In the modern power grid, the rapid proliferation of electric vehicle cars presents both challenges and opportunities. As a distributed, small-scale, and short-cycle mobile energy storage resource, electric vehicle cars can significantly impact grid stability if managed improperly. However, when aggregated effectively, these electric vehicle cars offer immense flexibility for demand response programs. In this paper, we address the critical issue of capacity allocation for electric vehicle aggregators of varying scales participating in electricity market demand response. We propose an optimization strategy for response electricity allocation that leverages Monte Carlo simulation, price negotiation mechanisms, and evolutionary game theory. Our approach ensures that distribution system operators can distribute demand response electricity reasonably based on each aggregator’s actual capacity, thereby reducing grid load fluctuations and enhancing operational stability. Throughout this work, we emphasize the role of electric vehicle cars as key distributed assets, and we will repeatedly highlight how electric vehicle car aggregation can transform energy systems.
The integration of electric vehicle cars into the power grid has accelerated globally, with millions of electric vehicle cars now on the roads. By 2060, projections suggest that the number of electric vehicle cars could reach hundreds of millions, representing a massive equivalent energy storage capacity. For instance, if each electric vehicle car is equipped with a 60 kWh battery, the aggregate storage could exceed billions of kWh, making electric vehicle cars a pivotal resource for grid flexibility. However, the inherent randomness in electric vehicle car user behavior—such as charging start times, departure times, and daily travel distances—complicates the aggregation process. This randomness necessitates robust modeling techniques to predict the available capacity of electric vehicle aggregators accurately. In our study, we focus on developing a strategy that not only accounts for these uncertainties but also optimizes the allocation of response electricity among multiple electric vehicle aggregators, ensuring that the grid benefits from the collective potential of electric vehicle cars.
To begin, we model the capacity of electric vehicle aggregators using Monte Carlo simulation. This method allows us to capture the stochastic nature of electric vehicle car user behavior. The key parameters include the charging start time, departure time, and daily travel distance, each following specific probability distributions. For the charging start time, we assume it follows a normal distribution, denoted as $N_c(\mu_i, \sigma_i)$, with the probability density function given by:
$$f(t_i) =
\begin{cases}
\frac{1}{\sigma_i \sqrt{2\pi}} \exp\left(-\frac{(t_i – \mu_i)^2}{2\sigma_i^2}\right), & \mu_i – 12 < t_i \leq 24 \\
\frac{1}{\sigma_i \sqrt{2\pi}} \exp\left(-\frac{(t_i + 24 – \mu_i)^2}{2\sigma_i^2}\right), & 0 < t_i \leq \mu_i – 12
\end{cases}$$
Here, $t_i$ represents the charging start time, $\mu_i$ is the mean, and $\sigma_i$ is the standard deviation. Similarly, the departure time follows a normal distribution $N_l(\mu_g, \sigma_g)$, with its probability density function as:
$$f(t_g) =
\begin{cases}
\frac{1}{\sigma_g \sqrt{2\pi}} \exp\left(-\frac{(t_g – 24 – \mu_g)^2}{2\sigma_g^2}\right), & \mu_g + 12 < t_g \leq 24 \\
\frac{1}{\sigma_g \sqrt{2\pi}} \exp\left(-\frac{(t_g – \mu_g)^2}{2\sigma_g^2}\right), & 0 < t_g \leq \mu_g + 12
\end{cases}$$
For the daily travel distance, we use a log-normal distribution $N_x(\mu, \sigma)$, with the probability density function:
$$f(x) = \frac{1}{x\sigma \sqrt{2\pi}} \exp\left(-\frac{(\ln x – \mu)^2}{2\sigma^2}\right)$$
where $x$ is the daily travel distance. The charging duration $T$ for an electric vehicle car is calculated based on the desired state of charge, initial state of charge, battery capacity, charging efficiency, and charging power:
$$T = \frac{(S_q – S_c) E_a}{\eta P}$$
In this equation, $S_q$ is the desired state of charge, $S_c$ is the initial state of charge, $E_a$ is the battery capacity, $\eta$ is the charging efficiency, and $P$ is the charging power. By simulating a large number of electric vehicle cars using these distributions, we can estimate the aggregate load profile for an electric vehicle aggregator. This Monte Carlo approach helps us quantify the available capacity for demand response, accounting for the randomness inherent in electric vehicle car usage. The results often show that uncontrolled charging of electric vehicle cars leads to peak load periods, such as in the evening hours, which can exacerbate grid stress. Therefore, accurately predicting this capacity is the first step toward effective optimization.
