Chapter B6

Process design and control structure evaluation and screening using nonlinear sensitivity analysis Panos Seferlisa and Johan Grievinkb a.CERTH - Chemical Process Engineering Research Institute (CPERI),

P.O. Box 361, 57001 Thermi - Thessaloniki, Greece b Department of Chemical Technology, Faculty of Applied Sciences,

Delft University of Technology, Julianalaan 136, 2628 BL, Delft, The Netherlands Published in Computer Aided Chemical Engineering, Volume 17, 2004, Pages 326-351. The Integration of Process Design and Control doi: 10.1016/S1570-7946(04)80065-8 ISBN: 978-0-444-51557-5 Publisher: Elsevier B.V.

Chapter B6

Process design and control structure evaluation and screening using nonlinear sensitivity analysis Panos Seferlisa and Johan Grievinkb a.CERTH - Chemical Process Engineering Research Institute (CPERI),

P.O. Box 361, 57001 Thermi - Thessaloniki, Greece b Department of Chemical Technology, Faculty of Applied Sciences,

Delft University of Technology, Julianalaan 136, 2628 BL, Delft, The Netherlands 1. INTRODUCTION

An optimisation-based method for the integrated process and control system design (Ref. 1-5) aims at the simultaneous determination of the flowsheet configuration, the equipment design parameters, the plantwide control structure, and the controller tuning parameters. Such an approach relies on the definition of a set of candidate flowsheet and control structure configurations in the form of a plant and control system superstructure, which involves numerous discrete and continuous variables representing the entire variety of design decisions. Furthermore, appropriate nonlinear process models should enable the accurate prediction of the steady state and dynamic behaviour of the candidate flowsheets to a set of representative and meaningful disturbance scenarios. Hence, the solution of the complete optimisation problem for a plantwide application accounting for all feasible design alternatives and possible combinations between potential manipulated and controlled variables may be proved extremely challenging in terms of both problem complexity and required computational effort. In this chapter, a number of techniques based on nonlinear sensitivity analysis of the static and dynamic plant controllability properties are introduced that facilitate the process design and control in a fully optimised and integrated way.

The chapter presents a procedure for the evaluation and screening of alternative process flowsheet and control structure configurations in a rigorous, effective and systematic way. It utilises an effective decomposition of the process and control system design task, which in sequence allows the enhancement of the process controllability properties. The quality of the steady state behaviour in response to the deleterious effects of multiple simultaneous process disturbances and model parameter variations is an essential prerequisite for good economic

performance during operation. Disturbance rejection sensitivity for a candidate process flowsheet is investigated in association with a number of decisions that involve the plantwide control objectives for the process, the prioritisation of these control objectives and the available resources for control purposes, and the input-output structure of the candidate control system. The method employs rigorous and detailed mathematical models that intend to capture nonlinear effects arising from the physical system and intensified by the interactions of multiple and simultaneously acting disturbances. Disturbance directionality and magnitude are two key factors that are thoroughly investigated and explored. The calculation of a static controllability performance index representative of the static disturbance rejection characteristics becomes the basis for the assessment and ranking of the alternative flowsheets and control structures. Obviously, poorly (i.e. beyond a defined acceptable level) performing combinations would be automatically disregarded from any further investigation.

Dynamic characteristics of the candidate process design play the most dominant role in the achievable dynamic performance of the control system. The performance of the control system is inherently affected by the dynamics of the designed process. Process dynamic characteristics are influenced not only by process design decisions but also by model parameters and disturbance variations. Disturbance rejection sensitivity analysis determines the margins for the process design from undesired dynamic behaviour (e.g., unstable, undesired oscillatory or sluggish response). Nonlinear process models act once again as the basic tools for the identification of the process dynamic components during a disturbance scenario that involves multiple disturbances of finite magnitude. Notably sensitive dynamically designs to model parameter variations become less attractive for selection.

Manipulation of the design decision variables using nonlinear sensitivity information becomes an additional instrument for the design engineer in order to modify and improve the desired static and dynamic properties of the plant. Flowsheet design parameters are associated either with the structure of the flowsheet (e.g., selection of units, connectivity of units and so forth) or the equipment design specifications. The disturbance rejection exercises and their influence on the static and dynamic controllability indices reveal the relationship between the indices and the design parameters. Proper modification of the design parameters utilizing nonlinear sensitivity analysis calculations can enhance the static and dynamic controllability properties of the design while maintaining the economic attractiveness of the final design.

The chapter is organized as follows: Section 2 begins with an overview of previous work (2.1), then introduces the formulation of the economic (2.2) static (2.4) and dynamic (2.5) controllability criteria within an optimisation framework, and finally presents the guidelines for the selection of the candidate input-output control structures in section 2.3. The nonlinear disturbance sensitivity control (DiSC) problem is thoroughly examined in section 3.1. The assessment and rank ordering of alternative flowsheet and control structure configurations using controllability performance indices is the subject of section 3.2. The improvement of the controllability properties through the solution and analysis of the design sensitivity control problem (DeSC) is presented in 3.3, while the overall screening procedure is summarized in

3.4. Section 4 analyses and discusses a number of stimulating and representative examples while concluding remarks appear in section 5.

2. DESIGN CRITERIA

2.1. Previous work

The key principle behind steady state disturbance rejection analysis for a plantwide control structure is that a control system that requires smaller steady state effort, measured in terms of changes in the manipulated variables, to compensate for a given set of anticipated disturbances is superior to one that requires a much larger effort. A characteristic example is the identification of the “snowball” effect in systems with recycles due to positive feedback induced through the recycle stream (Ref. 6). The principle is closely related to the relative disturbance gain (RDG) (Ref. 7) that further investigates the interactions between the possible pairings of controlled and manipulated variables. However, good steady state disturbance rejection properties do not necessarily imply good dynamic performance. Therefore, the dynamic behaviour of the plant design and control system requires further investigation through dynamic controllability analysis and dynamic simulations (Ref. 8).

Disturbance sensitivity has been used in the evaluation of alternative control structures (Ref. 9), the investigation of disturbance directionality effects at different frequencies (Ref. 10) and the quantification of the process feasibility window through steady state operability analysis and geometric indices (Ref. 11). Although linear plant models offer an easy generalisation of the results the complexity and nonlinearity of real processes make the use of rigorous and accurate nonlinear models a necessity (Ref. 12-15). Furthermore, the large dimensionality of the model parameters and disturbance space in industrial applications made the implementation of decomposition techniques quite attractive (Ref. 13). A common element on most of the referenced approaches that utilise disturbance sensitivity is the fact that the results are independent from the selection of a control algorithm.

