As the global push toward carbon neutrality intensifies, integrating electric vehicles (EVs) into modern power systems becomes critical. Existing EV charging strategies often overlook the segmental regulation capability of charging power, limiting economic and environmental benefits. This study proposes a novel Stackelberg game framework to optimize EV charging scheduling through dynamic pricing and flexible power adjustment, validated by significant cost reduction and grid stability improvements.
1. Problem Statement and Motivation
Current EV charging practices face two primary limitations:
- Inflexible Charging Protocols: Most strategies (e.g., constant-power, interruptible charging) neglect the potential for segmental power regulation. This restricts the ability to dynamically align EV charging loads with renewable generation peaks and grid conditions.
- Suboptimal Pricing: Pricing mechanisms often rely on fixed time-of-use (TOU) tariffs without incorporating real-time carbon trading costs or dynamic interactions between stakeholders, hindering cost efficiency and carbon reduction.
Our research addresses these gaps by introducing:
- Segmental Charging Power Regulation: Enabling EVs to select discrete charging power levels within their operational range, enhancing scheduling flexibility.
- Carbon-Aware Dynamic Pricing: Integrating carbon trading costs into real-time pricing signals within a multi-agent Stackelberg game structure.
- Efficient Meta-Model Solution: Utilizing an improved Kriging metamodel with Particle Swarm Optimization (PSO) to handle the computational complexity introduced by numerous segmental variables.
2. System Architecture and Agent Models
The microgrid comprises three key agents interacting within a hierarchical Stackelberg game:
- Microgrid Operator (MGO – Leader): Purchases electricity from the main grid and Distributed Resource Aggregator (DRA), operates gas turbines, sets dynamic EV charging prices (πtEVπtEV) and DRA selling prices (πtDAπtDA), and aims to minimize operational costs and load variance.
- EV Aggregator (EVA – Follower): Manages a fleet of EVs, responding to πtEVπtEV by optimizing individual EV charging schedules (start time, duration, segmental power level) to minimize total charging costs, including carbon trading components.
- Distributed Resource Aggregator (DRA – Follower): Manages photovoltaic (PV) generation and energy storage systems (ESS), responding to πtDAπtDA by optimizing PV/ESS dispatch to minimize operational costs.
2.1 EV Aggregator (EVA) Model
2.1.1 Segmental Charging Power Regulation Strategy
Each EV ii connects to the grid during an interval [tic,tid][tic,tid]. The available charging power is selectable from discrete levels:Pi,tEV∈{PEV0,PEV1,…,PEVmax}(e.g., 1, 2, …, 7 kW)Pi,tEV∈{PEV0,PEV1,…,PEVmax}(e.g., 1, 2, …, 7 kW)
The strategy dynamically sets the charging start time (ti′ti′), charging duration (TiTi), and power level (Pi,tEVPi,tEV) to satisfy the State-of-Charge (SOC) requirement while minimizing cost and grid impact.
- SOC Constraints:Simin≤Si(tid)≤SimaxSimin≤Si(tid)≤SimaxWhere SiminSimin and SimaxSimax are the minimum (e.g., 60-80%) and maximum (e.g., 80-100%) required SOC upon departure, SicSic is the initial SOC at connection.
- Charging Delay Coefficient: Determines the latest possible start time for a chosen power level to meet SiminSimin:Dil=(ti−l)PiEVηΔt−(Simin−Sic)CEVDil=(ti−l)PiEVηΔt−(Simin−Sic)CEVWhere:
- ll: Potential delay time slot index.
- ti=tid−ticti=tid−tic: Total grid connection time.
- ηη: Charging efficiency (e.g., 0.95).
- CEVCEV: EV battery capacity (e.g., 35 kWh).
- ΔtΔt: Time interval.
A feasible delay ll and power level PiEVPiEV must satisfy Dil+1<0Dil+1<0 and Dil≥0Dil≥0.