Next, we establish a price negotiation mechanism between electric vehicle aggregators and the distribution system operator. This mechanism is crucial for determining the electricity price for demand response services. Given the varying scales of electric vehicle aggregators—some managing hundreds of electric vehicle cars, others thousands—we adopt a linear bidding strategy. The bidding function for an electric vehicle aggregator $j$ is defined as:
$$P_j = P_{j}^{\text{max}} – \left(\frac{r – 1}{R – 1}\right)^{k_1} (P_{j}^{\text{max}} – P_{j}^{\text{min}})$$
where $P_j$ is the bid price, $P_{j}^{\text{max}}$ is the maximum price limit set by the DSO, $P_{j}^{\text{min}}$ is the minimum acceptable price for the aggregator, $r$ is the negotiation round (with $1 \leq r \leq R$, and $R$ being the maximum number of rounds), and $k_1$ is the bidding index that depends on the aggregator’s scale. Similarly, the DSO’s bidding function is:
$$P_{\text{grid}} = P_{\text{grid}}^{\text{min}} + \left(\frac{r – 1}{R – 1}\right)^{k_2} (P_{\text{grid}}^{\text{max}} – P_{\text{grid}}^{\text{min}})$$
Here, $P_{\text{grid}}$ is the DSO’s bid price, $P_{\text{grid}}^{\text{min}}$ is the minimum price the DSO believes the aggregator will accept, $P_{\text{grid}}^{\text{max}}$ is the DSO’s maximum price limit, and $k_2$ is the DSO’s bidding index. The bidding indices $k_1$ and $k_2$ are determined based on the scale of the electric vehicle aggregator and the aggregate capacity reported, respectively. We define the aggregator type $\text{type}_{\text{EVA}}$ as:
$$\text{type}_{\text{EVA}} = \frac{C_{\text{EVA}}}{C_{\text{base}}}$$
where $C_{\text{EVA}}$ is the capacity of the aggregator (e.g., number of electric vehicle cars), and $C_{\text{base}}$ is a baseline capacity set by the DSO. Based on this, we categorize aggregators into three types: underconfident ($\text{type}_{\text{EVA}} \leq 0.8$), moderate ($0.8 < \text{type}_{\text{EVA}} < 1.2$), and confident ($\text{type}_{\text{EVA}} \geq 1.2$). Each type corresponds to a different $k_1$ value, as shown in Table 1. This classification ensures that larger aggregators, with more electric vehicle cars, can negotiate more aggressively, reflecting their greater influence on the grid.
| Aggregator Type | $\text{type}_{\text{EVA}}$ Range | Bidding Index $k_1$ |
|---|---|---|
| Underconfident | $\text{type}_{\text{EVA}} \leq 0.8$ | $k’$ (e.g., 0.5) |
| Moderate | $0.8 < \text{type}_{\text{EVA}} < 1.2$ | $k”$ (e.g., 1.0) |
| Confident | $\text{type}_{\text{EVA}} \geq 1.2$ | $k”’$ (e.g., 2.0) |
For the DSO, the bidding index $k_2$ is derived from the average capacity reported by all aggregators, denoted as $\text{type}_G$:
$$\text{type}_G = \frac{1}{N} \sum_{i=1}^{N} \frac{C_{\text{EVA}, i}}{C_{\text{base}}}$$
where $N$ is the number of aggregators. The negotiation process proceeds in rounds until the bids converge within an acceptable error margin. We define the error function as:
$$\frac{2 |P_j – P_{\text{grid}}|}{|P_j + P_{\text{grid}}|} \leq 10\%$$
If this condition is met, the negotiation is successful, and the final price is set. This mechanism ensures a fair and transparent price discovery process, accounting for the diverse scales of electric vehicle aggregators and their electric vehicle car fleets. By incorporating these bidding strategies, we enable both parties to reach mutually beneficial agreements, which is essential for encouraging participation from electric vehicle car owners and aggregators alike.
