Eindhoven University of Technology MASTER Model Predictive Control of a binary high-purity distillation column Oonincx, M.M.E.M.

Eindhoven University of Technology

MASTER

Model Predictive Control of a binary high-purity distillation column

Oonincx, M.M.E.M.

Award date:

1996

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This document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required minimum study period may vary in duration.

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Eindverslag van de eerste fase opleiding INFORMATIETECHNIEK Final report of the graduate project

INFORMATIONTECHNICS

Model Predictive Control of a binary high-purity distillation colum

M.M.E.M. Oonincx

Industrial supervisor: Ir. J.H.J.M. Bazelmans DSM-Servïces, Geleen (NL) University supervisors: Dr. Ir. A.J.W. van den Boom

Prof. Dr. Ir. P.P.J. van de Bosch

faculty of electrical engineering and information technics Technical university, Eindhoven

Abstract

Model predictive control is a control method that uses a model to predict the future consequences of control actions. At every time step, this prediction is used to compute the optimal control action, with respect to a certain criterion. Contrary to most conventional control methods, model predictive control is able to handle constraints on the process variables. This report is the result of an investigation about the applicability of model predictive control on a high purity, binary distillation column.

Itappeared that the high purity, binary distillation process, is strongly interactive, badly scaled and highly nonlinear. The statie behaviour of an extensive dynamic simulator based on a real column is significantly different from the statie behaviour of that real column. The dynamie responses of the outputs to combined actions of the inputs can be intens.

Conventional advanced process control is applied to the simulator in the fonn of a double quality controller. This decoupling ofthe process works well in practice, but on the simulator it is not able to control all the outputs at the same time.

A linear model predictive controller is applied to a simpIer computer model and appeared to be succesful. Tuning the controller parameters is very time consuming. Even more success was obtained when a nonlinear model predictive controller was implemented. Such a

controller uses a nonlinear model to predict the effect of past and future control actions.

Finding the optimal control actions is still a linear optimization problem, where the linear model is apdated. These adjustments are based on so called error through linearization.

From experiences with this application and reading many articles one can conclude that model predictive control is a powerfull, general method. lts advantages, compared to

conventional control, lie in controlling multi-input-multi-output processes with constraints on the inputs and outputs. When there is a model-plant mismatch, the process is ill-conditioned or the process contains nonminimum phase behaviour, some precautions are to be taken, otherwise the model predictive controller will try to invert the process, which could lead to instability. When the process is highly nonlinear, some nonlinear model predictive controller should be considered.

Contents

1 Introduction , 5

2 Model predictive control 6

2.1 History and idea of MPC . . . .. 6

2.2 Description of MPC 6

2.2.1 Modelling the plant and the observer . . . .. 8 2.2.2 The predictor . . . .. 8

2.2.3 The optimizer 9

2.3 MIMü processes 10

2.3.1 Some remarks , 10

2.3.2 An example 11

2.4 Nonlinear MPC 14

2.4.1 Non-linear optimal control 14

2.4.2 Linearized optimal control 15

2.4.3 Iterative QDMC 16

2.5 An overview 18

3 Binary, high purity distillation 19

3.1 What is binary, high-purity distillation? 19

3.2 The processes 19

3.3 The equipment 22

3.4 The environment 25

3.5 The simulator 25

4 Modeling a binary, high purity distillation column 27

4.1 Introduction . . . .. 27

4.2 Statie models in Aspen 28

4.3 Statie behaviour ofthe C2-splitter and the simulator , 33

4.4 Dynamie modeling 36

4.5 Dynamie behaviour of the simulator and the Matlab-model . . . .. 40

5 Controlling a binary, high-purity distillation column , 42

5.1 Tuning basic controlloops 42

5.2 Double quality control 44

5.3 Linear Model Predictive control 51

5.4 Nonlinear Model Predictive control 52

6 Applicabilty of MPC in chemical industry 54

6.1 In general . . . .. 54

6.2 MIMü and nonlinear processes 54

7 Conclusions, discussion and further investigation 56

Bibliography ⁰ • • • • • • 0 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 0 • • • • • 57 AAninvestigation to literature ⁰ • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 0 61 B Stepresponses ... ⁰ • • • • • • • • • • • • • • • • • • • • 0 • • • • • • • • • • • •0 • 0 • • • • • • • • • • • • • • • • • • • 69

C DQC results on the simulator ^{0 • • 0 • • • • • • • • • •} 78

D MPC results on Matlab-model ^{0 • • • • • • •} 87

I Introduction

This report is the result ofthe author's final project as a graduate student in technical physics and infonnation technies at the Eindhoven University of Technology. The project was carried out at DSM in Geleen, from November 1994 until September 1995.Itwas part of an investigation about the applicability of Model Predictive Control at DSM, which was carried out at the department CPC-APC (Centre for Process Control- Advanced Process Control).

Model Predictive Control (MPC) is widely used in oil industry, but not yet in chemical industry. Within CPC-APC, there was a need to know more about the applicability of MPC for chemical plants. Some theoretical knowledge was present already, and the next step was to apply MPC at a plant, in order to learn more about the practical advantages and difficulties of MPC. A high purity distillation column separating ethane and ethylene was chosen as an example. This had several reasons. First, it seems to be a good example for MPC. Important features of MPC, such as constraint handling, play a role. Another advantage was that there is a rigorous simulator of the column available. The MPC controller could be tested on this simulator first, before it would be applied to the real column. Another motivation to use this column is that there is a need for improvement ofthe control strategy. Especially in the case when the working area changes, the current control method is not able to keep the distillation produets at their specifications.

The aim of this project was to implement an MPC controller at the simulator of the distillation column, and to compare its perfonnance to the current controller. In a first approach in coorporation with Adersa, a French engineering company specialized in MPC, a linear model predictive controller named Hiecon was implemented. Hiecon controls the process, but the results are not satisfactory. During experiments it was found that the

distillation process was highly nonlinear. Statie and dynamic models were obtained to capture the nonlinearities. In this report the statie and dynamie behaviour of the distillation process as weIl as some advanced controllers are designed. The results of the controllers are presented and some general conclusions about the applicability ofMPC at DSM were to be drawn.

