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Selecting the Best Distillation Control Structure
C()P~ ri g-Iu
© I F.-\C I h Il;t 111 il " ;111 <1 COllt rol (11
C he mica l Reacto r :>' (D\' ( :( )Rl) + ·;-':~II. \L!; \"tr il hi.
T he :\<" her IOl "d " I ~ I ~~ I
SELECTING THE BEST DISTILLATION
CONTROL STRUCTURE S. Skogestad, E. W. Jacobsen and P. Lundstrom (;lil'llli((l/ F IIgi//I'nillg, .\'lIr;'·"gillll 1II ,llilll l,' 0( Tl'I'lillll/lIg:r (.\ T II I, .\' -ill] -/ F I'II /II/I/I'illl, .\'lIn,'tl\'
Abst r ac t Til e ~cll,rt i o ll of an appro pri;tll' control structur e (collfigll ratioll) is the IIlOs t important dccision \\'\it'1l desi!-;Ili!lg di :,t illatioll control s ~·st(,llls.
The steady state ItG :\ is (onllIlon ly lI sed ill inou stry as
a toul for sckct illg tlte t)('~t stnlrtllrc. I II thi s paper it is stressed that deci sio ll s regarding cOlltroll€'r
de,i),;n s hould be ba sed o n the initial re; pon se ( bi gh -fre<]uency bebavior) rather t ban th e steady state, 1I 0II'l'I'('(', for 1II0s t di s tillation cOllfigurations tbe stc'ady statl' HGA turns out to be a good indicator of (' xl'ectl'd control qualitl'. 0ue countNcxa mple is the Dll-configuration II'hicb has iufinite s teady'tOltC' HG _-\ - \'alues , II OII'l'\'N, thc HG,\ -\'"lu e5 at bigber frequ e ncy are close to onc , "lid good control performan ce is possi ble,
Ba sed on a frequenc~'-dependent nc A-analysis and optimal P I controller designs_ th e ( L / D )( \ . / 13)COli fi g uratio n is found to be tbc bes t choice for tll'o-point compositioll co ntrol. Tbe traditiollal L\'cOllfi gu rati oll pe rfo rm s II111 Clt poorl'r. hut is preferab le if ollc-point control is used,
1
Introduction
Co nside r t he s im p le tll'o- prod ll c, di"illa t ioll (OllIlIIn ill Fi i\ , I, From a cont rol point it lIIay bc \'i(,lI'"d as ~ X ,S systc>m, Th" op tima l controll er s l,olll,1. based 011 ,, 11 a\'ai l"bl" illformation (lIIea s urement s. procl'ss mod el. l'xpeci('cI d is turballccs) ",allipul a te 111/ .' j input s (L.\'.\',(,.O.fl) ill ord c>r t o kecp the ,'i O"'p"\; (I<,\'pl , ill top and bottom. pres",,'c_ top alld bottom composition ) ;IS clos<, as poss ible to thcir dc; ired \,;dlles , lI olI'e\'cr. fell' colllmll s. if all\', are controlled II s ill g a full ,) x.'j co nt ro ll('1'. In order '0 si III pl i f." ,I,, ' design and get a (ontrol S\'s tl'lll whi ch i:; :-;illlplC'1' to ulld l' rsta nd. la ret u nc , to make failure t o lcra nt and wh ich is in se nsitiH.' to
plant operation , a decentralized control system based on sin gl ,· loops is used in practice. Since the levels and press ure ha\'e to be controll ed at a ll times to ensure s ta ble operation. and becauS/' t hese control loops arc essellt ially inclep<' II dent of the (ollll""i l ion co nt ro l. the Ic\"cl and pre ss ure cOlltrol sy:.;t cm is d e:-; i~lI rd firs \. Engineers often do a good crfort in des ig ni ng thi s s ub s ~'" ten reliably. hut Ill
:1 1;)
p (M v)
v L-_--t:k}_B.,
xB
Figure 1. T wo p roduc t d istillation column with single feed and t o t a l condenser ,
:11{i
S. Skogcstad. E. W. Jacobscll and P. Lundstriil1l
@ 6
D, Yo
of the seven example columns :\-G studied by Skogestad alld ~Iorari (1988a). Steady-state data for these columns are gi\'en in Table 1. For all examples. we assume constant molar flows. binary mixtures, constant relatil'e volatility and perfect control of pressure and le\·els. However. in contrast to Skogestad and Morari (1988a) we shall here include the liquid flow dynamics in the column. The steady state holdup on all trays. including reboiler and condenser. is .If;! F = 0.5 min. The steady state gain matrix for the L\'-configuration is shown in Table 2 for all columns. \'ote that scaled (relative. logarithmic) compositions have been used: (I )
@
Shown in Table 2 are also the time constant TI.T·, alld Ih which are used in the simplified dynamic model discussed below. ef. = :\'TL is overall liquid lag fr01l1 the top to the bottom of the col· umn. TL is computed from linearizing the Francis weir fOrIlllIla. Assuming in addition half of liquid Ol'er weir we get
v B,
'---D
TL
Xs
Figure 2. Two product distillation column with LV- configuration.
= (d.\l i ) = ~~ dL i
\.
3 L
(2 )
The following configurations are considered in the paper: I.\, (often denoted "energy balance" or "indirect material balance"). DV ("material balance"). DIl. (L/D)\, ("Ryskamps schl'llle") and (L/D)(V/B) ("double ratio").
dependent variables to use for compositiou control. The issue has been discussed for quite sOllle time by industrial people (eg., Shinskey, 19G7. 198·1). but has ollly recentl,' been treated in any Table 1. Steady-state data for distillation column examples. ,-\11 detail by academic researchers (eg., Wailer, 198G , Skogestad and columns have liquid feed ('IF = I). Morari,1987a). Shinskey relies on t.he steady state \'alue of t.he RGA for se- Column er 1'1 NF 1 - YD ZF xB D/F L/F lecting the best control structure. The steady state RGA contains no information about disturbances and dynamic behavior. both A 0.5 21 0.01 0.,s00 2.700 1.5 40 0.01 of which are crucial for e\'aluating control performance. so one B 0.1 1..5 40 21 0.01 0.01 0.092 2.329 would not expect that. this measure by itself is sufficient to make 0.5 C 21 0.10 1.5 40 0.002 0.55,s 2.737 conclusions with regards to control performance. The fact that D 0.65 1.12 110 39 0.005 0.10 0.61·1 11.802 this measure t urns out to be very useful for distillation colulllus E 0.2 15 5 0.0001 0.05 0.158 0.226 5 must therefore be a result of fortunate circumstances. One obF 0.5 15 10 0.0001 0.0001 0.500 0.227 5 jective of this paper is to investigate this in 1Il0re detail. G 0.5 1.5 80 40 0.0001 0.0001 0.500 2.635 One counterexample to the usefullness of the steady state RGA is the DB-configurat.ion. It involves using distillate product D and bottom product Il for composition control. This control scheme has pre\'iously been laheled ' impossible' by most distil- Table 2. Data used in simple model of distillation columns, eq. 8. lation experts (eg .. Shins!;ey. 19~q. p. 15·1) because its RGA is infinity at steady state. ·Yet. Finco et al. (1989) have recent.I,· Column GLl·(O)s eL TI T2 shown both with simulations and with actual implementation ( 87.8 -8G.4 ) that the scheme is indeed workable. We shall investigate the A 194 !.'i 2.4G 108.2 -109.6 reasons for this in more detail in this paper. In the paper three different modes of operation are considered 74 79 -17!.7) B 2.50 1.5 2.86 . with respect to the difference between the control configurations: 90.191 -90.5 1. Open-loop (manual) operation ( IG.023 -IG.O) C 2·1 10 2.44 2. One-point composition control 9.29 -10.7 3. Two-point composition control. ( 2~.5S5 -24.2 ) Although most industrial columns are operated with oneD 1.5~ 30 1.·5~ 21.270 -21.3 point composition control. the emphasis will be on the last mode. ( 203.4 -131..5) The reason is that this is the most difficult problem and has large E 82 30 11.06 22.47 -22 ..5 potential for economic savings. Although energy savings are of( 10740 -10730 ) ten mentioned. the increased yield of valueable product by keepF 299G -1 7.3-1 92.57 -92G7 ing both compositions at their optimal values (thereby avoiding overpurification at the one cnd of the column and loss of valuable ( 8648.94 -8G46 ) G 20333 30 .5.06 product at the other end) is probably far more important. The 11347.06 -11350 first mode (open-loop) is not too interesting by itself, but it is of interest for the other two modes because low open-loop sensitivity to disturbances (good 'built-in' disturbance rejection) introduces 2 Modelling. less need for feedback control and makes the remaining control problem simpler. Because of the large number of trays in some of the columns The analysis demonstrates that steady state data may be en(110 trays for column D which give a 220th order model if both tirely misleading for the enluation of control performance. For composition and liquid flow dynamics are included) we choose to feedback control, it is the initial response. corresponding to the use the simple two time- constant dynamic model presented by time constant of the closed-loop system, which is of primary imSkogestad and Morari (1988a). Using a simple model also makes portance. it easier for the reader to interpret and check the results. The two Example columns. Throghout the paper we shall make use time-constant model is deri ved assuming the flow and compostion
C
Selecting the Best Distillatioll COlltrol Structure dynamics be decoupled, and then the two separate model s for the composition and flow dynamics are simply combined, In reality, the flow dynamics do affect the composition dynamics and the model will be somewhat in error. In particular , the time constant Tl may be different (larger) from th" one shown in Table 2 which was obtained by Skogestad and :'!orari (1988a), However. the res ults in thi s paper tUI'l\ out to be o nly weakly dependent on '" Let dLT = dL and dl'T represent sJIlall changes in th"liquid and vapor flows in the top of the column, and let dLB and dl'lJ = dl' be changes in the bott o m, In addition to constant molar flows wc make the foll owing assumptions I. Imm ed iate \'apor response (perfect pressure control)
2, Liquid flow fr om top to bottoJll as:'; lag s in seri('s and \'apor has no dfect on holdup (,I )
where
!J ds)
= 1/( 1+ (8d\
(,) )
)--)'
:3, Perfect control of reboiler aud condenser le \,(,1
dl'lJ(s)
