Optimal Averaging Level Control for Multi-Tank Systems

Copyright e IFAC Advanced Control of Olemical Processes. Kyoto. Japan. 1994

OPTIMAL AVERAGING LEVEL CONTROL FOR MULTI· TANK SYSTEMS N. YEt, TJ. McAVOyt, MJ. PIOVOSOt AND K.A. KOSANOVICHt

t Department of OIemical Engineering, University of Maryland, CoUege Part, MD 20742, USA t E.l. Du Pmt d; Company WilmingtOll, Delaware 19880, USA

Abstract. In this paper earlier results on optimal averaging level control (McDonald et al., 1986) are extended to cover the case where two or more tanks are connected in series. It is shown that the optimal averaging level control approaches can be derived for a multiple tank system if the total volumetric capacity of the system is taken into accounL Intermediate flows between tanks are arbitrary. provided they satisfy the level constraints. The optimal solutions that consider tanks together produce beUer flow filtering than the optimal averaging level control approaches where each tank is treated individually. Key Words. Level control; Optimal control; Predictive control; Multiple tanks; Industrial test problems.

1. INTRODUCTION

ung and Luyben (1979) pointed out the difficulty of level control for tanks in series with PI controllers. In a PI level control system, flow fluctuations increase as a disturbance moves down the tanks in series. This work extends the two algorithms proposed by McDonald et al. (1986) to multi-tank cases.

In many cases the goal of liquid level control is to provide flow smoothing of the outlet flow from a vessel as opposed to tight level control. Such flow smoothing is called averaging level control. Averaging level control makes use of the surge capacity of a vessel to damp out fluctuations in its exit flow so that the fluctuations have minimal effect on down stream units.

Following the introduction, an optimal control problem for minimizing the MRCO of the last tank in a multi-tank system is formulated and solved. Similar to the solution of a single tank, an optimal feedback averaging level control approach and an optimal predictive control approach for tanks in series are proposed. Then simulation results are presented for step disturbances and compared with the results of the optimal averaging level control for separate tanks. Guidelines for application of optimal averaging control when multiple tanks are involved are presented. Finally, in addition to an ideal two tank system, the optimal averaging level control approaches are applied to a recently published industrial test problem, the Tennessee Eastman control problem (Downs & Vogel, 1993).

A conventional approach to the averaging level control problem has involved the use of PI controllers with the control parameters detuned. Nonlinear control has been found to be superior to PI control for the averaging level control problem (Cheung & Luyben, 1980). McDonald et al. (1986) solved an optimal level control problem which minimizes the maximum rate of change in the outlet flow (MRCO) when the input flow undergoes a step upset. The solution of the optimal problem resulted in two optimal level control approaches, one is a nonlinear feedback control law which ramps up the exit flow and another is optimal predictive averaging level control. Both of the optimal averaging level control approaches give significant improvement over PI averaging level approaches. Campo and Morari (1989) applied model predictive contralto the optimal averaging level control problem formulated by McDonald et al. (1986). In their approach, generalized level control constraints are used which affect the trade-off between the incompatible objectives of good flow filtering and rapid settling time.

2. OPTIMAL AVERAGING LEVEL CONTROL FOR TANKS IN SERIES For a multiple tank system, the objective of the averaging level control is usually to smooth the variation in the final outlet flow of the system. Often intermediate flows between tanks are not a major concern. For a system with multiple tanks it is shown that the optimal solution for the outlet flow in the last tank is a ramp. For a system as shown in Fig. 1 with two tanks, if the optimal feedback or optimal predictive averaging level controllers (McDonald et al., 1986)

Most of the research on averaging level control focuses on good control for a single tank. In many real processes, more than one tank in series is used. Che413

The proof of this optimal control solution is given in the Appendix. The outlet flow from the second tank for a step disturbance in the inlet flow of the first tank is a ramp. The intermediate flow between the tanks can be any function which satisfies the two conditions (Eq. (12» . It can be proved that if the intermediate flow follows the optimal solution of a single tank (McDonald et al., 1986), the two conditions are satisfied (Appendix).