- Charging End Time & Duration:tiend=min{tid,(Simax−Sic)CEVPiEVηΔt+tic+l}tiend=min{tid,PiEVηΔt(Simax−Sic)CEV+tic+l}Ti=tiend−tic−lTi=tiend−tic−l
2.1.2 Dynamic Charging Price Constraints
The EVA purchases power from the MGO and DRA. The dynamic charging price πtEVπtEV is constrained by power balance and price limits:PtEV=PtDA+PtMG(Power Balance)PtEV=PtDA+PtMG(Power Balance)∑t=1TPtEVπavgEV=∑t=1TPtDAπtDA+∑t=1TPtMGπtMG(Cost Recovery)t=1∑TPtEVπavgEV=t=1∑TPtDAπtDA+t=1∑TPtMGπtMG(Cost Recovery)πtmin≤πtEV≤πtmax(Price Bounds)πtmin≤πtEV≤πtmax(Price Bounds)
Where πavgEVπavgEV is the average charging price, πtMGπtMG is the MGO selling price (assumed equal to grid TOU tariff), πtDAπtDA is the DRA selling price, πtminπtmin and πtmaxπtmax are price bounds derived from TOU tariffs ± an adjustment (e.g., ±0.1 ¥/kWh).
2.1.3 EVA Objective Function: Minimize Total Charging Cost
The cost includes electricity purchase and carbon trading revenue:minJEV=∑t=1T(PtEVπtEV−CtEV)minJEV=t=1∑T(PtEVπtEV−CtEV)
- Carbon Trading Revenue (CtEVCtEV): Earned by reducing emissions compared to gasoline vehicles.EtEV=PtEVΔtβtqEV(EV Charging Emissions)EtEV=PtEVΔtβtqEV(EV Charging Emissions)βt=PtMT+PtGPtMT+PtG+PtPV+∣PtESS,d∣(Fossil Energy Ratio)βt=PtMT+PtG+PtPV+∣PtESS,d∣PtMT+PtG(Fossil Energy Ratio)EtGV=PtEVΔtLEVqGV(Equivalent Gasoline Vehicle Emissions)EtGV=PtEVΔtLEVqGV(Equivalent Gasoline Vehicle Emissions)RtEV=EtGV−EtEV(Emission Reduction)RtEV=EtGV−EtEV(Emission Reduction)CtEV=kRtEV(Carbon Revenue)CtEV=kRtEV(Carbon Revenue)Where:
- βtβt: Proportion of fossil fuel-based power supplying EV charging.
- PtMT,PtGPtMT,PtG: Gas turbine and grid purchase power.
- PtPV,PtESS,dPtPV,PtESS,d: PV output and ESS discharge power.
- qEV,qGVqEV,qGV: EV and gasoline vehicle emission coefficients.
- LEVLEV: EV efficiency (km/kWh).
- kk: Carbon price (¥/kg).
Table 1: Key EVA Carbon Trading Parameters
| Parameter | Description | Value | Unit |
|---|---|---|---|
| qEVqEV | EV Emission Coefficient | 0.9019 | kg/kWh |
| qGVqGV | Gasoline Vehicle Emission Coeff. | 0.197 | kg/km |
| kk | Carbon Price | 0.25 | ¥/kg |
| LEVLEV | EV Efficiency | 5 | km/kWh |
2.2 Distributed Resource Aggregator (DRA) Model
2.2.1 DRA Selling Price Constraint
To incentivize participation, the DRA selling price is bounded:πtDA,min≤πtDA≤πtDA,max(e.g., 0.49 to 0.57 ¥/kWh)πtDA,min≤πtDA≤πtDA,max(e.g., 0.49 to 0.57 ¥/kWh)
2.2.2 DRA Objective Function: Minimize Operational CostminJDA=∑t=1T[CtESS−πtDA(PtPV+PtESS)−CtPV]minJDA=t=1∑T[CtESS−πtDA(PtPV+PtESS)−CtPV]
- ESS Operating Cost (CtESSCtESS):CtESS=λESS∣PtESS∣ΔtCtESS=λESS∣PtESS∣ΔtWhere λESSλESS is the ESS cost coefficient (e.g., 0.1 ¥/kWh).