= dl'(.<) = dLB(S)
~
dB (s)
(i)
To generalize these "quilt ions to the case of not constant molar flows and to the case whell cilallges ill \'apoJ' flow affect holdups (.\ in Skogestad and :'!orari (19S8a) is nonzero) additional paramet ers bave to be introduced in equat iolts 3 and 4, IX-configuration, The simp lified model of the LV-configuration t he n becomes LV ~dL I+TI8
dXB
+
LV
I
I' ) _~ dl'
I+T28
LV
~gd,5)dL 1+
(LI' gII +gI2
+ ( g21LV + g22LV
I+TI8 LI'
_ ~
1 + T2S
TI S
1+
)
dl' (8)
TI S
gh
l where ' denote the s teady s tate gains for the LV-configurati on, These are given in Tabl e 2 for scaled compositions,
Models for other configurations, :'10dels for other configurations are easily obtained from eq, 8 by choosing two independe nt \'ariables ill and "2 for composition control and expn'ssing dL and dl' in te rllls of these \'ariables using equations 3 to i, This gi\'cS
( rIld~)
= ,\I"',"'(s) 1.1
(rlIII) dill
and dl (1 - gds))
= -g/.l s) dD ~ dB,
:117 We obtain
~l) ~1
( 14)
:\ot<' that (15)
and lim,_ogL(s) = l. Th e elements in gain matrix for the DB· configuration \\'ill therefore approach infinity at la\\' frequency (stea dy s tate) and it will al so become singular. The physical interpretation is that a decrease in, for example, D with B con· stan t. will imm ediately yield a corresponding increase in L, This increase will yield a corresponding increase in V, which subse· quently \\'ill increase L even more , etc, Consequently, the effect is that the internal flows eventually will approach infinity, ~!ath· ematically, there is an integrator at low frequency: As s - 0 we ha\'e dL = dV = -(dD + dB)/OLS, Several authors (eg, lIiiggblom and \\'aller ,19S6; Skogestad and :'!orari, 1987b; Skogestad.l9SS) discuss ho\\' to obtain steady sta te models for variou s confIgurations, but these transformations do not include the flow d,'namics which are crucial for the dy· namic behavior. Example columns, Accumcy of simplifie d Illodel fol' CollL",n A, To illu strate the accuracy of the simplifIed model the singular \'alu es a nd UG.-\· values as a function of frequency are compared with exact valu es in Fig, 3, The exact values were obtained by linearizing th e full 8 2th order model \\'hich has t\\'o stat es (composition and holdup) for each tray, The hydraulic tray lag in the full- order mod el is 'L = 8L /S = O,06min, From Fig, 3 we conclude that th e simplifi ed mod"l seems to capture the true behavior quite well, Diffel'ent conjigll1'flti01l8 fol' colllllln A, Fig, 4 show the gain elements gi ) as a function of frequency for four configurations, ""ote in particnlar how the flow dynami cs decoupl" the dynamics at high freqnency, RGA-I'!llues of Columns A-G, Fig, 5 shows the 1,1·elemen t of the UGA as a function of freqnency for five different configurations, \Ve shall discu ss these \'alnes in more detail in Section 5,1. Fig, ,) should be compared \\'ith s imilar figures given in Skogestad and ~!orari (1988a) for the case with no flow dynamic s ( that is, dLB = dL), The introduction of flow dynamics are seen to change the results dramatically, Firstly, the response becomes decoupled at high frequency which makes UGA=1. Second I,', the importance of the time constant T2 associated with internal flow s seems to disappear almost completely,
(9 )
and th e linear model for the new configuration
d!Jo) ( d"'B
= (;"'"'(s)
(dILl) dll 2
( 10)
becomes (11)
If ratios are used as inclepCIl(\ent \'ariables these must fir st be expressed as a function of the actual flows. for example with "1
= L/D, d( L/ D)(s) = (1/ D )dL(s) - (L/ D2 )dD( s)
( 12)
DV-configuration, In this case \\'ith perfect level control we have dL = -dD + dl ' , that is, 01' M LI ,
=
(-01
( 13 )
DB-configuration, In this case dl ' = fJL(s)dL - dB \\'here dL = dV - dD, Eliminating dV yields dL ( 1 - gLls)) = -dD - dB
Figure 3, Singular values and .\11 as a function of frequency for simplified model and full model. Dotted line: Simplified model.
S. Skogestad. E. \\' . .lacobsen and P. Lundstriim
LV
10'
le.:!
,. ·~----~~7~~-----+-----+----~ -----~----~
1-... .....
------, -,::- ~----!.
~
~ \
1 ••
1.
l
10
~
1.
~~-
:1-::._-___ .. ~.::..:::- .:: : ~~ :=--""-,
,, ,
-2- -- -- -- -
..
:u,~ - c.
1
=:_~- ~~ ---
. -=-~ -=-.4:=- 3
--, ' , ,, ,,,, - , '"''
--, """ '.' , , "'"
1.- b
.'"
10
)
'"
I l":!
I"
.7'> - -'
1
OV ,. '-r------,------.------,------.-----,r-----, 1 0° -r------~--~~------t------t----~r-----~
3
.....
4
~. ..jr-_-__- I __ -+__ +=-"">o __.
, l~- · _t------t_----_+------t_----_+----~'~\r_----~
---. --.--.-~.,.,
\.., - 1
,
10
I;
................", . _ ....",,., -----'--'-" " _" 1 '" ~ 1""/ 1""
,,,, _. _. __ . __ ._____ 10'
" " _--, ...., ' " ' 1" I
.....,-,- " , ,, 1"
___ C_____ ._ _ ____ .__
-.-_~-~-('-=L"-I =0--)..( ~ v I--=B-,-)~_-,-_---, .....
I ~? - ~------ r_----- ~------I-----~-------r_----~
..
-~
----......--.
--- - - --.---- :::."~~~~ ~F=---
1('1- t -
_ "
... ,,"
1,,6
_ ,- "
1 .. 5
.. "
__ ..-- . , ,, ,, _
10~
,_" ." "
._-._" " " '- , , " , "
\0 2
10 - )
10 1
1~ - ·_t------t_----_+------+_----_+~\--_+--~~ \
--,- , ., " '" - , -e " "" -,- , -, "'" 3
10
?
10
I
~~:'J.,"'t--r-~n' 1~ 1
2
"'............ ,-~~ le- 2_t------t_----_+------+_----_+-'~~~·~--~ - +-=~~~'~-~
__ __I .. ________ ______ __
\
\
10- ~-t------+------+------+------+----~\rt----~ \ \
10-
6 1')
--:-"" '"""11 - ' , . ... ,," - , - ,-.,'''' ----,-rT" r " r--,--r-n·, n~ -4
10
3
10 2
1"
1
l()
1
W Figure 4. Cain elelll ents as a fnnction of frequency for colulll n A using fOllr different con fig ura tion s: LV. DV, (L/D)(V/B) and DB.
3
Open-loop (manual) operation.
The term "open -Ioop" shollld here be put in quotes becau se we are not talking abou t an uncontrolled collllll n. but assullle the levels and pres snre arc perfectl\' controlled and co nsid e r th e effect of the remaining ind"Ill'ndent \'ariables on the compositions. For exa mple. the open -loop operation of the LV -con figuration assullles le\'e1 s and pressu re to be perfectly controlled with D.n and YT. and t he rcmainillg manipulated variables. Land Y. to be constant ( in manual). Th e use of a true open-loop 5 x 5 model where all five flow s L.V.D.n and VT are ind epende nt variables is co nsidered by Skogestad ( 1989). Cood open-loop performance is then achieved if the effect of di stu rbances 011 compositions are small , that is . if the 'built-in' di sturbance rejection is good. Disturbances include changes in feed conditions (F. ZF , qF) alld di sturbances on all manipulated flow s L,Y ,D ,B and V T . The steady- state effect of di sturbances on compos itions may be evaluated quit e accurately by considering the steady-state effect of the disturbances on 0 /13 or equivalently D/F. This follows since major effect on product compositions of any change in the column may be obtained by assuming the separation factor S const ant (Shinskey. 198·1). Differentiating the component material balance with S constant yi elds
F
'· ' l-r==+==:r---:~--'--I--I-r-l - _"'!- _ __ -~, 1
10 ~~----~----~~~~~~~,~----+_----+_----r_--~
~ ...... 5
3--,~
"
>~~
2
G
1. 3
---
- ..... , ~ ,
1.'
.....
"
;',,.4 " ....
~
5 ...... -
~
3 ".... ...
r-~
~
w Figure 5. All-value as a function of frequency for column s A-G using five different configurat ions. l.LV 2.DV 3.(L/D)(V /13) 4.( L/D)V 5.013
Selecting the Best Distillation Control Structure 1
dIB
-;-:-----,= -I", (Fdz F (1 - IB)l'B
(I -YD)YD
(YD -l'B)Fd(D / F))
( IG) where the impurity sum is defined as Is = D( 1 - YD )YD + B(l IB)I B. The effect of various disturbances on Fd( D / F) is gi "en in Table 3. Configurations with small entries in Table 3 are insens iti ve to disturbances at steady state and should be preferred. We see that the (L/O)(V /D )-configurat ion is a good choice when the reflux is large. Note that the DD-configuration is not included in Table 3 because 0 and B are not independent at steady state. In general, the values obtained with the above approximation are quite accurate , in particular, for the gain of the least pure product. For example , the effect of a change in feed rate on YD with constant L and V is approximately (eq.16)
BYD) (I-YD)YD(YD-XB ) (DD/F) ( 7iF Ly=-D(l-YD)YD+B(I-XB)XB F ~ LY (I" )
where from Table 3, F
( 3 ~fF) LY = 1- 'iF - D / F. For column .-\
the valu e obtained by this approximation is 0.~9 which compares quite well with the actual I'alue of 0.39~. For column C, which has an unpure top product. the values are 0. 882 and 0.883, Another interes ting point is that the I'alues in Table 3 happen to correlate closely with the RG.·\ for most configurations (Skogestad, 1988). This is one of the fortunate circumstances which make the RGA useful for dist ill ation configuration selection. The differences between the confignrations with respect to disturbance sens iti vity are smaller at high frequency. Consider Fig. 6 which shows the effect of a disturbanc e in F on the compositions for column 0 for three different configurations. Although the gain is entirely different at steady state (infinity for the DBconfiguratio n , zero for the (LjO)( V /D )-configuration) the high frequency gai n (in itia l effect on compositions) is quite similar.
e~~F)
Table 3. F
Ul,U2
= Linearized effect of flow disturbances
on Dj F when both composition loops are open. Configurat ion
Disturbance d
UIU2
dF
dVd
dDd
dBd
LV, {TV, {TL DX, BX,X X
l-riF-D/F -DjF B/F 0
I 0
0
0 0
i-J(
~X bV,&v D
-a.70 L'/F
fIT'7T5
mn?1i 0
1+ "
L~ , L~ LV
0 0
V
IHl ' 'ITr5B
I 0 BjF
-I -DjF
1
-!fMo l+U D
1
T+\'7B'
0
-ITV'lR
?
