Several aspects can be considered to manage the EVs charging in a distribution system, depending on the charging habits of vehicle owners and the automation level of the network. The following EVs charging conditions are established:
• It is assumed that EVs’ owners arrive at their homes at 18:00 and departure the next day at 07:00. During this frame of time the EVs are plugged to the network. Nevertheless, arrival time and departure time of EVs are described by probability distributions, hence, EVs do not arrive exactly at 18:00 or departure at 07:00. Therefore, the study period of this problem, known as T, will be from 17:00 to 08:00, in order to encompass the EVs that eventually arrive before 18:00 and/or departure after 07:00, depending on their arrival and departure times.
is previously defined by the network operator as a charging policy to be complied by the EVs’ users. Before the beginning of period T, the vehicle’s owner is encouraged to choose the priority subperiod under which his/her EV will be recharged, taking into account that the EV battery will be totally recharged during the chosen subperiod. Additionally, the selection of the EV charging subperiod depends on the level of urgency of the vehicle’s owner for its availability. Therefore, if it is required that the EV be recharged as soon as possible, high priority degree should be chosen, otherwise, another recharge subperiod may be chosen with a more favourable energy price. Another aspect consists in the variation of the starting time of high priority subperiod, which can be a little before 18:00, depending on the EV arrival time. For the low priority subperiod, the ending time can be a little after 07:00, depending on the departure time of the EV.
18:00 19:00 20:00 21:00 22:00 23:00 24:00 01:00 02:00 03:00 04:00 05:00 06:00 07:00
High
Priority MediumPriority PriorityLow
17:00 08:00
Figure 3.1: Priority subperiods for the EV charging
• The power delivered to EVs is controlled for each time period t (set to 5 min in this work), which implies the existence of a remote communication between the distribution utility and the EV charging infrastructure.
• Each node of the system is able to recharge just one EV, taking into account that its priority is to supply the demand connected to the node (as a residential load), regardless the energy required by the EV.
• EVs’ batteries have a State of Charge SOC determined by a probability distribution. For all EVs the minimum State of Charge SOCmin is 20%.
The nomenclature for indices, sets, parameters and variables involved in the proposed mathematical model are presented as follows:
Indices
l Network branch
n Network node
t Time interval of period T Sets
nl Number of network branches nn Number of network nodes
nt Number of time intervals within period T Parameters
C(t) Energy price during time interval t [U S$/kW h] δ(t) Duration of time interval t
Ed(n) Distance travelled by the EV [km]
Ev(n) Energy to be transferred to the EV
Evr(n) Range of the battery [km]
I(l)max Maximum current of branch l
iSOC(n) Initial state of charge of EV’s battery at the node n
K(n,t) Priority charging factor at node n during time interval t, in case an EV is connected to node n
η Penalty factor in case an EV was not fully recharged Pgmax Maximum power generated at the nodes
Pvmax Maximum power of EVs
R(l) Resistance of branch l SOCmin Minimum state of charge
Vmax Maximum voltage magnitude at the nodes
Vmin Minimum voltage magnitude at the nodes
Ω(l) Reactance of branch l Z(l) Impedance of branch l Variables
fq(l, t) Reactive power flow through line l during time interval t
i(l, t)sqr Square of the current flowing through branch l in time interval t
pd(n, t) Active power demanded by conventional loads at node n during time interval t
pg(n, t) Active power generated at node n during time interval t
qg(n, t) Reactive power generated at node n during time interval t
φ(n)) Missing energy for the full recharge of the EV at node n at the end of the study period T
pv(n, t) Active power demanded by the EV at node n in time interval t
qd(n, t) Reactive power demanded by conventional loads at node n during time interval t
SOC(n, t) State of charge of the EV connected at node n during time interval t tarr(n) Arrival time of the EV at node n
tdep(n) Departure time of the EV at node n
v(n, t)sqr Square of the voltage at node n during time interval t
The mathematical model presented as follows, is based on (Franco et al.), and involves several aspects of a coordinated smart EVs charging in a distribution system:
f = nt P t=1 nl P l=1 [C (t) δ (t)] [R (l) i (l, t)sqr] + nt P t=1 nn P n=1 [C (t) δ (t)] [pd(n, t) + pv(n, t)]− nt P t=1 nn P n=1 [δ (t) pv(n, t) K (n, t)] + η nn P n=1 ϕ (n, t)2 (3.1) s.t. nl P l=1 [fp(l, t)in] + pg(n, t) = pd(n, t) + pv(n, t)+ nl P l=1
[fp(l, t)out+ R (l)outi (l, t)sqrout] ∀n = 1, 2, ..., nn ∀t = 1, 2, ..., nt
(3.2) nl P l=1 [fq(l, t)in] + qg(n, t) = qd(n, t)+ nl P l=1
[fq(l, t)out+ Ω (l)outi (l, t)sqrout] ∀n = 1, 2, ..., nn ∀t = 1, 2, ..., nt
v (ns, t) sqr − v (nr, t) sqr = Z (l)2i (l, t)sqr+ +2 [R (l) fp(l, t) + Ω (l) fq(l, t)] ∀l = 1, 2, ..., nl ∀t = 1, 2, ..., nt (3.4) v (ns, t)sqri (l, t)sqr= fp(l, t)2+ fq(l, t)2 ∀l = 1, 2, ..., nl ∀t = 1, 2, ..., nt (3.5) nt X t=1 [δ (t) pv(n, t)] + ϕ (n) + iSOC(n) = Ev(n) ∀n = 1, 2, ..., nn (3.6) SOC(n, t) = SOC(n, t − 1) + δ (t) pv(n, t) ∀n = 1, 2, ..., nn ∀t = 1, 2, ..., nt (3.7) Vmin2 ≤ v (n, t)sqr≤ Vmax2 (3.8) 0 ≤ i (l, t)sqr ≤ I (l)2max (3.9) 0 ≤ pv(n, t) ≤ Pv max (3.10) 0 ≤ pg(n, t) ≤ Pg max (3.11)
Equation 3.1 represents a cost function which involves four terms. The first term is the energy losses cost of the system; the second term represents the energy cost drawn by the EVs and conventional loads (residential loads). An incentive cost by the EVs recharge is established in the third term considering the priority degree of recharge, and the last term corresponds to a penalty cost when one or several EVs are not fully recharged at the end of the study period T (composed by the nt subperiods). Model constraints are described by
3.2 to 3.7. Equations 3.2 and 3.3 represent the active and reactive power balances at each node. Voltage drops at the nodes and currents through the lines are implied in 3.4 and 3.5
respectively. Notice that nodes ns and nr are the send and receive nodes of the line l under
study. Equation 3.6 relates the total battery energy, the energy drawn by the EV in each time interval t of the study period T , and the missing battery energy in case this is not fully recharged. In 3.7 the SOC for the EV is computed for each time period t, taking into account the SOC of the preceding interval. The limits of nodal voltages, currents through the lines and maximum recharge power of EVs are set in 3.8 to 3.10 respectively. Equation3.11
limits the generated power at node n in a time interval t, which means that the distribution system under study may contain distributed generation. The subscripts in and out denote the flow entering and leaving a node respectively. Although the mathematical model has as output variable the recharge power of each EV for each time period, this is, pv(n, t), the optimal recharge rate is the summation of all the contributions of power delivered to the EVs, describing quantitatively the impact on the entire system.