A comparison between two methods of stochastic optimization for a dynamic hydrogen consuming plant

doi:10.1016/j.compchemeng.2014.02.004

Computers & Chemical Engineering

Volume 63, 17 April 2014, Pages 219-233

https://doi.org/10.1016/j.compchemeng.2014.02.004 Get rights and content

Highlights

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Two-stage and chance constrained optimization applied in a desulphurisation process.
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Both methods are formulated using a sequential approach for dynamic optimization.
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Implementation of two-stage results in open loop is made interpolating on scenarios.
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Chance constraints are solved with inverse mapping, formulated as nested estimation.
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Results of both methods are tested using Monte Carlo simulations.

Abstract

The following work shows the application of two methods of stochastic economic optimization in a hydrogen consuming plant: two-stage programming and chance constrained optimization. The system presents two main sources of uncertainty described with a binormal probability distribution function (PDF). Both methods are formulated in the continuous domain. For calculating the probabilistic constraints the inverse mapping method was written as a nested parameter estimation problem. On the other hand, to solve the two stage optimization, a discretization of the PDF in scenarios was applied with a scenario aggregation formulation to take into account the nonanticipativity constraints. Finally, a framework generalizing this solution based on interpolation was proposed. Both optimization methods, two-stage programming and chance constrained optimization, were tested using Monte Carlo simulation in terms of feasibility and optimality for the application considered. The main problem appears to be the large computation times associated.

Introduction

Uncertainty is always present in the operation of processes. Therefore, when optimal decisions have to be made, differences between the model and the reality must be considered in order to propose feasible actions. In this paper, two approaches have been used to deal with this problem for a hydrogen consumption process in a desulphurisation unit in which changes in the operating conditions take place.

In the classical approach of optimization, equations and parameters are considered totally known. However, when the computed solution is applied to the reality, frequently the value of the objective function is worse than expected and/or the constraints are violated (Birge and Louveaux, 1997, Rockafellar, 2001, Wendt et al., 2002). These problems can be attributed to the uncertainty that affects the system (Wendt et al., 2002). Usually, the behaviour of the uncertain parameters can be described using random variables named ξ, that belong to a probability space with a given probability distribution function (PDF).

One of the first attempts to solve optimization problems explicitly considering the uncertainty in the processes, appears in the work of Grossmann and co-workers introducing the concept of flexibility (Grossmann et al., 1983, Halemane and Grossmann, 1983). Also some works based on optimization under uncertainty using stages of decisions (Beale, 1955, Dantzig, 1955) were modified to be used in the process industry (Pistikopoulos and Ierapetritou, 1995, Rooney and Biegler, 1999, Rooney and Biegler, 2001, Rooney and Biegler, 2003). Additionally, this formulation has been used for discrete values of the random variables by means of the optimization over scenarios (Birge and Louveaux, 1997, Dupačová et al., 2000). According to some authors, the idea behind methods with stages of decision is not very adequate for process control, because of the interaction between the decisions in different stages, and the compensative decisions that must be taken (Arellano-Garcia and Wozny, 2009, Li et al., 2008, Wendt et al., 2002). That's why these authors propose the chance constrained formulation to be used in process optimization. For further information, an extensive recompilation can be found in the work of Sahinidis (2004).

In this work, both methods, two-stage optimization and chance constrained optimization, have been used for solving the stochastic optimization of a hydrogen consumption problem in a typical plant of a petrol refinery. In contrast to other approaches that appear in the literature, the stochastic dynamic optimization problem has been solved in the continuous time domain using a sequential approach. To do this, a control vector parameterization was used, combining optimization methods and dynamic simulation. Two changes have been proposed in the stochastic methods according to the continuous formulation. In the implementation step of the two-stage one, an interpolation method is presented to face the loss of generalization that takes place when scenarios are used to describe the continuous PDF and an open loop policy must be applied. In the same way, in the chance constraint method, a new approach for calculating the limits of the probability integrals as the solution of a parameter estimation problem has been proposed. The optimization results obtained with these methods were tested and analyzed using Monte Carlo simulations.