- PV Carbon Revenue (CtPVCtPV):CtPV=kϵPtPVΔtCtPV=kϵPtPVΔtWhere ϵϵ is the PV carbon quota coefficient.
- ESS Operational Constraints:0≤PtESS,c≤utPESS,max(Charging Power)0≤PtESS,c≤utPESS,max(Charging Power)0≤PtESS,d≤(1−ut)PESS,max(Discharging Power)0≤PtESS,d≤(1−ut)PESS,max(Discharging Power)StESS=St−1ESS+(ηcESSPtESS,c−PtESS,dηdESS)Δt(State of Charge)StESS=St−1ESS+(ηcESSPtESS,c−ηdESSPtESS,d)Δt(State of Charge)μminSESS,max≤StESS≤μmaxSESS,max(SOC Limits)μminSESS,max≤StESS≤μmaxSESS,max(SOC Limits)Where ut∈{0,1}ut∈{0,1} indicates charging (1) or discharging (0), ηcESS,ηdESSηcESS,ηdESS are charge/discharge efficiencies, SESS,maxSESS,max is max capacity, μmin,μmaxμmin,μmax are min/max SOC limits (e.g., 0.1, 0.9).
Table 2: Key DRA and ESS Parameters
| Parameter | Description | Value | Unit |
|---|---|---|---|
| SESS,maxSESS,max | Max ESS Capacity | 10,000 | kWh |
| PESS,maxPESS,max | Max ESS Charge/Discharge Power | 3,000 | kW |
| ηcESS,ηdESSηcESS,ηdESS | Charge/Discharge Efficiency | 0.95 | – |
| μmin,μmaxμmin,μmax | Min/Max SOC | 0.1, 0.9 | – |
| λESSλESS | ESS Cost Coefficient | 0.1 | ¥/kWh |
2.3 Microgrid Operator (MGO) Model
The MGO aims to minimize operational cost (J1J1) and load variance (J2J2), combined into a single objective:minJMG=ω1J1+ω2J2,ω1+ω2=1minJMG=ω1J1+ω2J2,ω1+ω2=1
- Operational Cost (J1J1):J1=∑t=1T[−PtEVπtEV+πtDA(PtPV+PtESS)+PtMTλMT+CtMT+CtG]J1=t=1∑T[−PtEVπtEV+πtDA(PtPV+PtESS)+PtMTλMT+CtMT+CtG]
- PtMTλMTPtMTλMT: Gas turbine fuel cost (λMTλMT e.g., 0.35 ¥/kWh).
- CtMT,CtGCtMT,CtG: Carbon trading costs for gas turbine and grid purchases.
- Load Variance (J2J2): Minimizes deviation from the average net load.J2=∑t=1T(PtL−PtPV+PtEV−Pave)2J2=t=1∑T(PtL−PtPV+PtEV−Pave)2Pave=1T∑t=1T(PtL−PtPV+PtEV)Pave=T1t=1∑T(PtL−PtPV+PtEV)Where PtLPtL is the base residential load.
- Carbon Trading Costs:
- Gas Turbine:QtMT=(EMT−eMT)PtMTΔtQtMT=(EMT−eMT)PtMTΔtCtMT=kQtMTCtMT=kQtMT(EMTEMT: Emission coeff., eMTeMT: Carbon quota coeff.)
