The rapid growth of electric vehicle adoption, particularly in China, has introduced significant challenges for power system optimization and scheduling. By the end of 2023, China’s electric vehicle fleet exceeded 20 million units, accounting for nearly half of the global total, and is projected to surpass 80 million by 2030. This surge in electric vehicle penetration brings unprecedented pressures on grid stability due to the spatiotemporal distribution of charging loads. Virtual power plants (VPPs) have emerged as a promising solution to aggregate distributed resources, including electric vehicles and photovoltaic systems, enabling participation in transmission grid dispatch through cloud-edge collaboration. However, the consistency in parameters such as charging/discharging power, efficiency, and battery capacity among large-scale homogeneous electric vehicles leads to identical edge optimization strategies in response to VPP control signals. This often results in iterative divergence or oscillations during collaborative scheduling, undermining computational efficiency and solution accuracy.
This paper addresses the issue of parameter consistency in electric vehicle clusters within VPPs by proposing a cloud-edge collaborative scheduling framework based on Lagrangian relaxation. The framework facilitates efficient coordination among massive electric vehicles, VPPs, and the transmission grid, while a perturbation function method is introduced to break parameter symmetry without altering the optimal solution structure. Theoretical analysis derives closed-form perturbation limits, and case studies based on real-world data from Shenzhen validate the method’s effectiveness and scalability. The approach not only accelerates convergence but also enhances the economic and operational efficiency of VPPs in managing China EV integration.
The integration of electric vehicles into power systems poses unique challenges due to their stochastic charging behaviors and high power demands. In China, the electric vehicle market has expanded rapidly, driven by government policies and technological advancements. The unordered charging of massive electric vehicles can lead to peak load spikes, grid congestion, and increased operational costs. Virtual power plants offer a decentralized approach to manage these distributed resources by leveraging cloud-edge computing architectures. The cloud layer handles high-level optimization and coordination with the transmission grid, while the edge layer processes real-time electric vehicle data and executes localized control strategies. This hierarchical structure reduces computational burden on the central grid, enhances privacy, and improves response times. However, the homogeneity in electric vehicle parameters—such as identical charging rates, efficiencies, and battery capacities—causes synchronization in optimization responses, leading to oscillatory behavior in iterative algorithms like Lagrangian relaxation. This paper delves into the mechanics of this issue and proposes a robust solution to ensure stable and efficient scheduling.
Problem Formulation: Parameter Consistency and Oscillation Mechanisms
Parameter consistency in electric vehicle clusters arises when multiple EVs share identical technical specifications, such as maximum charging power $P_{ ext{max}}^{c}$, discharging power $P_{ ext{max}}^{d}$, charging efficiency $\eta_c$, discharging efficiency $\eta_d$, and battery capacity $E_{ ext{max}}$. In optimization models, this symmetry creates equivalent basis structures in linear programming formulations, resulting in multiple feasible solutions with identical objective values. During distributed coordination, EVs respond uniformly to price signals or power dispatch commands from the VPP, causing overshooting or undershooting of total power adjustments. For instance, when a power deficit occurs, all electric vehicles with consistent parameters may simultaneously increase discharge power, leading to an excess supply. The VPP then adjusts signals to reduce output, but the synchronized response again creates a deficit, inducing oscillations. This phenomenon is exacerbated as the number of homogeneous electric vehicles increases, complicating convergence in cloud-edge iterative processes.
Consider a simplified model where two electric vehicles with identical parameters respond to a power shortage signal. Let $P_{ ext{def}}$ denote the deficit, and each EV adjusts its discharge power by $\Delta P$. Due to consistency, both EVs set $\Delta P = P_{ ext{def}}/2$, resulting in a total adjustment of $P_{ ext{def}}$. However, if the actual deficit is dynamic, this uniform response leads to iterative corrections. The oscillation can be modeled as a feedback loop where the system error $e_t$ at time $t$ follows $e_{t+1} = -e_t$ under certain conditions, preventing convergence. This issue is prevalent in China EV deployments where standardized models dominate the market, emphasizing the need for targeted solutions.
