- Email: [email protected]
Adaptation of output controllability index within dynamic operability framework
IFAC
Copyright © IF AC Dynamics and Control of Process Systems Jejudo Island, Korea, 200 I '
c:
0
[>
Publications www.elsevier.com/locatelifac
ADAPTATION OF OUTPUT CONTROLLABILITY INDEX WITHIN DYNAMIC OPERABILITY FRAMEWORK
Estiyanti Ekawati 1 and Parisa A. Bahd·
I School
ofEngineering, Murdoch University - Rockingham Campus Dixon Road, Rockingham WA 6168 A USTRALlA (*Email.·[email protected])
Abstract: This paper repo.rts.the adaptation of the Output Controllability Index (OCI) approach for the regul~tory case WIthin the Dynamic Operability Framework to facilitate controllability asses~ment III proce~s system design. The original framework requires the solution of NO~Ill~ ~rograrmrung ?r~blem on two optimization levels. This study replaces the multiple maXlmlzanon pro~lems III mner level and the whole inequalities in outer-level by a single ge~metnc calculan~n, a~d provides a controllability measurement involving all of the output vanables. The appbcablbty of the proposed approach is demonstrated on a nonlinear process system. Copyright © 2001 IFAC Keywo~ds :
process control, operability,
controllability, non linear programming,
uncertamty
I.
INTRODUCTION
The escalating demand of safe and efficient operation of process systems has increased the awareness of the importance of process control. It is understood that a well-designed process control system can guarantee optimum perfonnances, such as economic profit, safety and minimum dynamic error. The question is on how to assess the controllability in the design stage. This leads to rigorous assessment of process controllability, which by Skogestad and Wolff (1992) is defined as the ability to achieve a desired performance within various limitations on process operations, despite of external disturbances and uncertainty in design parameters, by using available input and manipulated variables. During the past fifteen years, there have been considerable efforts towards incorporating controllability assessment in process design and optimization. In effort to provide direct evaluation of process control economics, Narraway et. at. (1991) optimized the steady state objective function and the control cost subject to steady state and dynamic path constraints, assuming a single point disturbance affecting
685
the plant. The economic indicator was the required backoff amount from the active constraints to ensure feasible operation based on a linear model. In subsequent research, Narraway and Perkins (1994) extended the approach to nonlinear systems. Later, Bahri et. at. (1996) presented an iterative dynamic optimization algorithm to assess the dynamic operability based on back-off approach for both linear and nonlinear processes. This approach focused on two operability issues, which are flexibility and controllability. The flexibility problem was addressed by modeling disturbances and uncertainties as a set of step changes and automatically calculating their worst-case sets which violated system constraints. Bahri et. at. (1997) then included Integral Squared Error (lSE) as one of the system constraints for controllability measurement. Schweiger and F10udas (1998) utilized the ISE as a weighted point constraint to the optimization problem, and varied the weighting values to generate a set of trade-off solutions between economic and controllability objectives. The disturbance affecting the system, however, was only a single time function.
Recently, Vinson and Georgakis (2000) proposed the Output Controllability Index (OCI), as a steady state, input-output controllability measurement of a system. The measurement quantifies the system ability on reaching the full range of desired output values in the presence of expected process disturbances and within the limited range of its available inputs. They utilized the software developed by Veres et. al. (1995) to project the expected process disturbances space to system outputs space. Veres (1999) utilized the software to characterize model uncertainties as geometric shapes - such as polytopes and ellipsoids, transformed these shapes by linear operation of the model and assessed feasible models and tracked changes of a linear plant dynamics. This paper presents the extension of the dynamic operability assessment framework to incorporate the OCI approach for the regulatory case, specifically the geometric representation of the feasible operating conditions and system responses in process system design. Originally, the framework requires the solution of Nonlincar Programming (NLP) problem at two optimization levels. This study replaces the inner level from multiple constraint violations detection by convexhull detection and changes the inequalities in outer-level to one geometric calculation. The paper is structured as follows. The next section reviews the features and the formulation of Dynamic Operability Framework, followed by the rcviews and adaptations of the OCI approach in section 3. Section 4 presents the formulation of the proposed framework. In section 5, both original and proposed frameworks are applied to a case study. Section 6 summarizes and concludes the paper.
2.
DYNAMIC OPERABILITY FRAMEWORK
2.1 Features The features of the original Dynamic Operability Framework proposed by Bahri et al. (1997) is discussed below in a manner that facilitates the extension to incorporate a formal controllability assessment into the formulation. The principles of the original approach applied on a fixed structure are illustrated in Figure 1. The feasible operating condition is the area bounded by system constraints. It is typical that the nominal optimum operating point lies on at least one of the constraints. However, it is required to guarantee the feasible operation when disturbances enter the system, or when some uncertainties in the system parameters appear. For this purpose, the nominal operating point is moved inside the feasible region and an operating domain surrounding the point is defined. This domain represents all possible deviations from the nominal operating point due to disturbances and uncertainties. Whenever the operating domain lies entirely in the feasible operating condition, it is stated that the system is flexible. Furthermore, it is also typical to move the operating domain in the direction of
686
nominaloplimum
Sest feasib le operating condilions
Increasing Profit Control I Manip ulated Variab le 2
Fig. 1. The basic principles of Dynamic Operability Framework better profit. These efforts result in a design that is optimally flexible and economically viable. It is interesting to note from description above that flexibility requirements are the worst-case problem in controllability assessments. Controllability assessment is not only interested on the peak variability, but also on its quantity and its length of occurrences. These features represent the ability to reject the effect of disturbances in regulatory case. Therefore, there is a possibility to extend the flexibility study to assess more detailed problem in controllability, such as disturbance rejection problem. The potency can be examined further in the framework fonnulation.