I
0.!!. "
-I" / B
0 I"/B
D~
LV
-
--
/--'(V
One-point composition control
For one-point cOlltrol we s hall illtrodure the following cou\'ellt io n: The UI u2-configuration means that 1/1 is used for composition control and U2 is cons t ant (in manual). ~Io s t indu s trial column s use one-point control. usually of top composition. that is YD is controlled by 1/1 , The compositioll in the other elld of the column is then left uncontrolled or is poss iuly slowly adju sted by chang in g U2 ma nuall y. The automatic control of the one composition is quite simple, and th e choice of 1/1 is us ually 1I0t crucial. although reflux L or boilup \' is usually used because they directly affect composit ions. The more illlportant choice is which manipulat ed variable "2 is left in manual because this affect s the bcha\'ior of the uncontrolled cOlllpositioll. One should never choose 1/2 to ue D or II beC for 1/2 are for exalllpk L,V, L/O and V/B . and bl'cau se ofthl' strollg illt('ractioll iJl't\\'l'l'Il the top and the bottom of thl' COIUllllI Olll' g("IIl'rall," fiud s Ihat controlling th e one compositioll al.... o gin_ ' s rp<.tsollahly guod COII -
trol of the 'uncontrolkd ' cOlllpositioll. l'lIfurtullatell'. tl)('rl' is no simple llIethod ( like Table :3 for 01)('u.l oo p ) to fiud th(', bes l choice for U2.
H ow('\"cl'. ba sed OIl a large Ilumber of llullIeri -
cal evaluat ions of (DuD / Dd)" H, '" and (Dl·H / D") "L>. '" for "ari o ns disturbances d we foulld the 1)('s t choice for "2 to 1)(' L or \' if disturbances ill :F. Land \ . arC' cOllsidcrt'd. l :s iIlg Cl ratio. for example, u2=L/D. is of course best for ciisturi>all ces in F. 1I 0w ever, if the fecd rate is measured tlten usillg f('l'dforward act io ll and choosing 1/2 as L/F or V /F may be equally good, In many ca.ses the rea son for usillg ollc-point control is that the column is operating at maximulIl capacity. In this ca se III i s
not free to choose and will be equal to tlw flow wh ich is lim it ed (for examp le, V ifboilupor vapor flow s are lillliting. \'T ifcooling is limitin g) . Again, onc s hould not operatl' Ihe colullln s uch that or B are at their maxinlUIll ,'alues. The conclusion is that lire I.\' - or \·l. -ronfigurat io n ( poss ibl\' with a feedforward action froln F to 112 ) is th,' hest choice for onc-point composition conlrol. T hi s is probably thp rea SO li for why th is choice is the prderred ('ollfiguration ill illdu s try (how ever, as we shall sce below th e LV- configuration is not the hes t for two-point control). If reflux or boilup is large. then luel C011trol wit h 0 or D may be difficult. In thi s ca se th e DV- or BLconfigurations may be pre ferred for one-point control. The (L/O)(V/ll)-configuration is almost a s good as the IX· configuration witlr respect to composition CO!! I 1'01. gi"es bett e r level control if the reflux is large. and is ea sier tran sferred to two· point control. If one di s regarded the problem s of implementatioll (all flows L,V.O and B mu st be mea s ured or estimat ed) it wou ld have been tile best o\'erall choice.
o
"
Applies to steady state dVd = (1 - (V )dVd - (! - (LldLd ~v and (L represent deviations from constant molar Rows qF - fractIOn of liquid in feed X - any other manipulated input lL except D Band D subscript d denotes an additive disturbance 0;' this fl!~ V' = (1 - (v)V; L' = (I - (LlL; r' = 1 + L'/D + V;jB
,. 2~~----1'~~--'-
4
:1 I ~ I
______'-______r -____- ,
" -"
,.-1r L--."7'-'--'--t--
~9-21~.t.~~~1~.b3or~~,~.~2~~~1.~1~~~-~-~~ -~~~-~,~-~J
Figure 6 . Effect of disturbance in F On th e composition s for column D using three different configurations; LV, ( L/ D )(V / B ), DB . Solid line: (8Ybl8F) Dotted line : (8x~/8F).
5 5.1
Two-point control. Two-point control and the Rel a ti ve Gain Array
The 1, I-element of the RGA (denoted ),11 in the following) is shown as a function offrequency in Fig. ·5, Traditionallv. only t he steady state value of the RG.-\ has been con sidered (eg., ~lcA voy. 1983). However , newer results demonstrat e that the RGA as a funct ion of frequency yields even more information: Skogestad and ~ Ior ari ( 1987a) showed that large RGA-values indicate a plant which is very sensit ive to element -by- element uncertainty. and even more importantly, to input uncertaint~'. and that simple control st ru ctu res (decentralized controllers) should be used for su ch plants. Nett (1987) have demonstrated that RGA-\'alues close to 1 at frequencies corresponding to the closed-loop bandwidth means that sin gle- loops controlle rs may be designed independently. Therefore, a col umn will be easy to control wi t h single loops for a given config uration if the RGA is close to one at frequencies corresponding to the desired closed-loop bandwidth which in this case is about 0.1 m in-I. This statement will be the basis for the discussion that follows, Because of the liquid
S. Skogcstad. E. \\' . .Jacobscn and P. l.undstriim flow dynamics the plant will be triangular at high fr equency and the RGA will aproach one for all configurations. As one mea · sure of how easy a configuration is to control we shal l cons id er the frequency, "'I, where the RGA approaches one. \Ye shall see that thi s frequency turn s out to be c1osel!' related to the stead\' state RGA for most configurations (except for the DB- schelllej . Consequently the steady state beha\'ior reflects what happens at high frequency. This pro\'ides added justification for using the steady state RGA. The following five control configurations are included in Fig. 5: L V, D~', I '. DB. The resu lt s are discussed for each configuration: LV-configuration: All starts out at its steady state \alue until it reaches the first corner frequenC\' where it starts falling ofI' with a slope -Ion the logarithmic plol. Taking a close look one can fur column A.ll,D and G obscn'C' a slllall interlllediate rC'gioll when' the slope is somewhat less than -1. Thi s is the efrect of the time constant TZ as shown below. All then continne, falling ofI' with a -1 slope down to unity. For all tIle collllllns consi,lpred th,' frequency at which this happens is at
1J 1i-.1J
which is the frequenc!' at which t he response becomes dc'cou pled. The shape of the A1I-CUr\e can be di\'ided into three casps for the LV-structure: (1) rh :0; ~. The curve breaks off at l!Tl and there is then a flat region which s tart s at the frequency 1!T2' (2) BL "" ~. In thi s case the cur\e again brc'aks off al about l!TI , but now the flat region has di sappeared. (3) BL ~~. The break-off frequency is no longer at I!TI. but rather at WfV!AII(O). The three situations are illustrated in Figure 'j for column A at four different values of BL . Phy sically, difrerent \'alues of HL may be obtained by changing the plate geometry such that the holdup is more or less affected by the liquid flows. Note that increasing the influence of the flow dynamics, that is increasing TL, makes the decoupling take place at lower frequency and is thereby beneficial for control purposes. The reader should however notice that there usually is a close correlation between TI and TL, as they both depend 011 the amount of holdup in the column (Skogestad and ~Iorari, 19S8a). In fact, it mal' be shown that for well-designed columns we ha\-e TI "" Afr (O)e L and this is indeed seen to be the case with the example columns. 15 V-configuration: From Figure 5 wc obsen'e that the shap" of the RCA-curve for this configuratioll follows that of the LVconfiguration and that the frequency for crossing with olle is reduced by a factor corresponding to the ratio between their steadv state RGA-values, that is .
w~L/D)1
_ -
.Lt·
~I
A\7/ D )I '( 0) Afr(O)
( 19 )
where AlI(O) is the steadv-state IlG _-\-va lu e. -5~-configuration: TI;e same arguments as above apply to the this conhguratiOn and we have . ,lL/D)(I'/ B) _ . •LI' ~I
-
~l
A\7/ D )(I'/ B)(0) Af-rI'(O)
(20 )
iJ
The fact that the shape of the All-curve is similar for the L F. I' and 1J~ configurations assureS one that selecting the one with lowest All value at steady state guarantees the same to hold under dynamic conditions. DV-configuration: The All for the DV- configuration shows a different behavior than for the three discussed above. The main difference is of course that this configuration always has AlI(O) less than 1. The steady state value of the 1, 1-element in the RGA for this configuration is approximately 1!(1 + BXB! D(l - YD)). that is , it is close to one for columns with a pure bottom product (for columns with a pure top product the LB-configuration has RGA close to one). The RGA-value becomes one at the same frequency as for the LV-configuration, that is,
(:!I)
However, for columns with a pure top product the I(GA is close to one at all frequencies and this value is not too meaningful. One disadvantage with the DV-configuration for two-point control. even for columns with a pure bottom product, is that the RGA\'alues and thereby control beha\'ior may depend s trongly on the operating conditions. DB -configuration: The IlGA for the DEl -configuration is in finity at steady state. It fall s ofr with a -I slop e and the lowfrequency asymptote crosses one at (Skogestad et al.. 1989) 1 In S wfB "" ",fl' 1!(81'B) + 1!(D(l(22) YD)) LB multiplying ..:fl' is much kss than one for hi g h-purit!·
The term columns and for columns with large reflux. TlllIS. altho ugh the RGA for the OEl- configuration is much worse (higher) than for the LV-configuration at low frequency. il is significantly heller (closer toone) in the frequenc!' range important for feedback COlltrol, that is. in the fJ'()qllt'IlCY rallg<' frolll about 0.01 to 1 mill-I.
The obsen'ation of Finco et al. (10 ,~9) that the DEl-configuratioll gived better control performance thall the [.\'-confi guration fOI a propane- propylene column (simi lar to column D) is th,'rc,rore not surpr ising from the nGA-\·alues. In fart. from the nGA values all th e columns seelll to be quite simple to con trol usin g the DEl-configuration. Conclusion: The discussion abo\'e shows that there is a close correlation between the RGA at steady s tat e and it s value at all frequencies. This partly explain s why the use of steady stall' gain information for selecting the hest control configuration for two point cont rol of distillation colu lnll s IlIay be used with s uccess, cven if it from a control point of view is the frequenc!' (d!'namic) behaviour that is important. One interesting.but not loo surprising, result is that due to flow dynamics the value or '\1 1 becomes unity for moderate to high freqllencies for a ll configurations, which implies a decollpling of t he loops at these frequencies . This c1ecoupling is beneficial for control purposes.
