In this section, a model is presented to manage the charge and discharge of electric vehicles in the distribution network. In this model, the multi-objective optimization problem is designed to improve the network parameters and also the energy cost management in the distribution network. The objective functions of the problem are designed to simultaneously manage and navigate several basic network parameters.
The objective function of the optimization problem
The four main objectives, in the proposed management plan, are considered as follows:
- Minimizing the cost of energy supply needed for network load on the planning horizon
- Minimizing the cost of network losses in the planning horizon
- Energy management of exchanges between electric vehicles and the upstream network in the planning horizon
- Minimizing the deviation of the voltage from the nominal value in the planning horizon
The first two objectives are to manage the distribution network economically. The demand response program is used as an efficient tool to reduce the cost of energy supply. The third objective is to manage the penetration of electric vehicles in the distribution network and to minimize the cost of supplying the required energy to the first two objective functions. Finally, the fourth objective function is designed to improve the buses voltage as one of the basic parameters of the network and, as far as possible, to reach their nominal value. The objective function of the proposed model is expressed as follows.
Where,
is the time scheduling stage index, and are the bus indexes, is the electric vehicle index, is the phase index studied in the three-phase distribution network, is the time horizon studied set, is the network bus set, is the magnitude of nominal voltage domain, is the power price, is the active network demand response, is the loss rate of network active power, is the active exchange power with electric vehicle, is the network voltage, and P is the active power passing through the line.
The first part of the objective function represents the minimization of the voltage deviation of all buses at all simulation times (planning horizon) and for all the network phases of nominal value. The second part of the objective function addresses the cost of providing active load and the cost of network losses. The price element ρ is considered as a variable at different times. As energy prices change or the voltage of the buses diverges from the nominal value, according to the demand response strategy, some or all of the network loads respond to these changes and change their active and reactive power. It should be noted that the price of losses per kilowatt-hour is equal to the cost of load per kilowatt-hour. The third part of the objective is to model the purchase price of electricity by electric vehicles. are the objective function weighting coefficients, which are selected according to the operator's preference for operating the distribution network. The larger weight factor is multiplied into any part of the objective function which has higher priority
Constraints and limitations of the proposed model
The optimal response will be valid if it observes the network constraints. In other words, due to the limitations of the power transmission in the network and the technical constraints of different equipments, the optimization problem should be solved and the final solution must be extracted.
Power flow constraints
The equilibrium of produced and consumed power is the most important principle of stability in the power system. At this constraint, at each bus, the summation of inline power, load power, and upstream network injected power must be zero. The active and reactive power which passes from the bus to the bus is calculated by the following equation:
where
are
the rows of the
-th and
-th columns of the conductance and suspension matrix, respectively, and
Q is the reactive power passing through the line.
According to the above equations, in the proposed model, the summation of the injective power to the bus from all the lines connected to this bus is calculated by the following equations:
According to the equilibrium power constraint per bus, we have:
In the above equations, the equilibrium of power injected bylines and posts and consumed by the load (electric vehicles and loads) in each bus is evaluated. The power consumed per bus includes the load demanded and the power consumption of the electric vehicle. It should be noted that all network loads participate in the demand response program. is the total active power per network bus, is the total reactive power per network bus, is injectable power from substation per bus (upstream network injected power), and is reactive power injectable from substation per bus (injectable through Upstream network).
where is the phase confluence matrix of the electric vehicle and is the matrix of the confluence of the electric vehicle and bus.
Electric vehicle constraints
Electric vehicle constraints fall into three categories:
- Limitation of the interchange power of the vehicle with the network
- Technical limitation of vehicle battery
- Energy-related restrictions on vehicle batteries
The first category examines the interchange power between the vehicle and the network and the vehicle-network converter. According to the Balance Principle (KCL), the summation of exchanges between vehicle and network and the losses between vehicle and network converter must be zero. So we have:
where is the injected power by the network to the vehicle (converter input) and is the minus loss power at the converter output (electric vehicle battery injection). This schematic is if the electric vehicle is in the charge state and if it is in the discharge state. So at every hour for every vehicle we have:
Due to the efficiency between the vehicle and the network, its rate of losses is calculated as follows:
Depending on the power exchange infrastructure between the battery and the network, the maximum charge and discharge will be managed by the following equation:
In order to indicate the status of the charge and discharge of electric vehicles, binary variables are considered.
