With the rapid advancement of new energy technologies and increasing environmental requirements, electric cars have gradually replaced traditional fuel vehicles, gaining a larger market share. However, the large-scale disordered charging of electric cars has led to significant peak-valley differences in the power grid and insufficient distribution transformer capacity. This necessitates effective management of the charging and discharging processes of large-scale electric cars to enhance the stability of distribution grid operation and better support the integration of massive electric cars into the grid. In China, the proliferation of China EV has intensified these challenges, making it crucial to develop coordinated control strategies. This study designs a coordinated charging and discharging control method for electric car groups based on the Logit protocol dynamic game, aiming to optimize grid performance and economic benefits.
In the initial stage of evolutionary game theory, the electric car population randomly selects M strategies. The strategy set φ includes all operational parameters of electric cars across different time periods, expressed as:
$$ \phi = \{ s_1, s_2, \dots, s_m, \dots, s_M \} $$
$$ s_m = \{ E_{m,1}, E_{m,2}, \dots, E_{m,H} \} $$
Here, \( s_m \) represents the m-th strategy established randomly, and \( E_{m,H} \) denotes the charging and discharging power during period H for the m-th strategy. Using the Logit protocol, we set up an evolutionary game model modification protocol and establish the conditional transition probability as follows:
$$ \rho_{i}^{m,k} [p_i(t)] = \frac{ \exp \left( \frac{f_k^i(t)}{\theta} \right)^{-1} }{ \sum_{n=1}^{M} \exp \left( \frac{f_m^i(t)}{\theta} \right)^{-1} } $$
In this equation, \( \theta \) represents the noise level. Under the dynamic pricing of electric car aggregators, we establish a simulation model with the energy cost of electric car aggregators as the optimization objective. The two-stage game model for power demand response scheduling is illustrated in the process flow below.
The management area includes 2,000 electric cars, each with a battery capacity of 30 kWh. The electric car aggregator can handle a maximum load of 10 MW. The charging and discharging efficiency of the charging piles is set at 95%. The time-of-use electricity price parameters for electric cars are summarized in Table 1.
| Item | Time Period | Price (yuan/kWh) |
|---|---|---|
| Peak | 08:00–11:00, 15:00–21:00 | 1.178 |
| Off-Peak | 12:00–13:00, 23:00–07:00 | 0.425 |
| Normal | 07:00–08:00, 11:00–12:00, 13:00–15:00, 21:00–23:00 | 0.775 |
To evaluate the peak-shaving and valley-filling effects, we analyze load peak-valley differences and load variance. Two pricing scenarios are considered: Scenario 1, where the electric car aggregator leads and electric cars follow to compute charging and discharging prices, and Scenario 2, where the electric car aggregator sets prices based on real-time electric car load through population game theory. The calculation results for load indicators under various scheduling scenarios and methods are presented in Table 2.
| Parameter | Load Peak-Valley Difference (MW) | Load Variance (MW²) |
|---|---|---|
| Base Load | 48.63 | 263.35 |
| Disordered Charging | 50.06 | 258.94 |
| Ordered Charging | 49.27 | 243.61 |
| Scenario 1 ANN | 47.16 | 222.24 |
| Scenario 1 PSO | 49.26 | 209.67 |
| Scenario 1 Evolutionary Game | 41.22 | 182.45 |
| Scenario 2 ANN | 53.24 | 211.58 |
| Scenario 2 PSO | 49.15 | 202.64 |
| Scenario 2 Evolutionary Game | 39.62 | 178.58 |
By comparing the test results in Table 2, we assess the peak-shaving and valley-filling performance of electric cars. In Scenario 1, the electric car aggregator and electric cars adjust the charging and discharging control process to ensure that electric cars charge under conditions lower than the grid’s time-of-use electricity prices. This regulates the charging and discharging of electric cars, enabling more electric cars to perform control functions and effectively reduce the grid’s load peak-valley difference and variance. In Scenario 2, electric cars establish lower-cost charging and discharging strategies through evolutionary game theory, further reducing the charging and discharging electricity prices set by electric car aggregators. This ensures that the electric car optimization scheduling process obtains higher electricity quantities, allowing Scenario 2 to further reduce load peak-valley differences and load variances compared to Scenario 1. The widespread adoption of China EV highlights the importance of such optimizations.
For economic analysis, the Nash equilibrium solution determined by the two-stage game model achieves a balance between electric car charging and the power costs of electric car aggregators, thereby identifying the optimal electric car charging and discharging scheme. The economic results under various scenarios and scheduling methods are shown in Table 3.
| Parameter | Total Cost of Electric Cars (10,000 yuan) | Net Income of Electric Car Aggregator (10,000 yuan) |
|---|---|---|
| Disordered Charging | 2.25 | — |
| Ordered Charging | 1.78 | — |
| Scenario 1 ANN | 1.41 | 0.33 |
| Scenario 1 PSO | 1.26 | 0.31 |
| Scenario 1 Evolutionary Game | 0.89 | 0.29 |
| Scenario 2 PSO | 0.81 | 0.36 |
| Scenario 2 PSO | 0.73 | 0.37 |
| Scenario 2 Evolutionary Game | 0.52 | 0.33 |
According to Table 3, in Scenario 2, electric cars achieve the lowest total cost, while the electric car aggregator attains the highest net income. This indicates that after evolutionary game theory, electric cars obtain an ideal charging and discharging control scheme, which further reduces the charging cost of electric car aggregators. When entering the peak-shaving scheduling process, subsidies can be provided through discharging, effectively reducing charging costs. On the other hand, optimizing the charging and discharging scheduling scheme for electric cars can also reduce the energy costs of electric car aggregators, effectively increasing their conversion profits. The growth of the electric car market in China, particularly with China EV, underscores the significance of these economic benefits.