2.2 Problem formulation The implementation of the original framework leads to problems on characterizing the operating domain and assessing its feasibility . The problem was formulated as an iterative dynamic optimization algorithm with two main loops, i.e. the outer loop and the inner loop. The outer loop gives the optimal operating condition through the solution of dynamic NLP problem, and the inner loop investigates the feasibility of the operating condition found in the outer loop. The algorithm continues until the operating condition found in the outer loop meets the feasibility assessments. The mathematical formulations for these two optimization levels are as follows: Outer level:
min( z,e N ,x,):,p,t)
z S.t. h;(z,ek,x,):,p,t)=O iEE g;(z,e k ,x,):,p,t)5',O jE! ZE Z = (z : zl 5', z 5', ZU ),
(1)
ek E r = (e : el 5', e 5', eu )
Inner level: max g j(z*,e,x,X,p.t) 9
S.t. h;(z*,e,x,x,p,t)=O
eE r
iE E
(2)
Here, z is a vector of decision variables, x is a vector of state variables, eN is a vector of disturbances with nominal values,
e
k
is combination of disturbances
found in the previous inner loops, p is a vector of process parameters, E, I are set of indices of equality and inequality constraints respectively, r is a set of all possible realizations of the disturbances and z* is the optimal vector found in the previous outer loop . In the inner level, multiple constraint violations are detected. The maximization problems in (2) are as many as the constraints in the system. If there is no constraint violation in this step, the algorithm stops. Otherwise, the solutions ofthe inner loop are passed to the outer loop in the next iteration. The peak variability found in the inner level can be interpreted as the maximum absolute error and as the worst-case controllability measurement. If the evolution of the worst variability can be recorded against time, it provides links to known controllability measurement such as Integral Absolute Error (IAE). This problem requires a more flexible characterization of the variability. The following section describes how the adaptation of the OCI approach can provide such characterization.
3.
THE ADAPTATION OF OUTPUT CONTROLLABILITY INDEX
The Output Controllability Index (OCI) approach by Vinson and Georgakis (2000) is discussed below including its adaptation to extend the dynamic operability framework in the regulatory case.
3. J Definition To introduce their controllability concepts, Vinson and Georgakis developed several defInitions of variable set. The first set is the Available Input Set (AlS), which is the set of values that the input variable can take in the process. The second set is the Desired Output Space (DOS), which is the set of desired values of the output and measured variables. The third set is the Achievable Output Space (AOS), which are the output values that can be achieved based on the available input values. The AOS is calculated by translating the values in AlS using process model. The ratio of the magnitude of the intersection between DOS and AOS to the magnitude of the DOS determines the process controllability. Specifically, the Output Controllability Index (OCI) of the process is defined as:
OCI = J..L( A OS (') DOS) J..L(DOS)
(3)
EDS
DOS
Fig. 2. The Expected Disturbance Set transformed to Achievable Output Space AlS and DOS are defined by constraint inequalities in the system model. Therefore, the whole inequality can be treated simultaneously using a single geometrical calculation routine. .
3.2 Adaptation Dynamic operability framework focuses on the effect of the disturbances and uncertainties on system outputs. Therefore, it is preferable to use the set of disturbance values in place of AlS in the above definition. Here, Vinson and Georgakis definition of the Expected Disturbance Space (EDS), as the disturbance spaces that are expected to affect the process, is used and adapted to observe the effect of the disturbance set to system controllability in regulatory case. Thus, AOS is now calculated based on the values in EDS (AOSJ. Furthermore, this study views DOS as the representation of the feasible operating range of the process. If AOS d is inside DOS, it solves the flexibility problem. Therefore, it is more meaningful to define OCI based on the size of AOSd, rather than DOS. The size of AOSd itself represents the system controllability in regulatory case. For the above purpose, the definition of OCl for regulatory case (r-OCI) is adapted:
Originally, Vinson and Georgakis (2000) calculated OCI based on system steady state values. In the proposed approach, the convex-hull of AOSd and its variation over time is recorded. Time is an optional dimension of AOSd space and its exclusion leads to a flexibility case, where the convex-hull of AOS d is the projection of the worstcase responses in output space. The inclusion of time provides a General Integral Absolute Error (GIAE) value for the system, which involves the whole outputs and time space, instead of single output and time for conventional lAE.
