Switchability Analysis

Copyright e IFAC InLegration of Proc:eu Design and Control, Baltimore. Maryland. USA. 1994

SWITCHABILITY ANALYSIS

v. WHITEt, J.

D. PERKINSt and D. M. ESPIEt

tCentre for Proceu Sydems Engineering, Department of Chemical Engineering, Imperial College. Lo,.doll . S W7 28)1. tAi,. ProductlJ Plc, Mo/esey Road, Walton-on- ThamelJ, Surrey, KT12 4RZ.

Abstract. A method to evaluate the inherent switchabiJity of a process design, that is its ability to move between operating points. is proposed in this paper. This requires the solution of an optimal control problem to determine the optimum trajectory of the manipulated variables for the switch and the optimal value of any design parameters. End point constraints are required so that steady state is reached and path constraints to ensure that product purities are maintained throughout the switch .

Key Words. Cont rol Applications , Controllability, Distillation Columns, Optimal Control , Process COlltrol

1. INTRODl"<. -no:"

has increased and optimal control techniques improved to make the solution of larger dynamic optimisation problems possible. Moreover, path constraints, i.e. variables that are continuously constrained throughout a switch , can !lOW be dealt with efficiently using interior point constraints and an end point constraint limiting the int.egral of the square of the violations over the switch (Vassiliadis, 1993) .

In today's economic climate . chemical plants are often required to perforlll at multiple operating points with varyilt~ lJ.r,,"~hpul or product grades. Certain plaltl,.. fur Ilts\.ance service plants such as air s<'parallOlI units , must move between operating pUlllh . " .g to minimise power consumption , or follow luau variations as required by the demands of a downstream customer. e.g . O 2 demand frolll a stl:'t!l works . Since these switches occur frequently . it is important that the required dynamic performance be met by the plant design .

2.1 Switch ability Formulation

The problem to be solved can be posed following manner :

The objective of this research is to provide a measure of the inherent. switchabili t.y of a pro ces:; design . This will allow the elilllination of any underlying process and equipment constraillt.:; from the design , rather than merely mitigatin g the effects with an increasingly complex control scheme.

In

the

lllin J(u(t) . t . 0) ul/l./ .El

s. t.

{

f (i: (t ) , x ( t ), y( t ) , u ( t ) . t . 8) =0 l/dx(t). u(t) , t . 0) = u '7~(x(t), u(t) , t , 8) ~

( 1)

0

'2 . SWITCHABILITY ANAU'SIS To make this problem tractable , the time over which the problem is studied is split into Nswint intervals of length hJ . The control action is approximat.ed by a linear profil e ill each interval and is continuous . Hence th e actual value of

Optimal control was used by Howell (1984) to study the switchability of certain plants , that is the ability of a plant to move opt.imally between operating points . Since t.hen computer power 245

as a penalty term in the objective function (Sargent and Sulli van, 1979) . However, this does not guarantee that the solution will be within the path constraint bounds. Using the constraints detailed here the violations of the path constraints must be within a specified limit which is chosen to be insignificant . Within the dynamic model, Equation 6 can be implemented as follows :

control i in interval j at time t is given by :

(2)

where,

T

=

In Equation 1 f is the set of differential and algebraic equations that describe the process being studied , where x are the differential variables and Y are the algebraic variables . '71 and '72 are the equality and inequality constraints . In this impleme~tation, '71 and '72 are applicable only at discrete times in the set :

[',

lo

(max[O, (Yactual - Y1.mit)])2 dt ::;

(8)

Satisfying Equation 8 is a necessary condition for Equation 6. However this may interfere with the advantage of including the relaxation parameter
Design parameters, e, i.e. the optimisation parameters that are constant throughout the time of the dynamic subproblem. are also incorporated into the procedure to allow for design modifications to improve switchability. For example these are used below to determine the optimal number of trays in a distillation column which is required to perform a certain switch .

Since these constraints are continuous in their second derivatives they improve the conditioning of the optimisation problem .

