Shortest feasible paths with charging stops for battery electric vehicles

Abstract

We study the problem of minimizing overall trip time for battery electric vehicles in road networks. As battery capacity is limited, stops at charging stations may be inevitable. Careful route planning is crucial, since charging stations are scarce and recharging is time-consuming. We extend the Constrained Shortest Path problem for electric vehicles with realistic models of charging stops, including varying charging power and battery swapping stations. While the resulting problem is N P-hard, we propose a combination of algorithmic techniques to achieve good performance in practice. Extensive experimental evaluation shows that our approach (CHArge) enables computation of optimal solutions on realistic inputs, even of continental scale. Finally, we investigate heuristic variants of CHArge that derive high-quality routes in well below a second on sensible instances. Related Work. Classic route planning approaches apply Dijkstra's algorithm [18] to a graph representation of the transportation network, using fixed scalar arc weights corresponding to, e. g., driving time. For faster queries, speedup techniques have been proposed, with different benefits in terms of preprocessing time and space, query speed, and simplicity [2]. A* search [32] uses vertex potentials to guide the search towards the target. A successful variant, ALT (A*, Landmarks, Triangle Inequality) [27], obtains good potentials from precomputed distances to selected landmark vertices. Contraction Hierarchies (CH) [24], on the other hand, iteratively contract vertices in increasing order of (heuristic) importance during preprocessing, maintaining distances between all remaining vertices by adding shortcut arcs where necessary. The CH query is then bidirectional, starting from source and target, and proceeds only from less important to more important vertices. Combining both techniques, Core-ALT [4] contracts all but the most important vertices (e. g., the top 1 %), performing ALT on the remaining core graph. This approach can also be extended to more complex scenarios, such as edge constraints to model, e. g., maximum allowed vehicle height or weight [23]. More recently, techniques (including variants of CH and ALT) were introduced that allow an additional customization after preprocessing, to account for dynamic or user-dependent metrics [11,17,19]. Also, approaches towards extended scenarios exist, such as batched shortest paths [12] or shortest via paths [1,14]. Here, a common approach is to make use of a relatively fast target selection phase, precomputing distances to relevant points of interest to enable faster queries. However, these techniques were only evaluated for single-criterion search, where the distance between two vertices is always a unique scalar value. For multi-criteria scenarios, on the other hand, problem complexity and solution sizes increase significantly, and practical approaches are only known for basic problem variants [21,23,56]. For a more complete overview of techniques and combinations, see Bast et al. [2].Regarding rout...

A Method for the Optimization of Daily Activity Chains Including Electric Vehicles

“…They discussed the properties of charging functions, which are used to map an initial State of Charge (SoC) and the duration of charging sessions with a resulting SoC. The works by Baum et al [34][35][36], are very close to our approach in terms of methods and calculation mechanisms that were used to deal with the incorporation of electromobility. Additionally, useful information about the charging network, charging sessions, plugs, and strategies were also extracted from the work of Moghaddass et al [37].…”

“…In their next work conducted in 2016 [35], they proposed a framework with a more holistic approach to the problem, where parameters are included, such as the use and location of charging stations, turn costs, and the adjustment of speed in order to save energy. In their work conducted in 2015 [36], they introduced an approach named CHArge, which solves the EV routing problem in realistic settings. They discussed the properties of charging functions, which are used to map an initial State of Charge (SoC) and the duration of charging sessions with a resulting SoC.…”

Modeling and Engineering Constrained Shortest Path Algorithms for Battery Electric Vehicles

“…Labels (at v) added to the priority queue are also pushed into L uns (v). Every time the minimum element of L uns (v) changes (because an element is added or extracted), we check whether the new minimum element is dominated by any settled label in L set (v) and discard it in this case [7]. Dominance is tested as follows.…”

“…that incorporates current SoC at a vertex [7]. We represent π f (v, b) with a convex, piecewise linear function that maps SoC b ∈ [0, M ] at a vertex v ∈ V to a lower bound on remaining driving time.…”

“…However, energy-optimal routes often exhibit disproportionate detours, as using minor, slow roads can save energy due to less air drag [9]. Variants of the N P-hard Constrained Shortest Path (CSP) problem [26] overcome this by minimizing energy consumption without exceeding a given time limit [37] or finding the fastest route that does not exceed battery constraints [7,41]. Yet, time-consumption tradeoffs are not only affected by choice of route but also by driving behavior.…”