Acyclic modular flowsheet optimization using multiple trust regions and Gaussian process regression
Introduction
Process flowsheet optimization carried out using gradient-based methods requires that the gradient information for all the process models is available (i.e., equation oriented modeling). However, the convergence of the equation-oriented method is limited when comparing it to the sequential modular approach. This extra benefit in the convergence comes at the expense of the flowsheet’s gradient information, preventing gradient-based optimizers to be applied. In these cases, derivative-free techniques are often paired to the simulation models for optimization. But a flowsheet simulation itself may be computationally demanding, which calls for robust and sampling-efficient methods to drive the optimization. Existing approaches can be broadly classified into two categories. Global approaches proceed by constructing a surrogate model based on a flowsheet simulation before optimizing it, often within an iteration where the surrogate is progressively refined. A number of successful implementations rely on Gaussian processes (Caballero and Grossmann, 2008; Keßler et al., 2019) or artificial neural networks (Schweidtmann et al., 2019). By contrast, local approaches maintain an accurate representation of the flowsheet (or separate modules thereof) within a trust region, whose position and size are adapted iteratively. This procedure entails reconstructing the surrogate model as the trust region moves towards the local optimum. Applications of this approach to flowsheet optimization include the work by Eason and Biegler (2018) and Bajaj et al. (2018).
This paper leverages ideas from the real-time optimization field to optimize modular process flowsheets, by training separate Gaussian processes for each module with data generated by the process simulation in a trust region. Background material is reviewed next, before describing the new approach and illustrating it on an extractive distillation system.
Section snippets
Background
Gaussian process (GP) regression: Initially developed as an interpolation technique, GP regression can be used to describe an unknown function using noisy observations. GPs consider a distribution over functions and can be seen as a generalization of multivariate Gaussian distributions, f(⋅)~GP(m(•),k(⋅,⋅)). The mean function m(•) can express prior knowledge about f. The covariance function k(⋅,⋅) accounts for correlations between the function values at different points and has a great impact
Multiple trust regions and Gaussian processes for optimization
The modular flowsheet optimization problem can be stated as:
Each module i = 1… m (m denoting the last connected module) has inputs xi, process parameters θi, outputs yi, cost contribution gi,0, and design/operating constraints gi,j. The equality constraints cj describe the connections between modules. Notice that these connecting constraints are nonlinear in general—think for instance about the input
Extractive distillation system
The proposed methodology is applied to the minimization of the total annualized cost (TAC) of an extractive distillation system. An isomolar mixture of n-heptane and toluene is separated using phenol as the extractive medium. The input-output mappings are two separate Aspen HYSYS V9 simulations, corresponding to the two distillation columns in the process. These two columns are connected in a serial arrangement. Figure 1 shows a schematic representation of each individual column containing
Conclusion
In this work an algorithm to optimize process flowsheets was presented. The main advantage of the proposed method is that a complete flowsheet does not need to be simulated at any point. The modular flowsheet structure is exploited so that multiple GPs are trained independently for each individual module. These surrogates are built on separate trust regions and connected within an optimization subproblem using equality constraints corresponding to the connecting streams in the flowsheet. The
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