Cloud-Edge Collaborative Scheduling Framework
The proposed framework involves three layers: the transmission grid, VPPs, and electric vehicle clusters. The transmission grid operates a security-constrained unit commitment (SCUC) model to minimize generation costs while satisfying power balance and line flow constraints. VPPs aggregate distributed resources, including electric vehicles, photovoltaic systems, micro-turbines, and fixed loads, optimizing internal scheduling to minimize energy procurement costs. Electric vehicles individually optimize charging and discharging schedules to reduce electricity costs, subject to battery dynamics and user constraints. Cloud-edge collaboration enables bidirectional information exchange: the cloud layer (transmission grid and VPPs) sends price and power signals, while the edge layer (EVs) computes and returns optimal responses.
The transmission grid objective function is given by:
$$ \min \sum_{t=1}^{T} \sum_{g=1}^{N_G} \left[ c_{u,g} u_{g,t} + c_{d,g} d_{g,t} + a_{ ext{th},g} p_{ ext{th},g,t} + b_{ ext{th},g} (p_{ ext{th},g,t})^2 \right] $$
subject to:
$$ \begin{aligned}
&r_{g,t} P_{ ext{th},g}^{ ext{min}} \leq p_{ ext{th},g,t} \leq r_{g,t} P_{ ext{th},g}^{ ext{max}} \\
&r_{g,t} – r_{g,t-1} = u_{g,t} – d_{g,t} \\
&u_{g,t} + d_{g,t} \leq 1 \\
&\sum_{k=t}^{T_S} (1 – r_{g,k}) \geq T_S (r_{g,t-1} – r_{g,t}) \\
&|p_{ ext{th},g,t+1} – p_{ ext{th},g,t}| \leq s_{u,g} \\
&\sum_{g} p_{ ext{th},g,t} + \sum_{w} p_{ ext{wind},w,t} = \sum_{i} p_{ ext{vpp},i,t} + \sum_{m} p_{ ext{load},m,t} \\
&p_{l,t} = \sum F_l (p_{ ext{th},g,t} + p_{ ext{wind},w,t} – p_{ ext{load},m,t} – p_{ ext{vpp},i,t}) \\
&-P_{l}^{ ext{max}} \leq p_{l,t} \leq P_{l}^{ ext{max}}
\end{aligned} $$
where $u_{g,t}$ and $d_{g,t}$ are startup/shutdown variables, $p_{ ext{th},g,t}$ is thermal generation, and $p_{ ext{vpp},i,t}$ is the power exchange with VPPs.
The VPP scheduling model minimizes energy costs:
$$ \min \sum_{i=1}^{N} \sum_{t=1}^{T} C_t p_{t}^{s,i} $$
with constraints:
$$ \begin{aligned}
&0 \leq p_{t}^{ ext{pv},i} \leq P_{t}^{ ext{APV}} \\
&0 \leq p_{t}^{ ext{MT},i} \leq P_{t}^{ ext{MT}} \\
&p_{t}^{ ext{vpp},i} = p_{t}^{ ext{cha},i} + p_{t}^{ ext{load},i} – p_{t}^{ ext{dis},i} – p_{t}^{ ext{pv},i} – p_{t}^{ ext{MT},i} \\
&0 \leq p_{t}^{ ext{vpp},i} \leq p^{ ext{limit}}
\end{aligned} $$
Electric vehicle optimization for each vehicle $v$:
$$ \min \sum_{t=T_s}^{T} C_t (p_{v,t}^{ ext{cha}} – p_{v,t}^{ ext{dis}}) $$
subject to:
$$ \begin{aligned}
&0 \leq p_{v,t}^{ ext{cha}} \leq \alpha_v^c P_{v}^{ ext{cmax}} \\
&0 \leq p_{v,t}^{ ext{dis}} \leq \alpha_v^d P_{v}^{ ext{dmax}} \\
&\alpha_v^d + \alpha_v^c \leq 1 \\
&S_t = S_{t-1} + \frac{1}{E_v^{ ext{max}}} p_{v,t}^{ ext{cha}} \eta_v^c – \frac{1}{E_v^{ ext{max}}} \frac{p_{v,t}^{ ext{dis}}}{\eta_v^d} \\
&S_{T_v^a} = S_v^a \\
&S_{T_v^d} \geq S_v^d \\
&0 \leq S_t \leq S^{ ext{max}}
\end{aligned} $$
These models are solved distributively using Lagrangian relaxation, as detailed in the next section.