Here, J..L represents a function calculating the size of the space. 4. The advantage of this approach is that it characterizes each of the AlS, AOS and DOS as a whole polyhedral shape. This approach is in line with the geometrical approach in characterizing the disturbances and/or system uncertainties as polytopes or ellipsoids, which has been done for linear system by Veres (1999). In this case,
FORMULATION OF THE PROPOSED ALGORITHM
The main objective of the feasibility test in the inner level of Dynamic Operability Framework is to verify that the current z* is not violating inequality constraints gj(z*,e,x,x,p, t) for j = 1, ... ,J. For the proposed
687
approach, the whole set of inequality constraints is represented by ng-dimensional geometric shape DOS. Here, Ug is the number of output/measurement variables involved in the inequality constraints and time. EDS is represented by n.t-dimensional geometric shape (n.t is the number of disturbance variables) and AOS d is the projection of EDS to the output variables in ng dimension. Therefore, the size of AOSd, in terms of volume of an ng-dimensional shape, represents the system controllability.
Feed
,·,,~t,;{J · .. '"
~
.
The incorporation of the adapted OCI approach into the dynamic operability framework results in the following formulation: Outer level :
minet>(z,e k ,x,i,p,t) z S.t . h,(z,ek,x,i,p,t)=O iE E r -OCI(z,ek,x,x,p,t) = 1 z E Z = { z : / :5 z :5 ZU }
-I
CSTR 2
Product
Fig. 3. Case study: 2 CSTR + mixer
(5)
ek E r = {e : el :5 e :5 eu } Inner level:
max AOSd (z*,e,x,i,p,t)
e
Coolant 2
composition C f ,. The lower bOWld, nominal values and upper bound of their variations are [298, 300, 315] K and [19.5, 20, 21] moUm 3 respectively. The measured variables are the temperatures, the amount of cooling of each CSTR and the product composition, which are TI, Cod , T2, Coof and C2. The objective function is the net profit defined as:
(6)
et> =lO(Q}Cf +Q}Cf -0.3(Q} +Q} ))- ... (7) 0.0ICooI 1 -0.ICooI 2 -O.I(Q} +Q})
The advantages of the proposed approach over the original method are as follows:
The inequality constraints used for optimization of the process are as follows:
S.t.
1.
2.
h,(z*,e,x,i,p,t)=O
eE r
iE E
It facilitates the controllability assessment within operability assessment framework through the geometric characterization of the feasible operating region and the projection of disturbance set to output space. Thus, the flexibility becomes the special case of the controllability problem. The multiple maximization problems in the inner level and the inequalities in outer-level are replaced by single geometric maximization and equality constraint respectively.
5.
CASE STUDY
5.1 Process Model
To illustrate the applicability of the proposed formulation, a case study is presented (de-Hennin and Perkins 1991). It consists of two Continuous-StirredTank-Reactors (CSTRs) in series with an intermediate mixer introducing the second feed, as shown in Fig. 3. A single, irreversible, exothermic and first order reaction of A~B takes place in both reactors. The cooling jackets sUlToWlding each reactor provide heat control to the reactions. The nonlinear dynamic model of the system is derived from material and energy balances around the reactors, neglecting mixer dynamics. The design variables of the system are the feed into the CSTR1, Q/, and the feed into the mixer, Ql. The disturbances are the feed temperature T f and the feed
688
gl : Tl :5 350
g3 : Q} +Q} :50.8 2
gs : Cool :5 20
g7 : Q} ~ 0.05
g2 : T2 :5 350 l g4 : Cool :5 30 g6 : Q} ~0.05
(8)
2
gg : C :5 0.3
The plant is observed both for open loop and closed-loop cases. In closed-loop case, TI and Cool are controlled by manipulating rn,,1 and rn,,2, respectively, using multi loop PI controllers tWled by Ziegler-Nichols method. The controller gains and reset times are Kc 1=0.0023, 'ti l=0.1667 for first loop; and K/ = 0.01456, 'tj2=0.1668 for the second loop. The software used for the simulation and optimization is MATLABTM. The dynamic optimization has been done sequentially using the available integrators in MATLAB. The geometric calculation of DOS, AOS and OCI is done using Geometric Bounding Toolbox developed by Veres et. at. (1995).
5.2 Results and Discussion Both original and the proposed framework are applied to the case study. To provide similar measurement with the original approach, the proposed approach defines the AOS d as the convex-hull of the projection of EDS to output spaces in 5 dimensions, i.e. TI, CooI I, T2, Cool 2 and cl respectively. The EDS is discretized into mesh and AOS d is constructed by stepping all the mesh point.
On both algorithms, the first iterations use nominal operating values, and therefore supply optimal steady state values. In the inner level, the original approach delivers a part of critical disturbance combinations that cause the worst response, while the proposed approach gives all the critical disturbance combinations. These critical disturbances typically lie at the boundaries of parameter space, determining the number of disturbance combinations to consider in the next iteration. Consequently, the second iteration of the proposed approach solves the optimization problem. On the other hand, the original approach may still have to find another disturbance combination in the next iteration before it finds the feasible solution. This is verified in the open loop case, where the original approach solves the problem in 3 iterations, while the proposed approach needs 2 iterations. For the closed loop case, both approaches solve the problem in 2 iterations.