Figure 1: A rwo rank sysrem

are applied to each tank separately, with a step disturbance occuring in the inlet flow of the first tank, the outlet flow of the second tank is not a ramp anymore.

For a negative B, the equations become, t E [0 , t .......l

(13)

where The optimal averaging level control problem for a multi-tank system can be formulated including all the tanks together. Here a two tank system is considered. Results derived for the two tank system can be applied to systems with more than two tanks . For the two tank system (Fig. 1), if at time t = 0 there is a step disturbance B in the inlet flow of the first tank, qj(t), the optimal problem can be defined as min (max II'lo(t)J)

q.r ,q.(r)

r

0:=

t max =

2(AI (hlmin - hIs) + A2(h2max - h2s» B

and the conditions for q",(t) are the same as in Eq. (12).

Subject to: dhl (t) Al ---;Jt

=B + qs - q",(t)

dh2(t)

(2)

A2 ---;Jt = q",(t) - qo(t)

(3)

hlmin ~ hl(t) ~ hlmax

(4)

h2min ~ h2(t) ~ h2max

(5)

Based on the optimal solution for a two tank system, similar procedures to those given by McDonald et al. (1986) can be taken for deriving the optimal feedback and predictive averaging level control algorithms. Optimal feedback averaging level contro/. Adding Eq. (2) to Eq. (3) and eliminating the intermediate flow gives

The initial conditions are q",(O) = qo(O) = qs

(6)

hI (0) = hIs

(7)

h2(0) = h2s

(8)

dh2(t) dh l (t) Al ---;Jt + A2 ---;Jt = qs + B - qo(t)

t E [0 , t .......l

(16)

From Eq. (16) and Eq. (9), the feedback control law for a positive B is (see Appendix for a proof)

For a positive B, the optimal solution is given as qo(t) = qs + o:t

(15)

(1)

qo(t)

(9)

=qs+

where

1_ 0:=

-:-A~I::-(h_1(_t)_-~h_1s,,:-)_+~A_2(-:-:-h_2(-,-t)_-_h:-,-2s-:-) ) Al (h lmax - hIs) + A2(h2max - h2s) (17)

and for a negative B, then qo(t)

= qs+

(I» + A2(h2s - h2(t» ) Al (hIs - h lmin ) + A2(h2s - h2min)

1 _ Al (hIs - hI

and

(18)

Optimal predictive averaging level control. At 1= 10 substituting qj(lo) into Eq. (16) gives dh l (to) dh2(to) Al ~ + A2 ~ = qj(lo) - qo(to) 414

(19)

,,I----------------------------~

, ~---l-,

q.

~"'r--r

q . ~-----, , '--""r-,-'

~

q.

~

_I _I

Figure 3: Optimal avenging level control [or two tanks sepanate Figure 2: OptimalllVenging level control [or two tanks together

Assume that the inlet flow remains fixed in the future, then at time t (t > to), one can get dhl (t)

Al

dh2(t)

--;[l + A2 --;[l

=qj(to) - qo(t)

Subtracting Eq. (19) from Eq. (20) and letting B qj(to) - qo(to), one has Al

d(h l (t) - hi (to)) dl

10.20

(20)

....

~...

=

,.

,.

/ :.,.. ___; r - - - - - - - - i ...

.,'

~(~:'... .~~~:..

. . . . . . . . . . . . . . . YIU ........

0 0

10.10

. . . . . . . . . . . . . . . IW.~ . . . . .

ti:

~._. . . . ' . . UIIU

11 ~ 0

+

- h2(lo)) - .( ) B A 2 d(h2(t) dl - q. 10 +

......