Lagrangian Relaxation and Perturbation Function Method
Lagrangian relaxation is employed to decouple the multi-level optimization problem. The transmission grid-VPP coordination uses the Lagrangian function:
$$ \begin{aligned}
L(P, \lambda) &= \sum_{t} \sum_{g} \left[ c_{u,g} u_{g,t} + c_{d,g} d_{g,t} + a_{ ext{th},g} p_{ ext{th},g,t} + b_{ ext{th},g} (p_{ ext{th},g,t})^2 \right] \\
&\quad + \sum_{i} \sum_{t} c_{ ext{MT}} p_{t}^{ ext{MT},i} + \sum_{t} \lambda_t \left( \sum_{i} p_{t}^{ ext{vpp},i} – \sum_{g} p_{ ext{th},g,t} – \sum_{w} p_{ ext{wind},w,t} \right)
\end{aligned} $$
The dual function is:
$$ D(\Lambda) = \sum_{t} \sum_{g} \left[ c_{u,g} u_{g,t} + c_{d,g} d_{g,t} + a_{ ext{th},g} p_{ ext{th},g,t} + b_{ ext{th},g} (p_{ ext{th},g,t})^2 – \lambda_t p_{ ext{th},g,t} \right] – \sum_{t} \lambda_t \sum_{w} p_{ ext{wind},w,t} + \sum_{i} \sum_{t} V_t^i (\lambda_t) $$
where $V_t^i(\lambda_t)$ represents the VPP’s energy trading cost. The Lagrangian multipliers $\lambda_t$ are updated via the subgradient method:
$$ \lambda^{(k+1)} = \max \left\{ \lambda^{(k)} + \alpha^{(k)} \frac{g^{(k)}}{||g^{(k)}||_1}, 0 \right\} $$
with step size $\alpha^{(k)} = \frac{1}{w_1 k + w_2}$ ensuring convergence.
For VPP-EV coordination, the Lagrangian function is:
$$ D(\mu) = \sum_{i=1}^{N} \sum_{t=1}^{T} S_t^i – \sum_{t=1}^{T} \mu_t \left( p_{t}^{ ext{cha},i} + p_{t}^{ ext{load},i} – p_{t}^{ ext{dis},i} – p_{t}^{ ext{pv},i} \right) $$
To address parameter consistency, a perturbation function $\delta(v,t)$ is introduced to slightly alter parameters such as charging efficiency $\eta_v^c$ or discharging efficiency $\eta_v^d$ within bounds that preserve the optimal basis. The perturbed efficiency becomes:
$$ \eta_v^{c, ext{new}} = \eta_v^c + \delta(v,t) \sigma_{\eta} $$
where $\sigma_{\eta}$ is the perturbation variance, and $\delta(v,t)$ satisfies:
$$ \sum_{t} \delta(v,t) = 0, \quad \sum_{v} \delta(v,t) = 0 $$
This ensures fairness across time and vehicles. The perturbation limits are derived from sensitivity analysis in linear programming. For a linear program in standard form $\min \{ c^T x : Ax = b, x \geq 0 \}$, the optimal basis remains unchanged if perturbations in $c$, $b$, or $A$ satisfy:
For objective coefficient $c_j$ (e.g., electricity price $C_t$):
$$ \max \left\{ \frac{c_j – z_j}{y_{rj}} \mid y_{rj} < 0 \right\} \leq \Delta C_t \leq \min \left\{ \frac{c_j – z_j}{y_{rj}} \mid y_{rj} > 0 \right\} $$
For right-hand side coefficient $b_r$ (e.g., max power $P_{ ext{max}}$):
$$ \max \left\{ -\frac{\bar{b}_i}{(B^{-1})_{ir}} \mid (B^{-1})_{ir} > 0 \right\} \leq \Delta b_r \leq \min \left\{ -\frac{\bar{b}_i}{(B^{-1})_{ir}} \mid (B^{-1})_{ir} < 0 \right\} $$
For technical coefficient $a_{ij}$ (e.g., efficiency $1/\eta$):
$$ \max \left\{ \frac{c_j – z_j}{q_k} \mid q_k > 0 \right\} \leq \Delta a_{ij} \leq \min \left\{ \frac{c_j – z_j}{q_k} \mid q_k < 0 \right\} $$
where $z_j$ is the reduced cost, $y_{rj}$ is the pivot element, and $q_k$ is the shadow price. Perturbing efficiency parameters proves most effective, as it directly breaks symmetry without significantly affecting Lagrangian updates.
