timescaleseparation& the link between open-loop and closed-loop dynamics

7/24/2019 TimeScaleSeparation& the Link Between Open-loop and Closed-loop Dynamics

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16th European Sym posium on Com puter Aided Process Engineering

and 9th International Symposium on Process Systems Engineering

W. Marquardt, C. Pantelides (Editors)

1 4 5 5

2006 PubUshed by Elsevier B.V.

Time scale separation and the link between open-loop and closed-loop dynamics

Antonio Araiijo^, Michael Baldea*^, Sigurd Skogestad^ and Prodromes Daoutidis^

^Department of Chemical Engineering, Norwegian Universi ty of Science and Technology, Trondheim,

Norway

^Department of Chemical Engineering, Universi ty of Minnesota, Minneapolis, MN, USA

^Department of Chemical Engineering, Aristotle Universi ty of Thessaloniki , Thessaloniki , Greece

This paper aims at combining two different approaches ([1] and [2]) into a method for control structure

design for plants with large recycle. The self-optimizing approach ([1]) identifies the variables that must

be controlled to achieve acceptable economic operation of the plant, but i t gives no information on how

fast thes e variables need to be controlled and how to design the co ntrol system. A detailed controllabil ity

and dy nam ic analysis is generally needed for this. One alternativ e is the singular per turb atio n framework

prop ose d in [2] wh ere one identifies poten tial controlled a nd m ani pu late d variab les on different tim e scales.

The combined approaches has successfully been applied to a reactor-separator process with recycle and

purge.

K e y w o r d s :

singular perturbation, self-optimizing control , regulatory control , selection of controlled

variable.

1.

IN T R O D U C T I O N

Tim e scale sepa ration is an inherent prop erty of man y integrated process units and networks. Th e

time scale multiplici ty of the open loop dynamics (e.g. , [2]) may warrant the use of multi- t iered control

structures, and as such, a hierarchical decomposit ion based on t ime scales. A hierarchical decomposit ion

of th e control system arises from the genera lly sep arab le layers of: (1) Op tim al ope ratio n at a slower

time scale ( supervisory control ) and (2) Stabil ization and disturbance rejection at a fast t ime scale

( regulatory control ) . Within such a hierarchical framework:

a. The upper (slow) layer controls variables (CV's) that are more important from an overall ( long t ime

scale) point of view and are related to the o peration of the en tire plant. Also, i t has been shown

that the degrees of freedom (MV's) available in the slow layer include, along with physical plant

inputs, the setpoints (reference values, commands) for the lower layer, which leads naturally to

cascaded control configurations.

b .

The lower (fast) variables implements the setpoints given by the upper layer, using as degrees of

freedom (MV's) the physical plant inputs (or the setpoints of an even faster layer below).

c. W ith a reasonable t ime scale separa tion, typically a factor of f ive or more in closed-loop response

time, the stability (and performance) of the fast layer is not influenced by the slower upper layer

(because i t is well inside the bandwidth of the system).

d. Th e stabil i ty (an d performance) of the slow layer depends on a suitable control system being imple

men ted in the fast layer , but otherwise, assuming a reasonable t ime scale separa tion, it should

not depend much on the specific controller settings used in the lower layer.

e. Th e lower layer should take care of fast (high-frequency) distu rbanc es an d keep the sys tem reas onable

close to i ts optimum in the fast t ime scale (between each setpoint update from the layer above).

The present work aims to elucidate the open-loop and closed-loop dynamic behavior of integrated

plants and processes, with part icular focus on reactor-separator networks, by employing the approaches


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A. Araujo et al

of singular pertu rba tion analysis and self-optimizing control . I t has been found tha t the open -loop

strate gy by singular pertur bat ion analysis in general imposes a t ime scale separa tion in the regulatory

control layer as defined above.

2 .

S E L F O P T I M I Z I N G C O N T R O L

Self-optimizing control is defined as:

Self-optimizing control is when one can achieve an acceptable loss with constant setpoint values for the

controlled variables without the need to re-optimize when disturbances occur (real time optimization).

