The global transition towards sustainable transportation has catalyzed an unprecedented surge in Battery EV Cars. With annual growth rates exceeding 30%, the global fleet is projected to surpass 45 million vehicles, introducing significant and highly mobile charging demands. This rapid integration poses a formidable challenge to power grid stability, as uncoordinated charging, particularly during evening residential peaks, can exacerbate load fluctuations and increase local grid overloading risks. Conversely, the inherent mobility and storage capacity of battery EV cars present a substantial, yet underutilized, opportunity for demand-side flexibility and renewable energy integration. This work addresses the dual challenge of managing grid impact while unlocking this potential by proposing a novel coordinated charging-discharging strategy for battery EV cars. Our approach uniquely integrates a detailed travel chain-based behavioral model with a regionally dynamic pricing mechanism, all while explicitly quantifying and optimizing for multi-dimensional user satisfaction.
Existing strategies for guiding battery EV car charging often treat vehicles as static, flexible storage units, overlooking the rigid constraints imposed by their owners’ daily mobility patterns. Furthermore, while economic incentives are commonly employed, the subjective willingness of users to respond to these signals is seldom modeled in detail, leading to potentially impractical dispatch plans. To bridge these gaps, we propose a holistic framework that couples a spatiotemporal model of battery EV car movement derived from travel chain theory with a responsive, zone-based dynamic electricity pricing scheme. The pricing is dynamically determined by the net load (base demand minus renewable generation) in each zone. Crucially, we formulate a comprehensive user satisfaction model incorporating time window constraints, price sensitivity, and detour distance aversion. This satisfaction metric is embedded as a penalty term within an optimization model designed to minimize total system cost, ensuring that dispatch decisions are not only economically efficient but also align with user experience and practical feasibility.
Modeling Battery EV Car Mobility via Travel Chains
Accurate forecasting of charging demand from a fleet of battery EV cars necessitates a realistic model of their daily movement. We adopt the travel chain theory, which conceptualizes a user’s daily itinerary as a sequence of trips connecting various destinations, categorized primarily into Home (H), Work (W), and Other (O) zones. The chain is defined by both temporal and spatial stochastic variables.
Spatiotemporal Probability Distributions
The start time of the first trip of the day, \( T_{s1} \), is modeled using a Gamma distribution:
$$ f(T_{s1}) = \frac{T_{s1}^{9.881-1} e^{-T_{s1}/58.908}}{\Gamma(9.881) \cdot 58.908^{9.881}} $$
The driving duration for a single trip, \( t_d \), is modeled using a log-normal distribution, with parameters differentiated by trip type (I: H→non-H, II: non-H→non-H, III: non-H→H):
$$ f_i(t_d) = \frac{1}{t_d \sigma_i \sqrt{2\pi}} e^{-\frac{(\ln(t_d) – \mu_i)^2}{2\sigma_i^2}} $$
where the fitted parameters \( (\mu, \sigma) \) are (3.020, 0.775), (2.846, 0.815), and (3.040, 0.761) for types I, II, and III, respectively.
The dwell time \( \Delta t_{stay} \) at a destination is highly location-dependent. We model dwell times at W, H, and O zones using Generalized Extreme Value and Weibull distributions, respectively, based on empirical data.
The driving distance \( d_i \) is conditioned on the driving duration time window \( \Delta t_i \), following a normal distribution:
$$ f(d_i | \Delta t_i) = \frac{1}{\sqrt{2\pi}\sigma_i} e^{-\frac{(d_i – \mu_i)^2}{2\sigma_i^2}} $$
The parameters \( \mu_i \) and \( \sigma_i \) for different duration windows are summarized in the table below.
