Two operational scenarios for EV’s battery charging are considered in the application of the proposed energy loss allocation method:

#### 4.1. Scenario 1: Slow Charging at Off-Peak Load Conditions

In this case, slow battery charging stations operate from 00:00 to 08:00 with the five levels of penetration defined above. The optimization problem stated in Equations (

21)–(

23) was scripted in Matlab (version R2017, v.9.2) and solved by means of the

fmincon tool. The parameters of the admittance matrix were taken from OpenDSS simulation tool [

30].

When the solution algorithm converges, the state of the system for each level

$k=0,\dots ,5$ is given by

${x}_{t}^{k}$ =

$[{V}_{1,t}^{1},\dots ,{V}_{21,t}^{3};{\theta}_{1,t}^{1},\dots ,{\theta}_{21,t}^{3}]$ for

$t=1,\dots ,24$. Thereafter, power losses

$\Delta {P}_{t}$ per hour and per level are evaluated by Equation (

12) for each state of the system result

${x}_{t}^{k}$ for

$k=0,\dots ,5$. Level 0 corresponds to a grid operation with no EV penetration. Levels 1–5 correspond to the EV penetration from 5% to 25% in total daily consumed energy by EVs with respect to overall PQ load consumption.

The 24-h power loss curves by each level for the connection of EV loads under off-peak conditions are depicted in

Figure 3. Total real energy losses

$\Delta W$ to be allocated among network users is evaluated by EV penetration level using Equation (

9) and results are depicted in

Figure 4. This figure also shows the resultant load and loss factor. Load and loss factors are defined as the ratio between average and maximum values of demands and losses, respectively.

Results reveal how the progressive integration of slow charging stations at off-peak load conditions produces a flattening effect of the load curve. The load factor increased from 0.62 at level 0 to 0.77. However, the loss factor also increased from 0.48 to 0.61. This means that 24-h power loss curve is also becoming flat. As result energy losses rose in magnitude from 1.24 MW·h/day, 1.05% (level 0) to 1.70 MW·h/day, 1.08% (level 5). This result is important since despite energy losses grew almost 50% in magnitude, the relative energy losses remains constant around 1.05–1.08%.

The absolute value of Marginal Loss Coefficients

$\left|\right|ML{C}_{Di,t}^{p}\left|\right|$ are directly obtained for at each level

k, each bus

i, phase

p and time

t from

fmincon results via Lagrange multipliers as indicated in

Section 2.3.1.

Figure 5 displays the MLCs curves when EVs are charging at off-peak time. The MLCs are applied to agents connected at bus 21 along 24 h. Note that under lower EV penetration (5%, red curve), MLCs observed between 01:00 and 09:00 are significantly lower than ones achieved between 10:00 and 22:00. For a high EV penetration (25%, orange curve), the MLCs obtained between 01:00 and 09:00 are similar to those achieved between 10:00 and 22:00 when no EV is connected (around 0.02–0.03 along the day). Then, under EV charging at off-peak load conditions, the pattern of MLCs is somehow flat, similar to a uniform marginal coefficient. This uniform coefficient produces similar results of a roll-in embedded method applied to recover the power losses.

Figure 6 depicts a complete pattern for calculated MLCs by location and by time for each penetration level. It is worth to note in all phases that MLCs associated to high EV penetration (Level 5, 25%) cover more area (in time and location) than MLCs produced by lower levels.

Table 3 lists the general results for the allocated energy losses for aggregate PQ and EV loads under off-peak condition. The reconciliation factor

${k}_{r}$ was around 0.5 in all levels. Equations (

18) and (

28) were applied for the marginal and pro rata procedure, respectively. Energy losses range from 1339.08 kW·h/day at level 0 to 1696.64 kW·h/day at Level 5. At level 0, EVs do not exist then all losses are assigned to PQ loads. Regarding the allocation results, three facts can be highlighted:

EV charging stations operating under off-peak conditions and marginal loss allocation do not pay for additional energy losses. The marginal procedure assigns lower losses to EV than expected under a pro-rata procedure. This means that EV loads reach a small benefit by their produced losses at off-peak conditions. In fact, PQ loads do no take advantage of the marginal procedure being slightly penalized (they should pay for 1389 kW·h/day with respect to 1372 kW·h/day under the proportional approach).

