With the rapid growth of electric car adoption worldwide, particularly in regions like China EV markets, the integration of massive electric vehicles into power systems poses significant challenges for optimization and scheduling. The uncontrolled charging and discharging behaviors of electric cars can lead to grid instability, peak load surges, and inefficiencies in energy distribution. Virtual power plants (VPPs) have emerged as a promising solution to aggregate distributed resources, including electric vehicles, photovoltaics, and other flexible assets, enabling their participation in transmission grid dispatch. However, the homogeneity in parameters among large-scale electric cars—such as charging/discharging power, efficiency, and battery capacity—introduces consistency issues that cause iterative oscillations and non-convergence in distributed optimization algorithms. This paper addresses these challenges by proposing a cloud-edge collaborative scheduling framework based on Lagrangian relaxation, incorporating a perturbation function to accelerate convergence while maintaining optimal solutions. The approach is validated through case studies inspired by real-world data from Shenzhen, China, highlighting its effectiveness in enhancing scheduling efficiency for China EV integration.
The proliferation of electric cars has been driven by global efforts to reduce carbon emissions and transition to clean energy systems. In China, the electric car market has expanded dramatically, with over 20 million units deployed by 2023, accounting for nearly half of the global total. Projections indicate that China EV numbers could exceed 80 million by 2030, underscoring the transformative impact of electric cars on energy systems. However, the spatial and temporal distribution of electric car charging loads creates unpredictability, straining power grid operations. Virtual power plants offer a decentralized approach to manage these resources by leveraging cloud-edge collaboration, where cloud-based VPPs coordinate with edge devices like electric car charging stations to optimize energy flows. This paradigm shift alleviates computational burdens on central grids while improving responsiveness to dynamic demands.
The core of the cloud-edge framework involves a hierarchical structure: the transmission grid at the top level, VPPs at the intermediate level, and electric cars at the edge level. The VPP acts as an aggregator, managing resources such as distributed photovoltaics, micro-gas turbines, and electric car charging stations. Through information exchange, the VPP receives signals from the transmission grid and dispatches control commands to electric cars, optimizing their charging and discharging schedules. This collaboration reduces peak loads, enhances renewable energy utilization, and ensures grid stability. For instance, during periods of high solar generation, electric cars can store excess energy, which is later discharged during peak demand, thus supporting “peak shaving and valley filling” strategies. The model formulation for VPP participation in grid dispatch includes objective functions and constraints that minimize costs while satisfying operational limits.
The optimization model for the transmission grid employs a security-constrained unit commitment formulation, focusing on minimizing generation costs subject to constraints like power balance, ramp rates, and line flow limits. The objective function is expressed as:
$$ \min \sum_{t=1}^{T} \sum_{g=1}^{N_G} \left[ c_{u,g} u_{g,t} + c_{d,g} d_{g,t} + a_{\text{th},g} p_{\text{th},g,t} + b_{\text{th},g} (p_{\text{th},g,t})^2 \right] $$
where \( c_{u,g} \) and \( c_{d,g} \) are startup and shutdown costs for thermal unit \( g \), \( u_{g,t} \) and \( d_{g,t} \) are binary variables indicating unit status, \( a_{\text{th},g} \) and \( b_{\text{th},g} \) are cost coefficients, and \( p_{\text{th},g,t} \) is the power output. Constraints include:
$$ r_{g,t} P_{\text{th},g}^{\min} \leq p_{\text{th},g,t} \leq r_{g,t} P_{\text{th},g}^{\max} $$
$$ r_{g,t} – r_{g,t-1} = u_{g,t} – d_{g,t} $$
$$ u_{g,t} + d_{g,t} \leq 1 $$
$$ \sum_{k=t}^{T_S} (1 – r_{g,k}) \geq T_S (r_{g,t-1} – r_{g,t}) $$
$$ | p_{\text{th},g,t+1} – p_{\text{th},g,t} | \leq s_{u,g} $$
$$ \sum_{g} p_{\text{th},g,t} + \sum_{w} p_{\text{wind},w,t} = \sum_{i} p_{\text{vpp},i,t} + \sum_{m} p_{\text{load},m,t} $$
$$ p_{l,t} = \sum F_l (p_{\text{th},g,t} + p_{\text{wind},w,t} – p_{\text{load},m,t} – p_{\text{vpp},i,t}) $$
$$ -P_l^{\max} \leq p_{l,t} \leq P_l^{\max} $$
Here, \( p_{\text{vpp},i,t} \) represents the power exchange between VPP \( i \) and the grid at time \( t \), which is iteratively updated in the collaborative process.