| Duration Window \( \Delta t_i \) (min) | Type I \( \mu_i \) (km) | Type I \( \sigma_i \) (km) | Type II \( \mu_i \) (km) | Type II \( \sigma_i \) (km) | Type III \( \mu_i \) (km) | Type III \( \sigma_i \) (km) |
|---|---|---|---|---|---|---|
| (0, 10] | 2.20 | 1.02 | 2.49 | 2.20 | 1.02 | 2.49 |
| [10, 20] | 7.28 | 3.73 | 6.47 | 7.28 | 3.73 | 6.47 |
| [20, 30] | 13.32 | 6.14 | 11.75 | 13.32 | 6.14 | 11.75 |
| [30, 40] | 18.71 | 8.10 | 16.79 | 18.71 | 8.10 | 16.79 |
| [40, 50] | 24.65 | 10.69 | 21.80 | 24.65 | 10.69 | 21.80 |
| [50, 60] | 32.14 | 14.51 | 28.64 | 32.14 | 14.51 | 28.64 |
| [60, 80] | 36.87 | 16.95 | 33.48 | 36.87 | 16.95 | 33.48 |
| [80, 100] | 48.41 | 24.72 | 47.26 | 48.41 | 24.72 | 47.26 |
| [100, 150] | 68.31 | 27.11 | 71.51 | 68.31 | 27.11 | 71.51 |
Spatial Transition Probability
The sequence of destinations in a battery EV car’s travel chain can be modeled as a Markov process. The probability of transitioning to the next destination \( E_j \) depends only on the current destination \( E_i \):
$$ P(E_i \rightarrow E_j) = P(E_j | E_i) = p_{ij} $$
where \( p_{ij} \) is the spatial transition probability. Based on travel survey data, the primary travel chains and their observed proportions for private battery EV cars are:
| Travel Chain Pattern | Observed Proportion (%) |
|---|---|
| H-W-H | 52.2 |
| H-O-H | 23.3 |
| H-W-O-H | 24.5 |
Zone-Based Dynamic Charging Pricing Mechanism
Economic Bounds for Pricing
A viable pricing strategy must respect economic boundaries for both charging station operators and battery EV car users. The minimum acceptable charging price \( c_{min} \) ensures the operator covers costs, including wholesale electricity purchase \( C_p \), and is set above the grid’s valley tariff. The maximum acceptable price \( c_{max} \) from a user’s perspective is derived by comparing the total cost of charging a battery EV car against refueling a conventional gasoline vehicle, accounting for energy efficiency and the value of time lost during charging:
$$ c_{max} = \frac{P_{gas} \cdot E_{gas} – \tau \cdot T}{E_{ele}} $$
where \( P_{gas} \) is gasoline price, \( E_{gas} \) and \( E_{ele} \) are fuel and electricity consumption per km, \( \tau \) is the value of time, and \( T \) is extra time spent for charging.
Dynamic Time-of-Use Pricing Formulation
Instead of a fixed time-of-use (TOU) schedule, we propose a pricing scheme that responds dynamically to the real-time net load condition \( P_{eq}(t) \) (base load minus renewable generation) in each zone. The average net load \( P_{av} \) and the peak-to-valley difference \( P_d \) are calculated:
$$ P_{av} = \frac{1}{T} \sum_{t=1}^{T} P_{eq}(t), \quad P_d = \max_{t \in [1,T]} P_{eq}(t) – \min_{t \in [1,T]} P_{eq}(t) $$
The charging price \( c(t) \) at time \( t \) is then determined by a piecewise function relative to \( P_{av} \) and \( P_d \):
$$
c(t) =
\begin{cases}
c_f + (c_{max} – c_f) \frac{P_{eq}(t) – (P_{av} + P_d)}{P_{eq, max} – (P_{av} + P_d)}, & P_{eq}(t) \geq P_{av} + P_d \\
c_0 + (c_f – c_0) \frac{P_{eq}(t) – (P_{av} – P_d)}{2P_d}, & P_{av} – P_d \leq P_{eq}(t) < P_{av} + P_d \\
c_{min} + (c_0 – c_{min}) \frac{P_{eq}(t) – P_{eq,min}}{(P_{av} – P_d) – P_{eq,min}}, & P_{eq}(t) < P_{av} – P_d
\end{cases}
$$
Here, \( c_0 \) and \( c_f \) are base prices for the normal and peak periods, respectively. This formulation creates a sliding scale where prices are highest when the local net load is high (encouraging discharge or discouraging charge) and lowest when net load is low or negative (encouraging charge to absorb surplus renewable energy). This mechanism provides a clear economic signal for each battery EV car to engage in spatiotemporal energy transfer.
Iterative Price Adjustment for Congestion Management
To prevent over-concentration of battery EV cars in a single low-price zone, we implement an iterative feedback mechanism. The zone utilization \( \eta_{area}(t) \) is calculated:
$$ \eta_{area}(t) = \frac{P^{EV}_{cha, area}(t)}{Cap_{area}} $$
where \( P^{EV}_{cha, area}(t) \) is the total charging power of battery EV cars in the zone and \( Cap_{area} \) is the total charger capacity. If the utilization exceeds a threshold \( \eta_{max} \), the price for the next iteration \( k+1 \) is adjusted upward to dampen demand:
$$ c^{(k+1)}(t) = \min \left( c^{(k)}(t) + \theta \cdot \frac{\eta_{area}(t) – \eta_{max}}{1 – \eta_{max}} \cdot c^{(k)}(t), \, c_{max} \right) $$
where \( \theta \) is a price adjustment coefficient.
