE I: Design parameters of the distillation column To check that the distillation column is properly designed, the model is simulated at steady-state. In Fig. 2 the light fraction in the liquid at each theoretical stage xL,i is plotted against the number of theoretical stages. A correctly designed distillation column should show no pinch zone in this plot, since every stage must contribute to the separation between the two components in the mixture. Further note that the composition xL,30 at the feed stage NF = 30 has the same value as the light fraction zF in the feed, 0.5, which indicates that the feed stage has been correctly placed [9].

A. Linear analysis of the LV -configuration The non-linear system fDist is linearised around the steady state values xD = 0.99 and xB = 0.01. The resulting steadystate gain matrix Gss = C A−1 B is decomposed via a singular value decomposition (SVD), to obtain the high and low gain of the system and the accompanying directions of these gains (in steady state) Gss = UΣV ∗   −0.6617 −0.7498 U= −0.7498 0.6617   0.1964 0 Σ= 0 0.0016   −0.7067 0.7075 . V∗ = −0.7075 −0.7067

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0

10

20

30

40

50

60

Stage number

Fig. 2: Composition plot of liquid fraction of light component in the distillation column

(7)

From (7) we see that there is a large difference between the ’high gain’, or the highest singular value σ¯ (Gss ) = 0.1964 and the ’low gain’ or the lowest singular value σ (Gss ) = 0.0016. The condition number (CN), which is defined as the ratio between the highest and the lowest singular value of the steady-state gain matrix, is CN =

1

composition (x ) D

decomposition, of the LV -configuration and the (L/D)(V /B)configuration.

σ¯ (Gss ) = 122.75. σ (Gss )

To investigate the meaning of the high condition number of the LV -configuration, two random binary signals are applied to control inputs LT and VB . The resulting relative movement between xB and xD is plotted in Fig. 3. We note that xB and xD only move in the same direction, representing the high gain direction, while the low gain direction is almost nonexistent. Further, the plot shows that the relation between xD and xB is slightly curved. This is due to the non-linearity of the distillation column in high-purity regions. The consequence of the large difference in high and lowgain direction is that it is very difficult to excite the lowgain direction. This in turn means that is difficult to identify a black-box identification model, since the accuracy of the model relies on sufficient excitation in both output directions. This will be elaborated more in-depth in Section IV. B. Linear analysis of the (L/D)(V /B)-configuration

III. L INEAR ANALYSIS OF TWO DISTILLATION COLUMN CONFIGURATIONS

In this section, the distillation column designed in the previous section is analyzed for identification and control purposes. Both the LV -configuration and the (L/D)(V /B)configuration are analyzed. The LV -configuration is the most common configuration used in literature, see for example [7] and [5]. However, it is argued in [8] that the LV configuration is sub-optimal in high purity regions, and that the (L/D)(V /B)-configuration may yield better control performance. The purpose therefore in this section is to compare the linear properties, and then especially the singular value

As was done for the LV -configuration, the SVD of Gss is obtained Gss = UΣV ∗   −0.6672 −0.7449 U= −0.7449 0.6672   (8) 0.0641 0 Σ= 0 0.0080   −0.7022 0.7120 ∗ . V = −0.7120 −0.7022 Just as with the LV -configuration, the high gain is associated with a change in external flows, and the low gain is associated with a change in internal flows. From U and V ∗ we

For this reason the hypothesis is that the (L/D)(V /B)configuration will yield better closed-loop identification results, and subsequently also to better MPC performance when both these models are used for MPC control. In the first part of this section the closed-loop identification results are discussed. In the second part the MPC performance is discussed.

0.012

0.0115

0.011

xB

0.0105

0.01

0.0095

A. Closed-loop identification

0.009

0.0085 0.988

0.9885

0.989

0.9895

xD

0.99

0.9905

0.991

0.9915

Fig. 3: xD vs xB in the LV -configuration.

once again see that in the high gain direction, (LT /D) and (VB /B) move in opposite directions while xD and xB move in the same-direction, and in the low-gain direction, (LT /D) and (VB /B) move in the same direction and xD and xB in opposite directions. Further, note that CN(L/D)(V /B) =

σ¯ (Gss ) = 8.0125, σ (Gss )

which is 15 times smaller as compared with the LV configuration. Random binary signals are applied to the control inputs (LT /D) and (VB /B) and the relative motion between xB and xD is plotted, and is shown in Fig. 4. As expected from the lower condition number, the low-gain direction (where xB and xD move in opposite directions) is more pronounced. Thus, for the (L/D)(V /B)-configuration, it is easier to excite the low gain direction.