,u G -y- --,- -- -
~-- --
- -- - - - -
Figure 7. All as a function of frequenn' for column ..\ with four different \'alues of ilL. 1.0.1 min 2.1 I;,in 3,.}.6 miu 4.100
min 5,2
Comparison of control performance
In this section we compare configurations for colulllns A and D based on their achie\'able control performance using single-l oop PI controllers for composition contra\. Single-loop controllers are chosen because this is the preferred controller struct ure in indu stry. The controllers were tuned to optimize robust performance using the same weights for performance and uncertainty as used by Skogestad and ~Iorari (198Sb). This corresponds to- allowing 20% error on each manipulated input and a \'ariable time delay of up to 1 min , and the performance objective is that the worst case response should have a closed-loop time constant bet ter than 20 min. Furthermore, the worst- case amplification of high-frequency disturbances should be less than two. Note that the controllers were optimized with respect to set -point changes. and the analysis does not take into account that some configurations are less sensitive to disturbances than others (recall Fig. 6). Mathematically, these optimal tunings were obtained by·minimizing the structured singular value J1- as discussed in more detail by
Selecting the Best Distillation COlllrol Structure Skogestad and ~Iorari (1988b). Crudely. one could say that the value of 1' at a given frequency expresses the worst case weighted error. This error should be less than one at any frequency to satisfy our performance objective. Since single-loop controllers generally are insensitive to model errors we would not expect very different PI tunings if some other performance objective than 11 had been used. For two-point composition control using sillgle loops the con\'ention is that UI is used to co ntrol YD alld U2 for Ia. Four differe nt configurations (UIU2) are cOllsidered: LV. DV. Oil and (L/D)(\'/il). Thecontrollcrs param e te rs and the minimized peak II-values for the 8 cases a re sU lllmarized in Tabl e ~. The p-values are also shown as a function of frequency in Fig. 8. The res ults ill Table -I correlate \'ery well with what we expect from the HG,\-values at high frequency_ The p-analysi s show s that the best configuration. witll two PI controllers. is the (L/D)(V/B)-collfiguration. For thi s COli figuration the pea k I'-\'alue is about 0., for both COIUlllIlS. Thi s is significantly below onc, and indi ca tes that the IH'rfofmctnce requirement Illay
be tight e lled, that is, we may require a f'hter closed -loop time constallt thall 20 mill. In fa ct. the \'alue of 0.7 is gett.ing close' to th e th eo re ticalmillimulll of 0 ..) (the 0 ..5 comes from the requiremellt of di s turballce amplificatioll less thall 2 at high frequellcy). The II -value for the Oil-configuration is sli g htly worse than the (L/D)(V / n) -configuration for column A alld almost as good for column D. Thi s is also what one would expect from the HG_-\plots. The LV- and DV- co nfiguration s gi\'e considerably higher II-values. Again , thi s is in acco rdallce with th e IlGA-plot s. If 1' is \'ery diffe re nt from one (as for the (L/D)(V/il)- configuration) th e n from an e nginee ring point of \'i e w it may be better
make 11 for robu s t performance equal to one. Different designs can then be compared based on their maxilllum achie\'able bandwidth. Two disadvantages with this approach are: 1) If the 5)'Stem is not even robu stly stable. then it is impossible to achie\'e 1' for robust performance equal to onc by adjusting the performance weight. 2) It introduces an ollter loop in the ,,-calculations.
5.3
Simulations
Simulationsofthe LV-. DV-. (L / D)(\,/il)- and DB -co nfigurations for column 0 (a propane-propylene splitter) with a full-order nonlinear model and the PI controllers given in Table ~ are shown in Fig. 9. The correlation between these simulations and the 11values is quite good, and the simulations support the conclusions made above. However, one should note that the simulations shown here are for a specific choice of set points and mod el error (setpoint change in YD and no error on the input s and no time delay). Onc should not necessarily expect a correlation between a single simulation , and the worst case res pon se (worst case combination of setpoints and model errors) for which I' is a measure . For e xample, although the I'-\'alu es are almost identical. th e simulation s in Fig. 9 indicate that the Oil-c o nfiguration is bett er than the (L/D)(V /B)-configuration. 1I0wever. with some other choice of set point change and model error the resu lt may be differe nt. Thi s illustrates one of the main advantages of II-analysis as opposed to simulations: one does not have to sea rch for the worst case It finds it automatically.
to Hot fix th e performanc e wC'igilt. and instead \'ary so me pa·
rameter in thi s weight (eg .. so me llIeasure for the bandwidth) to
0 .996 Table 4. jl-optimal PI- se ttings for column A and 0; C(s) = k t~,T;'. Gains k y and le". are for scaled compositions (eq 1). Column
Configuration
It
Tl y
Th
min
min
11.21 28.37
7.27 23.70 1.10 30.02
~
rK 1//
L
0.991 0.676 1.083 0.829 1.090 0.723 1.0.53 0.74·\
I'
77 , ]1
le I
0.49
0.34 3.98 0.43 0.6.5
D,I' D,n L,II
D
L I' 0 18
D,II D, n
1~.13
31.78 \.'i.07 20.6.5 7.93 39.95
0.13
IA6
OA,
0.12
11.2, 0 ..51 24.48
138.3 0.32 1.26
0.11
~.2.5
0.43 0041
3.96 44.79 4.08 1.61
---
1- -
:J-
~- 1
I1
ky
..
0.995
L,V
A
'I
..c
0.10
..
..
s.
.
(min. )
{\2
~ ~4 ~
~
~ '-~
..
0.09 2'
6.
S.
...
(min. )
;, L
3 ----,-'~ /'
'
----~-
4
3
12.86~~r=~~~T-----+-----r---~
.. (min.)
Figure 9. Time reponses for YD, Ia and L for a setpointchange in YD, column D , using four different configurations: l.LV , 2.DV, 3 .(L/D)(V /B), 4.DB. Figure 8. Structured Singular Value , 1', as a function of frequency for column A and 0 using four different configurations. I.LV 2.DV 3.(L/D)(V /B) 4.DB
S. Skogestad. E. \\' . .l aCOhSl'll alld P. LUlldst riim
6
Conclusion
The results in this paper are summarized below. The arguments mainly refer to composition control. although comments on level control are included. LV-configuratioll . .-\ good choice for onc.point control. but is not recommended for two.point control because of se nsiti\·it.'" to disturbances (Table 3) and relativel.'" 1'001' control performance due to interactions between control loop s. DV-configuratioll. One-point control: D must alu'ays be used for automatic control (never in manual). \Ia.'" be better than L\' for columns with large reflux because top le vel control is simpler. Two-point control: Works poorly when bottom product is not purer than top: poss ibly a good choice for columns with a pure bottom product (has to be investigated further). Disad\'antage: Very poor performance if failur e leed s to D constant (for example measurement in top fails). DB-configuration. l'nacceptable performance if used for onepoint control. Two·point control: Good control quality. in partic. ular for column with high purity and/or larg<' reflux. Le\'el con· trol also favors this conflguration for columns with large reflux. The main disadvantage is that it lacks integrety; performance i, very poor if failure gi\'es D or 13 constant. In particular. one can not put one of the loop s in manual. (L/D)(I/B)-col!figllrntioll. Overall this is th e best best choiclfor all modes of ope-ration. The main disadvantage is the need for measurements of all flow s L.D .13 and V which makes it mo re failure sensitive and more difficult to implement. ( L/D)V-confiyuratioll. Behaves somewhere between L\' and (L/D)(V/B). One important lesson to learn from the results on the DB· configuration is that one should be careful about making conelusions with regards to control quality based on steady state arguments. It is a fact that integral control is impossibl e (leads to instability) if the gain matrix may change sign (determinant changes sign). ][owe\'er, for the DO-configuration the stea dy· state gain matrix is "simply" infinite and singular, and there is no possi bli ty for change in sign. Acknowledgement. This work was made possible by financial support from NTNF. NOMENCLATURE (also see Fig. I) . steady state gains for column I, = Dyo(1 - Yo) + Bxa(l - 2'a) - "impurity sum" L = LT - reflux flow rate (kmol/min) La - liquid flow rate into rcboiler (kmol/min) M; - liquid holdup on theoretical tray no. i (kmol) qF - fraction liquid inn feed RGA - Relative Gain Array. elements are >"J
g;j -
r (1-i
D B ) - separation factor S = l-YD r[J JI = Va - boilup from reboil e r (kmol/min)
JlT - vapor flow rate on top tra,'( kmol/min) xa - mole fraction of light component in bottom product yo - mole fraction of light component in di s tillate (top prod uct) ZF - mole fraction of light component in feed
Greek symbols Q
= (I Y~\7f;
x,) .
relative volatility
>. 11 (s)
= (1- 912(')921('))-1 - 1 , I-element in RGA. 911 (S)g22(S) W - frequency (min-1) a(G),Q.(G) - maximum and minimum singular values Tt - dominant time constant for external flows (min) T2 - time consant for internal flows (min) TL = (8M./8L)v - hydraulic time constant (min) (1£ = NTL - O\'eraIllag for liquid response (min) REFERENCES Finco, M. V. , W. L. Luyben and R. E. Polleck, 1989, "Control of Distillation Columns with Low Relative Volatility" , Ind.Eng. Chem. Res., 28, 76-83.
llaggblom, 1\. E. and 1\.' \"'. \\'allcr. 10~(i."Rclations bctw e~ !l steady-state properties of distillation control sys tem s tructures . PrEp1'i"ts D)'COIlD SO. 2·13-2-18. llournc-mouth. U\. Dec. 1986. \lcAvov. T.J .. 1983. Int emetio" Analysis. Instrument Society of America. Trianglc Park. \C, rSA. \ett, C. \., 198" Prescnted at 1987 American Control Confer· ence, ~Iinneapolis, \1;\ .. June 198,. Shinskey. F. G .. 196,. Process COllt"ol Systems. !\lcGraw- Hill. \ew York. Shinskev. F. G .. 198-1. Distil/ation Co,l/ml. 2nd Edi· tion, "lcGraw.][ill:\cw York Skogestad, S. and \1. ~lorari. 19Sh. "Con trol Configuration Se· lection for Di stilation Colulllns". AIChE Journal, 33.10,1620· 1635. Skogestad. S. and ~1. \Iorari. 1987b. "A Systematic Approach to Distillation Column Control". Di stil lation 87. Brighton. Pub· li shed in I. Chelll. E. SYlllposilUlI Suits. 104 . ..1.,1· 86. Skogestad. S. and \\. \Iorari. 19S7c "Implication of Large RGA· El e ments on Control Performancc" . 1",1. t-.' En!]. Chem. Res .. 26,11,2121-2330. Skogestad, S .. 1988. "Di sturbance Rejection in Distillation Columl"". Proc. CIIE.\fD.-lT.·\ SS. J(i';·371. 19th Europl'an Symp. on Cam· puter Application s in the Chcmical Indu stry. Goteborg. Sweden. June 1988 . Skogestad. S. and ~\. \lorari. 19.5S". "Understanding the Dy namic I3ehavior o f Distillat ion Columns" , Ind.Ellg. Chem. Res, 27. 10. 18·18-1 862 . Skogestad, S. and \1. \lorari .1988 b. "IX·cont rol of a lligh· Puri ty Distillation Column" , C;'OIl. F"g. Sci .. 43. 1. :33 .. 18 . Skogestad. S .. 1980 . "\Iodelling and COllt 1'01 of Distillation Columlls as a 5x5 system". Proc. C\CllI 89. 20th European Symp. on Computer Application s in the Chclllicallndllstry. Erlangen. BRD. April 1989. Skogestad, S., E.W.Jacob sc ll alld \1. ~I!orari. 1080 . "DB-colltrol of distillatioll col u III li S" • Work ill prog ress . Wailer , K. 108G. "Di st illat iOIl COllt 1'01 Systelll Strllct ures·' . Prep"i nt, D}'CORD SG, 1-10. BOllrtll'lllouth . rt\. Dee. 1086.