After establishing the price, we focus on the allocation of response electricity among multiple electric vehicle aggregators. This is where evolutionary game theory, specifically the replicator dynamics theory, comes into play. The DSO aims to maximize its utility while ensuring grid stability by allocating the total response electricity $C_{\text{all}}$ among the aggregators. Let $y_j$ represent the proportion of $C_{\text{all}}$ allocated to aggregator $j$, such that $0 \leq y_j \leq 1$ and $\sum_{j=1}^{J} y_j = 1$, where $J$ is the total number of aggregators. The contracted response electricity for aggregator $j$ is:
$$C_j^u = y_j \cdot C_{\text{all}}$$
The utility for the DSO from aggregator $j$, denoted as $U_j^{\text{DSO}}$, depends on whether the aggregator meets its contracted capacity. We define the contract completion ratio $r_j$ as:
$$r_j = \frac{C_j^{\text{ev}}}{C_j^u}$$
where $C_j^{\text{ev}}$ is the bid electricity submitted by aggregator $j$. The utility function is then:
$$U_j^{\text{DSO}} =
\begin{cases}
P_j C_j^u + S_j |C_j^u| – \alpha (C_j^u – C_j^{\text{ev}})^2, & r_j \leq 1 \\
P_j C_j^u + S_j |C_j^u|, & r_j > 1
\end{cases}$$
Here, $P_j$ is the negotiated price, $S_j$ is a positive revenue coefficient for the DSO (with $S_j > P_j$), and $\alpha$ is a penalty coefficient for underperformance. The term $S_j |C_j^u|$ represents additional benefits from the aggregator’s participation, while the penalty term discourages shortfalls. Based on replicator dynamics, the evolution of the allocation proportion $y_j$ over time $t$ is given by:
$$\frac{\partial y_j}{\partial t} = \delta y_j \left( \frac{U_j^{\text{DSO}}}{C_j^u} – \bar{U}^{\text{DSO}} \right)$$
where $\delta$ is the iteration step size, and $\bar{U}^{\text{DSO}}$ is the average utility per unit contracted electricity across all aggregators:
$$\bar{U}^{\text{DSO}} = \sum_{j=1}^{J} y_j \frac{U_j^{\text{DSO}}}{C_j^u}$$
To implement this dynamically, we discretize the equation:
$$y_j(z+1) = y_j(z) + \delta y_j(z) \left[ \frac{U_j^{\text{DSO}}(z)}{C_j^u(z)} – \bar{U}^{\text{DSO}}(z) \right]$$
where $z$ is the iteration index. The process iterates until convergence, defined as:
$$\left| \frac{U_j^{\text{DSO}}(z)}{C_j^u(z)} – \bar{U}^{\text{DSO}}(z) \right| < \epsilon_1$$
with $\epsilon_1$ set to a small value like 0.001. This iterative approach ensures that the DSO adapts the allocation based on the performance and capacity of each electric vehicle aggregator, leading to an equilibrium where utility is maximized. The use of evolutionary game theory is particularly apt for modeling the strategic interactions among multiple agents, such as electric vehicle aggregators competing for allocation, and it aligns well with the dynamic nature of electric vehicle car fleets.
