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Design Optimization under Parameter Uncertainty for General Black-Box Models Ipsita Banerjee and Marianthi G. Ierapetritou* Department of Chemical and Biochemical Engineering, Rutgers University, Piscataway, New Jersey 08854
Accurate knowledge of the effect of parameter uncertainty on process design and operation is essential for optimal and feasible operation of a process plant. Existing approaches for dealing with uncertainty at the design and process operations level assume the existence of a welldefined model to represent process behavior and, in almost all cases, convexity of the involved equations. However, most of the realistic case studies are described by nonconvex and, more importantly, poorly characterized models. Thus, a new approach is presented in this paper that is based on the development of input-output mapping with respect to a system’s flexibility and a design’s optimality. High-dimensional model reduction is used as our basic mapping methodology, although different mapping techniques can be employed. The result of the proposed approach is a parametric expression of the optimal objective with respect to uncertain parameters. The proposed approach is general to accommodate black-box models as it does not rely on the nature of the mathematical model of the process. Various examples are presented to illustrate the applicability of the proposed approach. 1. Introduction Uncertainties in chemical plants appear for a variety of reasons, both internal, such as fluctuations of values of reaction constants and physical properties, or external, such as quality and flow rates of feed streams. The need to account for uncertainty in various stages of plant operations has been identified as one of the most important problems in chemical plant design and operations.1-3 Two main problems are associated with the consideration of uncertainty in decision making: quantification of the feasibility and flexibility of a process design and incorporation of the uncertainty within the decision stage. The quantification of process feasibility is most commonly addressed by utilizing the feasibility function introduced by Swaney and Grossmann1 that requires constraint satisfaction over a specified uncertainty space, whereas the flexibility evaluation is associated with a quantitative measure of the feasible space. The flexibility index, as introduced by Swaney and Grossmann,1 represents the determination of largest hyperrectangle inscribed within the feasible region of the design. Other existing approaches to the quantification of flexibility involve deterministic measures such as the resilience index (RI) proposed by Saboo et al.4 and stochastic measures such as the design reliability proposed by Kubic and Stein5 and the stochastic flexibility index proposed by Pistikopoulos and Mazzuchi6 and Straub and Grossmann.7 Recently, Ierapetritou and co-workers8 introduced a new approach to the quantification of process feasibility based on the approximation of the feasible region by a convex hull. Their approach results in an accurate representation of process feasibility. However, it also relies on the utilization of process model and specific convexity assumptions. Existing approaches for quantifying the effects of uncertainty in process optimization include sensitivity * To whom correspondence should be addressed. Tel.: 7324452971. E-mail:
[email protected].
analysis and parametric programming techniques. Sensitivity analysis refers to post-optimality analysis that defines a range of parameter variation for which the identified solution remains optimal, whereas the theory of parametric programming provides a systematic method of analyzing the effect of parameter changes on the optimal solution of a mathematical programming model. For process engineering problems, sensitivity analysis has been widely used for the analysis of both linear and nonlinear continuous models.9,10 A review of parametric programming in linear models is given by Gal.11 Jongen and Weber12 reviewed the theoretical results of parametric nonlinear programming. Regarding design under parametric uncertainty, one of the proposed procedures is the deterministic approach,2,3,13,14 where the description of uncertainty is provided by specific bounds or via a finite number of fixed parameter values. An alternative approach is the stochastic approach,6,15 where uncertainty is described by probability distribution functions. A combined multiperiod/stochastic optimization formulation has recently been presented by Ierapetritou et al.16 and Hene et al.17 that combines the parametric and stochastic programming approaches to deal with synthesis/planning problems. Most