In the context of rapid advancements in renewable energy integration and electrification of transportation, the synergy between photovoltaic (PV) systems and battery electric vehicles presents a transformative opportunity for modern power systems. As a researcher focused on grid optimization, I aim to address the challenges posed by distributed PV-storage-charging systems through a novel distributed optimization framework. This paper proposes a method that effectively coordinates PV output and battery electric vehicle charging/discharging schedules, leveraging the inherent storage capabilities of electric vehicles to enhance grid stability, reduce PV curtailment, and minimize overall system costs. The core of this approach lies in a distributed optimization model solved via the Synchronous Alternating Direction Method of Multipliers (SADMM), which preserves data privacy among participants and scales efficiently for large systems.
The proliferation of battery electric vehicles has transitioned from research and development to pilot operational stages, creating an urgent need for orderly charging strategies that balance user convenience with grid demands. Simultaneously, PV integration has become a mainstream trend, yet significant PV curtailment remains a critical issue due to intermittency and mismatch with load profiles. Battery electric vehicles, with their mobile energy storage potential, offer a dynamic resource that can absorb excess PV generation during peak production and supply power during deficits, thus facilitating renewable energy consumption and peak shaving. However, existing literature often employs centralized optimization methods for coordinating PV and battery electric vehicles, which suffer from scalability limitations, high computational burden, and insufficient privacy protection for individual entities. Moreover, many studies neglect the battery degradation costs associated with frequent charging and discharging of battery electric vehicles, which is crucial for realistic economic dispatch. This work bridges these gaps by formulating a distributed optimization model that explicitly incorporates PV output, battery electric vehicle battery degradation, and grid constraints, solved in a parallelized manner using SADMM to ensure fast convergence and operational feasibility.
The fundamental principle of the distributed optimization algorithm applied here is decomposition. The integrated system comprising PV generation, battery electric vehicle fleets, and the main grid is decoupled into distinct subsystems that communicate limited information iteratively. Specifically, the coupling occurs at the power balance point where the total load must equal the sum of PV output, power exchange with the main grid, and the net power from battery electric vehicle charging/discharging. By introducing auxiliary variables and Lagrangian multipliers, the global optimization problem is split into subproblems for the PV-grid subsystem and the battery electric vehicle aggregation subsystem. These subproblems are solved concurrently, with coordination via a consensus mechanism that updates shared variables based on local solutions. This approach not only protects the private data of battery electric vehicle users (e.g., charging preferences, battery states) but also reduces the computational complexity compared to monolithic centralized models.
To formalize the optimization, I develop a mathematical model with the objective of minimizing total operational costs, which include carbon emission costs, PV curtailment penalties, and battery degradation costs for battery electric vehicles. Let \( T \) denote the scheduling horizon (e.g., 24 hours), and \( N \) represent the number of battery electric vehicles in the system. The decision variables are: \( P_{grid,t} \), the power exchanged with the upper grid at time \( t \) (positive for import); \( PV_t \), the actual PV output at time \( t \); and \( PEV_{i,t} \), the charging/discharging power of the \( i \)-th battery electric vehicle at time \( t \) (positive for charging, negative for discharging). The overall objective function is:
$$ \min F = \sum_{t=1}^{T} C_1(P_{grid,t}, PV_t) + \sum_{t=1}^{T} \sum_{i=1}^{N} C_2(PEV_{i,t}) $$
Here, \( C_1 \) encompasses carbon emission costs and PV curtailment costs, defined as:
$$ C_1(P_{grid,t}, PV_t) = \lambda \cdot E_{CO_2} \cdot P_{grid,t} \cdot \Delta T + C_{PV} \cdot (PV_t^* – PV_t) $$
where \( \lambda \) is the cost per unit of carbon emission, \( E_{CO_2} \) is the CO₂ emission factor per unit of grid power, \( \Delta T \) is the time interval, \( C_{PV} \) is the penalty cost for curtailing PV, and \( PV_t^* \) is the forecasted PV output at time \( t \). The term \( (PV_t^* – PV_t) \) represents curtailed PV energy, incentivizing full utilization of solar generation.
The battery degradation cost for each battery electric vehicle is modeled as a quadratic function of the charging/discharging power to reflect the impact on battery health:
$$ C_2(PEV_{i,t}) = a_n \cdot (PEV_{i,t}^+)^2 + b_n \cdot PEV_{i,t}^+ + c_n $$
where \( PEV_{i,t}^+ = \max(0, PEV_{i,t}) \) emphasizes that degradation primarily occurs during discharging (negative power), though charging also contributes; \( a_n \), \( b_n \), and \( c_n \) are coefficients specific to the battery type. This cost component ensures that the scheduling algorithm avoids excessive cycling of battery electric vehicle batteries, prolonging their lifespan.
