A steady-state process resiliency index for non-linear processes

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A STEADY-STATE PROCESS RESILIENCY INDEX FOR NON-LINEAR PROCESSES

Boris M. Solovyev and Daniel R. Lewin

Wolfson Department o/Chemical Engineering. Technion-IlT, Haifa 32000. Israel

Abstract: Preliminary process resiliency analysis is a necessary part of any modem process design. This paper extends the Disturbance Cost (DC) resiliency index, previously developed for linear plants, to non-linear systems, and demonstrates its application in the analysis of a continuous stirred tank reactor (CSTR). Copyright () 2001 IFAC Keywords: Nonlinear analysis, process controllability and resiliency, optimization.

1.

INTRODUCTION

Process resiliency in context of this paper implies the ability to maintain the process in the desired operation region in spite of disturbances, uncertainties and process upsets. Since adequate process resiliency is a necessary part of optimal process design (Seider et. aI, 1999), it is desirable to consider process resiliency assessment when fixing the process structure and determining the operating point. A number of attempts to quantify process resiliency have been reported previously, most of which are based on a linear process model (Skogestad and Morari, 1987, Stanley et al., 1985, Lewin, 1996, Cao et aI., 1997, Solovyev and Lewin, 2000). However, the actual process may exhibit highly non-linear behavior (Solovyev and Lewin, 2000), making linear analysis less effective. Several approaches intended to overcome the limitations of linear analysis have been reported recently (Seferlis and Grievink, 1999, Subramanian and Georgakis, 2000). The present paper proposes an alternative way to perfonn resiliency evaluation by using the "Disturbance Cost" philosophy (Lewin, 1996, Solovyev and Lewin, 2000). This defines the resiliency measure in terms of the control effort required to perfectly reject the worst-case disturbance, nonnalized with respect to the available range of control action afforded by the designed process. As a result, a DC value of unity

indicates control system saturation, and thus a process design that is characterized by DC values in excess of unity should be modified or dropped from the further investigation. The extension of DC to non-linear cases is proposed and demonstrated on resiliency analysis of a continuous stirred tank reactor (CSTR) based on a first-principles model (Subramanian and Georgakis, 2000). 2.

DISTURBANCE COST DEFINITION

The Disturbance Cost (DC) is defined as control effort required to reject the worst disturbance . Mathematically, it can be defmed as:

DC

= sup

in/ max d u i = l .... . n

else.

uinom- ui L uinom-ui

Subjecfto : /(x,u,d,v)=O y= g(x,u,d,v) /j$,Yj$.yJ . j:I , ...,m.

(I)

where x is the state variable space, u is the input space, d is the disturbance space, v is the design variables space, y is the output space, and the indices H,

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L and nom stand for high, low and nominal values, respectively, of the input and output variables, i and j, respectively. The expression in square brackets in Eq( 1) represents the normalized control effort, the minimization of which over the u domain being necessary when the input set that returns the process to the design operation region is not unique. It is important to note that multiple input sets that adequately return the process to the desired operation region can exist due to process non-linearity (input multiplicity) or as a consequence of partial control requirements. In this context, partial control ideas are of great practical use in keeping the process in the desired operation region rather than the desired (nominal) set point, especially where number of process outputs exceeds number of process inputs (Arbel et al., 1997). The maximization of the control effort on the d domain in Eq(l) provides a condition for the worst-case disturbance. A consequence of the normalization is that a DC value of unity indicates control system saturation, and thus a process design that is characterized by DC values in excess of unity should be modified or dropped from further investigation. Conversely, lower values of DC (below unity) imply designs that are easier to control and thus more resilient.

The aforementioned DC definition (Eq(I» can be reduced to that defined for the linear case (Solovyev and Lewin, 2000), assuming that the foHowing three conditions hold: 1. When the process is linear and square (i.e., the number of inputs equal to the number of outputs), 2. The nominal values of each process input and disturbance are located in the middle of their ranges (i.e., symmetry), 3. It is required to maintain the process at the nominal operation point.

(4)

3.

For nonlinear processes, the DC evaluation consists of a three steps: Step I.

For nominal disturbances and desirable operation point(s) find the set(s) of nominal inputs. This is carried out using a solver for sets of algebraic non-linear equations. In cases of input mUltiplicity, various initial guesses in different regions of the input space are required.

Step 2.

Check each input set identified in the previous step for output multiplicity, again using a solver for sets of algebraic nonlinear equations. The input sets that are characterized by output mUltiplicity must be investigated by more detailed dynamic analysis in order to determine size of attraction basins, conditions for limit cycles etc.

Step 3.

Evaluate DC (Eq(I» using an optimization solver.