Pro rata and marginal methods can produce a similar output when EV charging stations are operating under off-peak conditions. The share of energy losses attributable to EV loads (18%) are similar in both approaches: marginal and pro rata. This means that the MLCs are acting as a uniform factor capable to recover the cost of losses.

The EV share of losses is lesser than the EV share of consumption. For instance, at level 5 the ratio between EV and PQ loads consumption is 25%. The EV share of losses is lesser, 18%.

Payment for energy losses by EV location are calculated in a monthly basis using Equations (

19) and (

29) for marginal and pro rata procedure, respectively. Considering a flat energy price

$\rho $ of 0.05 USD/kW·h, left-hand chart of

Figure 7 shows how the marginal procedure penalize the slow EV charging stations connected from bus 15 to bus 21. A similar effect is also seen in PQ loads (right-hand chart of

Figure 7) connected from bus 15 to bus 21. In this scenario, marginal procedure is applying higher charges to loads (EV and PQ) connected at the end of the line.

Figure 7 also indicates that the application of MLCs for loads (EV and PQ) connected near to the origin have a lesser responsibility in the coverage of the entire energy losses.

#### 4.2. Scenario 2: Fast Charging at Peak Load Conditions

In this scenario, EV connection was implemented at peak load conditions: from 18:00 to 21:00 with a 7.5 kW charging station considering the same five levels of integration applied in the case of EV charging at peak load conditions (Scenario 1), that is 200, 400, 600, 800 and 1000 units until reach a penetration of 25% of base energy consumption along one day.

The 24-h power loss curves by each level for the connection of EV loads under peak conditions are depicted in

Figure 8. It is clear that the load curve becomes more sharp due to the progressive incorporation of slow EV charging stations.

Total real energy losses

$\Delta W$ to be allocated for each level among network users are indicated in

Figure 9. Unlike Scenario 1, results show how the progressive integration of fast charging stations at peak load conditions produces a significative distortion effect of the load curve. EVs are charging only from 17:00 to 22:00. Then, the load factor decreased from 0.62 at Level 0 to 0.42 at Level 5. In this case, average demand does not grow in the same extent than the maximum value. As a result, the load factor falls. The loss factor also fall from 0.48 to 0.26 at Level 5. This means that energy losses drastically rose in magnitude from 1.24 MW·h/day, 1.05% (Level 0) to 2.30 MW·h/day, 1.8% (level 5). In this circumstance, the effects of EV charging stations at peak load condition are too harsh.

In

Figure 10, the MLCs curves by EV penetration level at bus 21 along 24-h period is presented for the peak load conditions. It should be noted how marginal coefficients are able to reach high values 0.07 at peak time (18:00 and 21:00).

Figure 11 displays the MLCs curves when EVs are charging at 20:00. The MLCs are applied to agents connected from bus 2 to bus 21. As the load is increasing with the distance, the MLC magnitude at each bus also grows with the distance with respect to the reference bus. Then, closer loads to reference produce lower losses (and lower MLCs) than farther loads and therefore loads connected near to substation pay less for power losses than farthest loads.

In

Table 4, the energy allocation results for aggregate PQ and EV loads are presented for the peak load condition. The reconciliation factor

${k}_{r}$ also fluctuates around 0.5 in all levels. Equations (

18) and (

28) were applied for the marginal and pro rata procedure, respectively.

Unlike Scenario 1 where aggregate EV loads collect some marginal benefits due to the flattering effect over the load curve, Scenario 2 displays severe charges against EV loads due to energy losses associated with fast charging at peak conditions in all levels. If the marginal procedure is applied, PQ load should assume only a small part of the additional losses (23%). At Level 5, additional loses are 949 kW·h/day and PQ loads have to pay for 224 = 1563−1339 kW·h/day. Otherwise, if the pro rata procedure is applied, PQ loads should cover 46% of the incremental loads observed between Level 0 and Level 5.

