Multiple-Steady-States Identification in Homogeneous Azeotropic Distillation Using a Process Simulator

Kannan, A.; Joshi, Manish; Reddy, G. Ramalinga; Shah, Denish M.

doi:10.1021/ie049443s

Cited by 11 publications

(17 citation statements)

References 18 publications

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“…Both the conventional inside-out procedure [15] and the simultaneous correction procedure [57] were reported to miss the unstable steady-state solution, see [85] and [46] (all input variables specified; output multiplicity). However, all steady-state branches were computed either with the AUTO software package [20] or with an appropriate continuation method [39,46,85].…”

Section: Published Numerical Results With Continuation Methodsmentioning

confidence: 99%

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A robust approach for finding all well-separated solutions of sparse systems of nonlinear equations

2016

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Tearing is a long-established decomposition technique, widely adapted across many engineering fields. It reduces the task of solving a large and sparse nonlinear system of equations to that of solving a sequence of low-dimensional ones. The most serious weakness of this approach is well-known: It may suffer from severe numerical instability. The present paper resolves this flaw for the first time. The new approach requires reasonable bound constraints on the variables. The worst-case time complexity of the algorithm is exponential in the size of the largest subproblem of the decomposed system. Although there is no theoretical guarantee that all solutions will be found in the general case, increasing the so-called sample size parameter of the method improves robustness. This is demonstrated on two particularly challenging problems. Our first example is the steady-state simulation a challenging distillation column, belonging to an infamous class of problems where tearing often fails due to numerical instability. This column has three solutions, one of which is missed using tearing, but even with problem-specific methods that are not based on tearing. The other example is the Stewart-Gough platform with 40 real solutions, an extensively studied benchmark in the field of numerical algebraic geometry. For both examples, all solutions are found with a fairly small amount of sampling.

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Section: Published Numerical Results With Continuation Methodsmentioning

confidence: 99%

“…However, all steady-state branches were computed either with the AUTO software package [20] or with an appropriate continuation method [39,46,85]. The initial estimates were carefully chosen with the ∞/∞ analysis [14,38], and special attention was paid to the turning points and branch switching.…”

Section: Published Numerical Results With Continuation Methodsmentioning

confidence: 99%

A robust approach for finding all well-separated solutions of sparse systems of nonlinear equations

2016

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“…As for the simulation, conventional inside‐out12 and simultaneous correction13 procedures were reported to miss14, 15 the unstable steady‐state solution. In contrast, the general‐purpose solver IPOPT is able to converge to the unstable steady‐state solution from a poor starting point (given in mss2029).…”

Section: Resultsmentioning

confidence: 99%

“…Conventional inside‐out12 and simultaneous correction13 procedures were reported to miss14, 15 the unstable steady state solution of a homogeneous ternary azeotropic distillation problem (all input variables were specified; output multiplicity). All steady‐state branches were computed with an appropriate continuation method 14–17. A solution path was traced with respect to a user‐selected parameter (bifurcation parameter).…”

Section: Introductionmentioning

confidence: 99%

“…Physically meaningless solutions can be cut out a priori by imposing proper variable bounds. To the best of the authors' knowledge, the proposed algorithm is the first alternative to continuation methods that is capable to find all solutions to the homogeneous azeotropic distillation problem14–16 discussed above. The applicability of the proposed method is not limited to distillation.…”

Section: Introductionmentioning

confidence: 99%

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Computing multiple steady states in homogeneous azeotropic and ideal two‐product distillation

Baharev¹,

Kolev²,

Rév³

2011

AIChE Journal

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Multiple steady states are typically discovered by tracing a solution path, including turning points. A new technique is presented here that does not follow this approach. The original problem is solved directly, without tracing a solution path. The proposed branch-and-prune algorithm is guaranteed to find all solutions automatically. Core components of the framework are affine arithmetic, constraint propagation, and linear programming. The Cþþ implementation is available as an open-source solver and has an interface to the AMPL V R modeling environment. In certain difficult cases, only continuation methods have been reported to find the unstable solution automatically. The proposed method seems to be the first published alternative method in those cases. Although this article focuses mainly on distillation, the presented framework is fairly general and applicable to a wide variety of problems. Further, computational results are given to demonstrate this.

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A manifold-based approach to sparse global constraint satisfaction problems

2019

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We consider square, sparse nonlinear systems of equations whose Jacobian is structurally nonsingular, with reasonable bound constraints on all variables. We propose an algorithm for finding good approximations to all well-separated solutions of such systems. We assume that the input system is ordered such that its Jacobian is in bordered block lower triangular form with small diagonal blocks and with small border width; this can be performed fully automatically with off-the-shelf decomposition methods. Five decades of numerical experience show that models of technical systems tend to decompose favorably in practice. Once the block decomposition is available, we reduce the task of solving the large nonlinear system of equations to that of solving a sequence of low-dimensional ones. The most serious weakness of this approach is well-known: It may suffer from severe numerical instability. The proposed method resolves this issue with the novel backsolve step. We study the effect of the decomposition on a sequence of challenging problems. Beyond a certain problem size, the computational effort of multistart (no decomposition) grows exponentially. In contrast, thanks to the decomposition, for the proposed method the computational effort grows only linearly with the problem size. It depends on the problem size and on the hyperparameter settings whether the decomposition and the more sophisticated algorithm pay off. Although there is no theoretical guarantee that all solutions will be found in the general case, increasing the so-called sample size hyperparameter improves the robustness of the proposed method.

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Multiple-Steady-States Identification in Homogeneous Azeotropic Distillation Using a Process Simulator

Cited by 11 publications

References 18 publications

A robust approach for finding all well-separated solutions of sparse systems of nonlinear equations

A robust approach for finding all well-separated solutions of sparse systems of nonlinear equations

Computing multiple steady states in homogeneous azeotropic and ideal two‐product distillation

A manifold-based approach to sparse global constraint satisfaction problems

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