In the following, it is noted that all three steps outlined above are carried out using Mathematica® (Wolfram, 1999). Steps I and 2 are performed using the modified Newton method for a solution of a set of non-linear equations, available in Mathematica®, while step 3 (optimization) is carried out using the multi-start hill climbing method, available in the Global Optimization package written for Mathematica® by Loehle Enterprise (Loehle, 1998). As an illustration, the proposed approach is demonstrated on a first-principles model of a CSTR.

Conditions 1 and 3 imply that the input set is unique, thus optimization on the u domain in Eq( 1) is replaced by the direct computation of the set of inputs:

y=PU+Pdd=O~u=_p-lPdd

(2)

Here, the tilde symbol (-) indicates a deviation variable: (c) = (0)- (0 )nom' and P and Pd are process transfer matrices relating process outputs with process inputs and disturbances, respectively. The DC is therefore reduced to following maximization problem: 1

d

umax d

DC = sup max (abs(P- P

d

i=l ....n

Jl

IMPLEMENTATION STEPS

4.

RESILIENCY ANALYSIS OF CSTR (Subramanian and Georgakis, 2000)

A three-state CSTR model (perfectly mixed exothermic CSTR with a single irreversible reaction A-B) is described as: Ea

d(VC A ) -RT dt = QOC AO -QC A -V kO e CA dV

Tt=Qo-Q Ea

(3)

d(VT)

Mf

dt

pC p

--=QOTO-QT---VkOe

The index max stands for the maximum allowable variable change. The function if in Eq( I) is replaced by abs following Condition 2 (process symmetry). The symmetry assumption also leads to the simplification of Eq(3) to the form defined in Solovyev and Lewin (2000):

_ U A (T-T ) pC c p

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-RT

CA'"

Luyben (1993), which features high conversion, and Set 2, following Ray (1982), which features intermediate conversion and potential steady-state output multiplicity. The model parameters relating to both sets are summarized in Table I.

dTC Qc t ) U At) - - = - \ Tco -TC + \T-Tc dt Vc VCPC C pC

In the above, A is the heat transfer area, CA is the concentration of reactor outlet, Cp is the heat capacity, Ea is the activation energy, M! is the heat of reaction, ko is the pre-exponential factor, Q is the volumetric flow rate, R is the gas constant, T is the reactor temperature, U is the heat transfer coefficient, V is the volume and p is the density. It is further noted that the indices 0 and C represent feed and coolant properties respectively. The above model is expressed in dimensionless fonn by adopting the following replacements:

Set 1, (Luyben, 1993). The same seven designs investigated by Subramanian and Georgakis (2000) are treated here, analyzed according to five cases. These seven designs feature increasingly smaller reactor hold-ups, leading to larger nominal operating temperatures, while maintaining conversion constant at 95%. In the first case analyzed, it is assumed that hold-up and feed composition are constant (d, = I, U2 = 1), with a disturbance range of -0.15::; d 2 ::; -0.083, and an allowable input range of 0.3u'nom ::; u, ::; 1.05u'nom' The remaining cases investigate the impact of an additional disturbance variation, and for different range of manipulated variables and process requirements, as summarized in Table 2.

CA ~XICAre/,CAO ~XIOCAre/ , T~(X2 +l)Tre/ To ~(x20+1)Tref , TC ~(x3+I)Trej' Tco ~ (X30 + 1)Trej ,

Ea R

Qrej



~ yTrej , ko ~ --=:.--

Vrej

e-Y

pCp C pc ~--, V ~u2 Vref '

PC02

Q ~ qQref' Qo ~ qoQrej,Qc ~ u\QCrej, Vrej't QCrfj ~ qc Qre/ ,I ~-Qrej

After substitution of dimensionless variables, the model describing the reactor in the steady state is: qo(xlO - X\ )-u2 xJf(X2)::;; 0 QO(X20 -X2)-(X2 -X3XOB +U20S)+U2 X\ ~f(X2)::;;0

0\ {qcu\(X30 -X3)+ (X2 -X3XOB + u20S )o2}= 0 where:

The process outputs are A conversion, y,=I-x .. and reactor temperature, Y2 = X2, the disturbances are feed composition, d, = XIO and feed temperature, d2 = X20, and the manipulated variables are the coolant flow rate, u" and the reactor hold-up, U2. Following Subramanian and Georgakis (2000), two distinct parameter sets are investigated: Set I, following