© I F.-\C I h Il;t 111 il " ;111 <1 COllt rol (11
C he mica l Reacto r :>' (D\' ( :( )Rl) + ·;-':~II. \L!; \"tr il hi.
T he :\<" her IOl "d " I ~ I ~~ I
SELECTING THE BEST DISTILLATION
CONTROL STRUCTURE S. Skogestad, E. W. Jacobsen and P. Lundstrom (;lil'llli((l/ F IIgi//I'nillg, .\'lIr;'·"gillll 1II ,llilll l,' 0( Tl'I'lillll/lIg:r (.\ T II I, .\' -ill] -/ F I'II /II/I/I'illl, .\'lIn,'tl\'
Abst r ac t Til e ~cll,rt i o ll of an appro pri;tll' control structur e (collfigll ratioll) is the IIlOs t important dccision \\'\it'1l desi!-;Ili!lg di :,t illatioll control s ~·st(,llls.
The steady state ItG :\ is (onllIlon ly lI sed ill inou stry as
a toul for sckct illg tlte t)('~t stnlrtllrc. I II thi s paper it is stressed that deci sio ll s regarding cOlltroll€'r
de,i),;n s hould be ba sed o n the initial re; pon se ( bi gh -fre<]uency bebavior) rather t ban th e steady state, 1I 0II'l'I'('(', for 1II0s t di s tillation cOllfigurations tbe stc'ady statl' HGA turns out to be a good indicator of (' xl'ectl'd control qualitl'. 0ue countNcxa mple is the Dll-configuration II'hicb has iufinite s teady'tOltC' HG _-\ - \'alues , II OII'l'\'N, thc HG,\ -\'"lu e5 at bigber frequ e ncy are close to onc , "lid good control performan ce is possi ble,
Ba sed on a frequenc~'-dependent nc A-analysis and optimal P I controller designs_ th e ( L / D )( \ . / 13)COli fi g uratio n is found to be tbc bes t choice for tll'o-point compositioll co ntrol. Tbe traditiollal L\'cOllfi gu rati oll pe rfo rm s II111 Clt poorl'r. hut is preferab le if ollc-point control is used,
1
Introduction
Co nside r t he s im p le tll'o- prod ll c, di"illa t ioll (OllIlIIn ill Fi i\ , I, From a cont rol point it lIIay bc \'i(,lI'"d as ~ X ,S systc>m, Th" op tima l controll er s l,olll,1. based 011 ,, 11 a\'ai l"bl" illformation (lIIea s urement s. procl'ss mod el. l'xpeci('cI d is turballccs) ",allipul a te 111/ .' j input s (L.\'.\',(,.O.fl) ill ord c>r t o kecp the ,'i O"'p"\; (I<,\'pl , ill top and bottom. pres",,'c_ top alld bottom composition ) ;IS clos<, as poss ible to thcir dc; ired \,;dlles , lI olI'e\'cr. fell' colllmll s. if all\', are controlled II s ill g a full ,) x.'j co nt ro ll('1'. In order '0 si III pl i f." ,I,, ' design and get a (ontrol S\'s tl'lll whi ch i:; :-;illlplC'1' to ulld l' rsta nd. la ret u nc , to make failure t o lcra nt and wh ich is in se nsitiH.' to
plant operation , a decentralized control system based on sin gl ,· loops is used in practice. Since the levels and press ure ha\'e to be controll ed at a ll times to ensure s ta ble operation. and becauS/' t hese control loops arc essellt ially inclep<' II dent of the (ollll""i l ion co nt ro l. the Ic\"cl and pre ss ure cOlltrol sy:.;t cm is d e:-; i~lI rd firs \. Engineers often do a good crfort in des ig ni ng thi s s ub s ~'" ten reliably. hut Ill
:1 1;)
p (M v)
v L-_--t:k}_B.,
xB
Figure 1. T wo p roduc t d istillation column with single feed and t o t a l condenser ,
:11{i
S. Skogcstad. E. W. Jacobscll and P. Lundstriil1l
@ 6
D, Yo
of the seven example columns :\-G studied by Skogestad alld ~Iorari (1988a). Steady-state data for these columns are gi\'en in Table 1. For all examples. we assume constant molar flows. binary mixtures, constant relatil'e volatility and perfect control of pressure and le\·els. However. in contrast to Skogestad and Morari (1988a) we shall here include the liquid flow dynamics in the column. The steady state holdup on all trays. including reboiler and condenser. is .If;! F = 0.5 min. The steady state gain matrix for the L\'-configuration is shown in Table 2 for all columns. \'ote that scaled (relative. logarithmic) compositions have been used: (I )
@
Shown in Table 2 are also the time constant TI.T·, alld Ih which are used in the simplified dynamic model discussed below. ef. = :\'TL is overall liquid lag fr01l1 the top to the bottom of the col· umn. TL is computed from linearizing the Francis weir fOrIlllIla. Assuming in addition half of liquid Ol'er weir we get
v B,
'---D
TL
Xs
Figure 2. Two product distillation column with LV- configuration.
= (d.\l i ) = ~~ dL i
\.
3 L
(2 )
The following configurations are considered in the paper: I.\, (often denoted "energy balance" or "indirect material balance"). DV ("material balance"). DIl. (L/D)\, ("Ryskamps schl'llle") and (L/D)(V/B) ("double ratio").
dependent variables to use for compositiou control. The issue has been discussed for quite sOllle time by industrial people (eg., Shinskey, 19G7. 198·1). but has ollly recentl,' been treated in any Table 1. Steady-state data for distillation column examples. ,-\11 detail by academic researchers (eg., Wailer, 198G , Skogestad and columns have liquid feed ('IF = I). Morari,1987a). Shinskey relies on t.he steady state \'alue of t.he RGA for se- Column er 1'1 NF 1 - YD ZF xB D/F L/F lecting the best control structure. The steady state RGA contains no information about disturbances and dynamic behavior. both A 0.5 21 0.01 0.,s00 2.700 1.5 40 0.01 of which are crucial for e\'aluating control performance. so one B 0.1 1..5 40 21 0.01 0.01 0.092 2.329 would not expect that. this measure by itself is sufficient to make 0.5 C 21 0.10 1.5 40 0.002 0.55,s 2.737 conclusions with regards to control performance. The fact that D 0.65 1.12 110 39 0.005 0.10 0.61·1 11.802 this measure t urns out to be very useful for distillation colulllus E 0.2 15 5 0.0001 0.05 0.158 0.226 5 must therefore be a result of fortunate circumstances. One obF 0.5 15 10 0.0001 0.0001 0.500 0.227 5 jective of this paper is to investigate this in 1Il0re detail. G 0.5 1.5 80 40 0.0001 0.0001 0.500 2.635 One counterexample to the usefullness of the steady state RGA is the DB-configurat.ion. It involves using distillate product D and bottom product Il for composition control. This control scheme has pre\'iously been laheled ' impossible' by most distil- Table 2. Data used in simple model of distillation columns, eq. 8. lation experts (eg .. Shins!;ey. 19~q. p. 15·1) because its RGA is infinity at steady state. ·Yet. Finco et al. (1989) have recent.I,· Column GLl·(O)s eL TI T2 shown both with simulations and with actual implementation ( 87.8 -8G.4 ) that the scheme is indeed workable. We shall investigate the A 194 !.'i 2.4G 108.2 -109.6 reasons for this in more detail in this paper. In the paper three different modes of operation are considered 74 79 -17!.7) B 2.50 1.5 2.86 . with respect to the difference between the control configurations: 90.191 -90.5 1. Open-loop (manual) operation ( IG.023 -IG.O) C 2·1 10 2.44 2. One-point composition control 9.29 -10.7 3. Two-point composition control. ( 2~.5S5 -24.2 ) Although most industrial columns are operated with oneD 1.5~ 30 1.·5~ 21.270 -21.3 point composition control. the emphasis will be on the last mode. ( 203.4 -131..5) The reason is that this is the most difficult problem and has large E 82 30 11.06 22.47 -22 ..5 potential for economic savings. Although energy savings are of( 10740 -10730 ) ten mentioned. the increased yield of valueable product by keepF 299G -1 7.3-1 92.57 -92G7 ing both compositions at their optimal values (thereby avoiding overpurification at the one cnd of the column and loss of valuable ( 8648.94 -8G46 ) G 20333 30 .5.06 product at the other end) is probably far more important. The 11347.06 -11350 first mode (open-loop) is not too interesting by itself, but it is of interest for the other two modes because low open-loop sensitivity to disturbances (good 'built-in' disturbance rejection) introduces 2 Modelling. less need for feedback control and makes the remaining control problem simpler. Because of the large number of trays in some of the columns The analysis demonstrates that steady state data may be en(110 trays for column D which give a 220th order model if both tirely misleading for the enluation of control performance. For composition and liquid flow dynamics are included) we choose to feedback control, it is the initial response. corresponding to the use the simple two time- constant dynamic model presented by time constant of the closed-loop system, which is of primary imSkogestad and Morari (1988a). Using a simple model also makes portance. it easier for the reader to interpret and check the results. The two Example columns. Throghout the paper we shall make use time-constant model is deri ved assuming the flow and compostion
C
Selecting the Best Distillatioll COlltrol Structure dynamics be decoupled, and then the two separate model s for the composition and flow dynamics are simply combined, In reality, the flow dynamics do affect the composition dynamics and the model will be somewhat in error. In particular , the time constant Tl may be different (larger) from th" one shown in Table 2 which was obtained by Skogestad and :'!orari (1988a), However. the res ults in thi s paper tUI'l\ out to be o nly weakly dependent on '" Let dLT = dL and dl'T represent sJIlall changes in th"liquid and vapor flows in the top of the column, and let dLB and dl'lJ = dl' be changes in the bott o m, In addition to constant molar flows wc make the foll owing assumptions I. Imm ed iate \'apor response (perfect pressure control)
2, Liquid flow fr om top to bottoJll as:'; lag s in seri('s and \'apor has no dfect on holdup (,I )
where
!J ds)
= 1/( 1+ (8d\
(,) )
)--)'
:3, Perfect control of reboiler aud condenser le \,(,1
dl'lJ(s)
= dl'(.<) = dLB(S)
~
dB (s)
(i)