The structure of this paper is as follows: Section 2 presents a brief introduction to the stochastic optimization methods. Section 3 presents the operation of the hydrodesulphurisation unit and the application of the stochastic optimization methods to its optimal management. Next, Section 4 shows the outcomes of the optimization and discusses the results using Monte Carlo simulations with the generalization method proposed. The paper ends with some conclusions and comments about future work.

Section snippets

Optimization under uncertainty

In general, a problem of dynamic optimization under uncertainty can be summarized as: $\begin{array}{l} mi n_{u} f (x, u, ξ, t_{f}) \\ s . t . : \\ h (\dot{x}, x, u, ξ, t) = 0 \\ g (\dot{x}, x, u, ξ, t) \leq 0 \\ x \in X, u \in U, \forall ξ \in Ξ, t \in [t_{0}, t_{f}] \end{array}$

Where x is the vector of states, u are the decision variables, t is time and ξ represent the uncertain variables that have a random behaviour which can be described using a certain probability distribution function (PDF), named Ξ. The process model is given by the set of equations h, and the cost function to be minimized is represented

Process description

In modern petroleum refineries, the hydrodesulphurisation process (HDS) is used to remove sulphur from the hydrocarbons to fulfil environmental regulations. To do this, hydrogen is put in contact with the corresponding hydrocarbon in a fixed bed reactor with a specific catalyst (Bellos & Papayannakos, 2003). The optimal management of the hydrogen provided is very important in order to operate efficiently: if the quantity of hydrogen supplied is less than the minimum required, the expensive

Results and discussion

Several stochastic optimizations were carried out assuming at t₀ a step change of the flow and quality of the hydrocarbon stream that feeds the reactor: from $F_{H C}^{0} and ρ_{H C}^{0}$ to $F_{H C}^{f}$ and ξ₁ respectively. The values of the parameters used in the optimizations for both methods are the same ones used in the deterministic optimization (Table 1). The value of F_HC was available for the optimization, but not the ones for ξ₁ and ξ₂. Means ( $\bar{ξ}$ ), variances (σ) and correlation coefficient (r) of the PDF of

Conclusions and future work

From the results presented, it can be concluded that the use of stochastic optimization in uncertain processes can bring significant benefits in terms of feasibility and optimality. Two approaches have being considered studying its implementation and applicability. First, a two-stage optimization for an optimal process operation using the scenario aggregation method and the single shooting technique. Nevertheless, due to the large computation times, it is necessary to improve the dynamic

Acknowledgements

The financial support of Erasmus Mundus External Cooperation Window Lot 17 – Chile (EMECW Lot 17) program as well as the one of project DPI2012-37859 of the Spanish MINECO is greatly appreciated. The support from the European Union Seventh Framework Program under grant agreement no. 257462 HYCON2 Network of excellence and from the FONDECYT project 1090062 with international collaboration is also highly valued.

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View full text

A comparison between two methods of stochastic optimization for a dynamic hydrogen consuming plant

Highlights

Abstract

Introduction

Section snippets

Optimization under uncertainty

Process description

Results and discussion

Conclusions and future work

Acknowledgements

Computers & Chemical Engineering

Catalysis Today

Computers & Chemical Engineering

Computers & Chemical Engineering

Journal of Process Control

Computers & Chemical Engineering

European Journal of Operational Research

Computers & Chemical Engineering

Computers & Chemical Engineering

Chemical Engineering and Processing: Process Intensification

Electric Power Systems Research

A new method of interpolation and smooth curve fitting based on local procedures

Journal of the ACM

A model for the online optimization of integrated distillation columns under stochastic constraints

On minimizing a convex function subject to linear inequalities

Journal of the Royal Statistical Society Series B: Statistical Methodology

Introduction to stochastic programming

Numerical solution of initial-value problems in differential-algebraic equations