Disturbance sensitivity analysis has been embedded within an optimisation framework for the optimal selection of pairings between controlled and manipulated variables using steady state gain matrices (Ref. 8) that also incorporate integrating variables (Ref. 16). Elements for the dynamic performance of plantwide control systems have been elaborated in Ref. 17 through the minimisation of the deviations in the controlled and manipulated variables from target values using a linearised state-space model. Alternatively, the dynamic final settling time for the controlled variables defined the dynamic operability index for the assessment of process design in Ref. 15. The minimum additional surge capacity required for a given flowsheet and set of controlled and manipulated variables to achieve the dynamic objectives of the control system was the evaluation measure in Ref. 18.

The variety of hierarchical design procedures for plantwide control systems that employed modular analysis (Ref. 19-20), decomposition policies (Ref. 21-22), interaction analysis (Ref. 17), lexicographical analysis (Ref. 23), and economic potential analysis (Ref. 24) may lead to more than one control systems for the same plant configuration; therefore, a further evaluation

step is necessary with the assistance of accurate dynamic simulations. Furthermore, criteria for the control structure selection include structural output controllability (Ref. 25), and the economic performance for linear and nonlinear process models with PID control loops (Ref. 26-27). Multivariable control strategies in a centralised and decentralised fashion have been developed and employed in Ref. 28. The thermodynamic properties of the system and specifically the mechanisms of energy transfer to and from the system were elaborated to identify the dominant variables for the process and the inferential variables for the control objectives (Ref. 29).

2.2. Design on economic criteria

A hierarchical process design procedure as in Douglas (Ref. 30) sets the fundamental principles for the determination of a set of alternative flowsheet configurations. For each process flowsheet the optimal equipment sizing and steady state operating conditions are determined through the satisfaction of an economic criterion for a nominal set of model parameters and process related externally specified inputs (e.g., disturbances). The most commonly used economic criteria incorporate the annualised investment and operating costs. Nonlinear process models are employed for the prediction of the steady state behaviour of the plant, while a proper set of constraints would ensure the satisfaction of safety, environmental, product quality and operating specifications. In conclusion, the economic design optimisation problem takes the following form:

( )

( )( )

UL

ULref

ref

refxd

dddXXX

εdXgεdXh

εdX

≤≤

≤≤

≤=

=

00s.t.

Costs AnnualisedMin

,,,,

,,C,

(1)

C denotes the economic objective function, while h and g the equality and inequality constraints of the process model. Vectors X and d denote the process and design variables in the model, respectively. Vector εref denotes the model parameters and externally specified disturbances at a nominal (reference) value level. At this stage, vector X=[x y u]T contains all process variables without discriminating between state, x, controlled, y, and manipulated, u, variables. The degrees of freedom in Eq. (1) are the design variables d, and a subset of x (i.e. the independent set of process variables) that specifies the plant’s operating point. Upper and lower bounds on the process and design variables define the available operating space.

The solution of the mathematical problem of Eq. (1), (dopt, Xopt), depends on the values of the model parameters and exogeneous inputs (disturbances) to the process. If the optimal operating point lies at the intersection of process constraints, a most commonly situation, model uncertainty may cause the steady state actual operating point to violate process constraints. Therefore, the implemented optimal operating point should be compromised to

avoid violations of the feasible space (Ref. 31). However, if the optimal operating point has available degrees of freedom the influence of disturbances up to a maximum allowable magnitude can be accommodated by the system.

2.3. Control structure selection

The next step in the procedure is devoted to the determination of the candidate control structures as sets of input and output variables. Initially, the control objectives for each of the alternative flowsheets need to be precisely defined. Even though, most of the control objectives may be similar for the candidate process flowsheets (e.g., product quality specifications), the use of different unit operations will differentiate the safety and operating requirements. Typically, control objectives involve targets on product specifications, production levels, reactant conversion per pass, reactor selectivity, profitability, stability, flexibility, safety and so forth. It is apparent from this limited list that some objectives can be easily linked to process variables and operating conditions, while others require a more sophisticated and complex description. Therefore, control objectives can be classified as explicit and implicit. Explicit control objectives can be fully represented by the values of specific variables in the flowsheet. For instance, the composition of the final product stream describes the product quality specification. The satisfaction of the control objectives can be achieved by keeping the values of a proper set of controlled variables at a desired level (e.g., point target) or within a certain range (e.g., range target). Partial control (Ref. 32) that translates into controlling tightly only a subset of the output variables is also a plausible option. On the other hand, implicit control objectives are more abstract and inferred through, usually nonlinear, functions of a set of process variables and are rarely measured directly on-line. Typical examples for implicit control objectives are for instance the achieved conversion in a reactor, the reactor selectivity, and the overall profitability of the process.

The manipulated variables are easily identified for each flowsheet as all the legitimate (independent) control valves (Ref. 22). It is obvious, that the entire available input space for the process system should be utilised for each flowsheet configuration for the satisfaction of the control objectives. The most common situation is that the number of control objectives is larger than the number of independent inputs in the system, thus making impossible to completely satisfy all control objectives. In view of such an observation, flowsheet configurations that can provide a large number of independent handles for control (inputs) are more preferable. In the case that the number of input variables exceeds the number of control objectives and furthermore, the selected set has the ability to efficiently influence the complete set of controlled variables, there is plausible opportunity for further optimisation of the operating position.

2.3. Static controllability criterion

The systematic evaluation of the candidate control structures and process flowsheet configurations is performed in a way that remains independent of a selected control algorithm. The term control algorithm includes both the pairing between the input and output variables

and the postulated relationship between the input actions with respect to the error in the output variables. The main target is the evaluation of the candidate control structures such that any implemented control algorithm would result in adequate static and dynamic performance.

The formulation of the screening method for flowsheet configurations and control structures proceeds with the construction of a number of disturbance scenarios that are expected to influence the plant’s operating conditions. The greater is the knowledge of the nature, magnitude and directionality of the anticipated process disturbances, the greater becomes the reliability of the observed results and the merits of the screening method. The present method considers only disturbances of deterministic nature. Since the analysis is based on steady state behaviour the effects of time variations for the disturbances is not examined.

The static performance evaluation should be able to accommodate the large variety of control objectives. A quadratic performance objective function is introduced that penalizes deviations of the controlled variables from a target value in a least squares sense. Hence, the controlled variables are forced to either remain at a constant steady state value (set point) or vary within a specified region around a target value. Subsequently, it is desirable to identify a set of manipulated variables that require the least effort to compensate for the detrimental effects of process disturbances on the plant’s control objectives. Therefore, the sum of squares of changes from steady state optimal values for the manipulated variables is weighted in the objective function, as well. Large changes in the manipulated variables may imply large errors for the controlled variables during dynamic transition from one steady state to another. Such a case would require that the disturbance or set point variation is slow enough, compared to the plant’s dynamic speed of response, to allow sufficient time for the plant to reach a new steady state. However, the plant’s steady state effort to cope for the effects of disturbances is indicative for potential trouble in handling the situation even for variations of higher frequency content. Obviously, the full description for the compensation of high frequency disturbances would require the evaluation of the complete dynamic behaviour.

