This paper introduces the concept of affine reserve policies for accommodating large, fluctuating renewable infeeds in power systems. The approach uses robust optimization with recourse to determine operating rules for power system entities such as generators and storage units. These rules, or policies, establish several hours in advance how these entities are to respond to errors in the prediction of loads and renewable infeeds once their values are discovered. Affine policies consist of a nominal power schedule plus a series of planned linear modifications that depend on the prediction errors that will become known at future times. We describe how to choose optimal affine policies that respect the power network constraints, namely matching supply and demand, respecting transmission line ratings, and the local operating limits of power system entities, for all realizations of the prediction errors. Crucially, these policies are time-coupled, exploiting the spatial and temporal correlation of these prediction errors. Affine policies are compared with existing reserve operation under standard modelling assumptions, and operating cost reductions are reported for a multi-day benchmark study featuring a poorly-predicted wind infeed.Efficient prices for such "policy-based reserves" are derived, and we propose new reserve products that could be traded on electricity markets.

This paper considers the efficient operation of shared mobility systems via the combination of intelligent routing decisions for staff-based vehicle redistribution and real-time price incentives for customers. The approach is applied to London's Barclays Cycle Hire scheme, which the authors have simulated based on historical data. Using model-based predictive control principles, dynamically varying rewards are computed and offered to customers carrying out journeys, based on the current and predicted state of the system. The aim is to encourage them to park bicycles at nearby underused stations, thereby reducing the expected cost of redistributing them using dedicated staff. In parallel, routing directions for redistribution staff are periodically recomputed using a model-based heuristic. It is shown that it is possible to trade off reward payouts to customers against the cost of hiring staff to redistribute bicycles, in order to minimize operating costs for a given desired service level.Index Terms-Bicycle sharing, dynamic pricing, dynamic vehicle routing, model predictive control (MPC), shared mobility systems.

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We describe an approximate dynamic programming approach to compute lower bounds on the optimal value function for a discrete time, continuous space, infinite horizon setting. The approach iteratively constructs a family of lower bounding approximate value functions by using the so-called Bellman inequality. The novelty of our approach is that, at each iteration, we aim to compute an approximate value function that maximizes the point-wise maximum taken with the family of approximate value functions computed thus far. This leads to a non-convex objective, and we propose a gradient ascent algorithm to find stationary points by solving a sequence of convex optimization problems. We provide convergence guarantees for our algorithm and an interpretation for how the gradient computation relates to the state relevance weighting parameter appearing in related approximate dynamic programming approaches. We demonstrate through numerical examples that, when compared to existing approaches, the algorithm we propose computes tighter suboptimality bounds with less computation time. Paul N. Beuchat received the B.Eng. degree in mechanical engineering and B.Sc. in physics from the University of Melbourne, Australia, in 2008, and the M.Sc. degree in robotics, systems and control from ETH Zürich, Switzerland, in 2014, where he is currently working towards the Ph.D degree at the Automatic Control Laboratory. From 2009-2012 he was as a subsurface engineer for ExxonMobil. His research interests are control and optimization of large scale systems, with a focus towards developing approximate dynamic programming techniques for applications in the areas of building control, and coordinated flight.

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