At the VPP level, the optimization aims to minimize energy costs by scheduling distributed resources, including electric cars. The objective function for a VPP is:
$$ \min F(P) = \sum_{i=1}^{N} \sum_{t=T_s}^{T} C_t p_{s,i,t} $$
where \( C_t \) is the real-time electricity price, and \( p_{s,i,t} \) is the power exchanged with the grid. Constraints for distributed photovoltaics and micro-gas turbines are:
$$ 0 \leq p_{pv,i,t} \leq P_{APV,t} $$
$$ 0 \leq p_{MT,i,t} \leq P_{MT,t} $$
The net power output of a VPP is given by:
$$ p_{\text{vpp},i,t} = p_{\text{sta},i,t}^{\text{cha}} + p_{\text{sta},i,t}^{\text{load}} – p_{\text{sta},i,t}^{\text{dis}} – p_{pv,i,t} – p_{MT,i,t} $$
with limits on the exchange power:
$$ 0 \leq p_{\text{vpp},i,t} \leq p_{\text{limit}} $$
For electric cars, the individual optimization focuses on minimizing charging costs while meeting energy requirements:
$$ \min F(P) = \sum_{t=T_s}^{T} C_t (p_{v,t}^{\text{cha}} – p_{v,t}^{\text{dis}}) $$
subject to constraints on charging and discharging power:
$$ 0 \leq p_{v,t}^{\text{cha}} \leq \alpha_v^c P_v^{\text{cmax}} $$
$$ 0 \leq p_{v,t}^{\text{dis}} \leq \alpha_v^d P_v^{\text{dmax}} $$
$$ \alpha_v^d + \alpha_v^c \leq 1 $$
The state of charge (SOC) dynamics are modeled as:
$$ S_t = S_{t-1} + \frac{1}{E_v^{\max}} p_{v,t}^{\text{cha}} \eta_v^{\text{cha}} – \frac{1}{E_v^{\max}} \frac{p_{v,t}^{\text{dis}}}{\eta_v^{\text{dis}}} $$
with boundary conditions:
$$ S_t (t = T_v^{\text{arr}}) = S_v^{\text{arr}} $$
$$ S_t (t = T_v^d) \geq S_v^d $$
$$ 0 \leq S_t \leq S^{\max} $$
Aggregate constraints for electric car charging stations include:
$$ P_{\text{sta},i,t}^{\text{cha}} = \sum_{v \in \Pi_{\text{ev}}} p_{v,t}^{\text{cha}} $$
$$ P_{\text{sta},i,t}^{\text{dis}} = \sum_{v \in \Pi_{\text{ev}}} p_{v,t}^{\text{dis}} $$
$$ 0 \leq P_{\text{sta},i,t}^{\text{cha}} \leq P_{\text{sta},i}^{\text{cha},\max} $$
$$ 0 \leq P_{\text{sta},i,t}^{\text{dis}} \leq P_{\text{sta},i}^{\text{dis},\max} $$
The parameter consistency issue arises when multiple electric cars share identical parameters, such as charging efficiency \( \eta^{\text{cha}} \), discharging efficiency \( \eta^{\text{dis}} \), maximum power \( P^{\text{cmax}} \), and battery capacity \( E^{\max} \). This homogeneity leads to symmetric subproblems in distributed optimization, where electric cars respond identically to control signals, causing oscillations in iterative algorithms like Lagrangian relaxation. For example, if two electric cars with the same parameters are scheduled to discharge during a power deficit, they might both oversupply power, leading the VPP to adjust prices and reduce their output, resulting in cyclic behavior. This problem is exacerbated in large-scale China EV deployments, where uniformity in electric car models is common.
