The number of electric vehicles is increasing at a worldwide level. For instance, 12 000 electric cars were sold in the US country in January 2017, accounting for approximately 1% of the total US auto sales. This results into a Year over Year (YoY) increase of about 59% [1]. Similarly, the Chinese market had more than 32 000 new electric cars along the streets in March 2017, which corresponds to an 89% increase over the same month last year, with the YoY increase now at 31% [2]. Numbers in Europe are contradictory, as the penetration level strongly depends on the specific country. For instance, new registrations for battery powered cars increased in the first quarter of 2017 from 23703 to 32607 in the EU, Norway and Switzerland [3]. In particular, countries in Northern Europe are leading the way (e.g., UK, Netherlands, Scandinavian countries). For instance, Norway has the highest per capita number of all-electric cars in the world: more than 100 000 in a country of 5.2 million people [4].

The main reason behind such numbers is undoubtedly the growing concerns over environmental issues. To address them, governments have exerted a regulatory pressure to reduce urban pollution, in most cases through appropriate incentives to purchase electric vehicles, and to facilitate the creation of a charging infrastructure. The pressure is also expected to further increase the upcoming years, as some countries are taking even more strict measures. For instance, Paris, Madrid, Athens and Mexico City are planning to ban the most polluting cars and vans by 2025 to tackle air pollution, while they promote electric vehicles and cleaner transport [5]. Such restrictive measures are expected to further increase the penetration level of electric vehicles, and some projections (e.g., Bloomberg [6]) predict that electric vehicles might account for 35% of all new vehicle sales (worldwide) in 2040. Obviously, this implies that much larger penetration levels may be reached in specific areas.

The previous numbers and predictions are obviously creating some concerns regarding the ability of the power grid to accommodate such a new electric load. On the other hand, electric vehicles may be also regarded as an opportunity, as at the same time they might not just take electricity from the grid (G2V, as the power flows from the Grid to the Vehicles), but also occasionally inject power in the grid (V2G). In this sense, a fleet of electric vehicles plugged into the grid may be also regarded as a possible virtual battery connected to the grid.

Accordingly, the next sections investigate the mathematical problems arising in this context. In doing so, we shall mainly follow the recent reference [7]. Also, we shall intend the notion of Electric Vehicles (EVs) in the broadest sense, i.e., by EVs we shall both refer to Fully Electric Vehicles (FEVs) and to Plug-in Hybrid EVs (PHEVs).

Charging of Electric Vehicles

The charging problem can be modelled as (1):


where   is the power drawn by the i’th vehicle at time step k. If we assume that n(k) are plugged to the power grid at the same time, then the power drawn by the population of the fleet of vehicles cannot exceed a maximum value (i.e., ) that depends on the total power available for charging. Such maximum threshold depends itself on many variables, as on the amount of other loads connected to the same bus, the topology and the constraints of the power grid, possibly the power locally generated from renewable sources. At the same time, the second equation of (1) shows that the power drawn by the i’th vehicle cannot take arbitrary values, but is set to lie on a predetermined set of feasible charging powers. At this regard, it is possible to distinguish three main categories:

  1. On-off charging, where the value of  can either be 0 (no charging) or equal to the maximum value (e.g., 4kW in the case of standard single phase outlets). Examples in this direction can be found in [8-9].
  2.  Multi-level charging, where the value of  can be chosen among of a set of possible values. For instance, the vehicle may be charged in a fast mode at off-peak times, while it may be charged in a slow mode at on-peak times. Other intermediate charging speeds may be possible as well to further accomplish a larger variety of charging solutions.
  3. Continuously adaptable charging, when the value of  can continuously adjust its power rate, within the feasible set (i.e., between 0 – no charging -  and a maximum value given by the power rate of the outlet).