The evolutionary game model based on the Logit protocol involves a dynamic process where strategies evolve over time. The fitness function for each strategy can be defined as:
$$ f_m^i(t) = -C_m^i(t) + \beta \cdot B_m^i(t) $$
Here, \( C_m^i(t) \) is the cost for electric car i using strategy m at time t, \( B_m^i(t) \) is the benefit, and \( \beta \) is a weighting factor. The evolution of strategy proportions follows the discrete dynamic equation:
$$ p_m(t+1) = \frac{ p_m(t) \cdot \exp \left( \frac{f_m(t)}{\theta} \right) }{ \sum_{n=1}^{M} p_n(t) \cdot \exp \left( \frac{f_n(t)}{\theta} \right) } $$
This equation ensures that strategies with higher fitness are more likely to be adopted in the next time step. The noise parameter \( \theta \) controls the level of randomness in strategy selection; a smaller \( \theta \) makes the process more deterministic, while a larger \( \theta \) introduces more exploration.
In the context of China EV, the integration of large-scale electric cars into the grid requires sophisticated models to handle uncertainties. The charging and discharging power for each electric car can be modeled as a function of time and state of charge (SOC). Let \( SOC_i(t) \) be the state of charge of electric car i at time t, and \( P_i^{ch}(t) \) and \( P_i^{dis}(t) \) be the charging and discharging powers, respectively. The SOC update equation is:
$$ SOC_i(t+1) = SOC_i(t) + \frac{ \eta^{ch} P_i^{ch}(t) – \frac{ P_i^{dis}(t) }{ \eta^{dis} } }{ E_i^{max} } \Delta t $$
where \( \eta^{ch} \) and \( \eta^{dis} \) are charging and discharging efficiencies, \( E_i^{max} \) is the maximum battery capacity, and \( \Delta t \) is the time step. The constraints include:
$$ 0 \leq P_i^{ch}(t) \leq P_i^{ch,max} $$
$$ 0 \leq P_i^{dis}(t) \leq P_i^{dis,max} $$
$$ SOC_i^{min} \leq SOC_i(t) \leq SOC_i^{max} $$
These constraints ensure that the charging and discharging powers remain within feasible limits and the SOC stays within safe bounds. For the electric car aggregator, the total load at time t is the sum of all individual loads:
$$ L_{total}(t) = \sum_{i=1}^{N} \left( P_i^{ch}(t) – P_i^{dis}(t) \right) $$
The aggregator’s cost function includes energy purchase cost and any incentives provided to electric cars. Let \( C_{grid}(t) \) be the grid electricity price at time t, and \( I_i(t) \) be the incentive paid to electric car i. Then, the aggregator’s cost is:
$$ C_{agg}(t) = C_{grid}(t) L_{total}(t) + \sum_{i=1}^{N} I_i(t) $$
The objective of the aggregator is to minimize this cost over the scheduling horizon, while electric cars aim to minimize their own costs, which include electricity costs and incentives:
$$ C_i(t) = C_{grid}(t) \left( P_i^{ch}(t) – P_i^{dis}(t) \right) – I_i(t) $$
This leads to a game-theoretic formulation where the aggregator and electric cars interact strategically. The Logit protocol dynamic game provides a framework to model this interaction and converge to an equilibrium solution.
In practice, the implementation of such coordinated charging and discharging for electric car groups can significantly enhance grid stability and reduce operational costs. For China EV, this approach aligns with national goals of promoting clean energy and reducing carbon emissions. The use of evolutionary game theory allows for adaptive strategies that respond to changing grid conditions and user behaviors.
In conclusion, the study on coordinated charging and discharging of electric car groups based on the Logit protocol dynamic game yields the following beneficial results: First, it ensures that electric cars charge under conditions lower than the grid’s time-of-use electricity prices, securing higher electricity quantities during the optimization scheduling process, and achieving the effect of regulating electric car charging and discharging, thereby effectively reducing the grid’s load peak-valley difference and variance. Second, the electric car aggregator attains the highest net income; after evolutionary game theory, electric cars obtain an ideal charging and discharging control scheme, reducing the charging cost of electric car aggregators and effectively increasing their conversion profits. This research contributes to improving the charging and discharging efficiency of electric car groups and holds significant value for industrial energy saving and emission reduction, particularly in the context of China EV development.
Future work could explore the integration of renewable energy sources with electric car charging, consider more complex user behaviors, and extend the game-theoretic models to multi-agent systems. The continuous growth of the electric car market, especially in China, will require ongoing research to optimize grid integration and maximize benefits for all stakeholders.