To address this, we propose a distributed coordination algorithm based on Lagrangian relaxation, which decomposes the multi-level optimization into master and subproblems. The Lagrangian function for the transmission grid and VPPs is:
$$ L = \sum_{t} \sum_{g} \left[ c_{u,g} u_{g,t} + c_{d,g} d_{g,t} + a_{\text{th},g} p_{\text{th},g,t} + b_{\text{th},g} (p_{\text{th},g,t})^2 \right] + \sum_{t} \lambda_t \left( \sum_{i} p_{\text{vpp},i,t} – \sum_{g} p_{\text{th},g,t} – \sum_{w} p_{\text{wind},w,t} \right) $$
The dual function is then:
$$ D(\Lambda) = \sum_{t} \sum_{g} \left[ c_{u,g} u_{g,t} + c_{d,g} d_{g,t} + a_{\text{th},g} p_{\text{th},g,t} + b_{\text{th},g} (p_{\text{th},g,t})^2 – \lambda_t p_{\text{th},g,t} \right] – \sum_{t} \lambda_t \left( \sum_{w} p_{\text{wind},w,t} \right) + \sum_{i} \sum_{t} V_t^i (\lambda_t) $$
where \( V_t^i (\lambda_t) \) represents the energy transaction cost for VPP \( i \). For the VPP and electric car coordination, the Lagrangian function is:
$$ D(\mu) = \sum_{i=1}^{N} \sum_{t=1}^{T} S_t^i – \sum_{t=1}^{T} \mu_t \left( p_{\text{sta},i,t}^{\text{cha}} + p_{\text{sta},i,t}^{\text{load}} – p_{\text{sta},i,t}^{\text{dis}} – p_{pv,i,t} \right) $$
with \( S_t^i = \min \left[ C_t p_{\text{vpp},i,t} + \mu_t p_{\text{vpp},i,t} \right] \). The dual problem maximizes \( D(\mu) \), and the Lagrange multipliers are updated using a subgradient method:
$$ \lambda^{(k+1)} = \max \left\{ \lambda^{(k)} + \alpha^{(k)} \frac{g^{(k)}}{\| g^{(k)} \|_1}, 0 \right\} $$
where the step size \( \alpha^{(k)} = \frac{1}{w_1 k + w_2} \) ensures convergence.
The key innovation is the introduction of a perturbation function to break parameter consistency. By slightly perturbing parameters such as charging efficiency \( \eta^{\text{cha}} \), discharging efficiency \( \eta^{\text{dis}} \), maximum power \( P^{\text{cmax}} \), or electricity price \( C_t \), we disrupt the symmetry without altering the optimal solution structure. The perturbation for a parameter \( \theta \) (e.g., \( \eta^{\text{cha}} \)) is defined as:
$$ \theta_{\text{new}} = \theta + \delta(v, t) \sigma_\theta $$
where \( \delta(v, t) \) is a random function satisfying:
$$ \sum_{t} \delta(v, t) = 0, \quad \sum_{v} \delta(v, t) = 0 $$
and \( \sigma_\theta \) is the variance, constrained to ensure the perturbation remains within bounds that preserve the optimal basis. For example, the range for perturbing the cost coefficient \( C_t \) is derived as:
$$ \max \left\{ \frac{c_j – z_j}{y_{rj}} \mid y_{rj} < 0 \right\} \leq \Delta C_t \leq \min \left\{ \frac{c_j – z_j}{y_{rj}} \mid y_{rj} > 0 \right\} $$
Similarly, for right-hand side coefficients like \( P^{\text{cmax}} \):
$$ \max \left\{ -\frac{\bar{b}_i}{(B^{-1})_{ir}} \mid (B^{-1})_{ir} > 0 \right\} \leq \Delta b_r \leq \min \left\{ -\frac{\bar{b}_i}{(B^{-1})_{ir}} \mid (B^{-1})_{ir} < 0 \right\} $$
And for technical coefficients like \( \eta^{\text{cha}} \):
$$ \max \left\{ \frac{c_j – z_j}{q_k} \mid q_k > 0 \right\} \leq \Delta a_{ij} \leq \min \left\{ \frac{c_j – z_j}{q_k} \mid q_k < 0 \right\} $$
where \( q_k \) is the shadow price. This approach ensures that the perturbed parameters accelerate convergence while maintaining solution feasibility.