The remaining of this report is organized as follows. First, some theoretical backgrounds about MPC and its nonlinear variants are given in chapter 2. Then, in chapter 3, distillation is outlined. What are the processes and in what of an environment does distillation occur?In chapter 4 the process is modelled statically and dynamically. The behaviour of a high-purity, binary distillation column is studied. In chapter 5 a double quality controller, a linear and a nonlinear model predictive controller are applied to the column. Their results are presented and finally in chapters 6 and 7 some general conclusions are drawn about the applicability of MPC in chemical industry and the behaviour of the high purtiy distillation column.

Model predictive control

2.1 History and idea of MPC

The current interest in MPC can be tracked back to a set of papers that appeared in the late 1970s. In 1978 Richalet et al. [20] described successful applications of 'Model Predictive Heuristic Control' and in 1979 engineers from Shell outlined 'Dynamic Matrix Control' (DMC) [5]. Both algorithms use an explicit dynamic model ofthe plant to predict the effect of past and future actions of the manipulated variables on the output (Thus the name Model Predictive Control). The future moves ofthe manipulated variables are found by optimization with the objective of minimizing the predicted error subjected to operating constraints. The optimization is repeated at each sampling time based on updated information (measurements) from the plant.

2.2 Description of MPC

People who want a more detailed description are advised to read the report of Van der Burg [24]. The theory ofmpc and some single-input-single-output (SISO) examples are outlined there in detail. In this chapter only basic ideas and important equations are viewed.

Consider a process with one or more inputs, one or more outputs, and zero or more measured disturbances. In MPC, a discrete time model of the process is used to predict the outputs over a certain time horizon, for different sequences of control actions on the inputs.

Desired outputs (reference trajectory) are given, and the sequence of control actions that gives the 'best' results is chosen. To decide what the 'best' control sequence is a criterion is given. From this control sequence the first control action of each input is applied to the process. Then, at the next time step, new measurements are available, and the same thing is done again. Every time, only the first contoI action of the computed sequence is used. This is illustrated in Figure 2.1 on the next page: at time k, a control sequence u(k), u(k+ 1), ..., u(k+m-l) is determined such that the output over a horizon p tends to its setpoint (the dashed line) in a desired way. The input u(k) is applied to the process. At the next time point (the second graph), the output is measured, and it is different from what was predicted. A new control sequence is determined, using the new information such that the new predicted output (indicated with *, the old one is indicated with 0) tends to its setpoint in a desired way.

Actually, it seems a very natural thing to do: one looks at the consequences of each control action, and determines an action based on these predicted consequences. However, this requires a lot of computation time at every time step. Due to this, MPC has become more interesting with the development of faster computers.

MPC is especially useful for processes with constraints on the inputs or outputs. MPC can take these constraints directly into account. One should be careful in putting constraints on the variables, especially on the outputs, since it may make the problem infeasible, and it also takes more computation time.

time=k

u(k+i)

k k+m

time=k+l

k+p

- - - - -~;s-'-'-lII! ^lil! lIIl )IE y(k+i+l)

*

*

o

u(k+i+1)

k+l k+m+l k+p+l

Figure 2.1: Illustration ofModel Predictive Contra/.

In MPC three horizons have to be chosen: the model horizon n, the prediction horizon p and the control horizon m. The model horizon is the number of samples needed for a response of a stabie output to reach a steady value on a step action on an input. The prediction horizon is the number of samples for which the prediction is computed. This horizon should be chosen large enough to incorporate all important effects. The control horizon is the number of

samples over which the inputs may vary. After this period, the inputs are assumed to be constant.

A model predictive controller can be constructed of the following parts:

*

*

*

*

All of these parts will be outlined. The MPC algorithm will be fonnulated in state-space fonn. Later on it will be shown that an extension of linear MPC to nonlinear MPC is the replacement of some of these parts by their nonlinear versions. Note that the theory ofMPC controllers is not limited to single-input-single-output (SISO) processes, but is especially attractive for MIMO processes. In the describtion of MPC the assumption of applicability only to SISO processes is never made and if the text may suggests it one should bear in mind that the intention is controlling MIMO processes.

2.2.1 Modelling the plant and the observer

Suppose that predictions of the output i Yi(k), Yi(k+1), ..., Yj(k+n-l) are available at time k.

In these predictions the inputs, disturbances and measurements up to time k are taken into account. Let's name this set of outputs together with similar sets for the other outputs the state Y(k). After the optimizer has calculated the best control sequence the first control move àu(k) of each input has to be applied and the state has to be updated, taking the control actions into account. At the same time the prediction Y(k) has to be shifted, by multiplying with 'shift' matrix M, to get the prediction Y(k+ 1) containing the outputs at time k+ 1, k+2, ..., k+n.

In most of the original MPC fonnulations a stepresponse model of the plant is used to predict the future behaviour ofthe controlled variables. Stepresponse models give good insight in the process, although the number of parameters can be plentiful. For a MIMO system with nu inputs and nyoutputs we get a stepresponse matrix S with dimensions nyx nu in which the element Sjj is the response of output i caused by a unit step at input j.

State estimation is achieved by adding the difference between current measurements ymek) and the current predicted states y(k) multiplied by a filter F. In this report F contains the arbitrarily tuned 'filtergains' but it could also be a calculated Kalman filter. The current predicted state y(k) is just the first state of each output in Y(k). The matrix N selects these states.