The results from the original and the proposed algorithm are shown in Table I. The solutions of the problems in terms of Q~, Q: and from both approaches are the same, indicating that the proposed approach solves the flexibility problem similar to the original does. Furthermore, the proposed approach offers flexibility and controllability indices for the system. The indices are given as the volume of AOSd • The controllability index is a space bounded by variation of the absolute errors of the outputs over the time; therefore the volume is the GIAE for the system. The flexibility index is the projection of this space to output space. These indices are naturally higher in the dynamic open-loop case, indicating greater variability in the system. Given a proper weighting function, these indices can be transformed to the economic objectivesfor the system. Visualization of AOS d in 6-dimensional space is clearly impossible. Therefore, projections of AOSd to only 2dimensional spaces are presented here. Figures 4 and 5 show the evolution of AOS d over time, projected into two 2-dimensional variable spaces i.e. C2_T) and CoofT2. These selections are made because there are constraint violations on these spaces throughout the iterations. The figures also show the evolution of the projected EDS on the selected output spaces; from a point (initial values) to a nearly rectangular curve that grows and rotates due to interactions of the variables. The outer line with circle marks is the convex-hull of the responses, indicating a projected facet of AOSd• It is shown that constraint violations occurred on the first iteration of the proposed approach. The second iteration solves the problem and places the AOSd inside the DOS.
6.
CONCLUSION
This paper presents a new strategy for incorporating OCI within the dynamic operability framework for regulatory cases. The proposed approach utilizes a geometric representation of a feasible operating region and a projection of disturbance set into output spaces to assess the controllability. It provides a general controllability
689
Table I . The case study results
Steady State Optimwn Dynamic Open-loop Dynamic Closed-looE
Flexibility idx
Ctrlability idx
46.86
0.075
2.755
87.77
0.019
0.602
Q~
Q:
0.355
0.206
90.35
0.252
0.055
0.324
0.223
index, that is General Integral Absolute Error (GIAE) involving whole outputs and taking account of interaction between variables. The other advantage is the simplification of the optimization framework by replacing multiple maximization problems in inner level and inequalities in outer level with single geometric maximization and equality constraint, respectively. As illustrated, the approach works well for nonlinear case. 7.
REFERENCES
Bahri, P. A, A Bandoni and J. Romagnoli (1996). Operability Assessment III Chemical Plants. Computers and Chemical Engineering, 20 S787 - S 792. Bahri, P. A, J. A Bandoni and J. A. Romagnoli (1997). Integrated Flexibility and Controllability Analysis in Design of Chemical Processes. AICHE Journal, 43 (4), 997 - 1015. de-Hennin, S. R. and J. D. Perkins (1991). Structural Decisions in Online Optimization. Technical Report B93-37, Imperial College, London. Narraway, L. and J. Perkins (1994). Selection of Process Control Structure Based on Economics. Computers and Chemical Engineering, 18 (Supplementary), S511-S515. Narraway, L. T., J. D. Perkins and G. W. Barton (1991). Interaction between Process Design and Process Control: Economic Analysis of Process Dynamics. Journal ofProcess Control, 1243 - 250. Schweiger, C. A. and C. A Floudas (1998). Process Synthesis, Design and Control: A Mixed-Integer Optimal Control Framework. IFAC Conference on Dynamics and Control of Process Systems, Corfu, Greece. Skogestad, S. and E. A Wolff (l992). Controllability Measures for Disturbance Rejection. IFAC Workshop on Interactions between Process Design and Process Control. Veres, S. M. (1999). Iterative Identification and Control Redesign via Unfalsified Sets of Models: a Basic Scheme. International Journal of Control, 72 (10), 887 - 903 . Veres, S. M., D. S. Wall, A V. Kuntsevitch and S. Hermsmeyer (1995). Using GBT version 5.1 in Identification and Control. 1995 Mat/ab Conference. Vinson, D. R. and C. Georgakis (2000). A New Measure of Process Output Controllability. Journal of Process Control, 10 (2), 185 - 194.
DistUlbanc. EtJect 10 Temperalt.n ofCSTR1 & Composition ofCSlR2
0.3 y=OOS, b=A Sd
0.28 0.28
0.24 0.22
(l 0.2 0.16 0.16 0.14
0.12
0.1 330
335
>10
3'"
>15 T1
360
365
(a.i) Disturbance EfhtcI: 10 rampelilllunt dCSm, & Composit ion of CSTR2
0.1 330
3'"
345
335
T1
360
355
365
(a.ii) Oisturblnce Elf&c:t 10 Temperature and Cool ... in CSTR2
23
22 21
,.