To generalize these "quilt ions to the case of not constant molar flows and to the case whell cilallges ill \'apoJ' flow affect holdups (.\ in Skogestad and :'!orari (19S8a) is nonzero) additional paramet ers bave to be introduced in equat iolts 3 and 4, IX-configuration, The simp lified model of the LV-configuration t he n becomes LV ~dL I+TI8
dXB
+
LV
I
I' ) _~ dl'
I+T28
LV
~gd,5)dL 1+
(LI' gII +gI2
+ ( g21LV + g22LV
I+TI8 LI'
_ ~
1 + T2S
TI S
1+
)
dl' (8)
TI S
gh
l where ' denote the s teady s tate gains for the LV-configurati on, These are given in Tabl e 2 for scaled compositions,
Models for other configurations, :'10dels for other configurations are easily obtained from eq, 8 by choosing two independe nt \'ariables ill and "2 for composition control and expn'ssing dL and dl' in te rllls of these \'ariables using equations 3 to i, This gi\'cS
( rIld~)
= ,\I"',"'(s) 1.1
(rlIII) dill
and dl (1 - gds))
= -g/.l s) dD ~ dB,
:117 We obtain
~l) ~1
( 14)
:\ot<' that (15)
and lim,_ogL(s) = l. Th e elements in gain matrix for the DB· configuration \\'ill therefore approach infinity at la\\' frequency (stea dy s tate) and it will al so become singular. The physical interpretation is that a decrease in, for example, D with B con· stan t. will imm ediately yield a corresponding increase in L, This increase will yield a corresponding increase in V, which subse· quently \\'ill increase L even more , etc, Consequently, the effect is that the internal flows eventually will approach infinity, ~!ath· ematically, there is an integrator at low frequency: As s - 0 we ha\'e dL = dV = -(dD + dB)/OLS, Several authors (eg, lIiiggblom and \\'aller ,19S6; Skogestad and :'!orari, 1987b; Skogestad.l9SS) discuss ho\\' to obtain steady sta te models for variou s confIgurations, but these transformations do not include the flow d,'namics which are crucial for the dy· namic behavior. Example columns, Accumcy of simplifie d Illodel fol' CollL",n A, To illu strate the accuracy of the simplifIed model the singular \'alu es a nd UG.-\· values as a function of frequency are compared with exact valu es in Fig, 3, The exact values were obtained by linearizing th e full 8 2th order model \\'hich has t\\'o stat es (composition and holdup) for each tray, The hydraulic tray lag in the full- order mod el is 'L = 8L /S = O,06min, From Fig, 3 we conclude that th e simplifi ed mod"l seems to capture the true behavior quite well, Diffel'ent conjigll1'flti01l8 fol' colllllln A, Fig, 4 show the gain elements gi ) as a function of frequency for four configurations, ""ote in particnlar how the flow dynami cs decoupl" the dynamics at high freqnency, RGA-I'!llues of Columns A-G, Fig, 5 shows the 1,1·elemen t of the UGA as a function of freqnency for five different configurations, \Ve shall discu ss these \'alnes in more detail in Section 5,1. Fig, ,) should be compared \\'ith s imilar figures given in Skogestad and ~!orari (1988a) for the case with no flow dynamic s ( that is, dLB = dL), The introduction of flow dynamics are seen to change the results dramatically, Firstly, the response becomes decoupled at high frequency which makes UGA=1. Second I,', the importance of the time constant T2 associated with internal flow s seems to disappear almost completely,
(9 )
and th e linear model for the new configuration
d!Jo) ( d"'B
= (;"'"'(s)
(dILl) dll 2
( 10)
becomes (11)
If ratios are used as inclepCIl(\ent \'ariables these must fir st be expressed as a function of the actual flows. for example with "1
= L/D, d( L/ D)(s) = (1/ D )dL(s) - (L/ D2 )dD( s)
( 12)
DV-configuration, In this case \\'ith perfect level control we have dL = -dD + dl ' , that is, 01' M LI ,
=
(-01
( 13 )
DB-configuration, In this case dl ' = fJL(s)dL - dB \\'here dL = dV - dD, Eliminating dV yields dL ( 1 - gLls)) = -dD - dB
Figure 3, Singular values and .\11 as a function of frequency for simplified model and full model. Dotted line: Simplified model.
S. Skogestad. E. \\' . .lacobsen and P. Lundstriim
LV
10'
le.:!
,. ·~----~~7~~-----+-----+----~ -----~----~
1-... .....
------, -,::- ~----!.
~
~ \
1 ••
1.
l
10
~
1.
~~-
:1-::._-___ .. ~.::..:::- .:: : ~~ :=--""-,
,, ,
-2- -- -- -- -
..
:u,~ - c.
1
=:_~- ~~ ---
. -=-~ -=-.4:=- 3
--, ' , ,, ,,,, - , '"''
--, """ '.' , , "'"
1.- b
.'"
10
)
'"
I l":!
I"
.7'> - -'
1
OV ,. '-r------,------.------,------.-----,r-----, 1 0° -r------~--~~------t------t----~r-----~
3
.....
4
~. ..jr-_-__- I __ -+__ +=-"">o __.
, l~- · _t------t_----_+------t_----_+----~'~\r_----~
---. --.--.-~.,.,
\.., - 1
,
10
I;
................", . _ ....",,., -----'--'-" " _" 1 '" ~ 1""/ 1""
,,,, _. _. __ . __ ._____ 10'
" " _--, ...., ' " ' 1" I
.....,-,- " , ,, 1"
___ C_____ ._ _ ____ .__
-.-_~-~-('-=L"-I =0--)..( ~ v I--=B-,-)~_-,-_---, .....
I ~? - ~------ r_----- ~------I-----~-------r_----~
..
-~
----......--.
--- - - --.---- :::."~~~~ ~F=---
1('1- t -
_ "
... ,,"
1,,6
_ ,- "
1 .. 5
.. "
__ ..-- . , ,, ,, _
10~
,_" ." "
._-._" " " '- , , " , "
\0 2
10 - )
10 1
1~ - ·_t------t_----_+------+_----_+~\--_+--~~ \
--,- , ., " '" - , -e " "" -,- , -, "'" 3
10
?
10
I
~~:'J.,"'t--r-~n' 1~ 1
2
"'............ ,-~~ le- 2_t------t_----_+------+_----_+-'~~~·~--~ - +-=~~~'~-~
__ __I .. ________ ______ __
\
\
10- ~-t------+------+------+------+----~\rt----~ \ \
10-
6 1')
--:-"" '"""11 - ' , . ... ,," - , - ,-.,'''' ----,-rT" r " r--,--r-n·, n~ -4
10
3
10 2
1"
1
l()
1
W Figure 4. Cain elelll ents as a fnnction of frequency for colulll n A using fOllr different con fig ura tion s: LV. DV, (L/D)(V/B) and DB.
3
Open-loop (manual) operation.
The term "open -Ioop" shollld here be put in quotes becau se we are not talking abou t an uncontrolled collllll n. but assullle the levels and pres snre arc perfectl\' controlled and co nsid e r th e effect of the remaining ind"Ill'ndent \'ariables on the compositions. For exa mple. the open -loop operation of the LV -con figuration assullles le\'e1 s and pressu re to be perfectly controlled with D.n and YT. and t he rcmainillg manipulated variables. Land Y. to be constant ( in manual). Th e use of a true open-loop 5 x 5 model where all five flow s L.V.D.n and VT are ind epende nt variables is co nsidered by Skogestad ( 1989). Cood open-loop performance is then achieved if the effect of di stu rbances 011 compositions are small , that is . if the 'built-in' di sturbance rejection is good. Disturbances include changes in feed conditions (F. ZF , qF) alld di sturbances on all manipulated flow s L,Y ,D ,B and V T . The steady- state effect of di sturbances on compos itions may be evaluated quit e accurately by considering the steady-state effect of the disturbances on 0 /13 or equivalently D/F. This follows since major effect on product compositions of any change in the column may be obtained by assuming the separation factor S const ant (Shinskey. 198·1). Differentiating the component material balance with S constant yi elds
F
'· ' l-r==+==:r---:~--'--I--I-r-l - _"'!- _ __ -~, 1
10 ~~----~----~~~~~~~,~----+_----+_----r_--~
~ ...... 5
3--,~
"
>~~
2
G
1. 3
---
- ..... , ~ ,
1.'
.....
"
;',,.4 " ....
~
5 ...... -
~
3 ".... ...
r-~
~
w Figure 5. All-value as a function of frequency for column s A-G using five different configurat ions. l.LV 2.DV 3.(L/D)(V /13) 4.( L/D)V 5.013
Selecting the Best Distillation Control Structure 1
dIB
-;-:-----,= -I", (Fdz F (1 - IB)l'B
(I -YD)YD
(YD -l'B)Fd(D / F))
( IG) where the impurity sum is defined as Is = D( 1 - YD )YD + B(l IB)I B. The effect of various disturbances on Fd( D / F) is gi "en in Table 3. Configurations with small entries in Table 3 are insens iti ve to disturbances at steady state and should be preferred. We see that the (L/O)(V /D )-configurat ion is a good choice when the reflux is large. Note that the DD-configuration is not included in Table 3 because 0 and B are not independent at steady state. In general, the values obtained with the above approximation are quite accurate , in particular, for the gain of the least pure product. For example , the effect of a change in feed rate on YD with constant L and V is approximately (eq.16)
BYD) (I-YD)YD(YD-XB ) (DD/F) ( 7iF Ly=-D(l-YD)YD+B(I-XB)XB F ~ LY (I" )
where from Table 3, F
( 3 ~fF) LY = 1- 'iF - D / F. For column .-\
the valu e obtained by this approximation is 0.~9 which compares quite well with the actual I'alue of 0.39~. For column C, which has an unpure top product. the values are 0. 882 and 0.883, Another interes ting point is that the I'alues in Table 3 happen to correlate closely with the RG.·\ for most configurations (Skogestad, 1988). This is one of the fortunate circumstances which make the RGA useful for dist ill ation configuration selection. The differences between the confignrations with respect to disturbance sens iti vity are smaller at high frequency. Consider Fig. 6 which shows the effect of a disturbanc e in F on the compositions for column 0 for three different configurations. Although the gain is entirely different at steady state (infinity for the DBconfiguratio n , zero for the (LjO)( V /D )-configuration) the high frequency gai n (in itia l effect on compositions) is quite similar.
e~~F)
Table 3. F
Ul,U2
= Linearized effect of flow disturbances
on Dj F when both composition loops are open. Configurat ion
Disturbance d
UIU2
dF
dVd
dDd
dBd
LV, {TV, {TL DX, BX,X X
l-riF-D/F -DjF B/F 0
I 0
0
0 0
i-J(
~X bV,&v D
-a.70 L'/F
fIT'7T5
mn?1i 0
1+ "
L~ , L~ LV
0 0
V
IHl ' 'ITr5B
I 0 BjF
-I -DjF
1
-!fMo l+U D
1
T+\'7B'
0
-ITV'lR
?