- Grid Purchase (Coal-fired):QtG=[a1+b1PtG+c1(PtG)2−λGPtG]ΔtQtG=[a1+b1PtG+c1(PtG)2−λGPtG]ΔtCtG=kQtGCtG=kQtG
- Power Balance Constraint:PtEV+PtL=PtMT+PtG+PtPV+PtESSPtEV+PtL=PtMT+PtG+PtPV+PtESS
- Gas Turbine Constraints:PMT,min≤PtMT≤PMT,max(e.g., 0 – 18,000 kW)PMT,min≤PtMT≤PMT,max(e.g., 0 – 18,000 kW)∣PtMT−Pt−1MT∣≤RmaxPMT,max(Ramp Rate, e.g., Rmax=0.2)∣PtMT−Pt−1MT∣≤RmaxPMT,max(Ramp Rate, e.g., Rmax=0.2)
Table 3: Key MGO Carbon Trading and Equipment Parameters
| Parameter | Description | Value | Unit |
|---|---|---|---|
| EMTEMT | Gas Turbine Emission Coeff. | 0.95 | kg/kWh |
| eMTeMT | Gas Turbine Carbon Quota Coeff. | 0.798 | kg/kWh |
| λGλG | Grid Purchase Carbon Quota Coeff. | 0.648 | kg/kWh |
| a1a1 | Grid Emission Coeff. (Constant) | 36 | kg/h |
| b1b1 | Grid Emission Coeff. (Linear) | 0.38 | kg/kWh |
| c1c1 | Grid Emission Coeff. (Quadratic) | 0.0034 | kg/kWh² |
| PMT,maxPMT,max | Max Gas Turbine Output | 18,000 | kW |
| RmaxRmax | Max Gas Turbine Ramp Rate | 0.2 | p.u. |
3. Stackelberg Game Formulation and Solution
The interaction is modeled as a single-leader, two-follower Stackelberg game:Ω=⟨{MG,EVA,DA},{SMG(πtEV,πtDA),SEV(PtEV),SDA(PtPV,PtESS)},{JMG,JEV,JDA}⟩Ω=⟨{MG,EVA,DA},{SMG(πtEV,πtDA),SEV(PtEV),SDA(PtPV,PtESS)},{JMG,JEV,JDA}⟩
- Leader (MGO): Announces prices πtEV,πtDAπtEV,πtDA.
- Followers (EVA & DRA): React optimally by setting PtEVPtEV (via segmental charging schedules) and PtPV,PtESSPtPV,PtESS.
- Equilibrium: A Nash Equilibrium (NE) is reached when no agent can improve its objective by unilaterally changing its strategy, given the other agents’ strategies:{πEV∗,πDA∗=argminJMG s.t. EVA & DRA responsesPEV∗=argminJEV(πEV∗)PPV∗,PESS∗=argminJDA(πDA∗)⎩⎨⎧πEV∗,πDA∗=argminJMG s.t. EVA & DRA responsesPEV∗=argminJEV(πEV∗)PPV∗,PESS∗=argminJDA(πDA∗)
3.1 Solving the Game: Improved Kriging Metamodel
The segmental charging strategy introduces significant combinatorial complexity. Traditional methods like Genetic Algorithms (GA) struggle with convergence speed. We employ an Improved Kriging Metamodel integrated with Particle Swarm Optimization (PSO):
- Initial Sampling: Use Latin Hypercube Sampling (LHS) to generate initial sets of MGO’s price strategies (πtEV,πtDA)(πtEV,πtDA).
- Lower-Level Evaluation: For each sampled price set, solve the EVA and DRA optimization problems (followers’ response).
- Kriging Model Construction: Build a Kriging metamodel approximating the MGO’s objective function JMGJMG based on the sampled price points and corresponding objective values.
- PSO-Guided Refinement: Use PSO to search promising regions of the price strategy space identified by the Kriging model. Evaluate the true JMGJMG at these PSO-proposed points using the follower models.
- Model Update & Convergence: Add the new (price, JMGJMG) points to the training set. Update the Kriging model. Repeat steps 4-5 until convergence (minimal improvement in expected JMGJMG).
3.2 Equilibrium Existence Proof
The Stackelberg game admits a Nash Equilibrium under the following conditions, satisfied by our model:
- Convex Strategy Sets: The strategy spaces SMG,SEV,SDASMG,SEV,SDA defined by linear constraints (price bounds, power limits, SOC constraints, etc.) are non-empty, closed, bounded, and convex.