The steady state disturbance rejection problem is formulated within an optimisation framework. Given a set of structural and equipment design variables at their optimal values as calculated from the solution of Eq. (1) (e.g. number of stages in a distillation column, total volume of a reactor), dopt, a set of set points for the controlled variables, ysp, a set of optimal steady state values for the manipulated variables, uss, a set of model parameters and disturbances, ε, and symmetric weighting matrices for the deviations from target values for the controlled and manipulated variables, Wy and Wu, respectively, the following optimisation problem is constructed:

( ) ( ) ( ) ( )( )( )

ulululopt

ssuT

ssspyT

spuy

xxx,uuu,yyydd0εd,u,y,x,g0εd,u,y,x,h

uuWuuyyWyy

≤≤≤≤≤≤=

≤=

−−+−−=

,

f,

s.t.

Min

(2)

The optimal steady state values, uopt≡uss, for the manipulated variables, and the set points,

yopt≡ysp, for the controlled variables, are retrieved from the optimal Xopt=[xopt ysp uss]T vector calculated from Eq. (1) and remain unchanged during the solution of Eq. (2) for different parameter values, ε. However, a set point change can be simulated if ysp is considered a varying parameter itself. Upper and lower hard bounds for the controlled, y, manipulated, u, and the remaining process, x, variables that define the allowable ranges of variation are present.

The behaviour of the system can be conceived as the response to an unmeasured process disturbance governed by the respective objective function. Eq. (2) evaluates only the steady state effects of the disturbances on the flowsheet. In other words, it predicts the behaviour of an ideal multi-variable controller that simultaneously inspects deviations on controlled and manipulated variables and takes into consideration the nonlinear relations among the process variables. Eq. (2) handles explicitly manipulated variable saturation and allows the investigation of the integrity of a given control structure. Once a manipulated variable saturates (e.g., input reaches an upper or lower level) or fails to respond (e.g., control system failure), a degree of freedom for the control system is consumed. The minimisation of the objective function will then determine how to distribute the effort among the remaining manipulated variables to compensate for the effects of the disturbances according to the entries in Wu. The relative importance of the control objectives and the preference in the usage of each manipulated variable as reflected by the individual entries in the weighting matrices of the objective function will dictate the reaction of the system to the perturbation. For instance, a large weight on a diagonal entry of Wy forces the system to react decisively in order to eliminate the deviation in the corresponding controlled variables from the set point level at the expense of controlled variables with a smaller weight. Furthermore, the entries in the weighting matrices can impose different control objectives such as tight control of the controlled variables with relatively large weights or loose control with relatively smaller in magnitude weights. In general, the selection of individual entries in matrices Wy and Wu will be based on the aim to shift variability from the key performance and profit related variables (e.g., product quality variables, variables associated with safety requirements, use of expensive raw material) to the cheaper variables that a greater degree of variability is tolerated (e.g., utility system) (Ref. 33). In all cases hard variable bound define impregnable control limits.

The above formulation enables the study of control structures with unequal number of controlled and manipulated variables. As already mentioned, lack of sufficient input capacity may hinder the flowsheet from satisfying the underlined control objectives completely. The imposed ranking through the selected objective function would prioritise the importance of each control objective and guide the control system through the partial satisfaction of the control targets. The use of the input resources is closely associated with an economic factor (e.g., cost of steam and fresh material). Therefore, the most competitive way to compensate for the effects of disturbances relies on the efficient use of the available input capacity. In

cases, where an excess of manipulated variables is available, it is possible to drive the process to the most profitable, from an economic point of view, operating point (e.g., reduction of the usage of expensive resources).

The disturbance sensitivity control (DiSC) problem will enrich the designer engineer’s perception about the interactions between process design and control system performance in various ways: (i) Identify inadequate process designs and control structures that require large changes in the manipulated variables for small disturbance magnitudes. (ii) Calculate the capacity requirements for the process equipment or the range for manipulated variables in order to compensate for the effects of disturbances. (iii) Identify the inequality constraints and variable bounds that bottleneck the control system response and hinder its performance. (iv) Determine the feasibility region (i.e. the magnitude for the combined disturbance variation for which no feasible solution exists) for the imposed disturbance scenario rigorously. (v) Investigate the behaviour of the system under special circumstances such as input saturation, control failure, lack of input handles and non-square systems.

2.4. Dynamic controllability criterion

The static controllability criterion described in the previous section does not however consider the dynamic behaviour of the system explicitly. The required total transition time from one steady state operating point to another (e.g., set point tracking or disturbance rejection) and the type of dynamic response (e.g., oscillatory underdamped response, sluggish overdamped and so forth) are key factors that characterise the dynamic performance of the control structure. Variation during dynamic transition may cause violations of variable bounds and process inequalities. Constraint violation can be avoided if the static operating point is moved away from the optimal steady state point (back-off) to allow sufficient space for the expected dynamic variations (Ref. 34) or the proper surge capacity is included in the system to attenuate the undesired fluctuations (Ref. 18). The complete description and solution of the entire integrated problem of design and control thus becomes very difficult and tedious, especially when entire plant flowsheets are considered. In this section, a procedure is presented that allows the investigation of the dynamic characteristics of the system without using a full dynamic simulation for the plant. The technique acts as a preliminary screening mechanism of flowsheets and control structure configurations with undesired dynamics in a similar way as the static controllability analysis presented previously.

A linear model can be described in the following state-space form:

FεDuCxyEεBuAxx

++=++=& (3)

where vectors x, y, and u denote the state, input, and output variables, respectively. Vector ε includes the model parameters and externally specified inputs (e.g., disturbances).

The solution of the linear system of differential equations in Eq. (3) depends on the eigenstructure of matrix A governing both the stability and transient response characteristics

of the system. More specifically, system eigenvalues with large negative real parts give rise to fast dynamics that respond quickly to exogenous variations. Complex conjugate eigenvalues result in underdamped responses with oscillations. In conjunction with negative real parts close to the origin, the dynamic response becomes challenging from a control point of view. Eigenvalues with positive real parts lead to unstable open-loop dynamic behaviour, a usually undesirable situation because the control system must be designed with extreme caution to ensure stability and attain good performance. Eigenvalues with small negative real parts are responsible for sluggish dynamic modes. Usually feedback control systems aim among other objectives (e.g., stability, zero offset) to relocate the open-loop poles so that a faster closed-loop response is achieved. The system eigenvalues may be associated with specific states but in more complex systems such one-to-one association is not possible. In such cases, groups of eigenvalues are associated with respective groups of process states (Ref. 35).

For nonlinear systems the dynamic behaviour is retrieved from the solution of a set of nonlinear differential and algebraic equations.