To quantify this more precisely, we define the (economic) loss L as the difference between the actual

value of a given cost function and the truly optimal value, that is to say,

L{u,d) = J{u,d)-Jopt{d) (1)

During optimization some constraints are found to be active in which case the variables they are

related to must be selected as controlled outputs, since i t is optimal to keep them constant at their

setpoints (active constraint control) . The remaining unconstrained degrees of freedom must be fulf i l led

by selecting the variables (or combination thereof) which have the best self-optimizing properties with

the active constraints implemented.

3 .

T I M E S C A L E S E P A R A T I O N B Y S I N G U L A R P E R T U R B A T I O N A N A L Y S I S

In [2] and [3] it has shown that the presence of material streams of vastly different magnitudes (such

as purge streams or large recycle streams) leads to a t ime scale separation in the dynamics of integrated

process networks, featuring a fast t ime scale, which is in the order of magnitude of the t ime constants of

the individual process units, and one or several slow time scales, capturing the evolution of the network.

Using singular perturbation arguments, i t is proposed a method for the derivation of non-linear , non-stiff

reduced order m odels of the dynam ics in each t ime scale. This an alysis also yields a rat ional classif ication

of the available f low rates into groups of manipulated inputs that act upon and can be used to control

the dynamics in each time scale. Specifically, the large flow rates should be used for distributed control

at the unit level, in the fast time scale, while the small flow rates are to be used for addressing control

objectives at the network level in the slower time scales.

4 .

C A S E S T U D Y O N R E A C T O R S E P A R A T O R P R O C E S S

In this section, a case study on reactor-separator network is considered where the objective is to

hierarchically decide on a control structure which inherits the t ime scale separation of the system in

term s of i ts closed-loop characterist ics. This process was studied in [3], but for the present p aper the

expressions for the f lows F , L, P , and R and economic data were added.

4 . 1 . T h e

r e a c t o r - s e p a r a t o r p r o c e s s

The process consists of a gas-phase reactor and a condenser-separator that are part of a recycle loop

(see Figure 1) . I t is assumed that the recycle f low rate R is much larger than the feed flow rate Fo and

that the feed stream contains a small amount of an inert , volati le impurity yj^o which is removed via a

purge stream of small f low rate P. Th e objective is to ensure a stable operatio n while controll ing the

pur i ty of the product XB

fci

A first-order reaction takes place in the reactor, i.e. A-^ B. In the condenser-separato r , the interph ase

mole transfer rates for the components A, B , and / are governed by rate expressions of the form Nj =

p

Kja{yj

p ~ ^ j ) ^ ^

where Kja repre sen ts the mass transfer coefficient, yj the mole fraction in the gas

phase, X j the mole fraction in the hquid phase, P^ the sa turat ion vapor pressure of the component j ,

P the pressure in the condenser, and PL the l iquid density in the se parato r . A compressor drives the

flow fi:om the sep arato r ( lower pressure) to t he reactor . Moreover, valves with openings Zf, zi^ andZp

allow the f low thro ug h F , L, and P, respectively. Assum ing isotherm al opera tion (mean ing tha t th e

reactor and separator temperatures are perfectly controlled), the dynamic model of the system has the

form given in Table 1.


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Figure 1. Reactor-separator process.

4 . 2 .

E c o n o m i c a p p r o a c h t o t h e s e l e c t i o n o f c o n t r o l l e d v a r i a b le s : S e l f o p t i m i z i n g c o n t r o l

c o m p u t a t i o n s

The open loop system has three degrees of freedom at steady state, namely the valve at the outlet of

the reactor (2/) , the purge valve (^p), and the compressor power

iWs)-

Th e valve at the sepa rator outlet

{zi) has no steady state effect and is used solely to stabilize the process.

The profit (J) = {PL pp)L pw^s is to be maximized wherePL, PP, and pw are the prices of th e

liquid product, purge (here assumed to be sold as fuel), and compressor power, respectively.

The profi t should be maximized subject to the following constraints: The reactor pressure P reactor

should not exceed i ts nominal value and the product purity X bshould be at least at i ts nominal value.