of the existing approaches to modeling uncertainty in design/planning problems rely heavily on the nature of model equations and are restricted by the assumption of convexity or the specification of a number of uncertain parameters in the process model. What is lacking is a single method that can successfully predict uncertainty propagation in all models, irrespective of their nature or complexity, that is, a method that treats the process model as a black box and, given a set of input uncertainties, can predict the effect on the output. In this paper, the problem of design under uncertainty is viewed in a completely different way than it is in existing techniques. First, the system model is treated as a black box, and flexibility analysis is performed to evaluate the system’s feasible region using input-
10.1021/ie0202726 CCC: $22.00 © 2002 American Chemical Society Published on Web 12/02/2002
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output mapping. The second step is the solution of the design under uncertainty problem, which is treated similarly to the black-box model where feasibility and optimization analysis is performed iteratively using a mapping method. In this paper, the high-dimensional model representation (HDMR) technique is used as the mapping method between the variations in the input and the changes in the output. The HDMR method is a family of tools18,19 that prescribe systematic sampling procedures to map out relationships between sets of input and output model variables. The HDMR technique finds its application mostly in complex kinetics modeling,20 atmospheric chemistry,21 photochemical reaction modeling, etc., where it efficiently reduces the computational burden of the original model. The high-dimensional model representation is based on the realization that, for most physical systems, only relatively low order correlations of the input variables will have an impact on the output. Note that other mapping techniques such as the surface response method can also be utilized in the proposed methodology. The paper is organized as follows. Section 2 provides a brief description of the mapping method utilized, whereas section 3 presents a detailed description of the proposed methodology. A number of example problems presented in section 4 are used to illustrate the applicability, efficiency, and generality of the proposed approach, whereas section 5 summarizes the work and presents future directions. 2. Mapping Method: High-Dimensional Model Representation A traditional approach20 to mapping the behavior of a system with n input variables x1, ..., xn consists of sampling each variable at s points to assemble an interpolated look-up table that requires computational effort on the order of sn. Realistically, the number of sample points is approximately in the range of 10-20, and the number of variables n is 10-100 or larger. Viewed from this perspective, attempts to create a lookup table would be prohibitive. Furthermore, the evaluation of a new point by interpolation in an n-dimensional space would be exceedingly difficult. However, this analysis implicitly assumes that all n variables are important and, most significantly, that correlations among all variables at all orders jointly affect the system’s output. Hence, the output g(x) can be expressed as a hierarchical correlated function expansion in terms of the input variables as follows n
g(x1,x2,...,xn) ) f0 +
fi(xi) + ∑ fi,j(xi,xj) + ... + ∑ i)1 1eiejen f1,2,3,...,n(x1,x2,...,xn) (1)
where f0 is a constant; fi(xi) is the function describing the independent action of the variable xi on the output; fi,j(xi,xj) gives the correlated impact of xi and xj on the output; and so on, until the last term f1,2,3,...,n(x1,x2,...,xn), which incorporates any residual correlated behavior over all of the system variables. This expansion is of finite order and is always an exact representation of the model output. The fundamental principle underlying HDMR is that, using the model expansion (eq 1), the order of correlations between the independent variables that affect the
system’s output diminishes rapidly. Traditional statistical analysis of model behavior has revealed that a variance and covariance analysis of the output in relation to the input variables often adequately describe the physics (i.e., only low-order correlations describe the dynamics). On the basis of this observation, a secondorder HDMR expression has been shown to exhibit excellent performance for high-dimensional systems.19 The presence of only low-order variable cooperativity does not necessarily imply a small set of significant variables, nor does it limit the nonlinear nature of the input-output relationship. However, HDMR might not be of practical utility for systems where higher-order cooperativity among variables become significant, thereby requiring the inclusion of higher-order terms.18 The expression for second-order HDMR takes the form n