( ) 0=εduyxxh ,,,,,& (4) Differentiating Eq. (4) with respect to the state time derivatives, states, inputs, outputs and disturbances using the chain rule the following relation is derived:

0δεεhδu

uhδy

yhδx

xhxδ

xh

=∂∂

+∂∂

+∂∂

+∂∂

+∂∂ &&

(5)

Rearranging the terms in Eq. (5) the system can be brought to the form of Eq. (3):

∂∂

∂∂

∂∂

∂∂

∂∂

−=

−

δεδuδx

εh

uh

xh

yh

xh

δyxδ 1

&& (6)

The form of Eq. (6) depends on the characterisation of the input and output variables as

determined by the specifications of the selected control structure. A difficulty that arises in this point is related to the index of the differential-algebraic set of equations in Eq. (4) due to the selection of the input-output structure.

Model parameter variations and perturbations of the externally specified inputs will influence the position of the system eigenvalues through the linearised system in Eq. (6) (Ref. 36). Such a situation may lead to the appearance of dynamic modes that are responsible for the deterioration of the achievable control performance. The dynamic controllability criterion is the required tool for the investigation of the process dynamics transformation under the influence of multiple disturbances that accounts also for nonlinear interactions. Therefore, the static controllability optimisation problem in Eq. (2) is enriched as follows (Ref. 37):

( ) ( ) ( ) ( )

( )( )

( )( )( )

ulululopti

Ti

i

ssuT

ssspyT

spuy

xxx,uuu,yyydd01zz

0zIεd,u,y,x,A0εd,u,y,x,g0εd,u,y,x,h

uuWuuyyWyy

≤≤≤≤≤≤=

=−

==−≤=

−−+−−=

,

nz,,i

f

i

,

2

1

s.t.

Min

Kξ (7)

where z denotes the eigenvectors of matrix A and ξ the corresponding eigenvalues. The eigenvalues are scaled to unity magnitude to ensure their uniqueness. Matrix A is a function of all process and design variables, exogenous input variables and model parameters as derived in Eq. (6). Under certain conditions, variation of any of these variables would perturb the eigenvalues of the system in a continuous way. It should be noted here that Eq. (7) is valid for eigenvalues of algebraic multiplicity greater than one but limited to eigenvalues of geometric multiplicity equal to one.

Solution of Eq. (7) for different values for set ε, traces the optimal steady state response of the system and in addition the variation of the system eigenvalues, ξ. The benefits to the design engineer can be summarised as follows: (i) Identify situations where the system eigenvalues location gives rise to undesired dynamic characteristics and in extension difficulties in control. (ii) Calculate the impact of the disturbance scenarios on the dynamics (e.g., favourable or unfavourable influence). (iii) Determine eigenvalue sensitivity for large disturbance variations and active set changes. (iv) Calculate the margin of the system eigenvalues from a region that is considered as of acceptable dynamic response. (v) Identify design factors that have the greatest impact in changing the position of the eigenvalues (Ref. 37). 3. NONLINEAR SENSITIVITY METHOD 3.1 Disturbance sensitivity control (DiSC) problem

The static and dynamic behaviour of the candidate flowsheet and input-output control structure is obtained for multiple simultaneous model parameter and disturbance variations, ε, from the solution of Eq. (7). A large number of sampling points in the multi-dimensional parameter space would be required for a complete coverage of the entire disturbance space. System variability for all possible disturbance realisations becomes then an enormous task that would require significant computational resources. It is however noticeable, that some directions of perturbation in the disturbance space would upset the system in a more severe and profound way. The selection of the direction of variation is performed based on either process knowledge, or known correlation among process variables or parameters, or sensitivity information around the nominal operating point. Local sensitivity analysis of the modelling equations at the operating nominal point (i.e., solution of Eq. (1)) provides the

variability of the process variables and eigenvalues with respect to model parameters and disturbances.

( ) ( )[ ]Trefopt

εrefopt

εε εεXP ξ∇∇= (8) Singular value decomposition of Pε reveals the direction in the parameter and/or disturbance space that causes the largest changes in the process variables and system eigenvalues (Ref. 38). This is equivalent to a worst-case disturbance scenario where the main effort of the analysis is concentrated. The magnitude of variation is then adjusted along the specified worst direction of perturbation, θ, while a co-ordinate ζ, along this direction represents this magnitude.

The DiSC problem is equivalent to the solution of the parameterised set of the first-order Karush-Kuhn-Tucker (KKT) optimality conditions for the control problem of Eq. (7) for variable disturbance magnitude. The set is further augmented with the relations that govern the variations of multiple parameters or disturbances (Ref. 38), ∆ε, as follows.

( )( )( )

0

2

=

−−

−

∇+∇+∇

=

ζ

ξ

θΔε1zz

zIAgh

gμhλ

z,ε,d,u,y,x,F

iTi

i

A

A

TT

i

f

ξ,ζ (9)

The first entry in Eq. (9) represents the gradient of the Lagrangian function of the nonlinear programme of Eq. (7), with respect to vectors X=[x y u]T, d, and ε. Vectors λ and µ denote the Lagrange multipliers associated with the equality, h, and active inequality constraints, gA, respectively. The eigenvalue defining relations result in zero Lagrange multipliers because they do not affect the optimal solution of Eq. (7). Vector ∆ε denotes the relative changes of the model parameters or the externally defined disturbances from the nominal reference point (εref). The trajectory of the optimal solution is calculated for load and model parameter changes along predefined directions, θ, in the multidimensional disturbance space. Scalar ζ denotes the free continuation variable.

The size of matrix A depends on the number of state variables in the system. The main objective is to track the changes in a subset of all the system eigenvalues. The greatest interest is focused on the subset of eigenvalues that are responsible for unstable dynamics (positive real parts), sluggish responses (negative real parts close to the origin) and strongly oscillatory behaviour (conjugate eigenvalues with real parts close to the origin). It is assumed that the selected subset that comprises the slowest system eigenvalues remains the same during the disturbance scenario. The occurrence of pairs of conjugate complex eigenvalues requires the

use of separate defining equations for the real and imaginary parts in Eq. (9). This would double the eigenvalue-eigenvector defining equations for a complex eigenvalue pair compared to a real eigenvalue.

A unique solution for Eq. (9) at point s=(x,y,u,d,ε,z,ξ,ζ) exists if the Jacobian of F at point s is non-singular. In general, the Jacobian F is non-singular in the entire domain except a number of finite points that becomes singular. These singularities are related to either the optimality conditions or the eigenvalue problem. More specifically, violation of any of the linear independence constraint qualification, the strict complementarity condition or the second-order optimality conditions for the parameterised KKT set results in fold points, boundary points and active set changes in the optimal solution path (Ref. 38). On the other hand, the eigenvalue-tracking problem may result in fold points when the paths of two real eigenvalues intersect. At the point of intersection a real double eigenvalue is present; an eigenvalue with algebraic multiplicity equal to two and geometric multiplicity equal to one. The double eigenvalue may split again into either two real eigenvalues or a pair of conjugate eigenvalues. A complete overview of the types of singularities that arise in the asymmetric eigenvalue problem can be found in (Ref. 39). The tracking of the singular points of the optimality conditions and their impact on the eigenvalues is the main reason that the otherwise decoupled steady state controllability and eigenvalue problems are considered and solved simultaneously in Eq. (7) and (9).

