The Onset of Congestion in Charging of Electric Vehicles for Proportionally Fair Network Management Protocol
With the expected uptake of electric vehicles in the near future, we are likely to observe overloading in the local distribution networks more frequently. Such development suggests that a congestion management protocol will be a crucial component of futur
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1 Introduction With the expected uptake of electric vehicles in the near future, we are likely to observe overloading in the local distribution networks more frequently. Such development suggests that a congestion management protocol will be a crucial component of future technological innovations in low voltage networks. An important property of a suitable network capacity management protocol is to balance network efficiency and fairness requirements. Assuming a stochastic model, we study the proportional fairness (PF) protocol managing the network capacity in charging of electric vehicles. We explore the onset of congestion by analysing the critical arrival rate, i.e. the largest possible vehicle arrival rate that can still be fully satisfied by the network. We compare the proportionally fair management protocol with the max-flow (MF) management protocol. By numerical simulations on realistic networks, we show that proportional fairness leads not only to more equitable distribution of power allocations, but it can also serve slightly larger arrival rate of vehicles. We consider simplified setup, where the power allocations are dependent on the occupation of network nodes, but they are independent of the exact number of vehicles, and to validate numerical results, we analyse the critical arrival rate on a network with two edges, where the optimal power allocations can be calculated analytically.
2 Optimization Model We model the electrical distribution network as a directed rooted tree graph composed of the node set V and edge set E. Only the root node of the tree r ∈ V injects the power into the network and electric vehicles can be plugged into all other nodes. By ˇ Buzna (B) L. University of Žilina, Univerzitná 8215/1, 01026 žilina, Slovakia e-mail: [email protected] © Springer International Publishing Switzerland 2017 K.F. Dœrner et al. (eds.), Operations Research Proceedings 2015, Operations Research Proceedings, DOI 10.1007/978-3-319-42902-1_13
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the symbol ( j) we denote the subtree rooted in the node j ∈ V. An edge ei j ∈ E connects node i to node j, where i is closer to the root than j, and is characterised by the impedance Z i j = Ri j + i X i j , where Ri j is the edge resistance and X i j the edge reactance. The power loss along edge ei j is given by Si j (t) = Pi j (t) + i Q i j (t), where Pi j (t) is the real power loss, and Q i j (t) the reactive power loss. We model car batteries as elastic loads (i.e. able to absorb any value of power they are allocated). Electric vehicle l = 1, . . . , N (t) receives only active power Pl (t), where N (t) is the number of vehicles charging at time t. Value Δil (t) is one if electric vehicle l is charging on node i and zero otherwise. Vehicle l derives a utility Ul (Pl (t)) from the allocated charging power Pl (t). Let P( j) denote the active power, and Q ( j) the reactive power consumed by the subtree ( j) that include power consumed by all vehicles connected to the subtree and power losses dissipated on edges of the subtree. By the s
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