An Alternating Trust Region Algorithm for Distributed Linearly Constrained Nonlinear Programs, Application to the AC Optimal Power Flow

A novel trust region method for solving linearly constrained nonlinear programs is presented. The proposed technique is amenable to a distributed implementation, as its salient ingredient is an alternating projected gradient sweep in place of the Cauchy point computation. It is proven that the algorithm yields a sequence that globally converges to a critical point. As a result of some changes to the standard trust region method, namely a proximal regularisation of the trust region subproblem, it is shown that the local convergence rate is linear with an arbitrarily small ratio. Thus, convergence is locally almost superlinear, under standard regularity assumptions. The proposed method is successfully applied to compute local solutions to alternating current optimal power flow problems in transmission and distribution networks. Moreover, the new mechanism for computing a Cauchy point compares favourably against the standard projected search as for its activity detection properties

On dimension reduction in Gaussian filters

A priori dimension reduction is a widely adopted technique for reducing the computational complexity of stationary inverse problems. In this setting, the solution of an inverse problem is parameterized by a low-dimensional basis that is often obtained from the truncated Karhunen-Loeve expansion of the prior distribution. For high-dimensional inverse problems equipped with smoothing priors, this technique can lead to drastic reductions in parameter dimension and significant computational savings. In this paper, we extend the concept of a priori dimension reduction to non-stationary inverse problems, in which the goal is to sequentially infer the state of a dynamical system. Our approach proceeds in an offline-online fashion. We first identify a low-dimensional subspace in the state space before solving the inverse problem (the offline phase), using either the method of "snapshots" or regularized covariance estimation. Then this subspace is used to reduce the computational complexity of various filtering algorithms - including the Kalman filter, extended Kalman filter, and ensemble Kalman filter - within a novel subspace-constrained Bayesian prediction-and-update procedure (the online phase). We demonstrate the performance of our new dimension reduction approach on various numerical examples. In some test cases, our approach reduces the dimensionality of the original problem by orders of magnitude and yields up to two orders of magnitude in computational savings

Robust Mission Design Through Evidence Theory and Multi-Agent Collaborative Search

In this paper, the preliminary design of a space mission is approached introducing uncertainties on the design parameters and formulating the resulting reliable design problem as a multiobjective optimization problem. Uncertainties are modelled through evidence theory and the belief, or credibility, in the successful achievement of mission goals is maximised along with the reliability of constraint satisfaction. The multiobjective optimisation problem is solved through a novel algorithm based on the collaboration of a population of agents in search for the set of highly reliable solutions. Two typical problems in mission analysis are used to illustrate the proposed methodology

Computational methods for geochemical modelling: applications to carbon dioxide sequestration

Geochemical modelling is fundamental for solving many environmental problems, and specially useful for modelling carbon storage into deep saline aquifers. This is because the injected greenhouse gas perturbs the reservoir, causing the subsurface fluid to become acidic, and consequently increasing its reactivity with the formation rock. Assessment of the long term fate of carbon dioxide, therefore, requires accurate calculations of the geochemical processes that occur underground. For this, it is important to take into account the major water-gas-rock effects that play important roles during the gas storage and migration. These reactive processes can in general be formulated in terms of chemical equilibrium or chemical kinetics models. This work proposes novel numerical methods for the solution of multiphase chemical equilibrium and kinetics problems. Instead of adapting or improving traditional algorithms in the geochemical modelling literature, this work adopts an approach of abstracting the underlying mathematics from the chemical problems, and investigating suitable, modern and efficient methods for them in the mathematical literature. This is the case, for example, of the adaptation of an interior-point minimisation algorithm for the calculation of chemical equilibrium, in which the Gibbs energy of the system is minimised. The methods were developed for integration into reactive transport simulators, requiring them to be accurate, robust and efficient. These features are demonstrated in the manuscript. All the methods developed were applied to problems relevant to carbon sequestration in saline aquifers. Their accuracy was assessed by comparing, for example, calculations of pH and CO2 solubility in brines against recent experimental data. Kinetic modelling of carbon dioxide injection into carbonate and sandstone saline aquifers was performed to demonstrate the importance of accounting for the water-gas-rock effects when simulating carbon dioxide sequestration. The results demonstrated that carbonate rocks, for example, increase the potential of the subsurface fluid to dissolve even more mobile CO2.Open Acces