Note that the optimisation problem (1) can be interpreted as a resource allocation problem, where the shared resource (i.e., maximum power) must be shared among a set of vehicles. Note that in principle many different ways may be devised to share the power among the vehicles. While an obvious solution may be to give the same power to all vehicles, problem (1) becomes most interesting when one optimally shares the available power in order to further minimise a cost function of interest, i.e. (2),

In the optimisation problem (2), we further have that the cost functions  indicate some cost associated with receiving a given instantaneous power . In most cases; this may refer to the price for charging, to the overall time required for charging; or the inconvenience for the users associated with a given power; or the inconvenience for the grid (i.e., in terms of its overall availability to serve the requested total demand for charging); or may measure some CO2 emissions associated with a given charging (e.g., depending on whether power generated from renewable sources is available or not for charging EVs). Obviously, problem (2) may be also formulated as a maximisation problem, in the case one assumes that the functions   represent some utility of interest (e.g., revenues).

A number of algorithms have been proposed in the literature to solve problem (2). Different solutions may be categorised as:

Problem extensions

The optimisation problem (2) can be further refined and made more realistic along a number of different lines. In particular, the following problems are of interest in the related literature:



[1], last accessed: May 2017.

[2], last accessed: May 2017.

[3], last accessed: May 2017.

[4], last accessed: May 2017.

[5], last accessed: May 2017.

[6], last accessed: May 2017.

[7] E. Crisostomi, R. Shorten, F. Wirth and S. Stüdli, “Electric and Plug-in Hybrid Vehicle Networks: Optimization and Control”, CRC Press – Taylor & Francis, Series: Automation and Control Engineering, 2017.

[8] S. Stüdli, R.H. Middleton, and J.H. Braslavsky, “A fixed-structure automaton for load management of electric vehicles”. In Proc. IEEE European Control Conference (ECC), pages 3566-3571, Zurich, Switzerland,

July 2013.

[9] L. Yao, W. H. Lim, and T. S. Tsai, “A Real-Time Charging Scheme for Demand Response in Electric Vehicle Parking Station”, IEEE Transactions on Smart Grid, vol. 8, no. 1, pp. 52-62, 2017.

[10] S. Deilami, A.S. Masoum, P.S. Moses, and M.A.S. Masoum, “Real-time coordination of plug-in electric vehicle charging in smart grids to minimize power losses and improve voltage profile”, IEEE Transactions on

Smart Grid, vol. 2, no. 3, pp. 456-467, 2011.

[11] K. Clement-Nyns, E. Haesen, and J. Driesen, “The impact of charging plug-in hybrid electric vehicles on a residential distribution grid”, IEEE Transactions on Power Systems, vol. 25, no. 1, pp. 371-380, 2010.

[12] W. Su and M.-Y. Chow, “Performance evaluation of an EDA-based largescale plug-in hybrid electric vehicle charging algorithm”, IEEE Transactions on Smart Grid, vol. 3, no. 1, pp. 308-315, 2012.

[13] J.A. Peças Lopes, F.J. Soares, and P.M. Rocha Almeida, “Integration of electric vehicles in the electric power system”, Proceedings of the IEEE, vol. 99, no. 1, pp. 168-183, 2011.

[14] D. Wu, D.C. Aliprantis, and L. Ying, “Load scheduling and dispatch for aggregators of plug-in electric vehicles”, IEEE Transactions on Smart Grid, vol. 3, no. 1, pp. 368-376, March 2012.

[15] S. Stüdli, E. Crisostomi, R. Middleton, and R. Shorten, “A flexible distributed framework for realising electric and plug-in hybrid vehicle charging policies”, International Journal of Control, vol. 85, no. 8, pp. 1130-1145, 2012.

[16] E.L. Karfopoulos, K.A. Panourgias, and N.D. Hatziargyriou, “Distributed coordination of electric vehicles providing V2G regulation services”, IEEE Transactions on Power Systems, vol. 31, no. 4, pp. 2834-2846, 2016.

[17] L. Gan, U. Topcu, and S.H. Low, “Optimal decentralized protocol for electric vehicle charging”, IEEE Transactions on power systems, vol. 28, no. 2, pp. 940-951, 2013.