g(x1,x2,...,xn) ) f0 +
fi(xi) + ∑ fi,j(xi,xj) ∑ i)1 1eiejen
(2)
The critical feature of HDMR expansion is that its component functions, f0, fi(xi), fi,j(xi,xj), are optimal choices tailored to the specific function f(x) over the entire domain of x such that the high-order terms in the expansion are negligible. There are two commonly used HDMR expansions, cut HDMR and RS (random sampling) HDMR, which are two extreme cases of different HDMR expansions. Cut HDMR depends on the value of f(x) at a specified reference point x j , whereas RS HDMR depends on the average value of f(x) over the entire domain of x. In the present study, cut HDMR has been used, where the expansion is evaluated relative to the nominal point x j ) (xj1, xj2, ..., xjn) in the overall variable space. The f0 term is the model output evaluated at the nominal point. The higher-order terms are evaluated as cuts in the input variable space through the nominal point. Each first-order function fi(xi) is evaluated along its variable axis through the nominal point. Each second-order function fi,j(xi,xj) is evaluated in a plane defined by each binary set of input variables through the nominal point. The functions are defined as18
j) f0 ) g(x fi(xi) ) g(x j i,xi) - f0 fi,j(xi,xj) ) g(x j i,j,xi,xj) - fi(xi) - fj(xj) - f0
(3)
where the notation x j i indicates that all variables are at their nominal values except xi. Subtracting the lowerorder functions removes their dependence to ensure a unique contribution from the new expansion function. As a result, the expansion functions contain information only of the specified level of interaction, and they satisfy the null-point criteria
fi,j,...,l(xi,xj,...,xl)|xp)xjp ) 0 for p ∈ (i, j, ..., l)
(4)
These criteria ensure that the functions in eq 1 are orthogonal using a special inner product defined with respect to the nominal point. The functions in eqs 1 and 3 yield exact information about g(x1,x2,...,xn) along the cut lines, surfaces, and subvolumes for 2-D, 3-D, and 4-D cases, respectively, through the nominal points. The output response at any point x, away from the cuts, can be obtained by first interpolating each of the HDMR
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expansion terms in the look-up tables with respect to the input values of x and then summing the interpolated values of the HDMR terms from zero order to the highest required order. An important property of HDMR is that each component function of cut HDMR is composed of an infinite subclass of the full multidimensional Taylor series. Therefore, any truncated cut HDMR expansion gives a better approximation of f(x) than any truncated Taylor series because the latter contains only a finite number of terms of the Taylor series.24 3. Process Design under Uncertainty 3.1. Feasibility Analysis. In any production system, many key parameters can have significant uncertainty regarding their future values. This uncertainty can arise either from a lack of precise knowledge of its value, even though the parameter itself is deterministic, or from random fluctuations in the parameter’s value. Thus, in the choice of a particular process design, flexibility of the design is an important component, as it indicates the capability of a process to achieve feasible operation over a given range of uncertain conditions. To incorporate flexibility into the design of chemical processes, it is important to analyze whether the given design is feasible for operation over the range of variation of the uncertain parameter. Given a nominal value of an uncertain parameter, θN, and the expected deviation, ∆θ+ and ∆θ-, the flexibility test problem22 for a given design d consists of determining whether the inequalities fj(d,z,θ) e 0, j ∈ J, holds for all θ ∈ T ) [θ|θL e θ e θU]. This problem is posed as a standard optimization problem by defining a scalar variable u such that
ψ(d,θ) ) min u z,u
fj(d,z,θ) e u j ∈ J
subject to
(5)
For the process to be feasible in the parameter range of interest, T ) [θ|θL e θ e θU], it must be established that ψ(d,θ) e 0 for all θ ∈ T. To determine design feasibility without relying on a process model, HDMR is applied considering problem 5 as a black-box model. Evaluation of HDMR component function is performed according to eq 3 satisfying the null-point criteria of eq 4. In particular, the expansion for a fixed value of d is given by
fi(θi) ) ψ(θ h i,θi) - f0 (6)
The second-order HDMR expression is given by n
g(θ1,θ2,...,θn) ) f0 +
fi(θi) + ∑ fi,j(θi,θj) ∑ i)1 1ei