The model can be represented in the following state space fonn Y(k+l)= MY(k) + Sàu(k) + F[Ym(k) - y(k)]

y(k)= NY(k)

2.2.2 The predictor

(2.1) (2.2)

The objective ofthe predictor is to generate a vector Yp(k+llk) ofpredicted open-loop outputs over a horizon of p future time steps, the prediction horizon. This prediction vector is then used as an input to the optimizer. The predictor is described by the following equation:

(2.3) where Mp is also a shift matrix, which selects just the first Pi states of output Yi(Pi 5: nj).

2.2.3 Tbe optimizer

The optimizer finds a control sequence that optimizes an objective function. As objective function

J = min {

Ilr

Ilr

dU(klk)

is used where dU(klk) is the optimal control sequence computed at time k for m future input moves (mi is the input horizon for input i and mi s: min(p). Ym(k+ 11k) is a vector of outputs predicted at time k, over a horizon of p future time steps, including the effect of the m future input moves. R(k+ 11k) is a vector describing the desired output trajectory (setpoints). Finally

r

r

To this objective function several terms can be added. If the process allows it one can add limits or setpoints to the inputs as weIl. Even economic or environmental objectives can be added. The weigth matrices determine the 'value' of the different objectives.

When there are no constraints neither on the output nor on the input, the least squares solution to this problem can be analytically calculated. Only the first move of each input is implemented, and the resulting optimizer is a constant gain matrix

Kmpc,

du(k) = K_mpc [ R(k+11k) - Y (k_p +1Ik) ]

The algorithm is illustrated in Figure 2.2 on the next page.

In reallife processes however constraints are always present. Their importance has increased, because supervisory optimizing control schemes frequently push the operating point towards the intersection of constraints. The main attraction of MPC is that the engineer/operator can enter the constraints directly and the algorithm will find the best solution satisfying all of them.

(2.5)

One can distinguish two kinds of constraints, i.e. soft and hard constraints. A 'solution' of an optimizing problem with soft constraints is always found but the solution technique is sensitive to local minima. Anoptimizing problem with hard constraints can be solved. Hard constraints however might make the problem infeasible.

. Model : 1

,--

, - - ---~~----1 Y (k)