Linearisation of the nonlinear dynamic equations is performed at each trajectory point so the nonlinear interactions are taken into consideration. The computational technique can explicitly handle active set changes and hard bounds on all variables efficiently. Active set changes require the modification of the equation through the addition (if a bound or inequality constraint become active) or the removal (if a bound or inequality ceases to be binding) of the respective constraints. Optimality is ensured by inspection of the sign of the Lagrange multipliers associated with the active inequalities at every continuation point. The solution technique is quite efficient because an approximation of the optimal solution path of Eq. (9) is sufficient for the purposes of the problem.

A predictor-corrector type of continuation method as implemented in PITCON (Ref. 40) is used with ζ acting as the independent continuation parameter. Homotopy continuation methods specialised for the nonsymmetric eigenproblem as described in Ref. 41 and 42 can also be adapted. 3.2. Design assessment based on disturbance rejection sensitivity

The assessment of the flowsheet and control structure configurations ability to efficiently compensate for the effects of disturbances on the control objectives is based on the static controllability performance index ΩSC defined as follows:

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( )

p

i*

i,p

*i,p

*i,p

ii

p

*i

*i

*i

ii

p

*i

*i

*i

iSC fff

wy

yywu

uuwΩ ∑∑∑−

+−+−=0

00

00

0 ζζζζζζζ (10)

The controllability index accounts for the absolute relative changes of input and output variables augmented with a set of implicit control objectives represented by fp,i. The elements of fp,i are not directly controlled but rather indirectly with the proper selection of controlled variables, y, and are evaluated using the nonlinear model at every point in the optimal solution path. Deviations for the implicit control objectives from a desired level are an indicator for the ability of the selected measured controlled variables to accurately represent them. Hence, the performance index can attain broader meaning if the implicit control objectives are enriched with economic related terms. ΩSC is generally considered as a superset of the objective function in Eq. (7) when p=2 (i.e., Euclidean norm). The weighting terms, w(ζ), determine the significance of each calculated segment of the optimal solution path along the perturbation direction. For instance, a larger weight may be used for small perturbation magnitudes that are more likely to occur during plant operation.

As pointed out in section 2.3, a large value for ΩSC would imply large error in the controlled variables during the dynamic transition from one steady state operating point to another. This is not a sufficient condition and should not be the only criterion for selecting a proper control structure but rather a necessary condition for good achievable performance by the control system. The key objective remains the identification and screening of designs that possess undesirable characteristics that are however difficult to observe without thorough investigation.

Regarding the dynamic aspects of the candidate designs a dynamic controllability performance index ΩDC measures the margins from the region considered as an undesired dynamic behaviour (e.g., stability margins, margin from region that causes large oscillations) and defined as follows:

( ) ( )( )( ) p

bound*i

p

bound*i

iDC wΩ

ξξ

ξζξζζ

−

−= ∑

0 (11)

ξbound is a point in the boundary of the region for undesired dynamics (e.g., distance from the origin). Large relative variations for the process eigenvalues imply that the dynamic behaviour is very sensitivity to changes in the model parameters and exogeneous inputs. Such a case requires extreme caution if the margins at the nominal point from an undesired dynamic situation (e.g., unstable response or large oscillations) are small. Disturbance variations may also have a favorable impact of the system dynamics (i.e., move eigenvalues towards positions with improved dynamic features).

3.3. Design sensitivity control (DeSC) analysis

Disturbances can cause significant deterioration on the plant’s economic performance. Operating costs are affected due to the use of additional resources (e.g., raw material, utilities)

and the production of low valued product (e.g., off-spec product) in conditions of dynamic transition between steady states or at the preferred steady state operating point. The structural design characteristics of the plant, d, calculated from Eq. (1), have been held constant during the solution of the parameterised control problem of Eq. (9). At this stage, the design characteristics may act as an additional instrument in order to enhance the static and dynamic controllability performance properties of the plant. The goals can be summarised as follows: (i) Improve the static controllability performance index, ΩSC, and (ii) improve the dynamic behaviour with the proper placement of open-loop eigenvalues through the adjustment of the structural design characteristics of the design for a given disturbance scenario.

Along the optimal trajectory for the disturbance sensitivity control problem a second sensitivity problem is solved, namely the design sensitivity control (DeSC) problem. In this case, the local sensitivity of the process variables and system eigenvalues to infinitesimal changes of the design parameters (e.g., structural flowsheet characteristics, design equipment parameters), d, is calculated. More specifically, the sensitivity matrix for the operating conditions with respect to the design parameters, d, for the given disturbance scenario is derived. The sensitivity matrix, Pd, shown in Eq. (12), provides a measure of the relative change of the process variables, Lagrange multipliers and system’s eigenstructure for infinitesimal changes in the design parameters.

( ) ( ) ( ) ( ) ( )[ ]Tdddddd dzddμdλdXP ***** ∇∇∇∇∇= ξ (12)

Focusing on the more significant from an operation’s point of view X* and ξ* variables (even though Lagrange multipliers associated with binding inequality constraints are also important as they trigger active set changes) the detailed sensitivity matrix has the following form:

[ ]

T

T*d

*dd XP

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

=∇∇=

m

k

m

k

m

n

m

n

dd

dd

dx

dx

dx

dx

ξξ

ξξ

ξ

LMOM

L

LMOM

L

1

11

1

1

11

1

(13)

The asterisk indicates optimal values (i.e. at a solution point of Eq. (9)). The sensitivity matrix is usually scaled to facilitate the analysis. A commonly used scaling involves the expression of the matrix entries in terms of logarithmic sensitivities.

m,,jn,,idx

xd

dx

j

i*i

*j

j

i KK 11lnln

===∂∂

∂∂ (14)

The sensitivity information is calculated directly from the solution of Eq. (9) at a small additional computational cost, utilising the Newton step performed at every continuation stage (Ref. 43).

( ) ( ) ( )( )

−∇−∇

∇∇

∇

−=

∇∇∇∇∇

−∇−−∇∇∇

∇∇∇

1zzzIA

ghL

z

μλX

z0000zIAz00zIA

0000g0000h00ghL

Td

d

Ad

d

2Xd

*d

*d

*d

*d

*d

XX

AX

X

TAX

TX

2X

ξξξξ

(15)

L is the Lagrangian function as in Eq. (9) and X=[x y u]T. The matrix in the left side of Eq. (15) is derived from differentiation of Eq. (9) with respect to d. The computational cost for the calculation of the design sensitivity is associated with the solution of system in Eq. (15).