In addit ion, there are bounds on the valve openings which must be within the interval [0 1] and the

compressor power should not exceed i ts upper bound.

For optimization purposes the most important disturbances are the feed f low rate Fo, the feed compo

sitions 2/A,o5 2/B,o and ^/,o, the reaction rate fci, and the reactor temperature Treactor-

Two constraints are active at the optimal through all of the optimizations (each of which corresponding

to a diff 'erent disturbance), namely the reactor pressure Preactor at i t s upper boun d and the product pur i ty

X bat i ts lower bou nd. These consume two degree of freedom since i t is optim al to con trol them at their

setpoint (active constraint control) leaving one unconstrained degree of freedom.

To find the remaining controlled variable, it is evaluated the loss imposed by keeping selected variables

constant when disturbances occur and then picking the variable with the smallest average loss.

Accordingly, by the self-optimizing approach, the primary variables to be controlled are then y ~

[Preactor

Xb Wg]wi th the manipulat ions u = [zf Zp Wg]-

4 . 3 . S ingular perturbat ion approach for the s e lect ion of contro l l ed vs ir iab les

According to the hierarchical control struct ure design proposed by [2] based on th e t ime scale separa tion

of the system, the variables to be controlled and their respective manipulations are: MR (Preactor) ^

F {Zf)\ Mv (Pseparator) ^ R (Zp)] ML ^ L (zl); X b ^ MR^setpoint [Preactor,setpoint)] yi,R ^ P-

I t

is important to note that no constraints are imposed in the variables in contrast to the self-optimizing

control approach.

4 . 4 .

C o n t r o l c o n f i g u r a t io n a r r a n g e m e n t s

The objective of this study is to explore how the configurations suggested by the two different ap

proaches can be merged to produc e an effective control structur e for the system. Th us, as a star t ing

poin t, the following two original configura tions are prese nted :

1. Figure 2: This is the original configuration ([2]) from the singular perturbation approach.

2 . Figure3: This is the sim plest self-optimizing control configuration with co ntrol of the active con straints

{Preactor and X b) and self-optimizing variable W s-


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Araujo et ah

Table 1

Dynamic model of the reactor-separator network.

Differential equations

-

yA,R)-^^ = M^o{yA,o-yA,R) + R{y

-kiMRyA,R]

^ -^ w^[Fo {yi,o - yi,R) + Riyi - yi,R)]

^ = F-R-N-P

^ ^ ^lFiyA,R - yA) -NA+ yAN]

^ = lk[F{yi,R-yi)-Ni + yiN]

Algebraic equations

^reactor

^sevarat

MyRga

NA

=

KAa yA

- ^-^^XA) -

y ^

i^separator J i

Ni = Kia(yi - p

^^

^ xi)

^

\ ^separator ) i

NB

=

KBOL\yB -

p ^ ^

XB \ -

\ ^separator J i

N =

NA-\-NB+NI

^

^

^V fZf y/r reactor i'^separator

^ ^^ ^VlZly^

J^^separator

-t^^downstream

-t

^

^VpZpyJ rseparator -i downstream

R-

1 iRgasTs,

^ )

MR, M VI an d M L denote the molar holdups in the reactor and separator vapor and liquid phase, respectively.

Rgasis the universal gas constant; 7 = ^ is assumed constant; Cv /, Cvi, and Cvp are the valve constants;

Pdownstream

S

the prcssure dowustrcam the system; e the compressor efficiency; andPreactor.max is the maximum

allowed pressure in the reactor.

yjEu(cc)-

i ^

' S K

Fig ure 2. Orig inal configuration base d on singu- Figu re 3. Simplest self-optimizing con figuration

lar perturbation with control of X h^

Pseparator-,

and with control of X h^

Preactor,

an d

Wg.

yi,R-

None of these are acceptable. The configuration in Figure 2 is far from economically optimal and gives

infeasible operation with the economic constraints Preactor exceeded. On the other hand, Figure 3 gives

unac cepta ble dynam ic performance. Th e idea is to combine the two approach es. Since one normally

starts by designing the regulatory control system, the most natural is to star t from Figure 2. The f irst

evolution of this configuration is to change the pressure control from the separator to the reactor (Figure

4). In this case, bo th active constra ints {Preactoran d X b) are controlled in ad dit ion to impu rity level

in the reactor {yi^n). Th e final evolution is to change the prim ary con trolled variable from yi^R to the

compressor power Wg (Figure 5) . Th e dynam ic response for this configuration is very good and th e

economics are close to optimal.


