The rapid proliferation of battery electric vehicles (BEVs) represents a significant shift towards sustainable transportation. However, their uncoordinated integration into the power grid, characterized by high spatio-temporal uncertainty, poses substantial challenges, including the exacerbation of peak-valley load differences and potential local network congestion during peak hours. To harness the flexibility of battery electric vehicle charging as a grid resource, effective guidance strategies are essential. Concurrently, the widespread adoption of regenerative braking technology in battery electric vehicles contributes to energy conservation and carbon emission reduction. This paper proposes a comprehensive spatio-temporal guidance strategy for battery electric vehicle loads, incorporating energy recovery mechanisms and operating under a dynamic pricing framework. The strategy aims to optimize grid operation while minimizing user charging costs.
The foundation of accurate load modeling lies in understanding user travel patterns. We begin by constructing a detailed model of the urban transportation network, which serves as the spatial backbone for simulating battery electric vehicle movement. The network is mathematically represented as a graph:
$$G = (N(G), L(G), \psi_G)$$
$$N(G) = \{ (x_i, y_i) | i = 1,2, …, n \}$$
$$L(G) = \{ \langle d_i, d_j \rangle | d_i, d_j \in K \}$$
$$\psi_G = \{ e_{ij} | \langle d_i, d_j \rangle \in L \}$$
Where $N(G)$ is the set of road intersection nodes with coordinates $(x_i, y_i)$, $L(G)$ is the set of road segments connecting nodes $d_i$ and $d_j$, and $\psi_G$ is the adjacency matrix where $e_{ij}$ represents the length of the road between $d_i$ and $d_j$.
Urban areas are segmented into functional zones—primarily residential, work, and other areas (e.g., commercial, leisure)—using Kernel Density Estimation (KDE) to analyze Point of Interest (POI) data. The density at a point $p$ is calculated as:
$$K_{KDE}(p) = \sum_{i=1}^{M} \left[ \frac{1}{h^2} K\left( \frac{p – p_i}{h} \right) \right]$$
Where $h$ is the bandwidth, $M$ is the number of POIs within the threshold, and $K$ is the spatial weighting kernel function.
The daily movement of each battery electric vehicle is described by a travel chain $L_i$, which is a sequence of destinations and corresponding dwell times:
$$L_i = ((s_{i0}, t_{i0}), (s_{i1}, t_{i1}), …, (s_{in}, t_{in}))$$
Here, $s_{in}$ is the $n$-th node visited by user $i$, and $t_{in}$ is the dwell time at that node. The initial departure time is modeled using a three-parameter Weibull distribution mixture:
$$\Phi(T) = \sum_{i=1}^{3} \frac{\alpha_i}{\sqrt{2\pi}\varepsilon_i} e^{-\frac{(\frac{T}{60}-\kappa_i)^2}{2\varepsilon_i^2}}$$
Dwell times in different zones follow distinct probability distributions. For residential zones, a Weibull distribution is used:
$$f_r(R, \lambda_r, \theta_r) = \frac{\theta_r}{\lambda_r} \left( \frac{R}{\lambda_r} \right)^{\theta_r – 1} e^{-(R/\lambda_r)^{\theta_r}}$$
For work and other zones, a Generalized Extreme Value distribution form is applied after standardization $z = (R – \mu_z)/\sigma$:
$$f_w(z) = \frac{(1 – \xi_{w,z}z)^{-1+\frac{1}{\xi_{w,z}}}}{\sigma} e^{-(1-\xi_{w,z}z)^{1/\xi_{w,z}}}}$$
$$f_o(z) = \frac{(1 + \xi_{o,z}z)^{-1-\frac{1}{\xi_{o,z}}}}{\sigma} e^{(1+\xi_{o,z}z)^{1/\xi_{o,z}}}}$$
The proportions of different travel chain types, based on travel survey data, are summarized below:
| Travel Chain Type | Proportion |
|---|---|
| Residential-Work-Residential | 0.47 |
| Residential-Other-Residential | 0.15 |
| Residential-Work-Other-Residential | 0.12 |
| Residential-Other-Work-Residential | 0.13 |
| Other-Residential/Work-Other | 0.03 |
| Work-Residential/Other-Work | 0.03 |
A critical innovation in our load modeling is the explicit inclusion of braking energy recovery, a key feature of modern battery electric vehicles. This allows for a more precise calculation of net energy consumption during travel. The energy recovered by the battery $E_c$ is a fraction of the available kinetic energy minus losses to rolling resistance and air drag:
$$E_c = \omega (E_v – E_f – E_w)$$
$$E_v = 0.5 m (v_0^2 – v_1^2)$$
$$E_f = \int \delta m g v \, dt$$
$$E_w = \int A C_d \rho v \, dt$$
Where $\omega$ is a composite factor accounting for brake force distribution, charging efficiency, and drivetrain efficiency; $E_v$ is the total kinetic energy change; $E_f$ and $E_w$ are energy consumed overcoming rolling resistance and aerodynamic drag, respectively; $m$ is vehicle mass; $v_0$, $v_1$ are initial and final velocities; $\delta$ is the rolling resistance coefficient; $A$ is frontal area; $C_d$ is the drag coefficient; and $\rho$ is air density.