Reachmg The New Steady State . The aim of the switch is to move the plant from one steady state to another . Hence an end point constraint should be that the plant is at the new steady state . This is achieved in two ways . First, the last. two control points are constrained to the same value :

= U,(tr)' "Vi E {l , ... , Ne}

(7)

In Equation 7, (p is a small number that indicates the amount by which a path constraint. can be violated without being significant (Walsh , 1993) . An improvement to this is to add interior point constraints , which in this illlplementatioll are at the switching intervals:

The objective function to be minimised, J , is dependent upon the system under study and its required dynamic performance .

[1,(tr-l)

(p

3. AIR PLANT CASE STUDY In this case study the switchability of a cryogenic air separation unit, producing gaseous oxygen (GOX). was analysed in order to determine t.he factors that. would limit the speed at. which the plant could move between operating points . Therefore, we have determined the optimum trajectory of the control variables that would move the plant from one operating point to another as fast as possible without violating any operating constraints. Figure 1 shows the flowsheet of the plant with the controls indicated .

(4)

where Ne is the number of controls . Second , the deri vatives of all the state variables should be zero at the final time . Equation 5 is used to achieve this . where ( .. is sufficient.ly small.

(5)

2.2 Path COIIstralllts

3 I Problem FormulatlO1I

Path constraints are those t.hat can be posed in the following manner :

The object.ive funct.ion used. Equation 10 . wa'> the integral squared deviation of the main product flow (GOX) from its new desired value . Minimising this function with a free end time requires a fast transition to the new steady state.

Yactual(t) ::; Y1,m't , "VI E [0 . trJ

(6)

where Yllm't is the maximum allowed value for some variable Yactual· A common way to deal with path constraints has been to include them

J

246

['f

= la

(FGox .ss - FGOX)2dt

( 10)

=: - Control Variables .........---i __..!.!!r...... _Io--..... t----' ...,...GAN -.u.

__ .~_~ .. __ .....~.~....... .....~..~...

--

.. .. . 6. ~..... .• 6ft,.

::: :~~::: :: :: .... _c.I-.

Fig. 1. The Flowsheet of the Air Separation Unit The number of trays in a lump is usually around six , which is considered the best trade-off between speed and accuracy. The composition dynamics are then modelled as in Equation 12, where M/()/
The switch is also subject to constraints on product purities: 0.94 < yo,.

GOX

YO, . impurf GAN

Yo ,. Purt

GAN

< 0.96 } < 0.01 < U.UUl

( 11 )

Another path constraint was required since the models do not. take into account the effects of flooding . This condition should be avoided , hence the vapour rate at the bottom of the lowpressure column was constrained to avoid conditions that, on the real plant, would cause flooding .

d(M'o/ll/xoud

dt

It is recognised that any control study involving flow changes in the control of a unit operation. such as distillation, requires that liquid dynamics are included in the model (Skogestad , HJY:.!). Therefore, the usual assumption of consl,anl lump holdup is relaxed and an average tray M is Illodelled with hold up .M .!!!.lal&L N 'r. W'. . The liquid rate out of this average tray is then calculated using an inversion of the Francis weir equation .

The plant was moved from full throughput down to a new steady state at which the GOX product rate was reduced by 50% . The controls were bounded to 110% of their value at the maximum rate .

=

3 .2 Shortcut D,silllat,on Models When developing the model of the plant to be studied one must consider that it will form the function and gradient evaluation subproblems of a larger dynamic optimisation problem . Hence it must be of minimum complexity consist.ent wit.h a reliable analysis. Analyt.ical deri vati ves are desirable since these would improve the efficiency of the solution .

Since the above model was only used for the case of binary distillation, Smoker's equation (Smoker, 1938) was used for the calculation of Your. This has two advantages over other . iterat.lve. approaches . First. equation derivatives are available in analytical form and , second , the number of trays in the equation can be a nonint.egral, allowing it to becOJlle Cl cOlltinuous desigil paral1leter. This fact will be put to advautage in the next case study.