To validate our proposed strategy, we conduct simulations with three electric vehicle aggregators of different scales: EVA1 with 300 electric vehicle cars, EVA2 with 500 electric vehicle cars, and EVA3 with 1000 electric vehicle cars. We set key parameters based on realistic assumptions, as summarized in Table 2. These parameters reflect typical electric vehicle car characteristics, such as battery capacity, charging efficiency, and power ratings, which are crucial for accurate modeling.
| Parameter | Value |
|---|---|
| State of Charge Adjustable Range | 0.3 to 0.9 |
| Battery Capacity (kWh) | 60 |
| Charging Efficiency | 0.90 |
| Discharging Efficiency | 0.85 |
| Charging Power (kW) | 7 |
| Discharging Power (kW) | 6 |
For the probability distributions, we assume $N_c(17.6, 3.3)$ for charging start time, $N_l(9.2, 3.2)$ for departure time, and $N_x(3.8, 0.91)$ for daily travel distance. These values are derived from empirical studies on electric vehicle car usage patterns. The negotiation parameters for the aggregators are listed in Table 3, including their scales, $\text{type}_{\text{EVA}}$ values, bidding indices, and price limits. The DSO’s price limits are set based on market data, with a peak shaving energy price range from 1.491 CNY/kWh to 3.0 CNY/kWh.
| Aggregator | Number of Electric Vehicle Cars | $\text{type}_{\text{EVA}}$ | $k_1$ | $P_{j}^{\text{min}}$ (CNY/kWh) | $P_{j}^{\text{max}}$ (CNY/kWh) |
|---|---|---|---|---|---|
| EVA1 | 300 | 0.6 | 0.5 | 1.491 | 2.0 |
| EVA2 | 500 | 1.0 | 1.0 | 1.491 | 2.5 |
| EVA3 | 1000 | 2.0 | 2.0 | 1.491 | 3.0 |
Using Monte Carlo simulation, we first generate the uncontrolled charging load profile for the electric vehicle car fleets. The results, as shown in Figure 1 (though not displayed here, described textually), indicate that peak charging occurs around 18:00, with powers of 429 kW, 727 kW, and 1438 kW for EVA1, EVA2, and EVA3, respectively. This aligns with residential peak hours, creating a “peak on peak” effect that stresses the grid. The daily load curve exhibits significant peak-valley differences, highlighting the need for demand response. Our simulation of the negotiation process yields final prices for valley filling and peak shaving. For valley filling, EVA3, EVA2, and EVA1 settle at 0.585 CNY/kWh, 0.488 CNY/kWh, and 0.408 CNY/kWh, respectively. For peak shaving, the prices are 2.205 CNY/kWh, 1.938 CNY/kWh, and 1.705 CNY/kWh. These outcomes demonstrate that larger aggregators secure higher prices due to their greater capacity, incentivizing the aggregation of more electric vehicle cars.
Next, we apply the evolutionary game-based allocation strategy. Starting with equal proportions, the DSO iteratively adjusts $y_j$ based on the utility functions. The convergence of $y_j$ for valley filling response is illustrated in Figure 2 (described textually). After multiple iterations, the proportions stabilize: EVA3 receives 53%, EVA2 receives 29%, and EVA1 receives 18% of the total response electricity $C_{\text{all}}$. This allocation mirrors their capacities, with EVA3, having the most electric vehicle cars, obtaining the largest share. The convergence criteria are met within a reasonable number of iterations, confirming the stability and efficiency of our algorithm. This dynamic allocation ensures that each electric vehicle aggregator is assigned a response load commensurate with its actual capability, reducing the risk of underperformance or overcommitment.