The state of charge (SOC) of a battery electric vehicle at time $t$ is thus updated as:
$$E_{n, sur}^t = B – E_{n, con}^t + E_{n, c}^t$$
Where $B$ is the battery capacity, $E_{n, con}^t$ is the total travel consumption, and $E_{n, c}^t$ is the total recovered energy for vehicle $n$ at time $t$. The associated carbon emission reduction $\vartheta_{CO_2}$ from energy recovery is calculated using the grid emission factor $F$:
$$\vartheta_{CO_2} = F \sum_{n=1}^{N} E_{n, c}^{24}$$
To simulate the aggregate behavior of a large fleet of battery electric vehicles, we employ a Monte Carlo simulation. The process integrates the travel chain model, shortest-path calculation via the Floyd-Warshall algorithm, energy consumption and recovery models, and charging decision logic, iterating over all vehicles to build a spatio-temporal profile of charging demand.
The core of the guidance strategy is a dynamic pricing mechanism designed to influence user charging behavior. Unlike static or traditional time-of-use tariffs, our dynamic prices vary by both time and location (functional zone). A user’s decision to charge depends on remaining range and current price. We define a price influence coefficient $k$:
$$k = \frac{V_t}{\bar{V}}$$
Where $V_t$ is the electricity price at time $t$ and $\bar{V}$ is the average daily price. This coefficient adjusts the minimum SOC threshold $S’_{SOCmin}$ that triggers a charging need:
$$S’_{SOCmin} = \frac{S_{SOCmin}}{k}$$
A higher price increases $k$, thereby lowering the threshold $S’_{SOCmin}$, making users slightly more tolerant to lower SOC before charging due to cost sensitivity. The charging demand $E_{n,req}^t$ is then determined by comparing the current surplus energy $E_{n,sur}^t$ with this adjusted threshold and the energy $E_{n,next}$ needed for the next trip.
To reflect real-world user behavior where charging sessions are often continuous, we introduce a charging utility function $S$ that incorporates temporal adjacency:
$$S = \sum_{t=1}^{24} S_t(\eta_{t-1}, \eta_t, \eta_{t+1})$$
$$S_t(\eta_{t-1}, \eta_t, \eta_{t+1}) = V_t \eta_t + \lambda_1 \eta_{t-1}\eta_t + \lambda_2 \eta_t \eta_{t+1}$$
Here, $\eta_t$ is a binary variable indicating charging (1) or not charging (0) at time $t$. The terms $\lambda_1 \eta_{t-1}\eta_t$ and $\lambda_2 \eta_t \eta_{t+1}$ provide a reward for consecutive charging periods, modeling user preference for uninterrupted sessions. Dynamic programming is used to select the charging schedule that maximizes this utility function.
The interaction between the grid operator and battery electric vehicle users is formulated as a Stackelberg game (leader-follower game). The grid operator acts as the leader, setting dynamic prices to minimize load variance. The users act as followers, adjusting their charging schedules to minimize personal cost based on the announced prices.