The limitations for the energy balance of an electric vehicle are as follows:
In other words, the energy stored in the battery is equal to the primary energy in the battery (when the vehicle is connected to the network) plus the energy exchanged with the network. The minimum energy stored in the battery is also limited as follows:
Demand response program constraints
As explained in the previous section, two types of demand response programs including cost-sensitive and voltage-sensitive models are considered in the proposed model.
According to the above equations, active loads are involved in both load response (voltage-sensitive and cost-sensitive) programs. In this formula, k1, k2, and k3 are binary variables. If the load is involved in the demand response program, then k3 is equal to one, otherwise, it is equal to 0. If k1 is equal to one then the charge is cost-sensitive and if k2 is equal to one, it is voltage-sensitive. According to the following equation, reactive power is only voltage-responsive.
The values of α and β are determined by the type of load according to the reference [21]. The following equations manage the minimum and maximum participation of loads in the demand response program.
Network technical constraints
The network constraints which include the capacity of the lines and the voltage limitations of the buses are expressed in the following equations:
The values of and are calculated by equations (2) and (3). The constraint of the voltage difference between the phases is applied to avoid instability and unbalance reduction in the proposed scheme. In other words, the voltage difference between the three phases A, B and C in the three-phase distribution network is managed as follows:
Finally, according to the objective function, the amount of loss, active and reactive load of the whole system is calculated by the following equations:
Simulation and numerical results
In order to evaluate the effectiveness of the proposed approach, the model is implemented on GAMS software on the low voltage distribution network and evaluated by various tests described later.
Investigation of the effect of EVs penetration into the network without the proposed strategy
In this test, after connecting to the network, the EVs perform the charging process (G2V) immediately at maximum charge rate and they are fully charged after 4 hours based on the battery specifications. The number of vehicles connected to the network per hour is shown in Fig. 5. This test was conducted regardless of the smart charging and discharging strategy, and ignoring demand response.
The network power demand curve of the upstream network for power consumption is illustrated by changing the EV penetration level to values of 0%, 25% and 50%.
According to this figure, unmanaged charging of EVs increases the network peak and consequently increases the power demand during peak hours of the upstream network. According to the network capacity limit (upstream post), lines and buses voltage, with a 50% penetration level of electric vehicles, i.e 75 vehicles are connected to the network, the apparent power of the network reaches nominal power of 8 pu. In this situation, further vehicle connection is impossible. On the other hand, the network is at risk, and no capacity is left to cover any possible increase in load, thus the reliability of the network is reduced. Therefore, because of the significant increase in peak load, without a smart and controlled charging strategy, it is impossible to connect half of the vehicles. In order to connect these new loads, upgrading the network is needed, which requires considerable time and costs.
Naturally, as the peak load power increases, the bus voltage drop also increases relative to the EV-free mode. The effect of increasing the level of penetration of EVs in the network on the buses voltage drop is shown in Fig. 7. In this figure, bus number 1 is the reference bus and its voltage value is 1pu. The increase in voltage drop in this form is quite obvious.
Since the load distribution is unbalanced between different phases, the voltage drop of each phase in each bus varies from other phases. In other words, there is an imbalance between phases. The voltage drop of the different phases in each bus is shown in the Fig. 8. According to this figure, the worst case of voltage drop is related to bus 36, which is the farthest bus from the power supply, as shown in the Table 4 of values. This bus is considered in the following section as an indicator to investigate the effect of charge control of EVs on voltage drop.
Table 4
Voltage difference of bus 36 phases in different diffusion coefficients of EVs
Voltage difference | Penetration 0% | Penetration25% | Penetration 50% |
AB | 2.1 | 1.7 | 1.3 |
AC | 5.21 | 5.3 | 5.7 |
BC | 3.3 | 3.7 | 4.5 |
The amount of losses in the network is also shown in Table 5 for 24 hours. As expected, with the increase in loads due to the charging of EVs and consequently the increase in the line passing current, losses have also increased significantly.
Table 5
Losses (kwh) in the network at different diffusion coefficients of EVs
Penetration 0% | Penetration 25% | Penetration 50% |
39 | 47.8 | 58 |
The network was evaluated technically in the above description. Economically, according to Fig. 3, the cost of power supply needed by the network in 24 hours based on the level of EV penetration is presented in theTable 6. Certainly, as the cost of purchasing energy is high during peak hours of consumption, the cost of supply has also increased considerably with the increase in peak demand.