To assess the impact on grid stability, we compare the load profiles before and after implementing demand response with our optimization strategy. The uncontrolled charging scenario shows a peak load of 7187 kW at 18:00, which is 55% higher than the base load peak. After optimization, the peak load reduces to 6000 kW, a 19% decrease. This reduction significantly flattens the load curve, as depicted in Figure 3 (described textually). Moreover, we calculate the load variance for both scenarios. The variance under uncontrolled charging is substantially higher than under optimized demand response. Specifically, the variance drops by approximately 30% in our simulation, indicating improved grid stability. The reduction in fluctuations is critical for preventing equipment overloads and enhancing the integration of renewable energy sources, which often have intermittent output. By leveraging the flexibility of electric vehicle cars through aggregation, we can turn a potential burden into a valuable grid asset.
Our strategy’s effectiveness stems from several key features. First, the Monte Carlo simulation accurately models the randomness of electric vehicle car behavior, ensuring that capacity predictions are realistic. Second, the price negotiation mechanism accounts for the heterogeneous scales of electric vehicle aggregators, promoting fair and efficient market participation. Third, the evolutionary game-based allocation dynamically adjusts to performance, maximizing the DSO’s utility while maintaining grid balance. Throughout this process, the role of electric vehicle cars as distributed energy resources is emphasized; each electric vehicle car contributes to the aggregate capacity, and our strategy optimizes how this capacity is harnessed. We also consider practical constraints, such as the charging and discharging efficiencies of electric vehicle cars, which affect the net energy available for demand response.
In addition to the core methodology, we explore extensions and sensitivities. For instance, varying the parameters $\alpha$ and $S_j$ in the utility function influences the allocation outcomes. Higher $\alpha$ values penalize underperformance more severely, encouraging aggregators to bid conservatively. Conversely, higher $S_j$ values increase the DSO’s incentive to allocate more electricity to high-performing aggregators. We test these variations in our simulation and find that the equilibrium proportions shift accordingly, but the overall convergence remains robust. Furthermore, we examine the effect of different probability distributions for electric vehicle car parameters. Using alternative distributions, such as Weibull for travel distance, yields similar aggregate trends, confirming the versatility of our Monte Carlo approach. These analyses underscore the adaptability of our strategy to diverse scenarios involving electric vehicle car fleets.
The implications of our work extend beyond technical optimization. By facilitating the participation of electric vehicle aggregators in demand response, we promote the adoption of electric vehicle cars as grid-supportive assets. This can lead to economic benefits for electric vehicle car owners through incentives, reduced charging costs, and even revenue from grid services. For utilities, the strategy enhances operational efficiency and defers infrastructure investments by smoothing load profiles. Policymakers can use insights from our model to design regulations that encourage aggregation and market integration of electric vehicle cars. Moreover, as the number of electric vehicle cars grows globally, strategies like ours will be essential for managing the associated energy demands sustainably.
However, challenges remain. The uncertainty in electric vehicle car user behavior, though modeled, can still lead to prediction errors. Future work could incorporate real-time data and machine learning techniques to refine capacity forecasts. Additionally, the negotiation mechanism assumes full information transparency, which may not always hold in practice. Incorporating game-theoretic models with incomplete information could enhance realism. Another avenue is to consider the spatial distribution of electric vehicle cars and grid constraints, which affect aggregation potential. Despite these areas for improvement, our current strategy provides a solid foundation for optimizing electricity allocation among electric vehicle aggregators.
In conclusion, we have developed a comprehensive optimization strategy for response electricity allocation to electric vehicle aggregators under demand response mechanisms. Our approach integrates Monte Carlo simulation for capacity prediction, a linear bidding strategy for price negotiation, and evolutionary game theory for dynamic allocation. Simulation results demonstrate that the strategy allocates electricity proportionally to aggregator capacity, reduces grid load fluctuations by 19% in peak scenarios, and lowers load variance by 30%, thereby enhancing power system stability. The repeated emphasis on electric vehicle cars throughout this paper highlights their transformative potential in energy systems. As electric vehicle car adoption accelerates, such optimization strategies will be crucial for harnessing their collective power efficiently and reliably, paving the way for a more resilient and sustainable grid.