10.00

- qo ( I) (21)

According to Eq. (13), for a positive B, the optimal solution for this problem is given as qo(t)

~'

9 .90r......~......J.~~......L..~~...J...~~..J......~.........I

o

=qo(to)+

10

20 30 Tune (hours)

40

SO

2

B (/-l o ) (22) 2(AI (hlmax - hi (1 0 )) + A2(h2max - h2(to)))

Figure 4: Outlet /Iow [or a step disturbance with optimal avenging level control

As!!J = I-to --+ 0, qo(to) = 9.(1)1,0(1.,), then the optimal predictive control law for two tanks together can be written as qo(t)

=

first tank and its size is 0.2 (m 3 /hr). Fig. 4 shows the outlet flow of the second tank for all the control strategies. The parameter B in the optimal feedback control is the same as the disturbance size. The outlet flow of the first tank is the same for all the control schemes.

(qo(t) - qj(/))2

2(AI (h lmax - hi (t)) + A2(h2max - h2(t))) (23) If qo(t) - qj(t) < 0, then qo(t)

=

(24)

(qo(l) - qj(I))2

The MRCO with two tanks together is 0.006 (m 3/hr/hr), and with two tanks separate is 0.011 and 0.014 for the optimal feedback and predictive averaging level control respectively. Optimal averaging level control with two tanks together is better than with two tanks separate according to the MRCO.

2(AI (hlmin - hi (I)) + A2(h2min - h2(t)))

3. COMPARISON AMONG DIFFERENT OPTIMAL AVERAGING LEVEL CONTROL SCHEMES Optimal feedback averaging level control and optimal predictive averaging level control for two tanks are tested on the two tank system (Fig. 1). The parameters of the tank system are Al 1.0 (m 2 ), A2 1.5 (m2 ), qs 10.0 (m3 /hr), his 1.0 (m) and hls 1.5(m). The levels are constrainted as 0 S hi S 2 .0 and o S h2 S 3.0. For comparison, the optimal feedback averaging level control and optimal predictive averaging level control for a single tank are also applied to each tank separately. Fig. 2 illustrates the optimal averaging level control considering both tanks together. Fig. 3 describes the optimal averaging level control for separate tanks.

=

= =

4. APPLICATION IN AN INDUSTRIAL TEST PROBLEM

= =

The Tennessee Eastman control problem is one of the industrial test problems published recently which has captured wide attention both in academia and industry. Because of the complexity of the problem, many researchers have focused on designing a base control system for the process (McAvoy & Ye, 1993; Lyman & Georgakis, 1993). In (McAvoy & Ye, 1993), a base control system is designed which can reject all disturbances, follow all setpoint changes and satisfy all process constraints. The Tennessee Eastman process with the base control system is shown in Fig.

A step disturbance is added to the inlet flow of the 415

.cl------, r-----H-I I

F1

I

-----~ 11

.--...&..---,

TC.J

t---ti:--:---- -<3: B A

• •

:---~--I I I

F1

C

R::

I

~ 'A

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

10--

D

E

I

... -(3--: I I I I

l

R::

•

,

.-------IL.f'0.-: ~ c.oIioit I --....:.::J

I I

I

N.,)

S

A

B P

L

T Z

A Il A

I I

T

..,_-r--I

C T

~ cb : ________________________

I

-I

~

I

~------TI--~~~~I~----~--+

I L _______•~-------~I L I ________________________________________

I

~-----------------------------------------------------------_. FIgure 5: The Tennessee E.utman process and the base control system

S. Control of the process can be improved by adding advanced control on top of the base control system.

level control schemes on the separator and stripper level for a disturbance to the process which involves a random composition fluctuation in one of the feed streams to the process. The three averaging level control schemes are PI averaging level control on the separator and stripper, the optimal feedback averaging level control for the separator and stripper separately and the optimal feedback averaging level control for the separator and stripper together. The control parameters of the averaging PI level control are determined according to the method provided by Cheung and Luyben (1979). In Fig. 6, both the optimal feedback approaches have better flow filtering than the PI averaging level control. However in this case, control with the tanks treated separately is better than that with tanks together. Further analysis on the processes found that the separator and stripper are not an ideal two tank system, and the levels not only depend on the flows but also on the flow compositions. For a non-ideal two tank system, lumping the tank's capacity together might not be appropriate.

