In the smart grid, the intent is to use flexibility in demand, both to balance demand and supply as well as to resolve potential congestion. A first prominent example of such flexible demand is the charging of electric vehicles, which do not necessarily need to be charged as soon as they are plugged in. The problem of optimally scheduling the charging demand of electric vehicles within the constraints of the electricity infrastructure is called the charge scheduling problem. The models of the charging speed, horizon, and charging demand determine the computational complexity of the charge scheduling problem. For about 20 variants, we show, using a dynamic programming approach, that the problem is either in P or weakly NP-hard. We also show that about 10 variants of the problem are strongly NP-hard, presenting a potentially significant obstacle to their use in practical situations of scale.

In power systems, demand and supply always have to be balanced. This is becoming more challenging due to the sustained penetration of renewable energy sources and their inherit uncertain production. Because of the increasing amount of electrical vehicles (EVs), and the high capacity and flexibility of their charging process, EVs are a good candidate for providing balancing services to electric systems. We propose a stochastic optimization method for an EV aggregator that models the uncertainty of the imbalance price, the reserve prices and the probability of acceptance and deployment of reserves. The model results in an optimal charging and discharging strategy considering dayahead purchase, imbalance trading and reserve bids. Unlike previous studies, the reserve bids consists of both a quantity and an optimal price using a novel efficient formulation for the price bids. Experimental evaluation shows that the proposed stochastic optimization method results in lower costs than determin-istic and quantity-only bid solutions.

In between transportation services, trains are parked and maintained at shunting yards. The conflict-free routing of trains to and on these yards and the scheduling of service and maintenance tasks is known as the train unit shunting and service problem. Efficient use of the capacity of these yards is becoming increasingly important, because of increasing numbers of trains without proportional extensions of the yards. Efficiently scheduling maintenance activities is extremely challenging: currently only heuristics succeed in finding solutions to the integrated problem at all. Bounds are needed to determine the quality of these heuristics, and also to support investment decisions on increasing the yard capacity. For this, a complete algorithm for a possibly relaxed problem model is required. We analyze the potential of extending the model for multi-agent path finding to be used for such a relaxation.

Due to increasing numbers of intermittent and distributed generators in power systems, there is an increasing need for demand responses to maintain the balance between electricity generation and use at all times. For example, the electrification of transportation significantly adds to the amount of flexible electricity demand. Several methods have been developed to schedule such flexible energy consumption. However, an objective way of comparing these methods is lacking, especially when decisions are made based on incomplete information which is repeatedly updated. This paper presents a new benchmarking framework designed to bridge this gap. Surveys that classify flexibility planning algorithms were an input to define this benchmarking standard. The benchmarking framework can be used for different objectives and under diverse conditions faced by electricity production stakeholders interested in flexibility scheduling algorithms. Our contribution was implemented in a software toolbox providing a simulation environment that captures the evolution of look-ahead information, which enables comparing online planning and scheduling algorithms. This toolbox includes seven planning algorithms. This paper includes two case studies measuring the performances of these algorithms under uncertain market conditions. These case studies illustrate the importance of online decision making, the influence of data quality on the performance of the algorithms, the benefit of using robust and stochastic programming approaches, and the necessity of trustworthy benchmarking.