Bjarne Grimstad

Global optimization with spline constraints: a new branch-and-bound method based on B-splines

This paper discusses the use of splines as constraints in mathematical programming. By combining the mature theory of the B-spline and the widely used branch-and-bound framework a novel spatial branch-and-bound (sBB) method is obtained. The method solves nonconvex mixed-integer nonlinear programming (MINLP) problems with spline constraints to global optimality. A broad applicability follows from the fact that a spline may represent any (piecewise) polynomial and accurately approximate other nonlinear functions. The method relies on a reformulation–convexification technique which results in lifted polyhedral relaxations that are efficiently solved by an LP solver. The method has been implemented in the sBB solver Convex ENvelopes for Spline Optimization (CENSO). In this paper CENSO is compared to several state-of-the-art MINLP solvers on a set of polynomially constrained NLP problems. To further display the versatility of the method a realistic pump synthesis problem of class MINLP is solved with exact and approximated pump characteristics.

Global optimization of multiphase flow networks using spline surrogate models

Abstract A general modelling framework for optimization of multiphase flow networks with discrete decision variables is presented. The framework is expressed with the graph and special attention is given to the convexity properties of the mathematical programming formulation that follows. Nonlinear pressure and temperature relations are modelled using multivariate splines, resulting in a mixed-integer nonlinear programming (MINLP) formulation with spline constraints. A global solution method is devised by combining the framework with a spline-compatible MINLP solver, recently presented in the literature. The solver is able to globally solve the nonconvex optimization problems. The new solution method is benchmarked with several local optimization methods on a set of three realistic subsea production optimization cases provided by the oil company BP.

Towards an objective feasibility pump for convex MINLPs

This paper describes a heuristic algorithm for finding good feasible solutions of convex mixed-integer nonlinear programs (MINLPs). The algorithm we propose is a modification of the feasibility pump heuristic, in which we aim at balancing the two goals of quickly obtaining a feasible solution and preserving quality of the solution with respect to the original objective. The effectiveness and merits of the proposed algorithm are assessed by evaluation of extensive computational results from a set of 146 convex MINLP test problems. We also show how a set of user-defined parameters may be selected to strike a balance between low computation time and high solution quality.

A nonlinear, adaptive observer for gas-lift wells operating under slowly varying reservoir pressure

Abstract A nonlinear observer that estimates well flow rates and downhole pressure based on topside measurements only has been presented in the literature. As demonstrated in laboratory experiments the observer is suitable for a coupling with conventional PI control for stabilization of slug flow. The design of the observer is based on a nonlinear model with a linear inflow relation, and demands knowledge of several uncertain well parameters, including the reservoir pressure and production index. In this paper we present an adaptive extension to the nonlinear observer that eliminates one uncertain parameter by estimating the reservoir pressure. The extended observer is shown to be globally uniformly asymptotically stable and retain the properties of the original observer.

Petroleum production optimization – A static or dynamic problem?

Abstract This paper considers the upstream oil and gas domain, or more precisely the daily production optimization problem in which production engineers aim to utilize the production systems as efficiently as possible by for instance maximizing the revenue stream. This is done by adjusting control inputs like choke valves, artificial lift parameters and routing of well streams. It is well known that the daily production optimization problem is well suited for mathematical optimization. The contribution of this paper is a discussion on appropriate formulations, in particular the use of static models vs. dynamic models. We argue that many important problems can indeed be solved by repetitive use of static models while some problems, in particular related to shale gas systems, require dynamic models to capture key process characteristics. The reason for this is how reservoir dynamics interacts with the dynamics of the production system.

Optimization of a Simulated Well Cluster using Surrogate Models

Abstract In this paper we present an implementation of a partly Derivative-Free Optimization (DFO) algorithm for production optimization of a simulated multi-phase flow network. The network consists of well and pipeline simulators, considered to be black-box models without available gradients. The algorithm utilizes local approximations as surrogate models for the complex simulators. A Mixed Integer Nonlinear Programming (MINLP) problem is built from the surrogate models and the known structure of the flow network. The core of the algorithm is IBMs MINLP solver Bonmin, which is run iteratively to solve optimization problems cast in terms of surrogate models. At each iteration the surrogate models are updated to fit local data points from the simulators. The algorithm is tested on an artificial subsea network modeled in FlowManager ™ , a multi-phase flow simulator from FMC Technologies. The results for this special case show that the algorithm converges to a point where the surrogate models fit the simulator, and they both share the optimum.

A MIQCP formulation for B-spline constraints

This paper presents a mixed-integer quadratically constrained programming (MIQCP) formulation for B-spline constraints. The formulation can be used to obtain an exact MIQCP reformulation of any spline-constrained optimization problem problem, provided that the polynomial spline functions are continuous. This reformulation allows practitioners to use a general-purpose MIQCP solver, instead of a special-purpose spline solver, when solving B-spline constrained problems. B-splines are a powerful and widely used modeling tool, previously restricted from optimization due to lack of solver support. This contribution may encourage practitioners to use B-splines to model constraint functions. However, as the numerical study suggests, there is still a large gap between the solve times of the general-purpose solvers using the proposed formulation, and the special-purpose spline solver CENSO, the latter being significantly lower.