,~_u(k) ~

~~~ Plan t ~---~-r~

I :

! I I

~.---.---~ i

--I

i

I

·i -

!

Optimizer Predictor Observer

Figure2.2: The MPC algorithm based on stepresponses pictured in state space form.

2.3 MIMO processes 2.3.1 Some remarks

An advantage of MPC is that it can handle multi-input-multi-output (MIMO) processes.

One needs a model that describes all relations between inputs and outputs and MPC will control the process in an optimal way, making sure the constraints are respected.

Gelormino et al [13] report about a successful MIMO MPC application to achieve an economic and technical objective. A quotation: 'We describe the application ofMPC to a large-scale, constraint-dominated problem: the minimization of combined-sewer overflows (CSOs) in the Seattle metropolitan area. The key decision variables are flowrates at 23 locations throughout the sewer network. There are approximately 40 output variables that must he kept between lower and upperbounds. MPC reduces CSOs by 26%'.

Theoreticaly one can think of a MPC controller for a whole plant that predicts and controls.

The outputs of certain processes are the disturbances of other processes. If one could design a MPC controller that controls both processes, it could predict the disturbances of some

processes simply by predicting the outputs of other processes. What used to be an unknown disturbance now becomes a known variabie, which value is predicted over a time horizon. So MPC can take the necessary precautions knowing the major future disturbances. This is one advantage ofthe 'predictive' part of MPC.

There are cases with model-plant mismatch when a MPC controller does not perform weIl.

There always is a certain mismatch between a real process and a model. The mismatch can have various sources: uncertainties in the model parameters and the model structure, inaccuries of the actuators and measurement devices, etc.

Multi-variable systems introduce a special problem here, because the 'gain' of a multi- variabie process varies not only with frequency, but also with 'direction'.Itis shown that if a plant is ill-conditioned irrespective of scaling, that the control performance is strongly affected by input uncertainty, in particular, when the controller is trying to invert the plant.

The MPC controller is such a controller, especially, ifthe penalty weight on the input moves is low. Since there is always some input uncertainty, it should be clear that a MPC controller is potentially bad when used for an ill-conditioned plant. The binary, high-purity column will appear to be such an ill-conditioned plant.

2.3.2 An exam ple

Now let's look at a MIMü two by two process. The model and the process are both described with the same four stepresponses shown in Figure 2.3. For this process two MPC controllers are designed. In experiment 1 there are no constraints and there is almost no weight on the control moves. In experiment 2 the weight on the control moves is increased and the outputs and inputs are limited. The constraints are

o

y,

-0.I~Y2~0

I~ud ~ 0.2 for i= 1,2.

The filter F is not of much use here, since both the model and the measurements are 'perfect'.

Steprespol'L'ie ofyl on input uI Steprespol'L'ie ofyl on inputu2

40 30

10 20

sample

Steprespol'L'ie ofy2 on inputu2 40

10 20 30 sample

Steprespol'L'ie ofy2 on input uI -8

or--<"---~--~---'

-2 -4

-10

-12L--_~_ _~_ _~_---'

o

>. -6

0.8

0.6

S!.

0.4

0.2

-0.5

S!.

-1

40 30

10 20

sample

o

30 40 20

sample 10

ol---"-~_ _~_ _~_---'

o

Figure 2.3: The stepresponses used in the MIMO example.

First the setpoint is changed at timesample 2 from R

=

=

experiment 1 with dotted lines and the inputs and outputs of experiment 2 with normallines.

The setpoint changes are assumed not to be known in advance otherwise the MPC controllers could have given an even better result. At timesample 31 the setpoint changes to R= [2 0]'.

Response output variabieyl 2.5

....

---

2 r---·---"---":.~_{I, 1}

1 ,", /

1.5

; I "

>. , _{\ I}

,,

\

,

"

0.5

10 20 30 40 50 60 70 80 90 100

t(samples) Response output variabiey2 0.1

I

\

0.05 _"

"

"

0 r---'-ï-- - -- -- -^{I ,} ï^{L; ' _}- - - _:--

~ _,^", _,,

-0.05 _,' , ,

\ ,

", ^\ ,

-0.1 , ^{\ I}

-0.15

0 10 20 30 40 50 60 70 80 90 100

t (samples)

Figure 2.4: The responses ofthe outputs in experiments 1(:) and 2(-).

Action manipulated variabie ui

0 ,

-I ,

-

-2 ^I, _--_-

",

" "

-3 ^"-^-

-40 10 20 30 40 50 60 70 80 90 100

t (samples)

Action manipulated variabieu2

"

"

"

0

"

,

SI-I

,

~r^I

I ----... _ - - - -

-2 ^{- 1- , , -~,.}

-30 10 20 30 40 50 60 70 80 90 100

t (samples)

Figure 2.5: The inputs in experiments 1(.) and 2(-).

The setpoints are reached and the controller 2 even satisfies the constraints. The outputs and the inputs are much smoother applying controller 2. This is due to the constraints and the weighing of the change in inputs.Itis easy to implement constraints and weights in MPC and they are straightforward. A sharp reader might notice by looking at thestepresponses that they could describe a high-purity distillation column. The interactions between the inputs and outputs are significant. This is seen by the large control actions needed to reach the final setpoint. Interaction will also be a topic in chapter 4.

2.4 Nonlinear MPC

A question that arises is how to handle nonlinear systems. While we can deal with mild nonlinearities just by detuning linear controllers, it is likely that in the presence of strong nonlinearities nonlinear controllers offer distinct advantages. Though there are many important unresolved details, the conceptual extension ofthe MPC structure to nonlinear systems is straightforward. For nonlinear systems the assumption of an unmeasured additive disturbance acting at the process output is usually artificia1. Indeed, for nonlinear systems the issues ofmodel error (robustness) and unmeasured disturbances become indistinguishable.

All blocks in an MPC controller can be nonlinear. The process model is a simulation program where the nonlinear differential equations are solved on-line in parallel with the process. A general technique for the design of the nonlinear controller is not available to date.

Three attempts reported in the literature will be discussed next.

2.4.1 Non-linear optimal control

In analogy one can define the general nonlinear objective function

ti

min G[x(if)] +

J

u

subjected to

x =f[x(t),u(t)]dt • x(to) =Xo

h(x,u)= 0 g(x,u) ~ O.

(2.6)

(2.7) (2.8) (2.9) The solution to this problem when (2.8) is not present and g(x,u) varies only with u can be found in all classical references on optimal contro1. The variational methods become extremely complex when inequalities involving the states are present. Because of the computational complexity, these methods are not suitable for on-line use.

Itis more promising to discretisize the control vector with respect to time and to convert (2.6)-(2.9) into a nonlinear programming problem (NLP). Itis also possible to use a 'black box' simulation model instead of (2.7) and to compute the gradients necessary for the mathematical program numerically.

A powerful method for solving NLP problems is successive quadratic programming (SQP).

SQP approximates the objective function of an NLP problem by a locally quadratic function, and the constraints by locally linear functions. By solving this approximate problem using·