Table 6
cost of electricity ($) to supply the network load per day understudy of different penetration coefficients of EVs
0% penetration | 25% penetration | penetration 50% |
198.54 | 214.36 | 230.6 |
Investigating the effect of EVs penetration into the network without charge control strategy and with regard to demand response management
In this test, the effect of demand response management program on the network performance is evaluated. Since this study applies two types of voltage-sensitive and cost-sensitive loads, this section is divided into three sub-sections to investigate the effectiveness of each demand management program. In the first section, only voltage-sensitive loads are considered. In the second sub-section, price-sensitive loads and in the third sub-section, both types of loads are considered. The effect of each management strategy on the network parameters is discussed below.
Voltage- sensitive loads management
In this case, the coefficients k1 and k2 are considered to be 0 and 1 respectively. In this section, reactive and active loads are considered as decision variables in the optimization problem. The power demand variations of the network respect to the different penetration levels are shown in Fig. 9.
According to the figure, the management of voltage-sensitive loads lonely does not have much impact on the release of micro-grid capacity and the increase in EVs penetration coefficient. Managing these loads reduces the apparent network power slightly at all hours, and consequently increases the EVs penetration to 4%. Lack of network capacity for connecting all the electric vehicles, lack of free network capacity, and dramatically increased peak load are still evident. Therefore, managing these loads does not affect the daily load curve greatly.
Fig. 10 shows the effect of management of all this loads on the voltage drop of the network. According to this figure, the management of these loads has reduced the voltage drop of the network.
The voltage difference between the different phases is expressed in bus 36 in Table 7 and network losses is expressed in Table 8.
Table 7
Voltage difference of bus 36 phases in different diffusion coefficients of EVs
Voltage difference | Penetration 0% | penetration 25% | Penetration 50% | Penetration 54% |
AB | 2.1 | 1.7 | 1.3 | 1.3 |
AC | 5.21 | 4.1 | 5.3 | 5.3 |
BC | 3.3 | 2.4 | 4.1 | 4.1 |
Table 8
Losses (kwh) in the network at different diffusion coefficients of EVs
penetration 0% | penetration 50% | penetration 54% |
37 | 54 | 55 |
In order to better evaluate the effectiveness of voltage-sensitive load management, a comparison between the results of the previous section with this section at a constant penetration level of 50% for EV is presented in Table 9. According to this table, peak load power has been reduced by 2.5%, which has increased the EV penetration rate by 4%. On the other hand, network losses have been dropped by about 7 percent. Buses voltage drop and the voltage difference between phases, which reflects network imbalance, have also been reduced.
Cost-sensitive loads management
In this case the coefficients k1 and k2 are considered as 1 and 0, respectively. In this subsection, active load and energy price are considered as decision variables in the optimization problem. The management of the power demand variations of the network with respect to different penetration levels is shown in Fig. 11.
Table 9
Improvement in percentage of network parameters at 50% constant penetration coefficient
Title | Voltage difference between phase A and C (V) | Network bus voltage (pu) | Network losses (kWh) | Power (pu) |
Without demand response | 5.6 | 0.924 | 58 | 8 |
With demand response | 5.2 | 0.927 | 54 | 7.8 |
Improvement percentage | 7.15 | 0.32 | 7 | 2.5 |
According to this figure, the management of cost-sensitive loads has increased the penetration rate of electric vehicles up to 22%. This increased penetration states the efficiency of the proposed management approach on the reduction of peak load. However, the network still needs to be developed to supply all 150 electric vehicles, and the surplus power of the network, to cope with the possible power increase, which is equal to 0.
Fig. 12 shows the effect of the management of these loads on the voltage drop of the network. According to this figure, the management of these loads has reduced the voltage drop of the network. This reduction is less than the previous state (voltage-sensitive load management).
The voltage difference between the different phases is expressed in bus 36 in Table 10 and the network losses is expressed in Table 11.
Table 10
Voltage difference of bus 36 phases in different diffusion coefficients of EVs
Voltage difference | penetration 0% | penetration 25% | penetration 50% | penetration 73% |
AB | 2.1 | 1.3 | 0.4 | 0.5 |
AC | 5.21 | 4.1 | 4.9 | 5.7 |
BC | 3.3 | 2.9 | 4.5 | 5.3 |
Table 11
Losses (kwh) in the network at different DIFFUSION coefficients of EVs
0% penetration | penetration 50% | penetration 73% |
36 | 54 | 64 |
In order to perform a better evaluation of the efficiency constant of voltage-sensitive load management, a comparison between the results of the previous section with this section at a penetration level of 50% for EV is presented in Table 12. According to this table, peak load power has been reduced by about 12%, which has increased the EV penetration coefficient by 22%. This reduction of the peak is remarkable. Network losses, on the other hand, decrease to about 7 percent. Buses voltage drop and the voltage difference between phases which causes network unbalance have also been reduced.