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^^aw^

Figure 4. Modification of Figure 2: Consta nt pres- Figure 5. Final stru ctu re from modification of Fig-

sure in the reactor instead of in the sepa rator . ure 4: Set recycle {Ws) constant instead of the inert

composit ion (yi,R).

4 . 4 . 1 . S i m u l a t i o n s

Sim ulations are carried out so th e above configurations a re assessed for controllab ility. Tw o m ajor

disturbances are considered: a sustained reduction of 10% in the feed flow rate Fo at t = 0 followed by

a 5% increase in the setpoint for the product purity X ba^t t = 50h. The results are found in Figures 6

through 9 .

I' i

u .

6

.\n

50

Time h)

100 1

'

1

i 4

1]

1

|-4-

[ 7

t / [7

/

f i

Fig ure 6. Closed -loop respo nses for configuration in Figu re 7. Closed -loop responses for configuration

Fig ure 2: Profit = 43.13A:$//i and 43.32fc$//i (good in Figu re 3: Profit = 43.21/c$//i and = 43.02fc$//i.

but infeasible).

The original system in Figure 2 shows an infeasible response when i t comes to increasing the setpoint

ofXbsince the reactor pressure increases out of boun d (see Figure 6) .

W i t h Preactor Controlled (here integral action is brought about) by

Zf

(fast inner loop), the modified

configuration shown in Figure 4 gives infeasible operation for setpoint change as depicted in Figure 8.


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Araujo et al

T

5 , L

IT

v..

100 150

y^

i \

|5 [ V

o:

^

^ \

50 100 1.

Time (h)

Figu re 8. Closed -loop respo nses for con figuration

in Figure 4: Profit = 43.20/c$//i and =

AZmk%/h.

Fig ure 9. Closed-loo p respon ses for co nfiguration

in Figure 5: Profit = 43.21A;$//i and = 43.02fc$//i.

Th e proposed configuration in Figure 3, where the c ontrolled variables are selected based on economics

presents a very poor dynamic performance for setpoint changes inXhas seen in Figu re 7 due t o th e fact

that the fast modexi)is controlled by th e sm all flow rateZpand fast responses are obviously not ex pected,

indeed the purge valve {zp) stays closed during almost al l the transient t ime.

Finally, the configuration in Figure 5 gives feasible operation with a very good transient behavior (see

Figure 9) .

The steady state profi t for the two disturbances is shown in the caption of Figures 6 through 9.

5 . C O N C L U S I O N

Th is paper c ontras ted two different approach es for the selection of control configurations. Th e self-

optimizing control approach is used to select the controlled outputs that gives the economically (near)

optim al for the plan t. These variables must be controlled in the up per or intermediate layers in the

hierarchy. Th e fast layer (regulatory control layer) used to ensure stabil i ty and local disturba nce rejection

is then appropriately designed (pair inputs with outputs) based on a singular perturbation framework

proposed in [2] . The case study on the reactor-separator network i l lustrates that the two approaches may

be successfully combined.

R E F E R E N C E S

[1] S. Skogestad. Plantw ide control: Th e search for the self-optimizing control structu re. Journal of

Process Control, 10:487-507, 2000.

[2] M. Baldea a nd P . Dao utidis. Con trol of integrate d process networks - a multi- t ime scale perspective.

Com puters and Chemical Engineering. Subm itted, 2005.

[3] A. Kum ar a nd P. Daou tidis. Nonlinear dynam ics and control of process systems with recycle. Journal

of Process Control, 12(4):475-484, 2002.

timescaleseparation& the link between open-loop and closed-loop dynamics

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