Optimal control under uncertainty - Applied to upstream petroleum production
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Optimal control has been applied successfully in a wide range of applications, especially within the process industries in the form of Model Predictive Control (MPC). By employing a dynamic model to predict the response of the plant, optimal control inputs can be calculated. The performance is inevitably linked to the predictive capabilities of the model, and model uncertainty can be a critical issue. While optimal control is extensively used in the downstream petroleum industry, optimization based approaches in the upstream petroleum industry are still limited. One of the reasons for this, perhaps the most important, is the high level of uncertainty associated with models employed in the upstream industry. Obtaining an accurate description of a heterogeneous reservoir located far below the earth’s surface is naturally challenging. There are in general two approaches for handling model uncertainty; reducing it by further experiments and installation of sensory equipment, or finding a strategy to operate the system mitigating the effect of uncertainty. One does not exclude the other. This thesis is about the latter, and in particular; Develop methods and formulations for real time optimization taking uncertainty into account, with a special focus on reservoir management and production optimization. The inclusion of uncertainty in an optimal control problem increases the complexity significantly. While a deterministic model can be handled by optimizing over control trajectories, this will in general be suboptimal when uncertainty is considered, as feedback is disregarded over the prediction horizon. To incorporate feedback, control policies must be considered. However, that is often impractical, and it is usual to resort to optimizing over control trajectories instead (Mayne, 2014). This thesis is based on 6 articles, all focusing on optimal control of uncertain systems; two on production optimization, two on reservoir management, and two on control in general. Three of the articles, one for each application, provide a formulation optimizing over control policies. In (Hanssen and Foss, 2015a, Ch. 3), a policy for production optimization is expressed in terms of a list of set-points. The operator should adjust wells according to the list, until a constraint is reached or all wells are fully open. In (Hanssen et al., 2017, Ch. 4), a general formulation for optimizing over control policies in reservoir management is provided. The control policy is formulated as a set of implicit equations, solved as part of the simulation equations. This also allows for efficient calculation of gradients by the use of the adjoint method. In (Hanssen and Foss, 2015b, Ch. 7), a closed loop formulation is obtained by a multi-stage stochastic programming formulation. Both state and parameter updating is conducted by an Ensemble Kalman Filter (EnKF), resulting in an implicit dual controller. The approach is, however, computationally very extensive. Constraints are vital for many control problems, and the success of MPC is partially attributed to its ability of handling both input and output constraints. When uncertainty is considered, and feedback over the prediction horizon is disregarded, output constraints can lead to overly conservative solutions. Therefore, it is often desirable to somehow relax output constraints. In (Hanssen et al., 2015, Ch. 2), Conditional Value at Risk (CVaR) is used for handling output constraints in production optimization, allowing to control the level of conservativeness. A static model is employed. In (Hanssen and Foss, 2015c, Ch. 6), CVaR is used for output constraints in open loop predictions, with a gradual relaxation over the prediction horizon. This gradual relaxation is motivated by the increasing uncertainty over the prediction horizon, due to disregarding feedback. For uncertain systems, the selection of controlled variables can influence the inherent robust of a solution. In (Hanssen and Foss, 2016, Ch. 5), a systematic comparison of various controlled variables for reservoir management is conducted. The study is motivated by the ideas of Self Optimizing Control (SOC) (Skogestad, 2000), and the comparison indicates that optimizing over rates is more robust than optimizing over bottom hole pressures when the main uncertainty is permeability.