~ ~

Y

Os

Os

Step I in this case is trivial, since having defined Y2 , the system is linear with respect to u, and U2, and has a unique solution. Step 2 indicates a unique output for all seven designs, as expected from previous work (Russo and Bequette, 1995). The results of the nonlinear DC analysis are reported in Table 3, which are in good agreement with those of Subramanian and Georgakis (2000). The main result confinns that improved resiliency is obtained with larger reactor hold-ups (Cases 1, 4 and 5). It is interesting to note, however, that when the coolant flow rate has a relatively large range (Case 3), all seven designs are predicted to have same resiliency perfonnance by the DC index. Conversely, the consequence of limiting the reactor hold-up to its nominal value (Case 2), is loss of resiliency and the inability to reject disturbances, which also is in good agreement with previous results (Subramanian and Georgakis, 2000). Finally, the nonlinear DC analysis indicates the expected increased resiliency afforded by adopting a partial control strategy (Case 5) instead of attempting to control the reactor precisely at its set-points. It is noted that significantly smaller reactor hold-ups can be accommodated if a partial control strategy is adopted.

Table 1: CSTR Model Parameters (from Subramanian and Georgakis, 2000) Ray (1982) Luyben (1993) Ray (1982) Luyben (1993) 1.00 0.35 qo 1.00 1.33 1.00 1.00 XIO 0.60 1.00 -0.12 20.00 X20 0.00 25.18 -0.12 0.11 X30 -0.05 19.26 6.56 qc 0.44 1.00 91.42 Trcf 599.67 "R 0.06 11.43

447

Table 2: Disturbance and manipulated variable ranges and process requirements for Set 1 Case

I

d1 d2 U/U 1nom U/U2nom

-0.15 0.3

Yl

Y2

2

-0.083 -? 1.05 1 0.95 Y2nom -?

U2norn 16.21 9.75 5.97 2.35

Ylnom 0.95 0.95 0.95 0.95 0.95 0.95 0.95

.44 0.20

4

5

0.8 -? 1.2

0.8

1.2

0.8 -? 1.2

As Case 1

As Case 1

As Case 1

As Case 1

0.3 -? 3 0.25 -? 1 As Case 1 As Case 1

0.3 -? 3 0.25 -? 1.25 As Case 1 As Case 1

0.3 -? 1.25 0.25 -? 1.25

0.3 -? 1.25 0.25 -? 1.25 0.947 -7 0.953 Y2nom±0.003

-?

0.8

1.2

-?

As Case I As Case I

Table 3: Optimization results for analysis of Set I DC for cases

Design

Ulnom 5.95 3.18 2.17 1.32 1 .77 0.65

3

Y2nom -0.1 -0.083 -0.067 -0.033 0 0.033 0.067

2 0.615 0.62 0.63 0.66 0.71 0.77 0.85

4 0.98 0.99 1.00 1.04 1.09 1.17 1.29

3 0.84 0.84 0.84 0.84 0.84 0.84 0.84

00

00

00

00

00

00

00

5 0.32 0.40 0.74 0.88 0.97 1.07 1.19

Table 4: Optimization results for analysis of Set 2 Design DC for cases

Ul nom 2.30 1.64 0.80 0.25

U2nom 6.48 3.52 1.52 0.80

Ylnom 0.8 0.8 0.8 0.8

Y2nom 0.094 0.132 0.189 0.236

2 0.37 0.51 2.21 25.57

0.80 0.99 2.55 39.1

(a) Design 1

(b) Design 2

0.3

0.3

0.2

0.2

N

0.3

0.3

0.2

0.2

N

>.

>.

0.1

•

0.1 0 0

0.2

0.4

0.6

0.8

•

0.1

0 1

0.1

0

0 0

0.2

0.4

YI

0.6

0.8

1

YI

(c) Design 3

(d) Design 4

0.3

0.3

0.3

0.2

0.2

0.2

•

N

0.3

•

0.2

N

>.

>.

0.1

0.1

•

0.1

0.1

• 0

0

0 0

0.2

0.4

0.6

0.8

1

0

YI

0 .2

0.6

0.4

YI

Figure I: Step 2 - checking alternative designs for output multiplicity.

448

0.8

0 1

Set 2, (Ray, 1982, Russo and Bequette, 1995). Following Subramanian and Georgakis (2000), four different designs were investigated, again featuring increasingly smaller reactor hold-ups, leading to increasing values in the nominal operating temperature, while maintaining conversion constant this time at 80%. Two distinct operation cases are examined: Case 1: 0.85 ~ d 1 ~ l.15, -0.083 ~ d z ~ 0.083, 0.3 ~ Ul/Ul nom ~ 4, 0.25 ~ u/UZ nom :5; 1.3, Yl = 0.8, and Y2 = Y2nom' Case 2: 0.797 :5; Yl :5; 0.803, Yz = Y2nom ± 0.003, with the remaining variables in the same ranges as in Case 1. It is noted that Case 1 is associated with the set point control strategy, while Case 2 is an implementation of partial control. In contrast to the previous set, Step 2 indicates the existence of output multiplicity for the fourth design, as shown in Figure 1. The analysis results are summarized in Table 4, where the last entry is that associated with the high conversion steady state seen in Figure l(d). As in Set 1, the tendency of bigger reactors to have improved resiliency is preserved. For low temperatures and large reactor hold-ups, there is significantly improved resiliency when moving from a set point to a partial control strategy. However, not even the partial control strategy can make the system work for excessively small hold-ups.