20
~
~
18 17
16 15
" 13
300
305
310
315
320
325
330
335
340
345
T2
(b.i) OisltJl'bance Elect to Temperature and CooIMt In CSTR2 23,---r---r---r---r---r---r---r---r--,
y . DOS, b-AOSd
22 21
~~----------------------------~
r
" 8 17
16 15
13 300
/
305
m
ill
m
m
330
~
~
345
T2
(b.ii) Fig. 4 Projected AOS d on parameter spaces for open loop case (a) C2 - T2, iteration (i) and iteration (ii) (b) Cool2 - T2, iteration (i) and iteration (ii)
690
Copyright © IF AC Dynamics and Control of Process Systems Jejudo Island, Korea, 200 I '
c:
0
[>
Publications www.elsevier.com/locatelifac
ADAPTATION OF OUTPUT CONTROLLABILITY INDEX WITHIN DYNAMIC OPERABILITY FRAMEWORK
Estiyanti Ekawati 1 and Parisa A. Bahd·
I School
ofEngineering, Murdoch University - Rockingham Campus Dixon Road, Rockingham WA 6168 A USTRALlA (*Email.·[email protected])
Abstract: This paper repo.rts.the adaptation of the Output Controllability Index (OCI) approach for the regul~tory case WIthin the Dynamic Operability Framework to facilitate controllability asses~ment III proce~s system design. The original framework requires the solution of NO~Ill~ ~rograrmrung ?r~blem on two optimization levels. This study replaces the multiple maXlmlzanon pro~lems III mner level and the whole inequalities in outer-level by a single ge~metnc calculan~n, a~d provides a controllability measurement involving all of the output vanables. The appbcablbty of the proposed approach is demonstrated on a nonlinear process system. Copyright © 2001 IFAC Keywo~ds :
process control, operability,
controllability, non linear programming,
uncertamty
I.
INTRODUCTION
The escalating demand of safe and efficient operation of process systems has increased the awareness of the importance of process control. It is understood that a well-designed process control system can guarantee optimum perfonnances, such as economic profit, safety and minimum dynamic error. The question is on how to assess the controllability in the design stage. This leads to rigorous assessment of process controllability, which by Skogestad and Wolff (1992) is defined as the ability to achieve a desired performance within various limitations on process operations, despite of external disturbances and uncertainty in design parameters, by using available input and manipulated variables. During the past fifteen years, there have been considerable efforts towards incorporating controllability assessment in process design and optimization. In effort to provide direct evaluation of process control economics, Narraway et. at. (1991) optimized the steady state objective function and the control cost subject to steady state and dynamic path constraints, assuming a single point disturbance affecting
685
the plant. The economic indicator was the required backoff amount from the active constraints to ensure feasible operation based on a linear model. In subsequent research, Narraway and Perkins (1994) extended the approach to nonlinear systems. Later, Bahri et. at. (1996) presented an iterative dynamic optimization algorithm to assess the dynamic operability based on back-off approach for both linear and nonlinear processes. This approach focused on two operability issues, which are flexibility and controllability. The flexibility problem was addressed by modeling disturbances and uncertainties as a set of step changes and automatically calculating their worst-case sets which violated system constraints. Bahri et. at. (1997) then included Integral Squared Error (lSE) as one of the system constraints for controllability measurement. Schweiger and F10udas (1998) utilized the ISE as a weighted point constraint to the optimization problem, and varied the weighting values to generate a set of trade-off solutions between economic and controllability objectives. The disturbance affecting the system, however, was only a single time function.
Recently, Vinson and Georgakis (2000) proposed the Output Controllability Index (OCI), as a steady state, input-output controllability measurement of a system. The measurement quantifies the system ability on reaching the full range of desired output values in the presence of expected process disturbances and within the limited range of its available inputs. They utilized the software developed by Veres et. al. (1995) to project the expected process disturbances space to system outputs space. Veres (1999) utilized the software to characterize model uncertainties as geometric shapes - such as polytopes and ellipsoids, transformed these shapes by linear operation of the model and assessed feasible models and tracked changes of a linear plant dynamics. This paper presents the extension of the dynamic operability assessment framework to incorporate the OCI approach for the regulatory case, specifically the geometric representation of the feasible operating conditions and system responses in process system design. Originally, the framework requires the solution of Nonlincar Programming (NLP) problem at two optimization levels. This study replaces the inner level from multiple constraint violations detection by convexhull detection and changes the inequalities in outer-level to one geometric calculation. The paper is structured as follows. The next section reviews the features and the formulation of Dynamic Operability Framework, followed by the rcviews and adaptations of the OCI approach in section 3. Section 4 presents the formulation of the proposed framework. In section 5, both original and proposed frameworks are applied to a case study. Section 6 summarizes and concludes the paper.
2.
DYNAMIC OPERABILITY FRAMEWORK
2.1 Features The features of the original Dynamic Operability Framework proposed by Bahri et al. (1997) is discussed below in a manner that facilitates the extension to incorporate a formal controllability assessment into the formulation. The principles of the original approach applied on a fixed structure are illustrated in Figure 1. The feasible operating condition is the area bounded by system constraints. It is typical that the nominal optimum operating point lies on at least one of the constraints. However, it is required to guarantee the feasible operation when disturbances enter the system, or when some uncertainties in the system parameters appear. For this purpose, the nominal operating point is moved inside the feasible region and an operating domain surrounding the point is defined. This domain represents all possible deviations from the nominal operating point due to disturbances and uncertainties. Whenever the operating domain lies entirely in the feasible operating condition, it is stated that the system is flexible. Furthermore, it is also typical to move the operating domain in the direction of
686
nominaloplimum
Sest feasib le operating condilions
Increasing Profit Control I Manip ulated Variab le 2
Fig. 1. The basic principles of Dynamic Operability Framework better profit. These efforts result in a design that is optimally flexible and economically viable. It is interesting to note from description above that flexibility requirements are the worst-case problem in controllability assessments. Controllability assessment is not only interested on the peak variability, but also on its quantity and its length of occurrences. These features represent the ability to reject the effect of disturbances in regulatory case. Therefore, there is a possibility to extend the flexibility study to assess more detailed problem in controllability, such as disturbance rejection problem. The potency can be examined further in the framework fonnulation.