[18] S. Grammatico,Exponentially convergent decentralized charging control for large populations of plug-in electric vehicles”, in Proc. IEEE Conference on Decision and Control (CDC), pp. 5775-5780, Las Vegas, Nevada, December 2016.

[19] M. Gonzales Vaya, S. Grammatico, G. Andersson, and J. Lygeros, “On the price of being selfish in large populations of plug-in electric vehicles”, 54th IEEE Conference on Decision and Control (CDC), pp. 6542-6547, Osaka, Japan, December 2015.

[20] S. Grammatico, F. Parise, M. Colombino, and J. Lygeros, “Decentralized convergence to Nash equilibria in constrained deterministic mean field control,” IEEE Transactions on Automatic Control, vol. 61, no. 11, pp. 3315-3329, 2016.

[21] S. Zou, Z. Ma, X. Liu, and I. Hiskens, “An Efficient Game for Coordinating Electric Vehicle Charging”, IEEE Transactions on Automatic Control, vol. 62, no. 5, pp. 2374-2389, 2017.

[22] S. Stüdli, E. Crisostomi, R. Middleton, and R. Shorten. “Optimal real-time distributed V2G and G2V management of electric vehicles”, International Journal of Control, vol. 87, no. 6, pp. 1153-1162, 2014.

[23] Y. Ota, H. Taniguchi, J. Baba, and A. Yokoyama, “Implementation of autonomous distributed V2G to electric vehicle and DC charging system”, Electric Power Systems Research, vol. 120, pp. 177-183, March 2015.

[24] E. Ancillotti, R. Bruno, E. Crisostomi and M. Tucci, “Using Electric Vehicles to Improve Building Energy Sustainability”, in Proc. of IEEE International Electric Vehicle Conference (IEVC), Florence, Italy, 2014.

[25] D. Wu, H. Zeng, C. Lu, and B. Boule, “Two-Stage Energy Management for Office Buildings With Workplace EV Charging and Renewable Energy”, IEEE Transactions on Transportation Electrification, vol. 3, no. 1, pp. 225-237, 2017.

[26] C. Pang, P. Dutta, and M. Kezunovic, “BEVs/PHEVs as Dispersed Energy Storage for V2B Uses in the Smart Grid”, IEEE Transactions on Smart Grid, vol. 3, no. 1, pp. 473-482, 2012.

[27] H.K. Nguyen, and J.B. Song, “Optimal Charging and Discharging for Multiple PHEVs with Demand Side Management in Vehicle-to-Building”, IEEE Journal of Communications and Networks, vol. 14, no. 6, pp. 662-671, 2012.

[28] J. de Hoog, T. Alpcan, M. Brazil, D.A. Thomas, and I. Mareels, “Optimal charging of electric vehicles taking distribution network constraints into account”, IEEE Transactions on Power Systems, vol. 30, no. 1, pp. 365-375, 2015.

[29] G. Deconinck, K. De Craemer, and B. Claessens, “Combining Market-Based Control with Distribution Grid Constraints when Coordinating Electric Vehicle Charging”, Engineering, vol. 1, no. 4, pp. 453-465, 2015.

[30] A. Y. Saber and G. K. Venayagamoorthy, “Resource scheduling under uncertainty in a smart grid with renewables and plug-in vehicles”, IEEE Systems Journal, vol. 6, no. 1, pp. 103-109, March 2012.

[31] E. B. Iversen, J.M. Morales, and H. Madsen, “Optimal charging of an electric vehicle using a Markov decision process”, Applied Energy, vol. 123, pp. 1-12, 2014.

[32] S. Zao, X. Lin, and M. Chen, “Robust Online Algorithms for Peak-Minimizing EV Charging under Multi-Stage Uncertainty”, IEEE Transactions on Automatic Control, accepted for publication, 2017.


Dr Emanuele Crisostomi, University of Pisa