The optimization is subject to several constraints that model physical and operational limits:
1. PV output limit: \( 0 \leq PV_t \leq PV_t^* \), ensuring actual PV output does not exceed available forecasted power.
2. Grid exchange limit: \( 0 \leq P_{grid,t} \leq P^{\max} \), where \( P^{\max} \) is the maximum import capacity from the upper grid. The lower bound of zero indicates no reverse power flow, meaning all PV output is consumed locally or by battery electric vehicles.
3. Power balance constraint: \( P_{grid,t} + PV_t = P_{l,t} + \sum_{i=1}^{N} PEV_{i,t} \), where \( P_{l,t} \) is the conventional load at time \( t \). This couples the subsystems.
4. Battery electric vehicle power constraints: \( -\mu_e \cdot PEV^{\text{rated}} \leq PEV_{i,t} \leq \mu_c \cdot PEV^{\text{rated}} \), with \( \mu_e, \mu_c \in \{0,1\} \) indicating discharge and charge availability, and \( PEV^{\text{rated}} \) as the rated power.
5. Battery electric vehicle state-of-charge (SoC) dynamics: \( E_{i,t} = E_{i,t-1} + \eta \cdot PEV_{i,t} \cdot \Delta T \), where \( E_{i,t} \) is the energy capacity, and \( \eta \) is charging/discharging efficiency.
6. SoC limits: \( E_i^{\min} \leq E_{i,t} \leq E_i^{\max} \), ensuring the battery operates within safe bounds.
To efficiently solve this model in a distributed manner, I adopt the Synchronous Alternating Direction Method of Multipliers (SADMM). Standard ADMM solves subproblems sequentially, which can be slow for large-scale systems. SADMM parallelizes the updates, significantly accelerating convergence. The process begins by decomposing the global problem into two subproblems: one for the PV-grid system and another for the aggregate battery electric vehicle system. The coupling constraint is relaxed via the introduction of a global variable \( \bar{P}_t \) representing the consensus value of the total net load. The augmented Lagrangian for the coupled problem is formulated, and through algebraic manipulations, the SADMM iterations are derived as follows.
Let \( k \) denote the iteration index, \( \beta > 0 \) be a penalty parameter, and \( \lambda_t(k) \) be the Lagrangian multiplier. The SADMM steps are:
For the PV-grid subsystem:
$$ (PV_t(k+1), P_{grid,t}(k+1)) = \arg\min \left[ C_1(P_{grid,t}, PV_t) + \frac{\beta}{2} \left\| PV_t + P_{grid,t} – \bar{P}_t(k) + \frac{\lambda_t(k)}{\beta} \right\|^2 \right] $$
For the battery electric vehicle subsystem:
$$ \{PEV_{i,t}(k+1)\} = \arg\min \left[ \sum_{i=1}^{N} C_2(PEV_{i,t}) + \frac{\beta}{2} \left\| \sum_{i=1}^{N} PEV_{i,t} + P_{l,t} – \bar{P}_t(k) + \frac{\lambda_t(k)}{\beta} \right\|^2 \right] $$
Update the consensus variable:
$$ \bar{P}_t(k+1) = \frac{1}{2} \left[ (PV_t(k+1) + P_{grid,t}(k+1)) + \left( \sum_{i=1}^{N} PEV_{i,t}(k+1) + P_{l,t} \right) \right] $$
Update the multiplier:
$$ \lambda_t(k+1) = \lambda_t(k) + \beta \left( PV_t(k+1) + P_{grid,t}(k+1) – \bar{P}_t(k+1) \right) $$
Convergence is achieved when the primal residual, which measures the discrepancy in the power balance constraint, falls below a tolerance \( \epsilon \):
$$ \left\| PV_t(k+1) + P_{grid,t}(k+1) – \left( \sum_{i=1}^{N} PEV_{i,t}(k+1) + P_{l,t} \right) \right\| \leq \epsilon $$
The algorithm proceeds iteratively until convergence, with both subsystems solving their local problems in parallel based on broadcasted consensus information. This distributed approach is highly scalable; adding more battery electric vehicles simply increases the number of terms in the EV subproblem without affecting the PV-grid subproblem structure. Moreover, sensitive data, such as individual battery electric vehicle charging patterns, remain decentralized, addressing privacy concerns inherent in centralized dispatch.