Table 12
Improvement percent of network parameters at 50% constant penetration coefficient
Title | Voltage difference between phase A and C (V) | Network bus voltage (pu) | Network losses (kWh) | Power (pu) |
Without demand response | 5.6 | 0.924 | 58 | 8 |
With demand response | 4.8 | 0.93 | 54 | 7 |
Improvement percent | 14 | 0.76 | 7 | 11.85 |
Since the influence of electric vehicles has a great impact on peak load and on the other hand, the cost of the power supply is high during peak hours, the management of cost-sensitive loads, in comparison with voltage-sensitive loads, is much more effective on improving network parameters.
Simultaneous management of cost -sensitive and voltage- sensitive loads
In this case, the coefficients k1 and k2 are both equal to 1. Here, active load, reactive load, and energy price are considered as decision variables in the optimization problem. According to different penetration levels, managing power demand variations of the network according to different penetration levels is shown in Fig. 13.
According to this figure, the management of cost and voltage-sensitive loads has increased the penetration rate of electric vehicles up to 25%. This increase in penetration is higher than the previous two cases, and indicating DR's effectiveness in significantly reducing peak load. However, the network still must be expanded to supply all 150 electric vehicles, while the network surplus capacity to cope with potential increases is zero.
Fig. 14 shows the effect of the management of these loads on the voltage drop of the network. According to this figure, the management of these loads has reduced the voltage drop of the network. This reduction is less than the previous ones (managing voltage-sensitive and cost-sensitive loads alone).
The voltage difference between the different phases is stated in bus 36 in Table 13 and the network losses are stated in Table 14.
Table 13
Voltage difference of bus 36 phases in different diffusion coefficients of EVs
Voltage difference | Penetration 0% | Penetration 25% | Penetration 75% |
AB | 1.2 | 0.8 | 0.14 |
AC | 3.7 | 4.4 | 5.2 |
BC | 2.4 | 4 | 4.80 |
Table 14
Losses (kwh) in the network at different diffusion coefficients of EVs
0% penetration | 25% penetration | penetration 75% |
33 | 50 | 61 |
In Table 15, a comparison between the results of the previous section with this section at a constant penetration level of 50% for EV is performed to evaluate the effectiveness of managing voltage-sensitive loads. According to this table, peak load power has been reduced by about 14%, which has increased the EV penetration rate by 25%. This reduction in peak is considerable. Network losses, on the other hand, have decreased to 13 percent. This demonstrates the remarkable impact of load management on reducing network losses. Buses voltage drop and voltage difference between phases which indicate network unbalancing have also been reduced so that the voltage imbalance between different phases is improved up to 21%.
Table 15
Improvement percent of network parameters at 50% constant diffusion coefficient
Title | Voltage difference between phase A and C (V) | Network bus Voltage (pu) | Network losses (kWh) | Power (pu) |
Without demand response | 5.6 | 0.924 | 58 | 8 |
With demand response | 4.4 | 0.93 | 50 | 6.9 |
Improvement percentage | 21 | 0.76 | 13 | 13.35 |
As it can be seen from the results of this section, it is true that the DR program is very effective in improving network parameters but it alone cannot manage this level of penetration of EVs in the network. In the above description, the network was evaluated technically. Economically, according to the Fig. 3, the cost of power supply needed by the network in 24 hours based on the level of EV penetration is presented in Table 16. As the cost of purchasing energy is high during peak hours of consumption, due to transfer some part of the load from low-load to full-load, the cost of load supply is also reduced by applying a demand response management strategy.
Table 16
Cost of electricity ($) to supply the grid load per day under study of different diffusion coefficients of EVs
| Penetration 0% | Penetration 25% | Penetration 75% |
Without demand response | 198.54 | 360.6 | - |
With demand response | 188 | 223 | 242 |
Improvement percentage | 10 | 137 | - |
Investigating the effect of Evs penetration on the network by the proposed charge control strategy along with demand response management |
This test considers the strategy of charging and discharging electric vehicles along with managing loads. According to the proposed charging strategy, vehicles can be charged and discharged, which means they operate in V2G and G2V and they do not charge as soon as they are connected to the network. Charging and discharging operations are managed based on the network parameters and set of objectives in the previous section. Due to the high energy price during peak hours of consumption and increased losses and voltage drop due to high demand, the control strategy in these hours is to discharge operations and to charge vehicles during low load hours.
