4.2 Individual Economics: Linear Optimization Model
4.2.2 Optimization Model
The EV customer is modeled as a cost-minimizing entity which shifts his charg-ing times to the time slots of lower prices or ranks accordcharg-ing to the given (price) signal. The model considers the trips as mandatory constraints that have to be fulfilled. This means in particular that if the driving profile is feasible with the specified EV all trips are accounted for. In addition, the maximum driving speed of the vehicle the charging duration and the maximum battery capacity of the
ve-Table 4.1: Technical specifications of the BMW Mini E (BMW, 2009)
Parameter Description Value Unit
Power Consumption 0.14 (kWh/km)
Max Range 250 (km)
Top Speed 152 (km/h)
Maximum C-Rate 13 (1/h)
Full Charging Time (at 10.5 kW) 3 [h]
Storage Capacity (Usable) 35 (31.5) (kWh)
Charging Efficiency 93 (%)
Effective Charging Power 10.5 (kW)
hicle are considered. A first version of the model is first defined in (Dietz et al., 2011). The model is extended with respect to the objective function in the fol-lowing sections. In addition the existing building blocks concerning the battery modeling are described for reasons of completeness.
Assumptions and Parameters
The model builds on the following assumptions about the behavior of the EV customers and the availability of information which is employed for the opti-mization approach and the time frame of the analysis:
• People continue to use EVs like ICE vehicles.
• Time variable tariffs with hourly changing prices or charging signals are available to EV customers.
• Driving patterns and prices are ex-ante known for one week.
• EVs have an automated charging control device that calculates optimal charging times.
• The fulfillment of the mobility profile is guaranteed, even when charging times are shifted, under consideration of technical constraints.
• EVs are price takers and do not influence prices by their demand.
• EVs can only be charged at the owners home, and are plugged in as soon as they arrive there.
• The battery of the EV has to be fully charged at the beginning and the end of each week (SOC continuity).
• The optimization period is one week.
Table 4.2: Model Parameters
Model Parameter Overview Symbol Unit/Domain Usable capacity of the storage device C (kWh) Min. number of time slots to fully charge νc (#)
Charging efficiency ηc (%)
Storage cost ψ (EUR/kWh)
Price per energy unit in time step t pt (EUR/kWh) Charge parameter for time slot t ϕt (%)
Energy consumption in time slot ta dt (kWh) Energy level of the battery at time t Lt (kWh)
Rank of hour t rt (1 - 8760)
Location of the EV zt (0: not at home
1: at home)
a dt=kilometers driven in time step t (km) · power consumption per km (kWh/km)
Mathematical Description of Simple Charging
The first and most straight forward charging strategy is Simple or (as further de-noted) As Fast As Possible (AFAP) Charging. This strategy does not consider ex-ternal factors, but only the demand implied by the driven distance and specific energy consumption. The strategy thus recharges whenever this is possible (e.g.
the vehicle is at the home charging location) and can be formalized as follows:
ϕt=
1 : if SOCt+ C
νc ≤C and zt =1
C−SOCt C νc
: if SOCt+ νCc >C and zt =1 0 : otherwise
(4.1)
The costs resulting from this charging strategy are described as follows:
Cost=
∑
T t=1pt· C νc· ηc· ϕt
| {z }
Electricity Costs
+ C
νc· ψ · ϕt
| {z }
Battery Usage Costs
(4.2)
Cost=
∑
T t=1pt· C νc· ηc· ϕt
| {z }
Electricity Costs
(4.3)
The first term in the cost function in Equation 4.2 is due to the variable costs that are incurred for the purchase of (driving) electricity. The second term repre-sents the costs due to the usage of the battery storage. For the following analysis storage costs that are due to the energy throughput in the battery are not
con-sidered in the comparison of the operative costs between the different charging strategies, as the energy amount is equivalent for all charging strategies in the presented problem formulation. Thus the storage costs do not affect the oper-ative and in particular electricity purchase costs, since they are similar for all strategies (cf. Equation 4.3). The battery usage costs do vary between sociode-mographic groups. Section 4.2.7 will discuss the resulting weekly costs for differ-ent battery cost levels. In addition Section 4.3 will investigate the role of battery degradation costs related to energy throughput and charging power in more de-tail. The goals of this section is to provide insight on the individual operational costs of different charging strategies that will be described in more detail in the next paragraph.
Objective Function Smart Charging (SC)
The objective function of Smart Charging is to minimize the costs incurred, given a price for each time step of the optimization horizon of one week.
minϕ →Cost=
∑
T t=1pt· C νc· ηc· ϕt
| {z }
Electricity Costs
(4.4)
The term in the objective function corresponds to the operative formalization of AFAP charging, but this time the objective function value is minimized. When Smart Charging is compared with AFAP charging, the battery usage term can be neglected without loss of generality, as the storage cost and the total energy amount are the same for both strategies, assuming linear battery costs. The only deviation between the strategies thus occurs for the energy costs. This difference is caused by the shifting of charging times in the Smart Charging scheme. Other costs, like investment costs for the vehicle, are not considered in this approach since the focus is on operative decisions. Following the objective function the constraints in the following paragraph also apply.
Constraints
Equation 4.5 states that the SOC of the battery can not be higher than the actual capacity C and not lower than zero. The SOC is equal to the SOC in the previous time slot plus the amount of energy that has been charged into the battery minus the energy discharged for driving purposes. Equation 4.7 states that the amount
of energy charged into the battery is equal to the demand during the simulation period. This also implies that the battery is fully charged at the beginning and end of each week in the analysis.
C≥Lt−1+ C
νc· ϕt−dt
| {z }
SOCt
≥0, ∀t∈ [2, T]
(4.5)
C≥L1+ C
νc · ϕ1−d1
| {z }
SOC1
≥0, t=1
(4.6)
∑
T t=1C νc· ϕt =
∑
T t=1dt, ∀t∈ [1, T] (4.7)
pt, rt, dt, C, ηc, νc, ψ≥0, ∀t∈ [1, T] (4.8)
ϕt ∈ [0, 1]and ϕt≤zt, ∀t∈ [1, T] (4.9)
zt =
( 1 : EV at home within time step t
0 : otherwise (4.10)
t∈ [1, T] (4.11)
The simulation period is one week with T being 672 time slots, one time slot for every 15 minutes in one week. The analysis time frame is one year, consisting of 52 weeks and a total of 364 days, with data from 2007.