quadratic programming, a search direction is obtained. The objective function is minimized in this search direction, and the process is repeated at the new location until the optimum to the NLP problem is determined. Thus, SQP is an infeasible path optimization method.Itdoes not require that the constraints be satisfied (the model equations be solved) at each iteration, but finds the optimum and satisfies the constraints simultaneously.

Rhiel et al. [19] implemented a nonlinear modelbased inferential control scheme on a 42 tray, column with a binary system.

2.4.2 Linearized optimal control

Garcia [7] linearized a problem similar to (2.6)-(2.9) and solved the linear problem by the DMC/QDMC approach. As the state changes new stepresponse coefficients are obtained to update the linear model. The nonlinear model is used to perform the model prediction. That is, the nonlinear differential equations are integrated on-line, in parallel with the process, while the controller is a linear QDMC controller.

This method is a logical extension ofthe general description oflinear MPC.Itstarts with the description of a process by some nonlinear differential equations. The nonlinear model predicts the states by integration. Integration of the linearized equations at every scan time for a unit step change will produce the coefficients of S. Having both the predicted states and S the MPC problem can be solved and the optimal control moves can be calculated. This is still a QP problem !!. Figure 2.6 illustrates the algorithm.

In this Figure the usual MPC structure can be recognised. The dashed lines indicate updates ofthe block Kmpcand F. With an update ofKmpc we mean the update ofthe linear model (the stepresponse S) in the optimizer algorithm. The major disadvantage ofthe nonlinear QDMC algorithm by Garcia (from now on NLQDMC) is that it may not perform well in controlling integrating processes and may lead to instabilities when applied to open- loop unstable processes. Stability, better disturbance rejection and better performance are observed when Gw:cia's nonlinear version ofQDMC to open-loop unstable nonlinear

processes is extended with state estimation. The state estimation could be a Kalman filter (F) designed from the linearization of the nonlinear process. If the linearization changes every sample time the filter has to be updated.

Note that several approximations have been made to simplify the nonlinear problem. First, "

the superposition of past and future effects is not rigorously valid for nonlinear systems.

Secondly, the coefficients ofS are actually different for each ofthe future moves. The major advantage of the proposed algorithm compared to the nonlinear programming approache is that only a single quadratic program is solved on-line at each sampling time using the same techniques as for linear systems, which makes the proposed algorithm an attractive option for industrial implementation.

u(k) 1---1

~ -~~ Plant r--

I I

, .••...•_. __... . ----'I

. . . •-I-

Model

y. (k) m

-~--I

- j F f-.(- -- .

_ - - - J

1

lil I I I

I

i-~-~=li~-~a~1

I---'-~ model r---T~·~lNr----~-'

,_~I---1 l---r-.--~.-i

I I

K mpc

!"'<-'---

Optimizer

.•• t o . I

Y(k+ llk);-- ^I

.- .-- - Mp ~-

Predictor

Figure2.6: An imbedded non-linear model. The QDMC approach.

2.4.3 Iterative QDMC

Simminger et al. [22] go even further. They extend the linear K_mpc 'algorithm' into a nonlinear one. If a linear model is used for the future control moves the nonlinear contributions are not included. Let's model those contributions as good as possible. If a nonlinear model is available the errors made by linearization can be modeled as disturbance (Figure 2.7)

nonlinear

1-1

y

:/~

.---,--- I I"

~ mear

/1

---~~---

1---

k k+l

.-f-

Errors due to nonlinearities dn1

-'... --- ---r-

k+P

This is globally what Simminger proposed, but we slightly changed the procedure. The modified version will be explained from Figure 2.8. Since the effect of one control move cannot be calculated for reasons that will be explained in chapter 4, the effect of past control moves is obtained with a linear stepresponse model (switch A = 0). The same model is used in a QDMC algorithm finding the first 'optimal' control sequence.

r----

----,-__J

~----.: Plant

: i

L_

R

Linear model

, I

~---

Figure 2.8: Block algorithm ofthe algorithm.

This control sequence and the current disturbances are implemented and the predicted nonlinear output Y^mnl is calculated (switch B= 1). Simminger then subtracts the linear prediction which results in a 'disturbance' due to nonlinearities. Note however that he has the effect of past control moves calculated with a nonlinear model, so a more accurate state to start the optimization with. Since such a state is not available the errors due to 'past' and 'future' nonlinearities cannot be distinguised. To correct for both ofthem the linear effect of the control moves is subtracted from the predicted nonlinear outputYmnl, resulting in ypnl. ypnl is the effect of past control moves, current and past disturbances and even a first correction of the stepresponse model.

This predicted output is used for a second QDMC run (switch A= 1) resulting in a new 'optimal' control sequence. With this sequence a newYmnl and a new ypnl are calculated. This procedure should be repeated until a certain stop criterium is satisfied. The procedure is very time consuming and if one wants to use it on-line the stop criterium should be loose. The final 'optimal' control sequence is then applied to the plant (switch B⁼ 0).

Examples of Simmingers theory only handle CSTR-processes. The modified theory just treated is applied to a model of a binary, high purity distillation column. The results are found in chapter 5.

2.5 An overview

Attention has been given to five different MPC algorithms i.e.linear MPC, nonlinear optimal control, NLQDMC, iterative QDMC and modified iterative QDMC in order of appearance. In table 2.1 an overview is given of the way the different aspects of MPC are treated.

Table 2.1: An overview ofsome MPC controllers.

name past stepresponses optimizer time (s)

linear MPC linear once linear <20

modified iterative linear, nonlinear once linear, <600

QDMC corrected iterative

NLQDMC non-linear at every sample linear

iterative QDMC non-linear at every sample linear,

or once iterative

Non-linear non-linear of no interest non-linear

optimal control for this method

The first column contains the name of the algorithm. Then the way the effect of past control moves and disturbances is calculated is described. The third column denotes the times the stepresponses are indentified. Then the optimizer algorithm is classified. Finally

something is said about the time needed for a mpc-controller action if it is implemented in MATLAB and tries to control a high-purity distillation column.

Only the first two controllers are implemented. The controllers are ordered on the assumed time-consumption of controller steps, i.e. a linear MPC step takes about 20 seconds and a nonlinear optimal control step is dependent on the accuracy of the answer, but will take great time.Itis not clear if the most time-consuming is also the best performing algorithm, although nonlinear optimal control probably is.

Binary, high purity distillation