The product flow rate of the process is a key variable to be controlled. Large variation in the flow will affect the down stream distillation system. Minimizing the variation of the flow is one of the control goals. Averaging level control can be used to achieve this goal. Tight control on the reactor level is required because the reactor level has a large effect on the reaction kinetics. There is no such requirement for the separator and stripper levels. From Fig. 5, it can be seen that averaging level control on the separator and stripper levels can smooth the product flow. The optimal level control approaches, feedback or predictive, where the separator and stripper are treated separately or together, are tested on the process. Because fluctuating disturbances are involved in the process, the modified feedback averaging level control (Ye et al., 1993) is extended to the feedback averaging level control for multiple tanks. It was found that the predictive level control uses more of the tank capacity for small upsets with the result that larger upsets could not be filtered as well. The results produced by the modified optimal feedback approach were found to be superior and only these results are discussed here. The optimal feedback control approach is used for this application, also because it does not require the measurement of the inlet flows which are not available for both the separator and stripper.

S. CONCLUSIONS An optimal averaging level control problem for a multi-tank system to minimize the maximum rate of change of outlet flow (MRCO) of the system has been proposed and solved when a step disturbance is added to the inlet flow of the system. The optimal solution for multiple tank systems is superior to that with tanks treated separately. Optimal feedback and predictive averaging level control approaches have been derived for multiple tank systems. For step distur-

Fig. 6 shows the product flow rate with three different

416

See (McDonald et al., IYMb) tor a prool. Theorem 2 For the optimal problem

PlA...... x...-dc-.l

O'C-Il.-.....aA. ..... a....dc... .......) ..... w. .. C-- ... ~.... )

O"'aI~A,.

24

min (max Iqo(t)D

q.l ,q.(I)

(31)

1

Subject to:

22

21~~~~~~~~~~~~~~~

o

10

20

30

dhl(t) Al ----;it

=8 + q. -

dh2(t) A2 ----;it

=q".(I) -

q".(t)

(32) (33)

qo(l)

hlmin ~ hi (t) ~ him..

(34)

h2min ~ h2(1) ~ h2mu

(35)

with the initial conditions

Tunc (hours)

q",(O) = qo(O) = qs

(36)

=his

(37)

h2(0) = h2s

(38)

Figure 6: Averaging level control results for Ihe Tennessee &.slman process

hi (0)

where 8 > 0, I ~ 0, qs > 0, hlmin and h2min ~ hls ~ h2m.. ' the solution is

bances, both the optimal feedback and predictive averaging level control approaches for multiple tanks are demonstrated to have better flow filtering than those for separate tanks.

qo(l) qo(t)

The optimal feedback and predictive averaging level control approaches are tested on the Tennessee Eastman control problem. Both the optimal feedback approaches with tanks together and separate are better than the PI averaging level control for a fluctuating disturbance. Because the two tank system in the test problem is not an ideal two tank system, the optimal feedback averaging level control for multiple tank systems does not show its advantage over that for separate tanks.

=qs + 0'1 =q. + 8

~ his ~ h lmu ,

E [0, lma.r1

I

I> lmax

(39) (40)

where

Imu

= 2(AI (hl mu -

his) + A2(h2mu - hls))

8

(42)

and

Appendix Theorem 1 For the optimal problem min(max I qo(t) q.(I)

D

1

(25)

Subject to:

h.run

~ h. +

1 t A lo (8 + q. -

qo( r)dr ~ hmu

Adding Eq. (32) and Eq. (33) gives

=q.

qo(O)

Proof

(26) (27)

°

dhl (I) dh2(1) Al ----;it + A2""dl

where 8 > 0, I ~ 0, and q. > there is a solution to the problem given by qo(l)

=q. + 0'"1

qo(t)

I

= q. +8

(44)

It can be derived considering the level constraints that

E [0, 1';'..1

I> I';'..

=qs + 8 - qo(t)

(28) Al (hlmin - his) + A2(h2min - hls)

(29)

~

11

q. + 8-

qo(r)dT ~ Al (him.. - his) + A2(h2max - h2s)

with "

t mu

= 2A(hmu 8

(45)

hI) (30)

Substituting the tenn A(hmu - hs) in Theorem 1 with Al (hlmu -hl.r)+A2(h2mu -hls), the solution for the optimal problem is given in Eq. (39), Eq. (40) , Eq. (41) and Eq. (42).