- Unique Follower Response: For any given leader strategy (πtEV,πtDA)(πtEV,πtDA), the followers’ cost functions JEVJEV (linear in PtEVPtEV) and JDAJDA (linear in PtPV,PtESSPtPV,PtESS) are convex, guaranteeing unique optimal responses PtEV∗,PtPV∗,PtESS∗PtEV∗,PtPV∗,PtESS∗.
- Unique Leader Response: Given the followers’ optimal responses, the leader’s combined cost function JMGJMG (a weighted sum of linear and quadratic terms) is strictly convex in the price variables (πtEV,πtDA)(πtEV,πtDA), ensuring a unique optimal strategy for the leader.
4. Simulation Results and Analysis
A case study with 2,000 EVs, PV generation, ESS, and a gas turbine was simulated over 24 hours. EV connection/disconnection times and initial SOC followed defined probability distributions (Appendix A). Four scenarios were compared:
Table 4: Simulated Case Scenarios
| Scenario | Dynamic Charging Price? | Charging Strategy | Description |
|---|---|---|---|
| S1 | No | Uncontrolled | Plug-and-charge based on arrival time. Fixed TOU price. |
| S2 | No (Fixed TOU) | Segmental Power | Segmental power regulation allowed. Fixed TOU price. |
| S3 | Yes | Interruptible (7 kW) | Fixed 7 kW charging power (interruptible). Dynamic price set by MGO. |
| S4 | Yes | Segmental Power | Proposed: Segmental power regulation AND MGO dynamic price. |
4.1 Effectiveness of Segmental Power Regulation (S4)
- Dynamic Charging Price: The MGO dynamically adjusted πtEVπtEV (Fig. 3a), lowering prices during high PV output (08:00-17:00) to incentivize charging but slightly raising them within the low-price window (11:00-13:00) to prevent excessive concentration.
- Charging Load Shifting: S4 successfully shifted charging load away from grid peak periods (Fig. 3b). Unlike S2 (fixed TOU) which created a large peak at 11:00-13:00, and S3 (fixed 7kW) which caused spikes during the evening load peak, S4 resulted in a smoother profile better aligned with PV availability.
- SOC Trajectories: EVs utilizing segmental regulation (S4) delayed charging start times and used varying power levels, charging predominantly during high PV/low net-load periods (e.g., after 10:00) (Fig. 3c).
- Power Level & Duration Choice: Most EVs in S4 chose power levels between 3-7 kW and durations of 1-6 hours (Fig. 3d). Over 81% chose ≥5 kW, with 7kW for 3 hours being the most common (19.25%), indicating efficient demand fulfillment.
- SOC Satisfaction: Most EVs met or exceeded their minimum SOC requirement upon departure (Fig. 3e).
4.2 Economic and Low-Carbon Benefits
- EV Charging Cost: S4 achieved the lowest total EVA charging cost (Table 5), significantly lower than uncontrolled charging (S1) and outperforming S2 (segmental power, fixed price) and S3 (dynamic price, fixed power). The combination of dynamic pricing and flexible power regulation enabled more cost-effective scheduling.
- DRA Selling Price: The MGO dynamically set πtDAπtDA (Fig. 4a), increasing it during high demand/PV periods (07:00-17:00) to encourage DRA to sell more power rather than store it, reducing DRA operational costs.
- Carbon Emissions: Integrating carbon trading significantly reduced emissions in S2, S3, and S4 compared to S1 (Fig. 4b). S4 showed the strongest correlation between EV charging load and PV output, maximizing the use of clean energy and minimizing fossil-based generation for charging. EVA actively shifted charging to high-PV periods to gain carbon revenue.