What is to be controlled is a binary, high purity distillation column. In this chapter the process of distillation, some of the needed equipment and an environment are outlined. In this way one gets more acqainted with the nomenclature and the processes. This will come in hand when in chapter 4 the process is to be modeled.

3.1 What is binary, high-purity distillation?

Distillation is widely used in chemical and petroleum industries to separate mixtures of components into purer product streams. This separation is based on differences in 'volatilities' (tendencies to vaporize) among various chemical components.

One speaks of distillation if the vapour phase is created by adding heat to and evaporizing ofthe liquid phase. By absorption or rectification of a component out of a vapourphase the absortionmedium is fed back as liquid phase (reflux). If a component out of the liquid phase is removed by means of adding vapour phase (vapour generated by reboiling), one speaks of stripping.

The separation is based on the difference in composition between vapour and liquid at equilibrium. There is an enrichment ofthe most volatile component in the vapour phase.

Through condensating the vapourflow a liquid is obtained that is richer in the most volatile component than the original mixture. If a part of the gained distillate is vaporized again the vapour would become richer again. This principle is the essence ofwhat is called rectifying distillation.

In binary distillation two components are involved. High purity distillation means that the mixture has to be separated into very pure components. So binary, high purity distillation handles the highly separation through distillation of a mixture containing two components.

3.2 The processes

One ofthe basic laws of nature is that there must be a driving force to achieve motion.

Water flows downhill because of the force of gravity, just as electricity flows from high potential to low potential. Similarly, the driving force within a distillation system allows the separation of one component from another. This driving force for separation is the difference in vapour pressure between components in a system.

Consider a pure liquid at a temperature and pressure at which it is in equilibrium with its vapour. If equilibrium is defined as a state in which there is no driving force for change, then equilibrium can be used to define vapour pressure. This equilibrium pressure is called the vapour pressure of the liquid at the given temperature.

Vapour pressure and boiling point are closely related (Figure 3.1). The vapour pressure is the system pressure at which a pure component will boil at a given temperature. For pure water at 1DODe,the vapour pressure is 1atmosphere. Distillation occurs because of differences in component vapour pressures, for, at a fixed temperature, one component in a mixture has a higher vapour pressure and is more volatile.

Vapour pressure Benzene (more volatiIe /

,

/ / / / /

/ "

____ -- -- Toluene(lmvolatile)

o ---;-~-"'--'--,---,-,---~~,- ' - T - - - O.S

Vapour pressure (alm)

/

Vapour pressure of water

J

O.S Vapour pressure

(alm)

o SO 100

Temperature

r

r

Figure 3.1: Vapour pressure-temperature curveslor water, benzene and toluene.

The volatility of a component in a mixture is defined as Y^j*

K. -I (3.1)

x.I

in which

yt

(X, .. -

IJ K.

J

(3.2)

In case of binary systems the indices can be omitted and x and y are referred to the most volatile component. The relative volativity then becomes

ex = Y *(1 - x).

x(1 - Y *)

(3.3)

In principleIXis a function of temperature, pressure and composition, but oftenIXremains constant in a certain temperature range. For ideal systems we can apply Raoult's law

" 0 "

PI

=

=

in whichPI" is the partial pressure ofthe most volatile component,PlOis the vapourpressure of the pure component by the actual temperature andPIis the total pressure. For a binary system we can write for the total pressure

• " 0 0

P_t = PI + P2 = xPI + (1 - x)P2 (3.5)

After some calculations one can derive for the relative volality of an ideal binary system

ex

=

The relative volatility is an indicator for the ease of separation. The higher^IXthe easier we can separate the components from eachother. WhenIX= 1 separation ofthe two components by simple distillation is impossible. In Figure 3.2 the temperature-composition, pressure- composition and composition-compositiondiagram are plotted.

Vapour PreS9ure constant

Liquid

o X

Male fratlian 1 in liquid

..

"

~>

.5

---/'i

/ /

/ / 1/

/ /

/ /

"

~

ö /

I i / /

oL _

Temperature constant Liquid

~ - _ . _ - - , . _ . _ - - - _ .__. _ - - - - -

o X3 I

Mole fraction I

y-J

,BUbblc.points xv~P~~1 ^pi

I I •

-, I -- -- - - / -

J.

/f

~

p2 I I

: IDew.pointl Y VIpressurei Vapour bpl

Bubble-pointl xVItemp

" ----_._---_.._.._--^{- - - - ~}

o x,y 1

Mole fraction I bp2; IDcw.points y vs tcmp:

~_M

"h-/\,\ I K -

~, ^':

J: _'~..::-,

ol ^D

Figure3.2: T-x, P-x andy-x diagram/or a non-ideal normal system.

We can see that we are not dealing with an ideal system, because the line pressure versus x isn't a straight line, which it would be ifwe would apply Raoult's law. Suppose we heat up a liquid-mixture with a composition G at a given constant pressure (in a cylinder closed with a plunger). The Figure shows that the liquid boils at a temperature H (bubble point). The

composition ofthe first vapour that is created is J. Further heating will drag x from H to N and y in equilibrium with x from J to M. As long as the boiling point stays between H en M the mixture-state is in a boiling cycles and we have a liquid-vapour equilibrium. As soon as the last liquid with composition N evaporates, only vapour is left with a composition G. The other

way around: if we have a vapour-mixture 0 and we cool this down, then shall the first liquid- drop occur by a temperature M (dew point).

To give an impression how non-ideal (and thus how big the deviation from Raoult's law) the system can be the same diagrams as in Figure 3.2 are plotted, but now for an azeotropic mixture ofCSz-acetone (Figure 3.3). This isjust an impression. Such systems will not be considered in this report.

Pressure constant

I Vapour

i ^!

bp2.~.iDDoowW"IPoinll YVIlomp

i

~I\~~ ~ _{i: \:-}

.- ~

Bubble·points xVIItem~

Liquid

'---_...._ - - - ,

o x,y I

Mole fraction 1

p2

Temperature constant

Liquid

i

Bubble.pointsx:VIpressu+

I

!

/

! - -

!

Dew.points)' VI prcSlurej Vapour

L . ... •••. i

o x,y 1

Male fraction I

pi

-I

J

.s I

.~

J

~ ^{I /}

~ ^{! /} i ⁱ

I

:r

Q

iL__

,

o X

Mole fraction I in liquid

Figure 3.3: T-x, P-x and y-x diagramfor a non-ideal azeotropic system.

3.3 The equipment

In this paragraph the equipment needed for distillation will be viewed. We will start with trays, those are the places where the true separation is achieved and end in the next paragraph with the description of a plant that refines raw oil.

downcomer

Two stages in a distil/ation column outlet _

weir Iiquid active tray

area --"'+----~

Figure 3.4:

Figure 3.4 shows a section of a distillation column. Vapour generated below this section is transcending through the 'holes' in the active trayareas and the liquid holdups on the trays. Liquid formed above this section is flowing downstairs through the downcomers, over the trays and over the outlet weirs if the holdup is big enough. In the liquid holdup above the active tray area energy is exchanged and separation is achieved. The vapour that arises from the hold up contains more of the most volatile component than the vapour that enters this holdup.