2.2 Problem formulation The implementation of the original framework leads to problems on characterizing the operating domain and assessing its feasibility . The problem was formulated as an iterative dynamic optimization algorithm with two main loops, i.e. the outer loop and the inner loop. The outer loop gives the optimal operating condition through the solution of dynamic NLP problem, and the inner loop investigates the feasibility of the operating condition found in the outer loop. The algorithm continues until the operating condition found in the outer loop meets the feasibility assessments. The mathematical formulations for these two optimization levels are as follows: Outer level:
min( z,e N ,x,):,p,t)
z S.t. h;(z,ek,x,):,p,t)=O iEE g;(z,e k ,x,):,p,t)5',O jE! ZE Z = (z : zl 5', z 5', ZU ),
(1)
ek E r = (e : el 5', e 5', eu )
Inner level: max g j(z*,e,x,X,p.t) 9
S.t. h;(z*,e,x,x,p,t)=O
eE r
iE E
(2)
Here, z is a vector of decision variables, x is a vector of state variables, eN is a vector of disturbances with nominal values,
e
k
is combination of disturbances
found in the previous inner loops, p is a vector of process parameters, E, I are set of indices of equality and inequality constraints respectively, r is a set of all possible realizations of the disturbances and z* is the optimal vector found in the previous outer loop . In the inner level, multiple constraint violations are detected. The maximization problems in (2) are as many as the constraints in the system. If there is no constraint violation in this step, the algorithm stops. Otherwise, the solutions ofthe inner loop are passed to the outer loop in the next iteration. The peak variability found in the inner level can be interpreted as the maximum absolute error and as the worst-case controllability measurement. If the evolution of the worst variability can be recorded against time, it provides links to known controllability measurement such as Integral Absolute Error (IAE). This problem requires a more flexible characterization of the variability. The following section describes how the adaptation of the OCI approach can provide such characterization.
3.
THE ADAPTATION OF OUTPUT CONTROLLABILITY INDEX
The Output Controllability Index (OCI) approach by Vinson and Georgakis (2000) is discussed below including its adaptation to extend the dynamic operability framework in the regulatory case.
3. J Definition To introduce their controllability concepts, Vinson and Georgakis developed several defInitions of variable set. The first set is the Available Input Set (AlS), which is the set of values that the input variable can take in the process. The second set is the Desired Output Space (DOS), which is the set of desired values of the output and measured variables. The third set is the Achievable Output Space (AOS), which are the output values that can be achieved based on the available input values. The AOS is calculated by translating the values in AlS using process model. The ratio of the magnitude of the intersection between DOS and AOS to the magnitude of the DOS determines the process controllability. Specifically, the Output Controllability Index (OCI) of the process is defined as:
OCI = J..L( A OS (') DOS) J..L(DOS)
(3)
EDS
DOS
Fig. 2. The Expected Disturbance Set transformed to Achievable Output Space AlS and DOS are defined by constraint inequalities in the system model. Therefore, the whole inequality can be treated simultaneously using a single geometrical calculation routine. .
3.2 Adaptation Dynamic operability framework focuses on the effect of the disturbances and uncertainties on system outputs. Therefore, it is preferable to use the set of disturbance values in place of AlS in the above definition. Here, Vinson and Georgakis definition of the Expected Disturbance Space (EDS), as the disturbance spaces that are expected to affect the process, is used and adapted to observe the effect of the disturbance set to system controllability in regulatory case. Thus, AOS is now calculated based on the values in EDS (AOSJ. Furthermore, this study views DOS as the representation of the feasible operating range of the process. If AOS d is inside DOS, it solves the flexibility problem. Therefore, it is more meaningful to define OCI based on the size of AOSd, rather than DOS. The size of AOSd itself represents the system controllability in regulatory case. For the above purpose, the definition of OCl for regulatory case (r-OCI) is adapted:
Originally, Vinson and Georgakis (2000) calculated OCI based on system steady state values. In the proposed approach, the convex-hull of AOSd and its variation over time is recorded. Time is an optional dimension of AOSd space and its exclusion leads to a flexibility case, where the convex-hull of AOS d is the projection of the worstcase responses in output space. The inclusion of time provides a General Integral Absolute Error (GIAE) value for the system, which involves the whole outputs and time space, instead of single output and time for conventional lAE.