Case Studies and Results
Case studies are conducted using a modified IEEE 30-node transmission system and real-world electric vehicle data from Shenzhen. The system includes three VPPs with varying resources: VPP1 has 15 EV charging stations (100 chargers each), VPP2 has 5 stations, and VPP3 has 30 stations. Micro-turbine capacities and photovoltaic generation are set accordingly. Scheduling is simulated over 24 hours with 1-hour intervals, though the method applies to shorter scales like 15-minute dispatch.
The table below summarizes VPP configurations:
| Virtual Power Plant | Point of Common Coupling | Micro-Turbine Capacity | EV Charging Stations |
|---|---|---|---|
| VPP1 | Node 10 | 12 MW | 15 |
| VPP2 | Node 13 | 24 MW | 5 |
| VPP3 | Node 26 | 6 MW | 30 |
Thermal generator parameters are:
| Generator ID | Minimum Output (MW) | Maximum Output (MW) |
|---|---|---|
| G1 | 30 | 80 |
| G2 | 30 | 80 |
| G3 | 15 | 50 |
| G4 | 20 | 55 |
| G5 | 10 | 30 |
| G6 | 10 | 40 |
Without perturbation, the objective function oscillates significantly during iteration, failing to converge within practical limits. With perturbation applied to charging efficiencies ($\eta_v^c$ and $\eta_v^d$), convergence accelerates by over 40%, achieving stability within allowed error bounds. The following table compares computation times for different system scales:
| Case Study | Decentralized Scheduling Time (s) | Perturbed Scheduling Time (s) |
|---|---|---|
| IEEE 30-node (EV Coordination) | 94.055 | 51.107 |
| IEEE 30-node (Transmission Grid) | 1056.348 | 486.121 |
| IEEE 79-node (EV Coordination) | 149.414 | 68.682 |
| IEEE 79-node (Transmission Grid) | 3768.555 | 845.353 |
Orderly charging via VPP scheduling shifts electric vehicle loads to off-peak periods (e.g., late night) or high photovoltaic output times (e.g., midday), reducing peak demand and improving grid economics. In contrast, uncontrolled charging exacerbates evening peaks. The graph below illustrates daily charging/discharging profiles for two scheduled EVs (EV1, EV2) versus one unscheduled EV (EV3), showing how VPP control flattens the load curve.
Sensitivity analysis confirms that perturbation effectiveness depends on the parameter altered. Perturbing efficiencies yields the fastest convergence across various electric vehicle scales, while perturbing objective coefficients or power limits has diminishing returns as EV numbers grow. Different perturbation functions (e.g., Gaussian vs. uniform) with variances $\sigma = 10$ or $5$ produce similar acceleration, indicating robustness within derived limits.
Conclusion
This paper presents a cloud-edge collaborative scheduling method for virtual power plants that addresses parameter consistency in electric vehicle clusters. The Lagrangian relaxation framework enables efficient distributed optimization, while the perturbation function method breaks symmetry to eliminate oscillations, accelerating convergence by over 40% without compromising solution quality. Theoretical perturbation limits ensure optimal basis preservation, and case studies using China EV data from Shenzhen demonstrate scalability and effectiveness. Future work will explore adaptive step sizes in Lagrangian algorithms and multi-level scheduling for enhanced vehicle-grid integration. The approach significantly contributes to managing the growing impact of electric vehicles in China’s power systems, promoting stability and economic efficiency.