With model predictive controllers built for the distillation column configurations, the goal now is to identify a process model G(z, θ ) and noise model H(z, θ ) of the true system fDist in equations (2) and (3), using direct closed-loop prediction error identification. The main reason for using closed-loop identification over open-loop identification is that the distillation column, especially in the LV -configuration, is highly ill-conditioned. This means that it is very difficult to excite the low-gain direction, while identification in this direction is needed for proper multi-variable control. Using closed-loop identification partly alleviates this problem, which is also elaborated in [10]. The closed-loop identification experiment is performed in which the logarithmic purities XD and XB are controlled by two single loop PI controllers. The closed loop system is excited by introducing filtered white noise signals with a frequency band of [0, 0.1][rad/min] and a variance of σ 2 = 1 to the reference inputs of the controllers. Subsequently, the control inputs and logarithmic outputs of the system are measured. Further, the following disturbances are added to the feed rate F and feed composition zF:

vF = H1 (s)eF , vzF = H2 (s)ezF ,

Fig. 4: xD vs xB in the (L/D)(V /B)-configuration.

(9)

where vF and vzF are the disturbances that are added to the feed rate and feed composition channels, respectively. H1 (s) and H2 (s) are low pass filters that filter the white noise sequences eF and eZF , for which we have the following expressions:

IV. I DENTIFICATION AND CONTROL OF THE DISTILLATION COLUMN

In this section, models of the LV - and (L/D)(V /B)configuration of the distillation column are identified using closed-loop prediction error identification. The previous section has shown that the LV -configuration is ill-conditioned, which means that is hard to excite the low-gain direction.

σe2F = 4, σe2zF = 0.01, 0.69 , H1 (s) = 10s + 1 0.52 . H2 (s) = 15s + 1

(10)

evaluating the MPC cost-function, the control horizon is set at half the value of the prediction horizon: NC = 10. A quadratic performance index is chosen with the following weights     0.15 0 1 0 , R= Q= 0 0.15 0 1

∆ XD (sim) ∆ XD

1.5

1

0.5

0

-0.5

where Q is the weighting matrix for the output variables (XD and XB ) and R for the input variables LT and VB .

-1

-1.5

-2 1000

1200

1400

1600

1800

2000

2200

2400

2600

2800

3000

Time

(Logarithmic) Composition distillate product

(a)Measured LV -configuration and simulated model output 1.5 ∆ X

D

(sim)

∆ XD 1

0.5

0

-0.5

-1

1000

1200

1400

1600

1800

2000

2200

2400

2600

2800

3000

Time

(b) (L/D)(V /B)-configuration

Fig. 5: Comparison of output of identified model with true output for logarithmic output XD . Figs. 5 show comparisons between predicted outputs of the identified models and the actual outputs for both the LV and (L/D)(V /B)-configuration. Table II shows the results of the closed-loop identification in terms of variance accounted for (VAF) for both configurations: It should be clear that the (L/D)(V /B)-configuration shows better identification results than the LV -configuration. The reason for this is that at high purity control, the (L/D)(V /B)-configuration is less illconditioned than the LV -configuration, which means the low gain direction can be more accurately identified using closedloop identification.

C. MPC performance The control configurations that were identified in closedloop are both implemented in model-predictive controllers. Two cases are now used to compare the control performance. At first, the nominal performance is compared, and a quadratic performance function is used to see how well both model-predictive controllers reject disturbances on the feed rate F and feed composition zF . In the second case, model plant mismatch is introduced to the closed loop system, and a comparison is made between how well both model-predictive controllers are able to cope with model-plant mismatch. 1) Nominal performance: At first model-predictive controllers are run under nominal conditions, where reference values of xD = 0.99 and xB = 0.01 should be tracked, lower and upper constraints at xD = 0.989 and xB = 0.011. Further, the system is subject to feed and feed composition disturbances as defined in (9) and (10). In Figs. 7b and 7a, until t = 5000[min], the output xD is shown while being controlled under nominal conditions. A quadratic performance equation is now defined to compute the average variance over some time window Nt : J=

16

1 Nt

Nt

||xD (k) − xD,re f ||2

∑

(11)

k=t−Nt

×10-8

J(x D ) for LV-configuration J(x D ) for (L/D)(V/B)-configuration

14

12

10

J (xD)

(Logarithmic) Composition distillate product

Measured and simulated model output 2

8

VAF

(L/D)(V /B)-configuration 67.75%

LV -configuration 58.93%

6

4

TABLE II: VAF in closed-loop identification 2

B. Model predictive controller tuning In this section the closed-loop identified models are used to design model predictive controllers for the two control configurations. The performance of these controllers is compared under nominal conditions, and with model-plant mismatch. This section will further demonstrate that the (L/D)(V /B)configuration performs better in terms of disturbance rejection and sensitivity to model-plant mismatch. Supposing a desired closed-loop settling time of approximately tS = 100 min, the prediction horizon is taken as Ny = 20. In order to reduce the computational burden for

0

500

1000

Time

1500

2000

2500

Fig. 6: Performance function J applied to xD for the LV configuration and (L/D)(V /B)-configuration, where Nt = 1000 In Fig. 6, J has been plotted on the time interval t = [0, 2500][min]. As can be seen in the figure, in the (L/D)(V /B)-configuration the model-predictive controller is better able to reject the disturbance on the feed rate F and feed composition zF . A reason for this is that the (L/D)(V /B)-configuration is less ill-conditioned, and as such is less prone to feed rate/composition disturbances [10].