To validate the method, we conduct case studies using a modified IEEE 30-node transmission system and data from Shenzhen, China, reflecting diverse electric car scenarios. The system includes three VPPs with varying resources: VPP1 has high electric car penetration (15 stations with 100 charging points each), VPP2 emphasizes micro-gas turbines, and VPP3 balances photovoltaics and electric cars. The scheduling horizon is 24 hours with 1-hour intervals, though the method applies to shorter scales like 15 minutes for real-time adjustment. Key parameters for electric cars are based on real China EV data, with charging efficiency \( \eta^{\text{cha}} = 0.95 \), discharging efficiency \( \eta^{\text{dis}} = 0.95 \), and SOC limits \( S^{\max} = 1 \).
The results demonstrate that without perturbation, the objective function oscillates significantly, failing to converge within practical iterations. For instance, in a scenario with 100 homogeneous electric cars, the cost function fluctuates between 20.3 and 21.9 units over 40 iterations. After applying perturbations to charging efficiency parameters, convergence is achieved within 20 iterations, reducing oscillation amplitude by over 40%. The table below summarizes the convergence performance for different perturbation strategies:
| Perturbation Type | Convergence Iterations | Time Savings (%) |
|---|---|---|
| No Perturbation | >40 | 0 |
| Perturb \( C_t \) | 25 | 30 |
| Perturb \( P^{\text{cmax}} \) | 35 | 15 |
| Perturb \( \eta^{\text{cha}} / \eta^{\text{dis}} \) | 18 | 45 |
Additionally, we evaluate the impact on electric car scheduling. Under ordered charging controlled by the VPP, electric cars charge during low-price periods (e.g., nighttime) and discharge during peaks, reducing grid stress. For example, two electric cars (EV1 and EV2) are scheduled to charge at 2 AM and discharge at 7 PM, while an uncontrolled electric car (EV3) charges immediately upon arrival, exacerbating evening peaks. The VPP’s optimization increases photovoltaic utilization by 20% and reduces peak load by 15% in high electric car penetration scenarios.
The sensitivity analysis examines how perturbation magnitude affects convergence. Using different distributions for \( \delta(v, t) \) (e.g., Gaussian with \( \sigma = 10 \) or uniform), we find that convergence speed improves consistently as long as perturbations stay within derived limits. For larger electric car fleets (e.g., 500 vehicles), perturbing charging efficiency remains effective, whereas perturbing cost coefficients or power limits loses efficacy, highlighting the robustness of the proposed method for scaling China EV integration.
In conclusion, this paper presents a cloud-edge collaborative scheduling method for virtual power plants that addresses parameter consistency in electric car populations. By leveraging Lagrangian relaxation and a perturbation function, the approach eliminates iterative oscillations and enhances computational efficiency. The theoretical analysis provides closed-form bounds for perturbations, ensuring optimality preservation. Case studies from Shenzhen, China, validate the method’s practicality, showing significant improvements in convergence and grid performance. Future work could explore adaptive step sizes in Lagrangian updates or multi-level scheduling architectures to further optimize electric car management in evolving energy systems.