The sensitivity information can be decomposed in a set of dominant modes of variation for the process using singular value decomposition of Pd. A small perturbation in d, along the eigenvector direction that corresponds to the largest in magnitude singular value of matrix Pd (also an eigenvector of Pd

TPd), v1, reveals the dominant direction of variability in the system that causes the largest change in the process variables and open-loop eigenvalues in a least square sense (Ref. 13). Similarly, the eigenvector direction that corresponds to the second largest in magnitude singular value, v2, denotes the second most important direction of variability and so forth. The orthogonality property that holds for the eigenvectors of Pd

TPd, v1

Tv2=0, ensures the independence of the modes of variation. Therefore, the sensitivity factor affecting the static and dynamic behaviour of the process with respect to the design parameters can be projected in a low dimensional space defined by the dominant directions. The entries of the normalized eigenvectors v denote the contribution of each parameter to the given direction.

The local DeSC calculations and analysis are performed for various values of ζ, along the worst disturbance direction, θ. The DeSC sensitivity results calculated at sequential points along the optimal trajectory are homotopically equivalent provided that the active set remains unchanged. In such a case the dominant directions of perturbation are not expected to change substantially during the disturbance scenario for a mildly nonlinear system. However, changes in the active set, gA, may result in dramatic changes in the DeSC information and subsequently in the dominant directions and their relationship to the design parameters.

As soon as the most important design variables are identified (e.g. extra reactor capacity, additional stages in a distillation column) then the additional investment cost is devoted on the associated process units. Apparently, additional capacity will be translated to increased investment costs, however the objective function of Eq. (7) of the DiSC problem is then expected to improve for the same disturbance scenario. The compromise between the increment in the investment costs and the improvement in the controllability properties will determine the extent of additional capacity in the system.

The sensitivity matrix also involves the relative changes for the dynamic characteristics of the process as these are represented by the eigenvalues of the system. The design parameters can be then used to adjust the open-loop eigenvalues. This can be considered as a pole placement procedure where instead of the controller, the design itself provides the mechanism to alter the system dynamics.

3.5. Overall design procedure

The steps in the outlined evaluation, ranking and screening procedure are summarised as follows:

Step 1: Obtain a set of process designs that optimise a certain economic criterion. Step 2: Determine the control objectives and the limits for the desired dynamic

characteristics for the process system. Step 3: Determine a set of disturbance scenarios (e.g., worst-case scenario). Step 4: For each alternative flowsheet configuration and candidate set of controlled and

manipulated variables investigate the disturbance rejection sensitivity of the process and control system (DiSC). Evaluate controllability indices, ΩSC and ΩDC.

Step 5: Assess system performance. Three options are possible. Reject (continue with Step 8), accept (continue with Step 8) or modify design (continue with Step 6).

Step 6: Use sensitivity information along the solution path for the DiSC problem to retrieve information from the DeSC problem.

Step 7: Modify design at the expense of the economic criterion according to the guidelines extracted from the decomposed sensitivity information for the design parameters.

Step 8: Continue with another candidate flowsheet and control structure configuration. Go to Step 4.

Step 9: Terminate the procedure with eventually a much smaller rank ordered set of candidate flowsheets and control structure configurations based on static and dynamic controllability criteria.

4. DESIGN APPLICATIONS

4.1. Reactor system

An exothermic first-order reactive system in a sequence of jacketed CSTRs is considered. Several alternative process designs are constructed and studied with respect to their static and dynamic controllability properties to multiple and simultaneous process disturbances. The same system has been studied by numerous researchers (Ref. 14, 15, 44) and served as an illustrative example of process design and control interactions. The reaction is carried out in either a single reactor or two reactors in series (Fig. 1). The dynamic model (see Ref. 14, 15) contains four state variables per reactor; namely the reactor’s volume, concentration and temperature and the jacket temperature. Model parameters for the system are shown in Table

1. The steady state optimal operating points and the open-loop eigenvalues for several designs are shown in Table 2. The single reactor configuration is operated at three different reactor temperature levels (D1-D3). Alternative designs for the two reactors configuration involve the relationship between the volumes of the two tanks. More specifically, in D4 two tanks of equal volume are used, in D5 two tanks with volume ratio equal to 2:1 are used and in D6 the two reactor volumes are decided by the minimisation of the investment and operating costs. Table 1 Model parameters for reactor system. Parameters Values Parameters Values Reactor density 800 (kgr/m3) Feed temperature 320 K Reactant heat capacity 3,000 (J/kgr K) Coolant inlet temperature 290 K Coolant density 1,000 (kgr/m3) Heat of reaction 25,000 J/mol Coolant heat capacity 4,200 (J/kgr K) Heat transfer coefficient 1,250 J/hr m2 K Feed concentration 4,000 (mol/m3) Activation energy 10,000 J/mol Feed flowrate 0.1 (m3/hr) Pre-exponential term 0.94 hr-1

Table 2 Operating points for alternative designs. D1 D2 D3 D4 D5 D6 D7 TRX (K) 345 355 335 345 345 345 345 VRX (m3) 26.41 23.94 29.31 4.83 6.73 4.41 26.41 - - - 4.83 3.37 5.27 - TJ (K) 318.89 344.80 295.20 296.88 301.47 303.33 318.89 301.91 311.03 305.56 QJ (m3/h) 0.0125 0.0024 0.1136 0.0276 0.0211 0.0131 0.0125 - - - 0.0143 0.0057 0.0120 - Cost ($/y) 2163.3 1991.7 2772.0 1695.4 1606.59 1578.6 2163.3 Eigenv. -2.96 10-2 -3.27 10-2 -2.67 10-2 -3.41 10-2 -3.25 10-2 -3.46 10-2 -2.96 10-2 -1.09 10-3 -3.07 10-3 -2.12 10-3 -3.64 10-2 -4.00 10-2 -3.57 10-2 -1.45 10-3 -3.07 10-3 -6.31 10-4 -7.89 10-2 -1.03 10-2 -1.39 10-2 -1.12 10-2 -3.84 10-3 -1.01 10-2 -7.90 10-3 -9.42 10-3 -3.77 10-3 -3.90 10-3 -2.93 10-3 -2.73 10-3 -2.19 10-3 -3.46 10-3

Α à Β Α à Β

Α à Β

Fig. 1. Alternative designs for reactor system.