5.

CONCLUSIONS

This paper defines a resiliency measure, an extension to the previously presented Disturbance Cost, to handle non-linear applications. The nonlinear DC evaluation consists of three steps that have been successfully demonstrated on the resiliency analysis of a CSTR based on a first-principles model. It has been shown that the index highlights potential resiliency problems associated with design to be identified reliably. The approach also accommodates the analysis of the potential of a partial control strategy to improve process resiliency. While the approach has some similarities with those proposed by Vinson and Georgakis (1998) and Subramanian and Georgakis (2000), it mainly differs in the fact that it can handle input multiplicities explicitly, while those proposed by Georgakis and coworkers fail to correctly analyze resiliency in such situations. An associated feature is the ability to assess the impact of partial control, and again, our approach can do this, while theirs cannot. A final issue is associated with the computation load involved in resiliency estimation with the two alternative approaches. While efficient solvers are involved in the computation of the DC, the approach of Subramanian and Georgakis (2000) involves simply scanning the parameter space, which they themselves conceded to be computationally inefficient.

It should be pointed out that for complex plants, the development and solution of process models require significant engineering effort. This makes the analytical approach demonstrated here impractical to . apply in the early design stages, where the number of design alternatives may be large. As an alternative, a commercial process simulator can be used in place of, the first-principles model. Work is in progress to implement the approach using rigorous steady-state simulation using a commercial process simulator.

ACKNOWLEDGEMENT This research was supported by the Fund for the Promotion of Research at the Technion. REFERENCES Arbel, A., Rinard, I., H., Shinnar, R. (1997). "Dynamics and Control of Fluidized Catalytic Crackers. 4. The Impact of Design on Partial Control," Ind. Eng. Chem. Res., 36, 747-759 Cao, Y., Rossiter, D. and Owens, D. (1997). "Input Selection for Disturbance Rejection under Manipulated Variable Constraints," Camp. Chem. Engng, 21, S403-S408 Lewin D., R. (1996). "A Simple Tool for Disturbance Resiliency Diagnosis and Feedforward Control Design," Camp. Chem. Engng, 20, 13-25 Loehle, C. (1998), Global Optimization V2.1 s: User Manuals, Loehle Enterprises. Luyben W., L. (1993), "Trade-offs Between Design and Control in Chemical Reactor Systems," J. of Process Control, 3, 17-41 Ray, W., H. (1982), ''New Approaches to the Dynamics of Nonlinear Systems with Implications for Process Control System Design," Proc. of Chemical Process Control 2, 245, New York: United Engineering Trustees. Russo, L., P. and Bequette, B.W. (1995). "Impact of Process Design on the Multiplicity Behavior of a Jacketed Exothermic CSTR," AIChE Jnl. , 41,135-147. Seferlis, P. and Grievink, J. (1999). "Plant Design Based on Economic and Static Controllability Criteria", Proc. of the 5th 1nt. Con! of Foundations of Computer-aided Process Design, 346-350, American Institute of Chemical Engineers, NY Seider, W.D ., Seider, J., D., Lewin, D., R. (1999). Process Design Principles: Synthesis Analysis and Evaluation, John Wiley and Sons, NY Skogestad, S. and Morari, M. (1987). "Effect of Disturbance Direction on Closed-loop Performance," Ind. Eng. Chem. Res., 27, 18481862 Solovyev B. M. and Lewin, D. R. (2000). "Controllability and Resiliency Analysis for Homoge-

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neous Azeotropic Distillation Columns," Proc. of ADCHEM 2000, Vol. I, 69-74,Pisa Stanley, G., Marino-Galarraga, M., and McAvoy, T. J. (1985), "Shortcut Operability Analysis. 1. The Relative Disturbance Gain," Ind. Eng. Chem. Process Des. Dev., 24, 1181-1188 Subramanian, S. and Georgakis, C. (2000). "Steadystate Operability Characteristics of Reactors," Comp. Chem. Engng, 24, 1563-1568 Vinson, D. R. and Georgakis, C. (1998). "A New Measure of Process Output Controllability," Proc. OfDYCOPS'5, 700-709, Corfu. Wolfram, S. (1999). Mathematica 4: User Manuals. Wolfram Research. Inc.

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