At Scenario 1 (EVs are charging at peak load conditions) pro rata and marginal allocation results lead to similar pattern. However, at Scenario 2 (EVs are charging at peak load conditions) marginal and pro rata loss allocation produce dissimilar results. EVs must pay for additional energy losses. The marginal procedure assigns higher losses to EV than calculated by the pro-rata procedure. This means that EV loads are duly charged by their produced losses at peak conditions. In this case, PQ loads take advantage of the marginal procedure since they have not to pay for additional losses. The share of energy losses attributable to EV loads under marginal approach (32%) is significantly higher than the share obtained by the pro rata procedure (19%). If we consider a flat energy price

$\rho $ of 0.05 USD/kW·h, the left-hand chart of

Figure 12 shows how the marginal procedure strongly penalize EVs connected from the middle to the end of the circuit. Note how EVs connected from bus 9 to bus 21 are facing high charges due to increasing losses. Conversely, the right-hand chart of

Figure 12 visualizes how the marginal and pro rata procedures yield in similar charges. This means that there is not significative economical difference for PQ charges but strong incentives to EV loads to perform power loss reduction tasks.

#### 4.3. The Economical Effects in a Single EV Unit Under Off-Peak and Peak Load Conditions

Consider now the perspective of a single EV of 30kW·h capacity when Level 5 is reached (25% of penetration). If a fixed energy price $\rho $ of 0.05 USD/kW·h is considered, the overall charging or energy cost for the EV is 1.5 USD/day. At off-peak and peak conditions, the connection of a single EV unit has different outcomes.

On the one hand, under off-peak load conditions (slow charging from 01:00 to 09:00), when the EV is connected at bus 21, phase 1 (ending node) the payment for losses under marginal procedure is almost 0.43 USD/day. This amount corresponds to 29% of total payment for energy (1.5 USD/day). On the other hand, under peak load conditions (fast charging from 19:00 to 21:00) the payment for losses under marginal procedure is 0.64 USD/day (54% of total payment for energy). This result is important since the best economic solution for the EV is charging under off-peak conditions.

Consider now that the EV is connected at bus 2, phase 1 (very close to substation). In this case, both scenarios show the same result, the EV has to pay only 0.03 USD/day (2% of total payment for energy). This charge is very low when compared with charges applied to loads at the end of the feeder. Then, the incentive is to connect EVs as close as possible to substation since no additional losses are produced.

Economic results for the marginal allocation procedure evidence EV loads connected at farthest loads have to pay important shares due to incremental losses becoming an important incentive (mainly at peak conditions) to provide network support. Under standard pro rata approach, the overall cost is distributed among all loads in a proportional manner and no incentive is provided by time of use and location of the EV charger.

#### 4.4. The Impact of the EV Load Modeling on Loss Allocation Results

All results presented above were obtained assuming a specific EV load parametrization:

a = 0.9537,

b = 0.0463, and

$\alpha $ = −2.324 in Equation (

3). To evaluate the effects of the EV load model in the results, we ran the model under peak loading conditions (Level 5) varying

$\alpha $ from 0 (PQ load) to −8.0 and

b from 0.0 (PQ load) to 0.10. Results of the sensitivity analysis are depicted in

Figure 13.

If the EV load is regarded as constant PQ (

$\alpha $ = 0,

b = 0.0), the marginal loss factor at 20:00, bus 21, Phase 3 (

$ML{C}_{D21,20:00}^{3}$) is 0.014401769 and total losses to be allocated

$\Delta W$ is 2284.12 kWh/day. Conversely, if the EV parameters take a non-linear form

$\alpha $ = −8.0,

b = 0.1, the marginal loss factor at 20:00, bus 21, Phase 3 (

$ML{C}_{D21,20:00}^{3}$) is 0.014904362 and total losses to be allocated

$\Delta W$ increased 2317.50 kW·h/day. These variations on MLCs and energy losses represent 3.5% and 1.5% of the values achieved when loads were assumed as PQ constant, respectively. As a result, we observe significant differences on loss factors and overall energy losses to be allocated among the network users. The adoption of a correct EV load model for economic evaluation of the impacts on losses becomes an important issue to consider to guarantee the fairness of the allocation procedure. As there are several charging protocols of EV batteries [

37], future research on economical impacts of EVs on system losses should be devoted to include more detailed models.