The method used was that of tray lumping (Howell. 1984: Benallou et al .. 1986). where several trays are modelled as one section and the composition dynamics are modelled for the liquid composition leaving the bottom of the lump .

Feed and draw trays are modelled as individual trays to improve accuracy. The other trays are

247

0.0105

..~.01C1O

.. .... ..........

................ .. .. .

--.-.~

0.0016 Uppef Bound

o.ooao

LowetBound SoiUlion Prol..

0.0015 O.QOlO

OJM5 0.0075 OJMO

0.0070

0.0015 0'---1iOO±---1~~--1~1iOO~~~*--~~~~~~.

0~0~--~~~--1~~--1-~~--~-----~~--~~

Time (Seconds)

Time (Semnda)

(b) Impure GAN Product

(a) Main GOX Product

Fig . 2. Active Path Const.raints at the Solution of the Air Plant Case Study 4 .1 Problem Formulation

lumped as indicated and modelled as discussed above . The whole model has 38 state variables (N%) plus one for each of the path constraints and one for the objective function .

The aim here was to reach the new distillate composition as fast as possible. The objective function used to achieve this was:

3 .3 Discussion

J The resulting composition profiles at the solution are shown in Fig. 2 and the open loop control trajectories for the final solution are shown in Fig. 3. There are four control periods, the first of which is at its lower bound of 1 second .

['I

= la

(0.990 -

Xd

(13)

)2dt

During the switch between operating points must be constrained within bounds: Xb

= 0.001 ± 0.0002

Xb

(14)

Equation 14 leads to two path constraints treated as in Equation 7.

The results show that the GOX product reached its new flow rate in the first time interval, which was at its lower bound , i.e. there was no limit to the speed with which the GOX rate could be changed . The challenge for the optimiser was to find a set of control trajectories that maintained product purities throughout the switch . This indicates that there are no fundamental process constraints limiting the switch in t.he COX flow.

The optimum number of stages in the column is to be determined by design parameters that exploit the fact that Smoker's equation allows the number of stages to be non-integral. The design variables used are therefore :

4. BINARY COLllMN DESIGN The second example studied was that. of a distillation column required to switch between product grades. The feed was a 1100 kmols/hr stream containing 0.9 mole fraction benzene and 0.1 mole fraction n-octane . The first steady state was such that the distillation composition, Xd, was 0.999 and the bottoms composition , Xb , was 0 .001. For the second steady state Xd was dropped to 0 .990 but Xb was to be maintained at 0.001 . A constant relative volatility of 3.83 was assumed.

N umber of stages in the } rectifying section lumps

N

Number of stages in the } stripping section lumps The design reflux ratio

N,

r

RRd

The design reflux ratio is that at which the column is sized for operation and so is used to determine the column diameter , D t , required by the hydraulic calculations. i.e. D t fn( RRd). This function was determined by designing the column for the first steady state at multiple reflux ratios. Dt was then found to be an almost linear function of RR d .

=

248

1.S

I.S

, ,

1.0

Feed Ail Recycled GAN RelIux LIN P_GAN GOX Ptoduc1

"

-

Feed Ait Rec:ya.d GAN

ReI.... LIN

..... ::.. . ._._ ..

1.0 . ......... . ....

•..•.. ..'...,~.. ~;:-... ~.:.:;... ..•.. - .. -... .,........ . ..... "\ " .' , .' ..' , ,, ,, .' ,, ,

O.S

:::-

....

.....

P_GAH GOX PtodUC1

. . .. . . . .

'"

. . . . . ........... .

: :.::..•..::.:::.~:... ... ... ... ..

..•..•..•..•..•..

, , , ,

..

0.6

0.00~---'---~2---!3i--..... ~"'---~S

200

400

&00

Time (Seconda)

Time (Seconds)

(a) The First Few Seconds

(b) The Whole Switching Period

Fig. 3. Scaled Control Values for the Air Plant. Case Study Since this problem uses the number of stages as an optimisation parameter there are other constraints that apply :

0.999 } 0.001

procedure is one based upon over twice the minimum reflux ratio and very close to the minimum number of stages (Nmifl 11) . In fact, the design is very similar to one produced by a steady state optimisation with the objective of minimising the total column holdup .