Upper-Level (Grid Operator) Model:
The operator’s objective is to flatten the total load profile:
$$\min \gamma = \frac{1}{24} \sum_{t=1}^{24} (P_{z,t} – P_{av})^2$$
$$P_{z,t} = P_{0,t} + \sum_{n=1}^{N} P_{n,cha}^t$$
Where $\gamma$ is the load variance, $P_{z,t}$ is the total load at time $t$, $P_{0,t}$ is the base load, $P_{n,cha}^t$ is the charging load of vehicle $n$, and $P_{av}$ is the daily average load. Constraints include grid power limits $P_{sg}^t$, nodal voltage limits $U_{i,min} \leq U_{i,t} \leq U_{i,max}$, and price change limits (e.g., $|V_{t+1} – V_t| \leq \Delta V$).
Lower-Level (BEV User) Model:
Each user seeks to minimize their charging cost:
$$\min C = \sum_{n=1}^{N} \sum_{t} V_t E_{n,cha}^t$$
Subject to meeting their charging demand $E_{n,cha}^t \geq E_{n,req}^t$, respecting charging power limits $0 \leq E_{n,cha}^t / \Delta t \leq P_{cha,max}^n$, and battery capacity limits $E_{n,cha}^t \leq B$.
The bi-level model is solved using an iterative approach embedded within the Monte Carlo simulation. The Particle Swarm Optimization (PSO) algorithm is employed to find the optimal dynamic price signals that achieve the Stackelberg equilibrium. Key PSO parameters used are listed below:
| Parameter | Value |
|---|---|
| Population Size | 10 |
| Maximum Iterations | 200 |
| Learning Factor | 2.0 |
| Velocity Limit | [-2, 2] |
| Inertia Weight | 0.8 |
We conducted a case study with a fleet of 5,000 battery electric vehicles to validate the proposed strategy. The urban network of a city district was modeled, and POI data was classified to delineate residential, work, and other functional zones. Three scenarios were compared:
- Scenario 1: Uniform fixed pricing for all zones.
- Scenario 2: Traditional time-of-use pricing (high price during 08:00-22:00, low price otherwise) for all zones.
- Scenario 3: The proposed dynamic pricing strategy with zone-specific prices.
The results demonstrate the effectiveness of our strategy. The 24-hour total load curves for the three scenarios show that Scenario 3 achieves the flattest profile. The quantitative improvements are summarized in the following table:
| Scenario | Grid Load Peak-Valley Difference (kW) | Total User Charging Cost |
|---|---|---|
| 1 (Fixed Price) | 19,638.25 | 65,907.52 |
| 2 (Time-of-Use) | 9,272.95 | 50,029.13 |
| 3 (Dynamic Price) | 7,714.96 | 45,315.26 |
Compared to Scenario 1, the proposed strategy reduces the peak-valley difference by 60.71% and user costs by 31.24%. Compared to the conventional Time-of-Use pricing in Scenario 2, it achieves further reductions of 16.80% in peak-valley difference and 9.42% in cost.
The optimized dynamic electricity prices for different functional zones in Scenario 3 successfully guide charging loads to different time windows: residential loads concentrate during 20:00-06:00, work area loads during 16:00-20:00, and other area loads during 12:00-23:00. This spatio-temporal shift balances load across both time and space, preventing the creation of new peaks during low-price periods—a common drawback of simple Time-of-Use pricing.
The importance of incorporating energy recovery and the continuous charging utility function is highlighted through comparative analyses. Considering braking energy recovery corrected the net energy consumption, leading to a more accurate prediction of charging demand. In the simulated fleet, energy recovery saved 8,781.07 kWh, accounting for 11.14% of total travel consumption and reducing carbon emissions by approximately 4,977 kg. The inclusion of the continuous charging utility function increased the smoothness of the aggregated load curve, better reflecting realistic user behavior.
In conclusion, this paper presents a holistic spatio-temporal guidance strategy for battery electric vehicle loads. By integrating a high-fidelity travel and energy consumption model (including braking energy recovery) with a behaviorally realistic dynamic pricing response model within a Stackelberg game framework, the strategy effectively balances the objectives of grid load flattening and user cost minimization. The results confirm its superiority over fixed and traditional time-of-use pricing schemes. Future work will focus on incorporating additional behavioral factors, such as weather sensitivity and real-time traffic conditions, to further enhance the precision and robustness of battery electric vehicle load management for supporting the evolution of modern power systems.