Inspection of the steady state economic data reveal that a two-reactor system has significantly lower investment and operating costs. However, single reactor’s disturbance rejection performance is superior compared to the behaviour of the two reactors in series. Fig. 2 shows the behaviour of the static controllability index for variation of factors that simultaneously affect the heat transfer capacity of the system (e.g., heat transfer coefficient) and the total heat amount that required to be exchanged (e.g., inlet stream temperature, feed stream flowrate, heat of reaction). More specifically, the positive sign for parameter, ζ, indicates an increase in the inlet stream temperature, the feed flowrate and the heat of reaction and a decrease in the heat transfer coefficient. The single reactor not only exhibits a lower index than the two-tank configuration but also allows acceptable operation for a wider range of variation magnitudes. The finding is mainly attributed to the increased heat transfer capacity of the larger single reactor. The study is based on the assumption that a five degrees variation in the reactor temperature is acceptable and the maximum jacket coolant flowrate can be increased up to three times its normal operating value.

As expected the system eigenvalues for the single reactor are closer to the origin leading to an inherently slower response than the two-tank system. The difference in the dynamic characteristics is evident from the dynamic simulations in Fig. 3 for ζ values equal to 5.0. PID control loops have been placed for the maintenance of the reactor temperature at the desired level. The single tank response even though more sluggish, exhibits lower overshoot that the two-tank system. It should be noted here that the operating points (e.g., reactor set points) have been moved away from the nominal values due to the influence of the disturbance in accordance with the objective function of Eq. (7) with the single reactor operating much closer to the nominal point. The improvement of the speed of response for the single reactor can be achieved from the inspection of the sensitivity of the design parameters to the system eigenvalues. Reduction of the jacket volume (D7) would place the reactor and jacket energy balance eigenvalues further away from the origin thus increasing the system’s speed of response (Table 2). Furthermore, the jacket volume does not influence the steady state

operating point and therefore is an excellent candidate for the fine-tuning of the system’s dynamic properties. The improved dynamic performance is evident in the dynamic simulation shown in Fig 3b. The variation of the eigenvalues associated with the reactor and jacket energy balances for D1 and D7 is shown in Fig. 4. The faster dynamics are clearly depicted.

-25 -20 -15 -10 -5 0 5 10 150

0.5

1

1.5

2

2.5

Continuation parameter, zeta

Sta

tic c

ontro

llabi

lity

inde

x, o

meg

a

D1 D2 D3

-20 -15 -10 -5 0 5 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Continuation parameter, zeta

Sta

tic c

ontro

llabi

lity

inde

x, o

meg

a

D5 D6

D4

Fig. 2. Static controllability index for designs D1-D6.

0 0.2 0.4 0.6 0.8 1345

345.5

346

346.5

347

347.5

348

348.5

349

349.5

Time, hrs

Rea

ctor

tem

pera

ture

, K

2 Reactors

1 Reactor

0 0.2 0.4 0.6 0.8 1345

345.2

345.4

345.6

345.8

346

346.2

346.4

346.6

346.8

347

Time, hrs

Rea

ctor

tem

pera

ture

, K

(a) (b)

Fig. 3. Temperature control for: (a) ∆, o – first and second reactors in 2 CSTR and – single CSTR, (b) – base design, o – reduced jacket volume for single CSTR.

reduced jacket volume

-20 -15 -10 -5 0 5 10 15-6

-5

-4

-3

-2

-1

0x 10

-3

Continuation parameter, zeta

Eig

enva

lue reduced jacket volume

Fig. 4. Eigenvalue variation during disturbance scenario. 4.2. Reactor/separator/recycle system

The system initially described in Ref. 45 and shown in Fig. 5 comprises two reaction steps that take place in continuous perfectly mixed reactors. Component “C” is the product of the first reacting step, while by-product “D” further reacts in the second reactive step to form product “F” and the reactant “B” of the first reaction. The two reaction steps must remain well balanced in order to guarantee the uninterrupted operation of the plant. For instance, quenching of the second reaction step will deprive the system from “B”, thus affecting the extent of the first reaction. Components are ranked in descending order of volatility from “A” to “F”. Column C1 separates and recycles the light reactants to reactor RX1, while C2 purifies product stream “C”. Column C3 separates product “F” and recycles the rich in “B” stream back to RX1. The volatility of the components in the second reaction step (i.e., heavy product “F” and light product “B”) favours the integration of reaction and separation of the products in a single reactive distillation column (see dashed envelope in Fig. 5).

Four different flowsheet configurations (FC I-IV) are assessed. FC I and II use two and one CSTR, respectively, for the second reactive step and a conventional distillation unit (C3) for the second separation step. FC III and IV use a similar reactor sequence scheme but in conjunction with a reactive distillation column, C3. An optimal economic design is obtained using isothermal models for the reactors and stage-by-stage calculations for the reactive and conventional distillation columns using orthogonal collocation on finite elements techniques, so that a continuous representation for the number of stages is achieved. The flowsheet configurations with the reactive distillation column (FC III and IV) have a definite economic advantage over FC I and II, resulting in approximately 15% reduction of the total annualised costs. The reduction is achieved due to the significant decrease on the total required reacting volume for the second reacting step (the plates in the reactive column can be viewed as a sequence of CSTR).

F0AF0E

F0D

F0B

A+B à C+D D+E à B+F D+E à B+F

Product C

Product FB1 B2 B3

R1

R2

R3

D1

D2

D3

V1 V2 V3

Purge

Purge

RX1 RX2 RX3

C1 C2 C3

Fig. 5. Flowsheet configuration for the reaction/separation/recycle system.

4.2.1. DiSC problem The control objectives for the flowsheet are summarized as follows: • Maintain the desired product quality for final product streams D2 and B3. • Achieve high conversion in the reactors. • Maintain plant operation close to economic optimum.

A number of disturbance scenarios have been constructed for the specific flowsheet that aim to reflect realistic cases for the plant. Scenario 1 involves the simultaneous decrease of the activity of the chemical reactions (i.e., decrease of the pre-exponential kinetic coefficients) with a reduction of the purity level in the feed stream for component “A” (e.g., “B” concentration increases in the feed stream F0A). Scenario 2 involves the decrease of the activity levels for both reactions. Scenario 3 examines the ability of the system to balance the difference on the imposed activity for the two reacting steps. The final scenario 4 involves the loss of the vapour stream in the C3 reboiler from the set of the manipulated variables.

