The uncontrolled integration of massive electric vehicles (EVs) presents unprecedented challenges for power system optimization. Virtual Power Plants (VPPs), aggregating distributed resources like EVs and photovoltaics at the edge, offer significant potential for transmission grid dispatch. However, large-scale homogeneous electric vehicles exhibit parameter consistency in charging/discharging power, efficiency, and battery capacity. This leads to identical edge optimization strategies in response to VPP control signals, causing iterative oscillations or divergence during collaborative scheduling. This work proposes a cloud-edge collaborative framework using Lagrangian relaxation and a novel perturbation function to resolve oscillations while preserving optimality.
1. Cloud-Edge Collaborative Scheduling Framework
The hierarchical scheduling framework comprises:
- Transmission Grid: Minimizes generation cost (Eq. 1) subject to unit commitment constraints (Eqs. 2–9).
- VPP Layer: Minimizes energy cost (Eq. 20) while managing distributed resources (EVs, PV, microturbines).
- Electric Vehicle Layer: Minimizes user charging cost (Eq. 10) under SOC dynamics (Eqs. 14–17) and station limits (Eqs. 18–19).
Equation 1 (Transmission Grid Objective):min∑t=1T∑g=1N[cgsuugt+cgsddgt+agpgt+bgpgt2]mint=1∑Tg=1∑N[cgsuugt+cgsddgt+agpgt+bgpgt2]
where ugt/dgtugt/dgt = startup/shutdown variables, pgtpgt = thermal unit output, ag,bgag,bg = cost coefficients.
Equation 10 (EV User Objective):minF(P)=∑t=1TCt(putch−putdis)minF(P)=t=1∑TCt(putch−putdis)
where CtCt = real-time electricity price, putch/putdisputch/putdis = EV charging/discharging power.
Table 1: VPP-Edge Resource Interactions
| Layer | Exchanged Information | Optimization Goal |
|---|---|---|
| Transmission Grid → VPP | Electricity price (λλ), power limits | Minimize generation cost |
| VPP → Electric Vehicle | Lagrange multipliers (μμ) | Minimize EV user charging cost |
| EV → VPP | Charging/discharging schedules | Satisfy SOC constraints |
2. Parameter Consistency Problem in Electric Vehicles
Homogeneous electric vehicles (e.g., same ηch,ηdis,Pmaxch,Pmaxdisηch,ηdis,Pmaxch,Pmaxdis) generate identical response strategies. This symmetry creates:
- Equivalent optimal bases in subproblems.
- Iterative oscillation during distributed optimization (Fig. 1).
Table 2: Oscillation Cases Under Parameter Consistency
| Scenario | VPP Power Shortage Signal | EV Response | Outcome |
|---|---|---|---|
| Symmetric Parameters (EV1, EV2) | 3 kW shortage | Both discharge 3 kW → 6 kW surplus | Oversupply → Price adjust → Undersupply → Oscillation |
| Asymmetric Parameters | 3 kW shortage | Lower-cost EV discharges 3 kW | Stable convergence |
Mechanism: Identical electric vehicles respond identically to price signals, causing power overshoot/undershoot cycles.
3. Lagrangian Relaxation with Perturbation Function
3.1 Lagrangian Decomposition
The transmission-VPP-EV problem is decoupled using Lagrange multipliers (λ,μλ,μ):
Equation 25 (Transmission-VPP Lagrangian):minL=∑g,t[cgsuugt+⋯+bgpgt2]+∑tλt(ptgrid−ptVPP)minL=g,t∑[cgsuugt+⋯+bgpgt2]+t∑λt(ptgrid−ptVPP)
Equation 27 (VPP-EV Lagrangian):minD(μ)=∑u,tSut−∑tμt(ptPV+ptMT−ptload−ptch+ptdis)minD(μ)=u,t∑Sut−t∑μt(ptPV+ptMT−ptload−ptch+ptdis)
Subgradient updates (Eq. 29) ensure convergence.
3.2 Perturbation Function for Parameter Consistency
Introduce perturbations δ(ηch,ηdis)δ(ηch,ηdis) within theoretical bounds (Eqs. 36, 38, 41) to break symmetry without altering optimality:
Equation 36 (Perturbation Bound for Cost Coefficients):max(zj−cjyrj∣yrj<0)≤ΔCj≤min(zj−cjyrj∣yrj>0)max(yrjzj−cj∣yrj<0)≤ΔCj≤min(yrjzj−cj∣yrj>0)
Equation 38 (Perturbation Bound for RHS Constraints):max(−bˉi(B−1)ir∣(B−1)ir>0)≤Δbr≤min(−bˉi(B−1)ir∣(B−1)ir<0)max(−(B−1)irbˉi∣(B−1)ir>0)≤Δbr≤min(−(B−1)irbˉi∣(B−1)ir<0)
Key Insight: Perturbing charging/discharging efficiency (ηch/ηdisηch/ηdis) is most effective for accelerating convergence vs. perturbing prices or power limits.
4. Case Study: Shenzhen EV Data
4.1 Setup
- Test Systems: IEEE 30-bus (3 VPPs) & IEEE 79-bus (7 VPPs).
- Electric Vehicle Data: 100 chargers/station, Pmaxch=7kWPmaxch=7kW, ηch=ηdis=0.95ηch=ηdis=0.95, SOCmin=20%SOCmin=20%, SOCmax=95%SOCmax=95%.
- Perturbation: Gaussian noise δ∼N(0,σ2),σ=10δ∼N(0,σ2),σ=10.
Table 3: Convergence Acceleration with Perturbation
| Algorithm | IEEE 30-bus EV Time (s) | IEEE 30-bus Grid Time (s) | IEEE 79-bus EV Time (s) |
|---|---|---|---|
| Unperturbed Lagrangian | 348.51 | 486.12 | 845.53 |
| Perturbed Lagrangian | 94.06 (73% ↓) | 1.06 (99.8% ↓) | 149.41 (82% ↓) |
4.2 Impact on Grid Operation
Electric vehicle scheduling shifts charging to low-price/peak-PV periods (Fig. 2):
- Peak Shaving: Reduces evening grid load by 12.7%.
- PV Utilization: Increases from 68% (uncontrolled) to 89%.
Table 4: VPP Operational Comparison (24-h Schedule)
| VPP Type | PV Self-Consumption | Grid Purchase (Peak) | EV Discharge Utilization |
|---|---|---|---|
| High EV Penetration (D3) | 92% | 4.8 MW | 88% |
| Low EV Penetration (D1) | 71% | 8.2 MW | 52% |
4.3 Sensitivity Analysis
Perturbation Robustness: Convergence accelerates consistently under σ∈[5,10]σ∈[5,10] (Fig. 3).
Scale Adaptability: Efficiency perturbation remains effective at 8,000+ electric vehicles.
Equation 31 (Adaptive Step Size):α(k)=1w1k+w2α(k)=w1k+w21
Ensures convergence as k→∞k→∞.
5. Conclusions
- Parameter Consistency Mitigation: Perturbing electric vehicle charging/discharging efficiency (ηch/ηdisηch/ηdis) within derived bounds accelerates convergence by >40% while preserving optimality.
- Hierarchical Optimization: The Lagrangian-based cloud-edge framework enables privacy-preserving coordination between transmission grids, VPPs, and electric vehicles.
- Scalability: The method maintains efficiency for 8,000+ electric vehicles, supporting future mass EV integration.
Future Work:
- Heuristic step size optimization for Lagrangian multipliers.
- Multi-layer scheduling with dynamic electricity markets.