Multi-Dimensional User Satisfaction Model
The willingness of a battery EV car user to follow a dispatch schedule depends on subjective factors beyond pure economics. We quantify this through a composite satisfaction score \( S \) that integrates three key dimensions, allowing users to weigh them according to personal preference via weights \( \lambda_1, \lambda_2, \lambda_3 \) (with \( \lambda_1+\lambda_2+\lambda_3=1 \)).
1. Time Window Satisfaction (\( S_t \)): Models the user’s tolerance for waiting. It uses a sigmoid function for acceptable wait times \( t_{wait} \leq t_{max} \) and an exponential decay for unacceptable waits.
$$ S_t = \begin{cases}
\frac{1}{1 + \exp(-k_1 (t_{wait} – t_{max}/2))}, & t_{wait} \leq t_{max} \\
k_2 \cdot \exp(-(t_{wait} – t_{max})), & t_{wait} > t_{max}
\end{cases} $$
The average waiting time \( t_{wait\_av} \) is estimated using an M/M/S queuing model based on the arrival rate \( \lambda \) and service rate \( \mu \) of charging stations.
2. Price Sensitivity Satisfaction (\( S_c \)): Reflects the user’s sensitivity to the difference between the actual average charging price in the chosen zone and the price in their original zone.
$$ \bar{c}_{area} = \frac{\sum_{t=t_{arrive,area}}^{t_{end,area}} c_{area}(t)}{\Delta t_{stay,area}}, \quad S_c = \max \left(1 – \frac{\bar{c}_{actual} – \bar{c}_{base}}{c_{max}}, 0 \right) $$
A lower \( S_c \) indicates higher dissatisfaction due to paying a premium.
3. Detour Distance Satisfaction (\( S_d \)): Captures the inconvenience of deviating from the shortest path to reach a charging station. It is modeled with a cosine function up to a maximum acceptable detour \( d_{max} \).
$$ S_d = \begin{cases}
1 – \frac{1}{2} \left(1 – \cos\left(\pi \frac{d}{d_{max}}\right)\right), & 0 \leq d \leq d_{max} \\
0, & d > d_{max}
\end{cases} $$
The overall composite satisfaction for a given charging choice \( p \) at decision point \( m \) for battery EV car \( n \) is:
$$ S_{n,m,p} = \lambda_1 S_t + \lambda_2 S_c + \lambda_3 S_d $$
Integrated Charging-Discharging Dispatch Optimization Model
Objective Function
The core optimization model aims to minimize the total system cost, which includes the net charging cost for all battery EV cars and a penalty for user dissatisfaction. This dual objective ensures a trade-off between economic efficiency and user experience.
$$ \min \, G = \alpha \sum_{n=1}^{N} \frac{C_n}{F_{1max}} + \beta \sum_{n=1}^{N} \sum_{m} \sum_{p} \frac{(1 – S_{n,m,p})}{F_{2max}} $$
where \( \alpha + \beta = 1 \). \( C_n \) is the net electricity cost for the \( n \)-th battery EV car, \( F_{1max} \) is the total cost under uncoordinated charging (for normalization), and \( F_{2max} \) is the maximum possible dissatisfaction sum. The net cost \( C_n \) is:
$$ C_n = \sum_{t=1}^{T} \left( c_{area}(t) \cdot P^{EV}_{n,cha}(t) – c_{sell}(t) \cdot P^{EV}_{n,dis}(t) \right) $$
where \( c_{sell}(t) \) is the feed-in tariff for discharging.
Constraints
The optimization is subject to a comprehensive set of constraints ensuring feasibility, battery health, and travel requirements for every battery EV car.
1. Power and State Constraints:
$$ 0 \leq P^{EV}_{n,cha}(t) \leq P^{max}_{cha}, \quad 0 \leq P^{EV}_{n,dis}(t) \leq P^{max}_{dis} $$
$$ U^{cha}_n(t) + U^{dis}_n(t) \leq 1, \quad U^{cha}_n(t), U^{dis}_n(t) \in \{0,1\} $$
2. Battery State-of-Charge (SOC) Dynamics and Limits:
$$ SOC_n(t) = SOC_n(t-1) + \frac{\eta_{cha} P^{EV}_{n,cha}(t) \Delta t}{Q_n} – \frac{ P^{EV}_{n,dis}(t) \Delta t}{\eta_{dis} Q_n} $$
$$ SOC^{min} \leq SOC_n(t) \leq SOC^{max} $$
$$ SOC^{meet}_n = SOC^{min} + \frac{d_i \cdot E_{100}}{100 Q_n}, \quad \max(SOC^{exp}_n, SOC^{meet}_n) \leq SOC^{e}_n \leq SOC^{max} $$
$$ SOC_n(t) \geq SOC^{dis\_min} \quad \text{(to allow discharge)} $$
The constraint for \( SOC^{meet}_n \) ensures the battery EV car has enough energy to complete its next trip, while \( SOC^{exp}_n \) allows for a user-specified desired departure SOC.