Table 3 Design parameters for candidate flowsheet configurations at nominal operating point (DeSC design modifications in italics). FC I FC II FC III FC IV Reactor volume (m3) RX1 604.5 / 650.0 645.8 581.9 581.9 / 620.0 RX2 263.2 / 280.0 2118.9 11.0 37.0 RX3 407.2 0.0 19.9 0.0 Column stages C1 35 35 35 35 C2 40 40 39 39 C3 39 36 41 41 / 43 Stage holdup (m3) C3 - - 2.4 2.5 / 2.8 Column reboil ratio C1 1.66 1.49 1.77 1.77 C2 2.50 2.04 2.85 2.85 C3 1.56 1.87 1.60 1.64 D1 recycle (kmol/hr) 359.04 387.96 342.78 343.04 D3 recycle (kmol/hr) 134.60 179.23 110.65 110.95 Costs (×103 $/yr) 1,305.9 / 1,306.7 1,495.3 1,131.1 1,132.8 / 1,138.7

The candidate input-output control structures as proposed in Ref. 45 are briefly described

in Table 4. Sets of controlled and manipulated variables that lead to “snowball” effects due to positive recycle feedback can be eliminated quickly through the disturbance rejection sensitivity. For example, control structure “d” described in Table 4 maintains all recycle streams on level control, which results in extremely large changes in the flowrate of the recycle streams for relatively small changes in the flowrate of feed stream F0A. Similar small changes in F0A cause the recycle stream from column C1 to reactor RX1 to rapidly increase for control structures “c” and “e”. Generally, it is assumed that perfect control is achieved for all flow and level controllers in the plant. Controlled variables that correspond to composition variables are allowed to vary around a target value and within a range specified by upper and lower bounds.

Table 5 shows the cumulative performance index with uniform weighting for FC I-IV for control structures “a” and “b” that differ on the installation or not of a feed/purge stream system for component “D” (see dashed envelope at the bottom of column C2 in Figure 1) calculated at various disturbance magnitudes. The cumulative index, ΩSC, uses all the scaled controlled and manipulated variables specified for each control structure. The results reveal

that FC III-IV exhibit in general a superior disturbance rejection performance than FC I-II for most disturbance scenarios (except scenario 3) because of the higher average achieved concentration of reactants in the reactive distillation column. Consequently, the effects from the kinetic parameters variation on the product purity are absorbed more efficiently in the reactive distillation column. The analysis also suggests that the use of a feed/purge system for component “D” in control structure “a” may improve the disturbance rejection capabilities of the flowsheet, especially in the cases of a single reactor in the second reacting step and for larger disturbance magnitudes (ζ >20.0). In scenario 3, FC I and II perform better than FC III and IV, because the reduction of the kinetic parameter in the second reactive step affects not only the conversion in the reactors but also indirectly the separation through the equilibrium relations in the reactive column C3. The severity of the disturbance scenarios increases, as fewer manipulated variables are available. Saturation of key manipulation variables becomes evident as the magnitude of the parametric variation increases (scenario 4). 4.2.2 DeSC problem

Decomposition of the design sensitivity information for the disturbance scenarios reveals the key design parameters that affect the most the system’s static controllability properties. Volumes for RX1 and RX2 have the greatest impact on the dominant direction vector for FC I. On the other hand the stage holdup and the number of stages in the reactive column C3 are the most significant design parameters for FC IV, even though total holdup for RX1 and RX3 contribute as well. Both results pinpoint the design parameters that affect the extent of the reacting steps. Table 3 shows the modifications that are implemented in the flowsheets, and the new total costs (in italics). Table 5 clearly shows the improvement in the cumulative performance index achieved by the modified flowsheets under simultaneous kinetic parameter fluctuation, especially at larger perturbation magnitudes (flowsheets with asterisks). Results suggest that significant improvement is possible with only slight increase in the total costs for the plant. An increase of the total costs less than 1% results in a respective 25% and 18% improvement on the controllability performance at ζ=25.0 for FC Ia* and IVa*. Table 4 Control structures proposed in Ref. 45. ΜV CV Fixed variables Common in all control structures a F0A-D3-F0D MRX1-xR1,B-MB2 R1-B2 CV: MB1-MB3- MD1- MD2- MD3-

xB1,B-xD2,D-xB3,D-xR2,E-MRX2-MRX3 MV: B1-B3-D1-D2-F0B-V1-V2-V3-F0E-R2-R3

b F0A-D3-B2 MRX1-xR1,B-MB2 R1-F0D

c D3-F0D MRX1-MB2 F0A-R1-B2

d R1-D3-B2 MRX1-xR1,B-MB2 F0A-F0D

e R1-B2 MRX1-MB2 F0A-D3-F0D

Note: M denotes material inventory; symbols in capital denote stream flowrates (Fig. 5); x denotes component molar fractions for distillation and reactor outlet streams.

Table 5 Cumulative performance index calculated for different designs, controlled structures and disturbance scenarios at various disturbance magnitudes expressed in terms of ζ (modified designs with asterisks).

Scenario 1 Scenario 2 Scenario 3 Scenario 4 ζ 15.0 20.0 15.0 20.0 25.0 15.0 20.0 25.0 15.0 20.0

I a 1.63 3.16 0.15 0.46 1.74 0.13 0.41 0.99 0.22 0.77 I a* - - 0.12 0.37 1.31 - - - - - I b 1.12 3.17 0.18 0.51 1.85 0.13 0.42 1.13 0.23 0.77 II a 2.46 5.66 0.26 0.67 2.01 0.07 0.28 0.79 0.67 1.50 II b 2.65 8.07 0.29 0.73 2.45 0.08 0.33 1.06 0.20 0.61 III a 0.72 2.22 0.20 0.50 1.56 0.39 0.82 1.35 0.22 0.56 III b 0.74 2.19 0.18 0.49 1.43 0.30 0.84 1.86 0.21 0.52 IV a 0.75 2.35 0.19 0.52 1.59 0.38 0.85 1.52 0.23 0.65 IV a* - - 0.19 0.48 1.30 - - - - - IV b 0.74 2.23 0.17 0.51 1.78 0.30 0.89 2.11 0.37 0.79

CONCLUSIONS – SUMMARY This chapter presents the tools for the evaluation, rank orderingand screening of alternative

process flowsheet and control structure configurations in a systematic, rigorous and efficient way. Flowsheet and control structure configurations are analysed on the basis of: (a) static disturbance rejection characteristics utilizing nonlinear sensitivity techniques, and (b) sensitivity of process dynamics with respect to process disturbances and model parameter variations. A static controllability performance index calculates the impact of disturbances and model parameter variations on the steady state operation of the flowsheet. A dynamic controllability index evaluates the margins from an undesired dynamic behaviour for the process system. These two indices form the basic indicators for the rank ordering of the alternative design options. Designs may be rejected from any further consideration if poor controllability properties are identified. Designs may be modified efficiently using design sensitivity information in order to enhance the controllability properties of the plant. Overall, the outlined procedure exploits the predictive accuracy of nonlinear models both for steady state and dynamic behaviour, the prioritisation of multiple control objectives, the preference in the use of available resources, and the powerful properties of nonlinear sensitivity analysis. Furthermore, disturbance directionality and sensitivity information decomposition increase the efficiency of the analysis tools for the benefit of the design engineer.

Acknowledgement The work is financially supported by European Commission (GROWTH programme,

G1RD-CT-2001-00649).

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