Analogously the liquid that leaves this

holdup by passing the outlet weir is richer of least volatile component than the liquid that

enters the holdup through the downcomer. So on each tray the separation of the components is carried out a bit further.

The change in concentration at a stage is dependent on the concentrations in the stream that enter the stage. The difference in concentration is the source ofthe change (diffusion) as we shall see in chapter 4. The higher these differences in concentration are, the more effective a change in streams will be.

The vapour stream toot transcends and the liquid stream that flows down and their

concentrations determine the performance of a tray. The higher the streams, also called flows, the higher the total separation will be. That is of course if the trays operate in the normal working area. Ifthe flows take on extreme values different effects take place (see [14]).

The way the fluid and the vapour are mixed on a tray also depends on the magnitudes of the flows. One can read about it in the literature [12].Itmay be clear that when the process leaves the normal working area the tray-efficiency will be reduced and the separation will be less.Itis therefore important to stay in the normal working area.

---~--_._-

Overhead vapour

~\_{---- 1 \-<---,}^Reflux

- 2 !

Now let's consider the whole distillation unit including a column, a reflux vessel, a condenser and a reboiler. These are the major parts to do distillation and the configuration is drawn in Figure 3.5. The trays inside the column are numbered from the top downward starting with 1 and ending with the total number of trays n.

iCondenser

-(ç;;---

,~i Refluxvessel

I,

~V,kliiii;;r.

~.-~

---.

Lt D

Column xd

___ nrt

I

_ n!

,..----,

~ 9_R~;iler

i

B !Bottoms xb,

c Feed •

--~

F

At tray nf a mixture of components called feed enters the column. The amount of feedflow is denoted as F and the

composition as c. Since we will discuss only binary distillation all compositions will be reproduced as the concentration of the most volatile component.

At the top ofthe column the vapour Figure3.5: A distillation unit.

stream leaving tray 1, which is called

overhead vapour, is condensed in the condenser. Heat is transferred out ofthe condenser at a rateQc'The fluid created is gathered in a reflux vessel. From this vessel flow a product flow D and a refluxflow Lt. The product flow D with compositionXd'called distillate, flows to some destination elsewhere. The refluxflowLtis pumped back into the column on tray 1. The liquid flow inside the column largely depends on this reflux.

At the bottom of the column below tray n there is a liquid holdup. From this holdup a flow is tapped. A part of this flow is the product flow B with concentration xb,called bottoms. The remaining part is evaporated by transferring heat at a rate

Qr

vapour flow is led back into the columnjust beneath tray n and largely prescribes the vapour flow inside the column.

What is outlined is a standard distillation unit that can separate a (binary) mixture into its components. The column that is to be controled is the c451 distillationcolumn at the NAK4 of DSM (Figure 3.6). The c451 distillation column also called the C2-splitter is a 96-metre high distillation column containing 156 trays and is used for separating ethylene (C₂H₄₎ and ethane (C₂H6)·

Ethylene, the most volatile component is the main product. Most of it leaves the column at the top. A small part of the overhead vapour is the productstream D_v- The remaining part is condensed and led into the refluxvesse1. A distillate flow DI and a reflux flowL_tleave the refluxvesse1.

Ethane leaves the column at the bottom as a flow B. Not at the bottom but from a big 'tray' just above, a liquid flow is tapped, evaporated in two reboilers into vapour and fed back into the column just beneath tray 156. Because the capacity of the reboilers at the bottoms is not enough, an intermediate reboiler is added at tray 139.

Dl

The feed can enter the column at tray 126 and at tray 114. In practice the feedvalve at tray 114 is closed, so we assume a feedentrence only at tray 126.

Several quantities are measured and controlled in this unit. The most important measurements are:

*

*

*

*

*

*

Qr

B

Figure 3.6: The 'C2splitter' ofthe NAK4

Note that Xlis not analysed in the distillate but in the overhead vapour and that_Xbis not analysed in the bottoms but in the liquid at tray 156. The analysis takes 12 minutes forXband IS minutes for ^Xl.The main manipulated values are the flows Lt, Dl, Dv, Ir, Qr and B.

3.4 The environment

At the NAK4 crude oil, also called NAFTA is cracked and then refined into more valuable components. Some ofthe components are hydrogen, methane, ethane, ethylene, ethyne, acetylene, etc. The plant consists offurnaces, columns, reboilers, condensers, vessels, etc. The plant can be divided in three parts, i.e. a 'front-factory', a 'middle-factory' and an 'end-factory'.

In the front-factory are the furnaces in which the NAFTA is cracked. Then some major separations take place, like the separation of hydrogen, C₂-groups, C₃-groups, etc. In the middle-factory and end-factory the separation of components is carried out further. The c451 column is in the endfactory.

The feed ofthe distillation column c451 is the topproduct of the distillation column c431 that separates ethane and ethylene from heavier components. The bottoms ofthe c2-splitter contains the less valuable ethane. This is fed back to the fumaces and cracked again into more valuable components. The reboilers and condensers subtract and add energy to a

propylene (C₃₎ flow that acts as the heat system. A part ofthe vapourtopproduct is sent to the 'ISato' net (also called 'logistic'). The remaining part is complemented with evaporated fluid distillate to maintain a '5 ato' net. The remaining fluid distillate is cooled down further and sent to the fluid ethylene storage.

3.5 The simulator

To train personnel and to develop new control strategies an extensive dynamical

mathematical model has been made. The model is available on a mainframe computer. The model is so complex that it takes the same time to do a simulation as it does to do a real experiment. The reliability of the model was checked by process operators who normally operate the real situation.Itappeared to be a 'hifi' model.

The simulator gives thus a good representation of the real situation. Though there are some differences. The simulator only represents a part ofthe end-factory. i.e. from the column c451 to the distributions to the 5 ato net, the IS ato net and the liquid storage. The c2-splitter in the simulator contains only 105 trays (with an efficiency of 100%) instead of 156 as in the rea!

column. The assumption was that the efficiency ofthe trays in the rea! column is approximately 80%.

In the simulator it is possible to manipulate and measure the feedflow and

feedconcentration. In the real column feedflow and concentration are disturbances, which means that they are given and cannot be influenced.

VAX

Agnes model

,

---_.~-----~-

DEC stations