=

(15)

RRd

Here one can see a competition between the undesirable economics associated with the high reflux ratio and the desire to switch product composition as fast. as possible. However , if the design of the distillation column internals can be altered to reduce column holdup, without the need to reduce the number of trays (increasing RR), then a faster switch could be achieved without an increase in operating cost .

Figure 4 (a) shows the binary distillation column to be optimised with the lumped sections indicated . The initial values for the design parameters were determined from steady state design equations with RRd 1.2 X RRmifl (see Table 1) . The controls for this problem are:

=

RR( t)

Reflux ratio Boilup ratio

-

BR(t)

5. CONCLUSIONS 4 .2 DIscussIon

The technique described in this paper allows the analysis of a plant 's ability to move optimally bet.ween operating points : its switchability. Two exam pies are used as an ill ustration .

There are two types of results frolll thiS case study : Design parameters have been used to determine the optimum number of stages. and the feed location . to allow optimal switching between distillate compositions . See Table 1 for the optimal values . Steady State Design .

First , an air plant is required to reduce its GOX product rate as fast as possible . The analysis shows that the part of the plant which is modelled, the separation sequence , presents no limit to this objective.

Control TraJectones. The open loop control trajectories to bring about. the optimal switch are found for the optimal design. Figures 4 (d) and (e) show these trajectories and Figs . 4 (b) and (c) show the composition responses.

Second . a binary distillation sequence is required to move between distillate compositions . The optimality of this switch is shown to be improved by the reductiolJ of the total hold up 011 the trays .

The design found by the dynamic optimisation

249

1.000

Jo0._

0.0G0I

0.102

NrStages

0.20

NrStages

2

-----I0000-.....

0.0 0 ' - - -..... 0.0 0.5

3

1.0

(b) Distillate Composition 0.0013 JO.0012

N.Stages

'-'

--~

0.0011 0.0010

N.Stages

.... .....

--....

1.0

(d) Reflux Ratio

I ,. t e

I. 17

15

0.000II

14

0.0008

13 12 '--_ _ _ _Ioooo-.....__

0.0007 ' - - - - - - - - - . . ; - o 2 3 4

0 .0

0 .5

1.0

1.5

~

2.0

Tme(IGn)

TIM(Hou ..)

(a) The Design Problem

1.5

n.(-...)

TIM(Hou .. )

(e) Boilup Ratio

(c) Bottoms Composition

Fig. 4. The Binary Column Design Case

Initial Design Dynamic Optimum Minimised holdup at first steady state

RRd

Nr

N.

Total Stages

0.451 0.897

3.25 1.50

4.00 3.00

23 .0 14.0

Column Holdup (kmol) 65 .3 45.8

0.796

1.41

3.45

14 .7

44 .5

Table 1. Binary Column Design Parameters S. Skogestad. Dynamics and control of distillation columns - a critical survey. In I FA CSYPOSlltlll DYCORD+. Maryland. USA, 1992.

6. REFERENCES

A. Benallou, D. E. Seborg , and D. A. Mellichamp. Dynamic compartmental models for separation processes. AiChE Journal. 32(7) :1067 , July 1986 .

E. H. Smoker . Analytical determination of plates in fractionating columns. Trans. Am. Inst. Chem . Engrs., 34 :165, 1938 .

J . M. Howell . The Effect of Energy integratlOlI on the SWltchabi/ity of Chemical Processes . PhD thesis, University of London , 1984 .

V. Vassiliadis. Computational Solution of DynamIc Optimization Problems with General Differential - Algebraic Constraints. PhD thesis. University of London. 1993.

R. W . H. Sargent. and G . R . Sullivan. Development of feed changeover policies for refinery distillation units . ind. £ng . Chem . ProCtSS Des. Dev. , 18(1) :113 , 1979.

S. P. Walsh . Integrated DelHg1l of Elfiuent Treatmtllt Systems . PhD thesis, University of Lon don . 1993 .

250