3. Dwell Time and Charging Logic:
$$ t_{wait} + t_{charge} = \Delta t_{stay} $$
This ensures the actual charging time fits within the available parking duration.
Case Study and Analysis
A simulation study involving 100 battery EV cars was conducted across three zones (H, W, O) with distinct base load and renewable generation profiles. The dynamic pricing bounds were set as \( c_{min} = 0.4 \) ¥/kWh and \( c_{max} = 1.6 \) ¥/kWh. User satisfaction weights were initially set to \( \lambda_1=0.3, \lambda_2=0.4, \lambda_3=0.3 \), and the objective weights to \( \alpha = \beta = 0.5 \).
The resulting dynamic prices successfully mirrored the net load in each zone. The scheduling strategy effectively shifted charging load from peak net load periods to valley periods and encouraged spatial transfer. For instance, charging was guided from the evening peak in the H zone to periods with high photovoltaic output or lower overall demand in other zones. Crucially, the SOC trajectories for all simulated travel chains (H-W-H, H-O-H, H-W-O-H) strictly satisfied the minimum energy requirement for the next trip, demonstrating the model’s adherence to user mobility needs.
The performance of the proposed strategy was compared against two benchmarks: 1) Uncoordinated Charging (UC), and 2) Traditional Fixed TOU Pricing. The key results are summarized below:
| Scenario / Metric | Total User Net Cost (¥) | Equivalent Load Variance (MW²) | Average User Satisfaction |
|---|---|---|---|
| Proposed Strategy (\( \lambda_c=0.4 \)) | -244.3 (Profit) | 20.74 | 0.835 |
| Uncoordinated Charging (UC) | +626.8 (Cost) | 30.11 | 1.000 (by definition) |
| Traditional Fixed TOU | N/A (Higher cost than UC) | >30.11 (often worse) | Low (spatial mismatch) |
The results are compelling. Under the proposed strategy, the fleet of battery EV cars achieves a net profit (negative cost) through strategic charging during low-price periods and discharging during high-price periods. Simultaneously, the grid experiences a significant reduction in equivalent load variance (over 31% reduction compared to UC), leading to a flatter net load curve and improved stability. While user satisfaction naturally decreases from the theoretical maximum of uncoordinated charging (where no constraints are imposed), it remains at a high level (above 0.83), indicating the schedule is acceptable to users. The traditional TOU strategy performed poorly, often increasing costs and failing to manage spatial congestion, sometimes even exacerbating load variance by inducing discharge at non-optimal times.
Further sensitivity analysis on the satisfaction weights showed that increasing the price sensitivity weight \( \lambda_2 \) led to higher user profits, as the model prioritized finding lower-cost charging options, albeit with marginal impacts on load variance and overall satisfaction, which remained stable.
Conclusion
This research presents a comprehensive and practical framework for coordinating the charging and discharging of a large fleet of battery EV cars. By deeply integrating a realistic travel chain model of user mobility, a responsive zone-based dynamic pricing mechanism, and a quantifiable multi-dimensional user satisfaction model, the proposed strategy effectively addresses the tri-lemma of grid stability, economic efficiency, and user acceptance. The travel chain foundation ensures all dispatch plans respect the fundamental mobility constraints of each battery EV car. The dynamic pricing provides precise, location- and time-specific economic signals that guide battery EV cars to act as a distributed flexible resource, charging when and where renewable generation is high or demand is low, and discharging to support the grid during peaks. The explicit modeling of user satisfaction transforms subjective preferences into an optimizable metric, ensuring that the resulting schedules are not only technically and economically sound but also palatable to users.
Case studies confirm the strategy’s superiority over uncoordinated charging and rigid TOU schemes. It successfully flattens the grid’s equivalent load profile, reduces operational costs for the system, and—most notably—transforms battery EV car charging from a net cost into a net revenue stream for users, all while maintaining a high level of user satisfaction. This work demonstrates that with the right market and control architecture, the massive integration of battery EV cars can be transformed from a grid challenge into a powerful asset for a more sustainable, resilient, and economically efficient power system.