• place for something new, e.g. a new mpc version

VMS

Figure 3. 7: The way the simulator is build up and Us communication possibilities.

One can interact with the simulator via GIDS. GIDS is the userinterface between user and model (Figure 3.7). Via GIDS it is possible to operate all valves, setpoints, etc. There are also written some userprograms in FORTRAN that communicate with GIDS. In Figure 3.7 a section Hiecon is drawn which is the linear model predictive controller implemented by the French firm Adersa that is specialised in MPC. Furthermore one can see the interactive programs this controller needs to communicate with GIDS.

Itis also possible via GIDS to change or look at variables, such as sizes of vessels,

concentrations and temperatures. A program that saves simulation-results makes it possible to plot and study simulation runs off-line.

Modeling a binary, high purity distillation column

In controlling the simulator one has to know its statie and dynamie behaviour. Distillation process can have large time constants resulting in long settling times. This means that

experiments can last for days (even weeks). The simulator, which simulates more than only the column, needs the same time as the real process. In order to study the behaviour of the process and to design and test model predictive controllers some models were derived.

4.1 Introduction

This chapter describes the modeling ofthe binary, high purity distillation column. Several models and simulators will be treated so it is useful to name the different processes.Itall started with the real distillation column C451 at the DSM business unit NAK4. This column will be called the 'C2-splitter'. A high order dynamie model of the C2-splitter is available and will be called the 'simulator'. From both processes statie models are obtained. Those models are set up in the software package 'Aspen 9.0'. The model describing the C2-splitter will be called 'Aspen-C2-splitter'. The model describing the simulator will be called the 'Aspen- simulator'. Finally a nonlinear dynamie model of the simulator is programmed in the software package Matlab/Simulink. This model will be called the 'Matlab-model'.

An overview ofthe models is given in Figure 4.1. In this Figure the models (and the process) are ordered. The arrows in the Figure indicate the relations the models have. If an arrow is pointing from model A to model B, model B is a simplified description of modelA.

_____ I

~ ^simulator ~ MaUab-model F I

_J -~--. ^:

dynamie

- i I

~--~---i---I---

:1 Y

---- --- ,---, I

statie

: Aapen-C2-lIplitter: --~ Aspen-.lmulator: I

~------._-- -- --- - I

real proce..

Figure 4.1: An overview of(he used modeIs.

Though the Aspen-simulator is not a simplified descpription of the aspsen-C2-splitter, they are c10sely related. A model of the Aspen-C2-splitter was already at hand and with some modification and some tuning a duplicate model was fitted on the simulator. That is why the arrow from the Aspen-C2-splitter to the Aspen-simulator is dashed.

4.2 Statie models in Aspen

To ohtain good insight in the static hehaviour of a process, without excessive hurdening the process, a static computer model can he used. In such a simplified model modifications can easily he made and the static hehaviour can he studied.

For distillation processes static models rely on the principles of conserving mass and energy. Distillation is a process where no mass or energy is created, nor a process where mass and energy are transformed into eachother. One can look at the principles of conservation separately to form a static model.

Taking the principle of conservation of mass alone into account will he a first approach. A schematic plot of the distillation unit is drawn in Figure 4.2. Since only static resuits are considered holdups on trays are no longer important neither are the levels in the reflux vessel and the hottom. The refluxvessel and the hottom can actually he modelled as a tray.

In Figure4.2 a dashed line separating the distillation unit from the environment is drawn.

Mass conservation can he translated in 'what enters the unit also has to leave the unit'. Some relations hetween the flows entering and leaving the unit can he derived. The conservation law also applies to all the stages. So from all the stages relations can he derived. Finally there will he a complete set of equations.

---..

/---

"

/ \

/ \

; .... -~ ~/v" Q

I

.~.~ D, xd i

I--r I

i ; /

\ "...t., /

\

YI

B, xb

_

.._q----_._',.

---_. ".",,'

, I

I I I

- - - . - - I :

~7 ' :.

/<~

/ / / I

I

/

, - - - - / I I

I I! / ^{I I}

! ,I! I I i

, i ^!

j'

1 :

' i / ( x b , y b ) : :

i

,_.__.I ,~__~ I ....LJI i

F,^C I'"

I

I

!

I - - - I

~ ~'~Q

f-'C--~ : f

1

y

o o

1

Figure 4.2: A dis/illa/ion unit. Figure 4.3: A McCabe-Thiele diagram.

The solution of this set of equations will give the static mass hehaviour inside the unit. The concentrations at the trays can now he calculated. Strikingly is the importance of the ratio hetween the liquid flow and the vapour flow inside the column. These flows determine the concentrations on the stages. A McCahe-Thiele diagram visualizes the relations.

In Figure 4.3 such a McCabe-Thiele diagram for a unit without an intermediate reboiler is drawn. In this diagram the concentration in the liquid phase (x) in relation to the concentration in the vapour phase (y) is sketched. The curved line describes the relation between x and yin equilibrium. The x= y line is drawn as a reference to indicate the ease of separation. A staircase line gives the concentrations in the liquid and vapour holdups at the stages. The straight line starting in(Xb,yb) is called the bottomworkline. lts slope is determined by the ratio between the liquid flow and the vapour flow at the bottom. The straight line starting in(Xd,Yd) is called the topworkline and represents a similar relation but then for the top. The straight line starting at (c,c), with c the concentration in the feed, is called the q-line.

A detailed explanation of the terms and the construction of a McCabe-Thiele diagram can be found in [12]. The set of equations is solved iteratively. What one actually does is trying to fit a staircase line, like in Figure 4.3, into the space between the worklines and the equilibrium curve. The intersections of the staircase line with the worklines give the concentrations in the liquid phase at the stages. The intersections with the equilibrium curve give the concentrations in the vapour phase at the stages.

Itmay be c1ear that solving the equations is sensitive to errors and time consuming if the ease ofseparation(a) is low. Furthermore adding an intermediate reboiler will give an extra workline and the problem gets more complex. The worklines, which are until now assumed to be straight lines, become curved by adding energetic relations. To account for al these

modifications a license for a professional software package' Aspen' was obtained. Aspen solves a much more complex set of equations than outlined before. Nonlinearities in the equilibrium relations are also taken into account. Another advantage was that a fitted model of the C2-splitter was available. This model has been duplicated. The duplicate is altered and fitted on the simulator. A drawing ofthe worksheet in Aspen containing the Aspen-simulator is given in Figure 4.4.