2) Performance with MPM: Model-plant mismatch is investigate new state of the art re-identification techniques introduced at t = 5000 min by applying the following rotation in case of model-plant mismatch, such as the ones described to the controlled inputs before they enter the distillation in [5] and [13]. column: R EFERENCES      cos φ sin φ ∆LT ∆LTm = , LV [1] S. Amjad and H.N. Al-Duwaish. Model predictive control of shell − sin φ cos φ ∆VB ∆VBm benchmark process. In Proceedings of the 10th IEEE International      Conference on Electronics, Circuits and Systems (ICECS), United ∆(LT /D)m cos φ sin φ ∆(LT /D) , (L/D)(V /B) = Arabian Emirates, 2003. − sin φ cos φ ∆(VB /B) ∆(VB /B)m [2] U. Volk, D.-W. Kniese, R. Hahn, R. Haber, and U. Schmitz. Optimized multivariable predictive control of an industrial distillation column where φ = π8 [rad]. Fig. 7 shows the comparison: before considering hard and soft constraints. Control Engineering Practice, the rotation is applied, the variance of xD is only slightly 13:913–927. [3] M. Diehl, I. Uslu, R. Findeisen, S. Schwarzkopf, F. Allg¨ower, H.G. larger for the LV -configuration, but the difference in rejection Bock, T. B¨urner, E. D. Gilles, A. Kienle, J. P. Schl o¨ der, and E. Stein. performance is not dramatic. The situation is completely difReal-time optimization for large scale processes: Nonlinear model ferent when a rotational mismatch is applied at t = 5000 min. predictive control of a high purity distillation column. In Online Optimization of Large Scale Systems, pages 363–383. Springer Berlin The model plant mismatch has a larger effect on the LV Heidelberg, 2001. configuration as compared to the (L/D)(V /B)-configuration. [4] R. Kawathekar and J.B. Riggs. Nonlinear model predictive control of [5]

0.992

0.991

[6]

xD

0.99

0.989

[7]

0.988

0.987

0

1000

2000

3000

4000

5000

Time

6000

7000

8000

9000

10000

(a) (L/D)(V /B)-configuration with plant model mismatch

[8] [9] [10]

0.996

0.994

0.992

[11]

0.99

0.988

xD

[12]

0.986

0.984

[13]

0.982

0.98 0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Time

(b) LV -configuration with plant model mismatch

Fig. 7: Feed noise rejection in both configurations (nominal vs mismatch). V. C ONCLUSIONS AND FUTURE WORK In this work a benchmark distillation column was designed, based on the Matlab code of Sigurd Skogestad, and two different control configurations were compared. It was shown that while the LV -configuration is most often used in research concerning distillation column control, the (L/D)(V /B)-configuration performs better in terms of noise rejection and the handling of model-plant mismatch, due to being less ill-conditioned in high purity regions. In the future, the benchmark distillation column with the (L/D)(V /B)configuration should be considered as test environment to

a reactive distillation column. Control Engineering Practice, 15:231– 239, 2007. Christian A. Larsson. Application-oriented Experiment Design for Industrial Model Predictive Control. PhD thesis, KTH Royal Institute of Technology Stockholm, 2014. Per Erik Mod´en and Micheal Lundh. Performance monitoring for model predictive control maintenance. In Proceedings of the 12th European Control Conference (ECC), pages 3770–3775, Z¨urich, Switzerland, 2013. M. Annergren, D. Kauven, C.A. Larsson, M.G. Potters, Q. Tran, and ¨ L. Ozkan. On the way to autonomous control: a distillation column study. In Proceedings of the 10th IFAC International Symposium on Dynamics and Control of Process Systems, pages 713–720, Mumbai, India, 2013. Sigurd Skogestad. Dynamics and control of distillation columns: A tutorial introduction. Trans IChemE Part A, 75:539–562, 1997. Ivar J. Halvorson and Sigurd Skogestad. Distillation theory. Encyclopedia of Separation Science, pages 19–58, 1999. H.H.J Bloemen, C.T. Chou, T.J.J van den Boom, V. Verdult, and M. Verhaegen. Wiener model identification and predictive control for dual composition control of a distillation column. Journal of Process control, 11(6):601–620, 2001. Sigurd Skogestad and Ian Postlethwaite. Multivariable feedback control: Analysis and design. Wiley, 2005. Xavier Bombois, Max Potters, and Ali Mesbah. Closed-loop performance diagnosis for model predictive control systems. In Proceedings of the 13th European Control Conference, pages 264–269, Strasbourg, France, 2014. M.G. Potters, X. Bombois, and P.M.J. van den Hof. Maximizing the informativeness of experimental data of linear systems under input, output and parameter accuracy constraints. To be submitted, 2015.