Here, J..L represents a function calculating the size of the space. 4. The advantage of this approach is that it characterizes each of the AlS, AOS and DOS as a whole polyhedral shape. This approach is in line with the geometrical approach in characterizing the disturbances and/or system uncertainties as polytopes or ellipsoids, which has been done for linear system by Veres (1999). In this case,
FORMULATION OF THE PROPOSED ALGORITHM
The main objective of the feasibility test in the inner level of Dynamic Operability Framework is to verify that the current z* is not violating inequality constraints gj(z*,e,x,x,p, t) for j = 1, ... ,J. For the proposed
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approach, the whole set of inequality constraints is represented by ng-dimensional geometric shape DOS. Here, Ug is the number of output/measurement variables involved in the inequality constraints and time. EDS is represented by n.t-dimensional geometric shape (n.t is the number of disturbance variables) and AOS d is the projection of EDS to the output variables in ng dimension. Therefore, the size of AOSd, in terms of volume of an ng-dimensional shape, represents the system controllability.
Feed
,·,,~t,;{J · .. '"
~
.
The incorporation of the adapted OCI approach into the dynamic operability framework results in the following formulation: Outer level :
minet>(z,e k ,x,i,p,t) z S.t . h,(z,ek,x,i,p,t)=O iE E r -OCI(z,ek,x,x,p,t) = 1 z E Z = { z : / :5 z :5 ZU }
-I
CSTR 2
Product
Fig. 3. Case study: 2 CSTR + mixer
(5)
ek E r = {e : el :5 e :5 eu } Inner level:
max AOSd (z*,e,x,i,p,t)
e
Coolant 2
composition C f ,. The lower bOWld, nominal values and upper bound of their variations are [298, 300, 315] K and [19.5, 20, 21] moUm 3 respectively. The measured variables are the temperatures, the amount of cooling of each CSTR and the product composition, which are TI, Cod , T2, Coof and C2. The objective function is the net profit defined as:
(6)
et> =lO(Q}Cf +Q}Cf -0.3(Q} +Q} ))- ... (7) 0.0ICooI 1 -0.ICooI 2 -O.I(Q} +Q})
The advantages of the proposed approach over the original method are as follows:
The inequality constraints used for optimization of the process are as follows:
S.t.
1.
2.
h,(z*,e,x,i,p,t)=O
eE r
iE E
It facilitates the controllability assessment within operability assessment framework through the geometric characterization of the feasible operating region and the projection of disturbance set to output space. Thus, the flexibility becomes the special case of the controllability problem. The multiple maximization problems in the inner level and the inequalities in outer-level are replaced by single geometric maximization and equality constraint respectively.
5.
CASE STUDY
5.1 Process Model
To illustrate the applicability of the proposed formulation, a case study is presented (de-Hennin and Perkins 1991). It consists of two Continuous-StirredTank-Reactors (CSTRs) in series with an intermediate mixer introducing the second feed, as shown in Fig. 3. A single, irreversible, exothermic and first order reaction of A~B takes place in both reactors. The cooling jackets sUlToWlding each reactor provide heat control to the reactions. The nonlinear dynamic model of the system is derived from material and energy balances around the reactors, neglecting mixer dynamics. The design variables of the system are the feed into the CSTR1, Q/, and the feed into the mixer, Ql. The disturbances are the feed temperature T f and the feed
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gl : Tl :5 350
g3 : Q} +Q} :50.8 2
gs : Cool :5 20
g7 : Q} ~ 0.05
g2 : T2 :5 350 l g4 : Cool :5 30 g6 : Q} ~0.05
(8)
2
gg : C :5 0.3
The plant is observed both for open loop and closed-loop cases. In closed-loop case, TI and Cool are controlled by manipulating rn,,1 and rn,,2, respectively, using multi loop PI controllers tWled by Ziegler-Nichols method. The controller gains and reset times are Kc 1=0.0023, 'ti l=0.1667 for first loop; and K/ = 0.01456, 'tj2=0.1668 for the second loop. The software used for the simulation and optimization is MATLABTM. The dynamic optimization has been done sequentially using the available integrators in MATLAB. The geometric calculation of DOS, AOS and OCI is done using Geometric Bounding Toolbox developed by Veres et. at. (1995).
5.2 Results and Discussion Both original and the proposed framework are applied to the case study. To provide similar measurement with the original approach, the proposed approach defines the AOS d as the convex-hull of the projection of EDS to output spaces in 5 dimensions, i.e. TI, CooI I, T2, Cool 2 and cl respectively. The EDS is discretized into mesh and AOS d is constructed by stepping all the mesh point.
On both algorithms, the first iterations use nominal operating values, and therefore supply optimal steady state values. In the inner level, the original approach delivers a part of critical disturbance combinations that cause the worst response, while the proposed approach gives all the critical disturbance combinations. These critical disturbances typically lie at the boundaries of parameter space, determining the number of disturbance combinations to consider in the next iteration. Consequently, the second iteration of the proposed approach solves the optimization problem. On the other hand, the original approach may still have to find another disturbance combination in the next iteration before it finds the feasible solution. This is verified in the open loop case, where the original approach solves the problem in 3 iterations, while the proposed approach needs 2 iterations. For the closed loop case, both approaches solve the problem in 2 iterations.