I
0.!!. "
-I" / B
0 I"/B
D~
LV
-
--
/--'(V
One-point composition control
For one-point cOlltrol we s hall illtrodure the following cou\'ellt io n: The UI u2-configuration means that 1/1 is used for composition control and U2 is cons t ant (in manual). ~Io s t indu s trial column s use one-point control. usually of top composition. that is YD is controlled by 1/1 , The compositioll in the other elld of the column is then left uncontrolled or is poss iuly slowly adju sted by chang in g U2 ma nuall y. The automatic control of the one composition is quite simple, and th e choice of 1/1 is us ually 1I0t crucial. although reflux L or boilup \' is usually used because they directly affect composit ions. The more illlportant choice is which manipulat ed variable "2 is left in manual because this affect s the bcha\'ior of the uncontrolled cOlllpositioll. One should never choose 1/2 to ue D or II beC
trol of the 'uncontrolkd ' cOlllpositioll. l'lIfurtullatell'. tl)('rl' is no simple llIethod ( like Table :3 for 01)('u.l oo p ) to fiud th(', bes l choice for U2.
H ow('\"cl'. ba sed OIl a large Ilumber of llullIeri -
cal evaluat ions of (DuD / Dd)" H, '" and (Dl·H / D") "L>. '" for "ari o ns disturbances d we foulld the 1)('s t choice for "2 to 1)(' L or \' if disturbances ill :F. Land \ . arC' cOllsidcrt'd. l :s iIlg Cl ratio. for example, u2=L/D. is of course best for ciisturi>all ces in F. 1I 0w ever, if the fecd rate is measured tlten usillg f('l'dforward act io ll and choosing 1/2 as L/F or V /F may be equally good, In many ca.ses the rea son for usillg ollc-point control is that the column is operating at maximulIl capacity. In this ca se III i s
not free to choose and will be equal to tlw flow wh ich is lim it ed (for examp le, V ifboilupor vapor flow s are lillliting. \'T ifcooling is limitin g) . Again, onc s hould not operatl' Ihe colullln s uch that or B are at their maxinlUIll ,'alues. The conclusion is that lire I.\' - or \·l. -ronfigurat io n ( poss ibl\' with a feedforward action froln F to 112 ) is th,' hest choice for onc-point composition conlrol. T hi s is probably thp rea SO li for why th is choice is the prderred ('ollfiguration ill illdu s try (how ever, as we shall sce below th e LV- configuration is not the hes t for two-point control). If reflux or boilup is large. then luel C011trol wit h 0 or D may be difficult. In thi s ca se th e DV- or BLconfigurations may be pre ferred for one-point control. The (L/O)(V/ll)-configuration is almost a s good as the IX· configuration witlr respect to composition CO!! I 1'01. gi"es bett e r level control if the reflux is large. and is ea sier tran sferred to two· point control. If one di s regarded the problem s of implementatioll (all flows L,V.O and B mu st be mea s ured or estimat ed) it wou ld have been tile best o\'erall choice.
o
"
Applies to steady state dVd = (1 - (V )dVd - (! - (LldLd ~v and (L represent deviations from constant molar Rows qF - fractIOn of liquid in feed X - any other manipulated input lL except D Band D subscript d denotes an additive disturbance 0;' this fl!~ V' = (1 - (v)V; L' = (I - (LlL; r' = 1 + L'/D + V;jB
,. 2~~----1'~~--'-
4
:1 I ~ I
______'-______r -____- ,
" -"
,.-1r L--."7'-'--'--t--
~9-21~.t.~~~1~.b3or~~,~.~2~~~1.~1~~~-~-~~ -~~~-~,~-~J
Figure 6 . Effect of disturbance in F On th e composition s for column D using three different configurations; LV, ( L/ D )(V / B ), DB . Solid line: (8Ybl8F) Dotted line : (8x~/8F).
5 5.1
Two-point control. Two-point control and the Rel a ti ve Gain Array
The 1, I-element of the RGA (denoted ),11 in the following) is shown as a function offrequency in Fig. ·5, Traditionallv. only t he steady state value of the RG.-\ has been con sidered (eg., ~lcA voy. 1983). However , newer results demonstrat e that the RGA as a funct ion of frequency yields even more information: Skogestad and ~ Ior ari ( 1987a) showed that large RGA-values indicate a plant which is very sensit ive to element -by- element uncertainty. and even more importantly, to input uncertaint~'. and that simple control st ru ctu res (decentralized controllers) should be used for su ch plants. Nett (1987) have demonstrated that RGA-\'alues close to 1 at frequencies corresponding to the closed-loop bandwidth means that sin gle- loops controlle rs may be designed independently. Therefore, a col umn will be easy to control wi t h single loops for a given config uration if the RGA is close to one at frequencies corresponding to the desired closed-loop bandwidth which in this case is about 0.1 m in-I. This statement will be the basis for the discussion that follows, Because of the liquid
S. Skogcstad. E. \\' . .Jacobscn and P. l.undstriim flow dynamics the plant will be triangular at high fr equency and the RGA will aproach one for all configurations. As one mea · sure of how easy a configuration is to control we shal l cons id er the frequency, "'I, where the RGA approaches one. \Ye shall see that thi s frequency turn s out to be c1osel!' related to the stead\' state RGA for most configurations (except for the DB- schelllej . Consequently the steady state beha\'ior reflects what happens at high frequency. This pro\'ides added justification for using the steady state RGA. The following five control configurations are included in Fig. 5: L V, D~', I '. DB. The resu lt s are discussed for each configuration: LV-configuration: All starts out at its steady state \alue until it reaches the first corner frequenC\' where it starts falling ofI' with a slope -Ion the logarithmic plol. Taking a close look one can fur column A.ll,D and G obscn'C' a slllall interlllediate rC'gioll when' the slope is somewhat less than -1. Thi s is the efrect of the time constant TZ as shown below. All then continne, falling ofI' with a -1 slope down to unity. For all tIle collllllns consi,lpred th,' frequency at which this happens is at
1J 1i-.1J
which is the frequenc!' at which t he response becomes dc'cou pled. The shape of the A1I-CUr\e can be di\'ided into three casps for the LV-structure: (1) rh :0; ~. The curve breaks off at l!Tl and there is then a flat region which s tart s at the frequency 1!T2' (2) BL "" ~. In thi s case the cur\e again brc'aks off al about l!TI , but now the flat region has di sappeared. (3) BL ~~. The break-off frequency is no longer at I!TI. but rather at WfV!AII(O). The three situations are illustrated in Figure 'j for column A at four different values of BL . Phy sically, difrerent \'alues of HL may be obtained by changing the plate geometry such that the holdup is more or less affected by the liquid flows. Note that increasing the influence of the flow dynamics, that is increasing TL, makes the decoupling take place at lower frequency and is thereby beneficial for control purposes. The reader should however notice that there usually is a close correlation between TI and TL, as they both depend 011 the amount of holdup in the column (Skogestad and ~Iorari, 19S8a). In fact, it mal' be shown that for well-designed columns we ha\-e TI "" Afr (O)e L and this is indeed seen to be the case with the example columns. 15 V-configuration: From Figure 5 wc obsen'e that the shap" of the RCA-curve for this configuratioll follows that of the LVconfiguration and that the frequency for crossing with olle is reduced by a factor corresponding to the ratio between their steadv state RGA-values, that is .
w~L/D)1
_ -
.Lt·
~I
A\7/ D )I '( 0) Afr(O)
( 19 )
where AlI(O) is the steadv-state IlG _-\-va lu e. -5~-configuration: TI;e same arguments as above apply to the this conhguratiOn and we have . ,lL/D)(I'/ B) _ . •LI' ~I
-
~l
A\7/ D )(I'/ B)(0) Af-rI'(O)
(20 )
iJ
The fact that the shape of the All-curve is similar for the L F. I' and 1J~ configurations assureS one that selecting the one with lowest All value at steady state guarantees the same to hold under dynamic conditions. DV-configuration: The All for the DV- configuration shows a different behavior than for the three discussed above. The main difference is of course that this configuration always has AlI(O) less than 1. The steady state value of the 1, 1-element in the RGA for this configuration is approximately 1!(1 + BXB! D(l - YD)). that is , it is close to one for columns with a pure bottom product (for columns with a pure top product the LB-configuration has RGA close to one). The RGA-value becomes one at the same frequency as for the LV-configuration, that is,
(:!I)
However, for columns with a pure top product the I(GA is close to one at all frequencies and this value is not too meaningful. One disadvantage with the DV-configuration for two-point control. even for columns with a pure bottom product, is that the RGA\'alues and thereby control beha\'ior may depend s trongly on the operating conditions. DB -configuration: The IlGA for the DEl -configuration is in finity at steady state. It fall s ofr with a -I slop e and the lowfrequency asymptote crosses one at (Skogestad et al.. 1989) 1 In S wfB "" ",fl' 1!(81'B) + 1!(D(l(22) YD)) LB multiplying ..:fl' is much kss than one for hi g h-purit!·
The term columns and for columns with large reflux. TlllIS. altho ugh the RGA for the OEl- configuration is much worse (higher) than for the LV-configuration at low frequency. il is significantly heller (closer toone) in the frequenc!' range important for feedback COlltrol, that is. in the fJ'()qllt'IlCY rallg<' frolll about 0.01 to 1 mill-I.
The obsen'ation of Finco et al. (10 ,~9) that the DEl-configuratioll gived better control performance thall the [.\'-confi guration fOI a propane- propylene column (simi lar to column D) is th,'rc,rore not surpr ising from the nGA-\·alues. In fart. from the nGA values all th e columns seelll to be quite simple to con trol usin g the DEl-configuration. Conclusion: The discussion abo\'e shows that there is a close correlation between the RGA at steady s tat e and it s value at all frequencies. This partly explain s why the use of steady stall' gain information for selecting the hest control configuration for two point cont rol of distillation colu lnll s IlIay be used with s uccess, cven if it from a control point of view is the frequenc!' (d!'namic) behaviour that is important. One interesting.but not loo surprising, result is that due to flow dynamics the value or '\1 1 becomes unity for moderate to high freqllencies for a ll configurations, which implies a decollpling of t he loops at these frequencies . This c1ecoupling is beneficial for control purposes.