Table 5: Operational Cost Comparison (¥)
| Scenario | EV Aggregator (EVA) Cost | Dist. Res. Aggr. (DRA) Cost | Microgrid Operator (MGO) Cost |
|---|---|---|---|
| S1 | 21,899 | -21,619 | 82,590 |
| S2 | 21,280 | -31,818 | 85,669 |
| S3 | 19,130 | -31,699 | 85,620 |
| S4 | 18,589 | -31,410 | 82,499 |
4.3 Peak Shaving and Valley Filling Benefits
- Total Load Profile: S4 demonstrated the best peak shaving performance (Fig. 5a). Uncontrolled charging (S1) exacerbated the morning peak. S2 shifted load but created a large midday peak due to fixed low TOU prices. S3 mitigated the midday peak via dynamic pricing but caused evening spikes due to high fixed charging power. S4 effectively avoided concentrated charging during low-price periods and high net-load periods (e.g., evening peak reduced by >2000 kW vs S3).
- Peak-to-Valley Difference Rate Reduction: S4 achieved the largest reduction (11.6%) compared to the uncontrolled baseline S1 (Table 6). This highlights the superior load-flattening capability of the combined segmental power and dynamic pricing approach.
Table 6: Peak-to-Valley Difference Rate Improvement vs S1
| Scenario | Peak-to-Valley Rate (%) | Improvement vs S1 (%) |
|---|---|---|
| S2 | 39.8 | 8.9 |
| S3 | 40.9 | 7.6 |
| S4 | 36.9 | 11.6 |
4.4 Algorithm Performance
The improved Kriging metamodel significantly outperformed a standard Genetic Algorithm (GA):
- Faster Convergence: Kriging converged in an average of 66 iterations, compared to 267 iterations for GA.
- Better Solution Quality: Kriging found a solution reducing MGO cost by ¥1,059 compared to GA (Table 7).
Table 7: Algorithm Performance Comparison
| Algorithm | MGO Cost (¥) | Avg. Iterations |
|---|---|---|
| Genetic Alg. (GA) | 83,558 | 267 |
| Kriging Metamodel | 82,499 | 66 |
5. Conclusion
This study successfully developed and validated a hierarchical Stackelberg game framework for optimizing electric vehicle charging within a microgrid featuring distributed resources and carbon trading. Key contributions and findings are:
- Enhanced EV Charging Flexibility: The proposed segmental electric vehicle charging power regulation strategy allows EVs to dynamically adjust both charging power levels and duration within their grid connection window. This flexibility is crucial for effective demand response, enabling EVs to act as more responsive grid assets.
- Superior Economic and Environmental Performance: The synergistic combination of segmental charging power flexibility and carbon-aware dynamic charging pricing significantly reduced costs for all agents, particularly the electric vehicle aggregator (15.1% reduction compared to uncontrolled charging). Simultaneously, it facilitated greater utilization of renewable energy (PV), leading to a substantial decrease in electric vehicle charging-related carbon emissions (11.6% reduction in grid peak-to-valley difference rate).
- Effective Multi-Agent Coordination: The Stackelberg game effectively modeled the strategic interactions between the Microgrid Operator (leader), Electric Vehicle Aggregator, and Distributed Resource Aggregator (followers). Integrating carbon trading costs directly into the dynamic pricing mechanism aligned economic incentives with low-carbon objectives across the system.
- Efficient Computational Solution: The improved Kriging metamodel coupled with Particle Swarm Optimization proved highly effective in solving the complex optimization problem involving numerous discrete segmental power choices and dynamic pricing variables. It achieved faster convergence (66 vs. 267 iterations) and better solutions (lower MGO cost) compared to a standard Genetic Algorithm.
This research demonstrates that unlocking the segmental power regulation capability of modern electric vehicle chargers, combined with intelligent dynamic pricing within a coordinated market framework (Stackelberg game), is essential for maximizing the economic viability and environmental benefits of large-scale electric vehicle integration into the future smart grid. Future work will explore real-time implementation challenges and the integration of vehicle-to-grid (V2G) capabilities.