The results from the original and the proposed algorithm are shown in Table I. The solutions of the problems in terms of Q~, Q: and from both approaches are the same, indicating that the proposed approach solves the flexibility problem similar to the original does. Furthermore, the proposed approach offers flexibility and controllability indices for the system. The indices are given as the volume of AOSd • The controllability index is a space bounded by variation of the absolute errors of the outputs over the time; therefore the volume is the GIAE for the system. The flexibility index is the projection of this space to output space. These indices are naturally higher in the dynamic open-loop case, indicating greater variability in the system. Given a proper weighting function, these indices can be transformed to the economic objectivesfor the system. Visualization of AOS d in 6-dimensional space is clearly impossible. Therefore, projections of AOSd to only 2dimensional spaces are presented here. Figures 4 and 5 show the evolution of AOS d over time, projected into two 2-dimensional variable spaces i.e. C2_T) and CoofT2. These selections are made because there are constraint violations on these spaces throughout the iterations. The figures also show the evolution of the projected EDS on the selected output spaces; from a point (initial values) to a nearly rectangular curve that grows and rotates due to interactions of the variables. The outer line with circle marks is the convex-hull of the responses, indicating a projected facet of AOSd• It is shown that constraint violations occurred on the first iteration of the proposed approach. The second iteration solves the problem and places the AOSd inside the DOS.
6.
CONCLUSION
This paper presents a new strategy for incorporating OCI within the dynamic operability framework for regulatory cases. The proposed approach utilizes a geometric representation of a feasible operating region and a projection of disturbance set into output spaces to assess the controllability. It provides a general controllability
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Table I . The case study results
Steady State Optimwn Dynamic Open-loop Dynamic Closed-looE
Flexibility idx
Ctrlability idx
46.86
0.075
2.755
87.77
0.019
0.602
Q~
Q:
0.355
0.206
90.35
0.252
0.055
0.324
0.223
index, that is General Integral Absolute Error (GIAE) involving whole outputs and taking account of interaction between variables. The other advantage is the simplification of the optimization framework by replacing multiple maximization problems in inner level and inequalities in outer level with single geometric maximization and equality constraint, respectively. As illustrated, the approach works well for nonlinear case. 7.
REFERENCES
Bahri, P. A, A Bandoni and J. Romagnoli (1996). Operability Assessment III Chemical Plants. Computers and Chemical Engineering, 20 S787 - S 792. Bahri, P. A, J. A Bandoni and J. A. Romagnoli (1997). Integrated Flexibility and Controllability Analysis in Design of Chemical Processes. AICHE Journal, 43 (4), 997 - 1015. de-Hennin, S. R. and J. D. Perkins (1991). Structural Decisions in Online Optimization. Technical Report B93-37, Imperial College, London. Narraway, L. and J. Perkins (1994). Selection of Process Control Structure Based on Economics. Computers and Chemical Engineering, 18 (Supplementary), S511-S515. Narraway, L. T., J. D. Perkins and G. W. Barton (1991). Interaction between Process Design and Process Control: Economic Analysis of Process Dynamics. Journal ofProcess Control, 1243 - 250. Schweiger, C. A. and C. A Floudas (1998). Process Synthesis, Design and Control: A Mixed-Integer Optimal Control Framework. IFAC Conference on Dynamics and Control of Process Systems, Corfu, Greece. Skogestad, S. and E. A Wolff (l992). Controllability Measures for Disturbance Rejection. IFAC Workshop on Interactions between Process Design and Process Control. Veres, S. M. (1999). Iterative Identification and Control Redesign via Unfalsified Sets of Models: a Basic Scheme. International Journal of Control, 72 (10), 887 - 903 . Veres, S. M., D. S. Wall, A V. Kuntsevitch and S. Hermsmeyer (1995). Using GBT version 5.1 in Identification and Control. 1995 Mat/ab Conference. Vinson, D. R. and C. Georgakis (2000). A New Measure of Process Output Controllability. Journal of Process Control, 10 (2), 185 - 194.
DistUlbanc. EtJect 10 Temperalt.n ofCSTR1 & Composition ofCSlR2
0.3 y=OOS, b=A Sd
0.28 0.28
0.24 0.22
(l 0.2 0.16 0.16 0.14
0.12
0.1 330
335
>10
3'"
>15 T1
360
365
(a.i) Disturbance EfhtcI: 10 rampelilllunt dCSm, & Composit ion of CSTR2
0.1 330
3'"
345
335
T1
360
355
365
(a.ii) Oisturblnce Elf&c:t 10 Temperature and Cool ... in CSTR2
23
22 21
,.
20
~
~
18 17
16 15
" 13
300
305
310
315
320
325
330
335
340
345
T2
(b.i) OisltJl'bance Elect to Temperature and CooIMt In CSTR2 23,---r---r---r---r---r---r---r---r--,
y . DOS, b-AOSd
22 21
~~----------------------------~
r
" 8 17
16 15
13 300
/
305
m
ill
m
m
330
~
~
345
T2
(b.ii) Fig. 4 Projected AOS d on parameter spaces for open loop case (a) C2 - T2, iteration (i) and iteration (ii) (b) Cool2 - T2, iteration (i) and iteration (ii)
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