,u G -y- --,- -- -
~-- --
- -- - - - -
Figure 7. All as a function of frequenn' for column ..\ with four different \'alues of ilL. 1.0.1 min 2.1 I;,in 3,.}.6 miu 4.100
min 5,2
Comparison of control performance
In this section we compare configurations for colulllns A and D based on their achie\'able control performance using single-l oop PI controllers for composition contra\. Single-loop controllers are chosen because this is the preferred controller struct ure in indu stry. The controllers were tuned to optimize robust performance using the same weights for performance and uncertainty as used by Skogestad and ~Iorari (198Sb). This corresponds to- allowing 20% error on each manipulated input and a \'ariable time delay of up to 1 min , and the performance objective is that the worst case response should have a closed-loop time constant bet ter than 20 min. Furthermore, the worst- case amplification of high-frequency disturbances should be less than two. Note that the controllers were optimized with respect to set -point changes. and the analysis does not take into account that some configurations are less sensitive to disturbances than others (recall Fig. 6). Mathematically, these optimal tunings were obtained by·minimizing the structured singular value J1- as discussed in more detail by
Selecting the Best Distillation COlllrol Structure Skogestad and ~Iorari (1988b). Crudely. one could say that the value of 1' at a given frequency expresses the worst case weighted error. This error should be less than one at any frequency to satisfy our performance objective. Since single-loop controllers generally are insensitive to model errors we would not expect very different PI tunings if some other performance objective than 11 had been used. For two-point composition control using sillgle loops the con\'ention is that UI is used to co ntrol YD alld U2 for Ia. Four differe nt configurations (UIU2) are cOllsidered: LV. DV. Oil and (L/D)(\'/il). Thecontrollcrs param e te rs and the minimized peak II-values for the 8 cases a re sU lllmarized in Tabl e ~. The p-values are also shown as a function of frequency in Fig. 8. The res ults ill Table -I correlate \'ery well with what we expect from the HG,\-values at high frequency_ The p-analysi s show s that the best configuration. witll two PI controllers. is the (L/D)(V/B)-collfiguration. For thi s COli figuration the pea k I'-\'alue is about 0., for both COIUlllIlS. Thi s is significantly below onc, and indi ca tes that the IH'rfofmctnce requirement Illay
be tight e lled, that is, we may require a f'hter closed -loop time constallt thall 20 mill. In fa ct. the \'alue of 0.7 is gett.ing close' to th e th eo re ticalmillimulll of 0 ..) (the 0 ..5 comes from the requiremellt of di s turballce amplificatioll less thall 2 at high frequellcy). The II -value for the Oil-configuration is sli g htly worse than the (L/D)(V / n) -configuration for column A alld almost as good for column D. Thi s is also what one would expect from the HG_-\plots. The LV- and DV- co nfiguration s gi\'e considerably higher II-values. Again , thi s is in acco rdallce with th e IlGA-plot s. If 1' is \'ery diffe re nt from one (as for the (L/D)(V/il)- configuration) th e n from an e nginee ring point of \'i e w it may be better
make 11 for robu s t performance equal to one. Different designs can then be compared based on their maxilllum achie\'able bandwidth. Two disadvantages with this approach are: 1) If the 5)'Stem is not even robu stly stable. then it is impossible to achie\'e 1' for robust performance equal to onc by adjusting the performance weight. 2) It introduces an ollter loop in the ,,-calculations.
5.3
Simulations
Simulationsofthe LV-. DV-. (L / D)(\,/il)- and DB -co nfigurations for column 0 (a propane-propylene splitter) with a full-order nonlinear model and the PI controllers given in Table ~ are shown in Fig. 9. The correlation between these simulations and the 11values is quite good, and the simulations support the conclusions made above. However, one should note that the simulations shown here are for a specific choice of set points and mod el error (setpoint change in YD and no error on the input s and no time delay). Onc should not necessarily expect a correlation between a single simulation , and the worst case res pon se (worst case combination of setpoints and model errors) for which I' is a measure . For e xample, although the I'-\'alu es are almost identical. th e simulation s in Fig. 9 indicate that the Oil-c o nfiguration is bett er than the (L/D)(V /B)-configuration. 1I0wever. with some other choice of set point change and model error the resu lt may be differe nt. Thi s illustrates one of the main advantages of II-analysis as opposed to simulations: one does not have to sea rch for the worst case It finds it automatically.
to Hot fix th e performanc e wC'igilt. and instead \'ary so me pa·
rameter in thi s weight (eg .. so me llIeasure for the bandwidth) to
0 .996 Table 4. jl-optimal PI- se ttings for column A and 0; C(s) = k t~,T;'. Gains k y and le". are for scaled compositions (eq 1). Column
Configuration
It
Tl y
Th
min
min
11.21 28.37
7.27 23.70 1.10 30.02
~
rK 1//
L
0.991 0.676 1.083 0.829 1.090 0.723 1.0.53 0.74·\
I'
77 , ]1
le I
0.49
0.34 3.98 0.43 0.6.5
D,I' D,n L,II
D
L I' 0 18
D,II D, n
1~.13
31.78 \.'i.07 20.6.5 7.93 39.95
0.13
IA6
OA,
0.12
11.2, 0 ..51 24.48
138.3 0.32 1.26
0.11
~.2.5
0.43 0041
3.96 44.79 4.08 1.61
---
1- -
:J-
~- 1
I1
ky
..
0.995
L,V
A
'I
..c
0.10
..
..
s.
.
(min. )
{\2
~ ~4 ~
~
~ '-~
..
0.09 2'
6.
S.
...
(min. )
;, L
3 ----,-'~ /'
'
----~-
4
3
12.86~~r=~~~T-----+-----r---~
.. (min.)
Figure 9. Time reponses for YD, Ia and L for a setpointchange in YD, column D , using four different configurations: l.LV , 2.DV, 3 .(L/D)(V /B), 4.DB. Figure 8. Structured Singular Value , 1', as a function of frequency for column A and 0 using four different configurations. I.LV 2.DV 3.(L/D)(V /B) 4.DB
S. Skogestad. E. \\' . .l aCOhSl'll alld P. LUlldst riim
6
Conclusion
The results in this paper are summarized below. The arguments mainly refer to composition control. although comments on level control are included. LV-configuratioll . .-\ good choice for onc.point control. but is not recommended for two.point control because of se nsiti\·it.'" to disturbances (Table 3) and relativel.'" 1'001' control performance due to interactions between control loop s. DV-configuratioll. One-point control: D must alu'ays be used for automatic control (never in manual). \Ia.'" be better than L\' for columns with large reflux because top le vel control is simpler. Two-point control: Works poorly when bottom product is not purer than top: poss ibly a good choice for columns with a pure bottom product (has to be investigated further). Disad\'antage: Very poor performance if failur e leed s to D constant (for example measurement in top fails). DB-configuration. l'nacceptable performance if used for onepoint control. Two·point control: Good control quality. in partic. ular for column with high purity and/or larg<' reflux. Le\'el con· trol also favors this conflguration for columns with large reflux. The main disadvantage is that it lacks integrety; performance i, very poor if failure gi\'es D or 13 constant. In particular. one can not put one of the loop s in manual. (L/D)(I/B)-col!figllrntioll. Overall this is th e best best choiclfor all modes of ope-ration. The main disadvantage is the need for measurements of all flow s L.D .13 and V which makes it mo re failure sensitive and more difficult to implement. ( L/D)V-confiyuratioll. Behaves somewhere between L\' and (L/D)(V/B). One important lesson to learn from the results on the DB· configuration is that one should be careful about making conelusions with regards to control quality based on steady state arguments. It is a fact that integral control is impossibl e (leads to instability) if the gain matrix may change sign (determinant changes sign). ][owe\'er, for the DO-configuration the stea dy· state gain matrix is "simply" infinite and singular, and there is no possi bli ty for change in sign. Acknowledgement. This work was made possible by financial support from NTNF. NOMENCLATURE (also see Fig. I) . steady state gains for column I, = Dyo(1 - Yo) + Bxa(l - 2'a) - "impurity sum" L = LT - reflux flow rate (kmol/min) La - liquid flow rate into rcboiler (kmol/min) M; - liquid holdup on theoretical tray no. i (kmol) qF - fraction liquid inn feed RGA - Relative Gain Array. elements are >"J
g;j -
r (1-i
D B ) - separation factor S = l-YD r[J JI = Va - boilup from reboil e r (kmol/min)
JlT - vapor flow rate on top tra,'( kmol/min) xa - mole fraction of light component in bottom product yo - mole fraction of light component in di s tillate (top prod uct) ZF - mole fraction of light component in feed
Greek symbols Q
= (I Y~\7f;
x,) .
relative volatility
>. 11 (s)
= (1- 912(')921('))-1 - 1 , I-element in RGA. 911 (S)g22(S) W - frequency (min-1) a(G),Q.(G) - maximum and minimum singular values Tt - dominant time constant for external flows (min) T2 - time consant for internal flows (min) TL = (8M./8L)v - hydraulic time constant (min) (1£ = NTL - O\'eraIllag for liquid response (min) REFERENCES Finco, M. V. , W. L. Luyben and R. E. Polleck, 1989, "Control of Distillation Columns with Low Relative Volatility" , Ind.Eng. Chem. Res., 28, 76-83.
llaggblom, 1\. E. and 1\.' \"'. \\'allcr. 10~(i."Rclations bctw e~ !l steady-state properties of distillation control sys tem s tructures . PrEp1'i"ts D)'COIlD SO. 2·13-2-18. llournc-mouth. U\. Dec. 1986. \lcAvov. T.J .. 1983. Int emetio" Analysis. Instrument Society of America. Trianglc Park. \C, rSA. \ett, C. \., 198" Prescnted at 1987 American Control Confer· ence, ~Iinneapolis, \1;\ .. June 198,. Shinskey. F. G .. 196,. Process COllt"ol Systems. !\lcGraw- Hill. \ew York. Shinskev. F. G .. 198-1. Distil/ation Co,l/ml. 2nd Edi· tion, "lcGraw.][ill:\cw York Skogestad, S. and \1. ~lorari. 19Sh. "Con trol Configuration Se· lection for Di stilation Colulllns". AIChE Journal, 33.10,1620· 1635. Skogestad. S. and ~1. \Iorari. 1987b. "A Systematic Approach to Distillation Column Control". Di stil lation 87. Brighton. Pub· li shed in I. Chelll. E. SYlllposilUlI Suits. 104 . ..1.,1· 86. Skogestad. S. and \\. \Iorari. 19S7c "Implication of Large RGA· El e ments on Control Performancc" . 1",1. t-.' En!]. Chem. Res .. 26,11,2121-2330. Skogestad, S .. 1988. "Di sturbance Rejection in Distillation Columl"". Proc. CIIE.\fD.-lT.·\ SS. J(i';·371. 19th Europl'an Symp. on Cam· puter Application s in the Chcmical Indu stry. Goteborg. Sweden. June 1988 . Skogestad. S. and ~\. \lorari. 19.5S". "Understanding the Dy namic I3ehavior o f Distillat ion Columns" , Ind.Ellg. Chem. Res, 27. 10. 18·18-1 862 . Skogestad, S. and \1. \lorari .1988 b. "IX·cont rol of a lligh· Puri ty Distillation Column" , C;'OIl. F"g. Sci .. 43. 1. :33 .. 18 . Skogestad. S .. 1980 . "\Iodelling and COllt 1'01 of Distillation Columlls as a 5x5 system". Proc. C\CllI 89. 20th European Symp. on Computer Application s in the Chclllicallndllstry. Erlangen. BRD. April 1989. Skogestad, S., E.W.Jacob sc ll alld \1. ~I!orari. 1080 . "DB-colltrol of distillatioll col u III li S" • Work ill prog ress . Wailer , K. 108G. "Di st illat iOIl COllt 1'01 Systelll Strllct ures·' . Prep"i nt, D}'CORD SG, 1-10. BOllrtll'lllouth . rt\. Dee. 1086.