Proof 417

is a realization of the optimal solution given by Eq. (39), Eq. (40), Eq. (41) and Eq. (42).

The change of q". is arbitrary only subjecting to the constraints on the levels, i.e. hlmin -

-1

Al

Proof Substituting Eq. (39) into Eq. (44) gives

his ~

l'

qs + B - q".( r)dr ~ hlmu - his (46)

0

AI

d(hl - his) A d(h l - his) dl + 2 dl

=B -0'1 ,

lE [O , lmIJ (60)

Integrating Eq. (60) results in Al (hi

- his) + A2(h2 - h2s)

=BI -

1 20'r2

(61)

Eliminating I with Eq. (39) gives A (h

Corollary 1 If

I

=qs + all q".(I) =qs +B

q".(I)

I E [O,lmud

(48)

I>

(49)

ImuI

h) + A (h Is

2

AI(hl mu

=

0'1

B2 2A ( I hlmu -

his)

,

(mul

=

2AI (hlmlx -

his)

=

B

(50)

Proof Substituting Eq. (48) into Eq. (32) and integrating from 0 to Imlxl gives 1 {'-I (51) Al (B - 0'1 r)dr =hlmu - his

-1

Al

0
o

l'

ImuI

(52)

(B - 0'1 r)dr

0

=hl max -

his

I

>

Imaxl

(53)

So hlmin -

his ~

Jot

1

Al

(qj(t) - q".(I»dl ~ hlmu - his

(54)

It can be easily proved that Imu

~ Imlxl,

Lyman, P. R. & Georgakis, C. (1993). "Plant-wide control of the Tennessee Eastman problem." Submit to Computers and Chemical Engineering. McAvoy, T. J. & Ye, N. (1993). "Base control for the Tennessee Eastman problem." To be appear in Computers and Chemical Engineering. McDonald, K., McAvoy, T., & Tits, A. (1986). "Optimal averaging level control." AlChE Journal , 32:(1),75. Ye, N., McAvoy, T. J., Kosanovich, K. A., & Piovoso, M. J. (1993). "Optimal averaging level control for the Tennessee Eastman problem." Presented at the 1993 Annual AIChE Meeting, S1. Louis, Missouri, USA.

and

o

q". - qodr

<

h2mu -

h2s

0
Imu

(56) (57)

So

Corollary 2 qo(t)

B

=qs+

(1 _

rl-_----:-AI~(Io~I(:-:-'}-"'7Io~I,)~+A7""2-:7(1o"'72(,7""}_7'"Io,"'7)' AI (10 1_ - II I,)+Az(II z_ -112,)

(63)

Chem. Engng., 17:(3).

then (55)

o ~ A21 Jt

(qo - qs)2 B2

Campo, P. J. & Morari, M. (1989). "Model predictive optimal averaging level control." AlChE Journal, 35:(4),579. Cheung, T. F. & Luyben, W. L. (1979). "Liquidlevel control in single tanks and cascades of tanks with proportional-only and proportional-integral feedback controllers." fnd. Eng. Chem. Fund., 18:(1), 15. Cheung, T. F. & Luyben, W. L. (1980). "Nonlinear and nonconventional liquid level controllers." Ind. Eng . Chem. Fund., 19, 93. Downs, J. J. & Yogel, E. F. (1993). "A plantwide industrial process control problem." Compulers

and

Jf' qs+B-q".dr < hlmax-hls

2(qo - qs) B

References

Jo

1

-hls)+A2(h2mu -h2s)

Eq. (63) holds for I > Imu as well. Eq. (63) can be solved to give Eq. (59).

q". satisfies the conditions in Eq. (43)

Al

h) _ B(qo - qs) (qo - qs)2 2s. 0' 20'

2-

(62) With substitution of 0' (Eq. (41», Eq. (62) becomes

with

o~

I-

(59) 418