It should be noted that charging and discharging operations depend on the vehicle being connected to the network. Therefore, the vehicle may not be possible to charge in some hours of the day despite the low-load network, because it is not connected to the network. As explained in the previous section, the study is based on the assumption that the vehicles leave the network between 5 am and 10 am and they navigate the route at this time.
In addition to the charging and discharging strategy, price and voltage- sensitive loads have also been used to improve the network parameters in this test. Therefore, the decision variables in this test, in addition to the active and reactive loads, are also the active power stored in the EVs battery.
Given the 100% penetration coefficient (150 vehicles), the total number of vehicles connected to the network for different hours of the day is shown in Fig. 15.
The start time of the simulation process in this test is 10 am. Since the objective function consists of three parts, voltage drop and imbalance between phases, cost of power consumption and losses and cost of energy supply of EVs, therefore, the weighting coefficients of the objective function must also be adjusted. These coefficients are set so that the amount of changes in each part of the objective function is equal to each other. In other words, the degree of importance of all three parts is the same. For this purpose, the distance between the minimum and maximum values of each part of the objective function must be equal to the other parts. The weighting coefficients are obtained from the following equations, where and are the minimum and maximum values of the objective function, respectively, and the indices 1, 2, and 3 represent the deviations of voltage, charge supply and loss cost, and energy supply cost of EVs, respectively. So, the weighting coefficients are obtained from the following equations.
According to the above explanations and relationships, the coefficients and are considered as 1, 0.12 and 0.17, respectively. The network demand load with the maximum diffusion coefficient of the EVs and considering the load response is shown in Fig. 16.
According to this figure, by applying the proposed strategy, unlike the two previous tests, it is possible to connect all vehicles (100% penetration coefficient) without imposing very high costs for network development. In addition, the effect of the proposed strategy in flattening the load curve is quite evident.
Depending on their presence, EVs charge between hours 12 and 17. They perform discharge operations in the full-load hours to reduce power costs and help improve network parameters. This has caused a reduction of the peak load consumption or clipping peak of the load curve due to EV discharge during peak hours, and valley filling due to discharge during low and intermediate load hours. By this proper management, not only is the need for network expansion to supply new loads is eliminated but also some network capacities remain empty despite the connection of all EVs. This residual capacity increases the reliability of power supply, especially in conditions of unexpected increases in demand.
Table 17
Voltage difference of 36 bus phases at 20 h (peak load) and energy losses at 100% EVs penetration coefficient
Energy losses (kWh) | Phase to Phase | Voltage difference (V) |
BC | AC | AB |
40 | 1.5 | 3 | 1.2 | |
The value of the voltage amplitude is shown in Fig. 17as a per unit in the various buses. Table 17 also shows the phases voltage and the amount of losses. According to these results, despite the 100% penetration of electric vehicles in the network and the significant load being added at different times, the rate of loss, voltage drop and imbalance between phases have decreased as peak power consumption. The amount of loss at 0% penetration rate was 39 kWh (according to the first test), with 150 vehicles connected to the network, the amount of losses was only about 2 kWh, indicating the efficiency of the proposed model in reducing losses. It should be noted that according to the first test, if the proposed strategy was not used, only 75 vehicles would increase the amount of losses by about 20 kWh.
In addition to the losses, even with 150 vehicles connected to the network, the imbalance between phases is significantly reduced compared to 0 penetration levels.
In the above description, the network was evaluated technically. Economically, according to the figure, the cost of power supply needed by the network in 24 hours for various tests and based on the maximum EV penetration level is presented in Table 18. The results of this table illustrate the effectiveness of the proposed method. Given that the proposed strategy plays an effective role in flattening the load curve (shifting the load from full load to intermediate and low load) and on the other hand, the cost of purchasing energy is high during peak hours, the cost of supplying the load will reduce a lot by 150 vehicles connected to the network and a significant increase in load, much less. This is the concept of turning the challenge into an opportunity. The results showed that by properly managing the penetration of electric vehicles along with responsive loads, not only the network parameters were not compromised and the penetration of all the vehicles was managed without the need for network development, but also by applying the proposed strategy, the network parameters were improved.
Table 18
Comparison of cost and maximum penetration rate of EVs and maximum peak load in different tests
Without demand response program (First test) | Amount of the network peak (pu) | Cost of energy supply ($) | Maximum penetration rate |
Alone with the proposed response program (Second Test, Part 3) | 8 | 530.6 | 50% 75 EVs |
With the proposed strategy (charge and discharge management and responsive demands) (Third Test) | 8 | 242 | 75% 112 EVs |
Without demand response program ((First Test | 5.05 | 185 | 100% 150 EVs |