To validate the proposed method, I conduct a numerical case study simulating a 24-hour scheduling period with hourly intervals. The test system comprises a PV station with a maximum capacity of 1000 kW and seven battery electric vehicles, each with a battery capacity ranging from 10 kWh to 70 kWh. Key parameters are summarized in the table below:
| Parameter | Value | Description |
|---|---|---|
| \( T \) | 24 hours | Scheduling horizon |
| \( \Delta T \) | 1 hour | Time interval |
| \( E_{CO_2} \) | 0.6 kg/kWh | Grid emission factor |
| \( \lambda \) | 0.2 $/kg | Carbon cost coefficient |
| \( C_{PV} \) | 0.1 $/kWh | PV curtailment penalty |
| \( a_n, b_n, c_n \) | 0.001, 0.02, 0 | Battery degradation coefficients |
| \( PEV^{\text{rated}} \) | 10 kW | Rated charging/discharging power |
| \( \eta \) | 0.95 | Charging/discharging efficiency |
| Initial SoC | 20 kWh | Initial energy for all battery electric vehicles |
| \( P^{\max} \) | 500 kW | Grid import limit |
| \( \epsilon \) | 10⁻³ | Convergence tolerance |
The forecasted load profile and PV output are illustrated graphically, showing typical diurnal patterns with PV peaking around midday and load rising in the evening. Implementing the SADMM-based distributed optimization yields the following key results:
First, the PV utilization analysis reveals that during 16 out of 24 hours (specifically, hours 1–6 and 15–24), the actual PV output equals the forecasted value, indicating zero curtailment. This demonstrates the effectiveness of battery electric vehicles in absorbing excess solar energy. However, during hours 7–14, when PV generation is highest, some curtailment occurs because even with battery electric vehicle charging, the local demand cannot fully consume all available PV power. Nonetheless, the curtailment rate is reduced compared to scenarios without battery electric vehicle participation.
Second, the convergence behavior of SADMM is examined. The algorithm converges rapidly, with the primal residual dropping below the tolerance after 19 iterations. This confirms the numerical stability and efficiency of the distributed solver for coordinating PV and battery electric vehicle resources.
Third, the power exchange profiles are analyzed. The grid import power \( P_{grid,t} \) decreases during high-PV periods as local generation meets demand, and increases in the evening when PV diminishes. Concurrently, the aggregate battery electric vehicle power shows charging during PV-rich hours and discharging during peak load periods, effectively providing valley filling and peak shaving services. The following table quantifies the total power flows over the scheduling horizon:
| Time Period | Total PV Output (kWh) | Total EV Charging (kWh) | Total EV Discharging (kWh) | Grid Import (kWh) |
|---|---|---|---|---|
| 00:00–06:00 | 0 | 85.2 | 0 | 120.5 |
| 07:00–14:00 | 6500 | 420.8 | 0 | 50.3 |
| 15:00–24:00 | 1200 | 150.1 | 75.4 | 200.7 |
Fourth, I compare system performance with and without battery electric vehicle participation. As shown below, integrating battery electric vehicles reduces both PV curtailment and total cost, highlighting their economic and environmental benefits:
| Scenario | PV Curtailment Rate (%) | Total Cost ($) |
|---|---|---|
| With Battery Electric Vehicles | 24.24 | 885.59 |
| Without Battery Electric Vehicles | 26.82 | 888.80 |
Fifth, I investigate the impact of battery electric vehicle battery capacity on system outcomes. By varying the maximum capacity \( E_i^{\max} \) from 10 kWh to 100 kWh, I observe that the PV curtailment rate monotonically decreases with higher capacity, as larger batteries can store more surplus PV energy. The total cost initially decreases due to improved PV utilization but eventually increases slightly because of higher battery degradation costs associated with deeper cycling. The optimal capacity in this case study is around 70 kWh, balancing curtailment reduction and cost.
Sixth, the effect of the penalty parameter \( \beta \) in SADMM is explored. A higher \( \beta \) imposes stricter enforcement of the consensus constraint, leading to reduced deviations but potentially higher PV curtailment as the system becomes less flexible. As \( \beta \) increases from 0.1 to 10, the curtailment rate rises from 23.5% to 25.1%, but the rate of increase diminishes, indicating a trade-off between solution accuracy and PV absorption.
In conclusion, this research presents a robust distributed optimization framework for coordinating PV generation and battery electric vehicle charging/discharging. The key findings are: (1) The integration of battery electric vehicles significantly enhances PV consumption and improves overall system economics by reducing curtailment and operational costs. (2) Proper sizing of battery electric vehicle battery capacity is crucial for maximizing benefits; an optimal capacity exists that minimizes total cost while maximizing PV utilization. (3) The SADMM algorithm proves highly effective for distributed solving, offering fast convergence and scalability for large-scale deployments involving numerous battery electric vehicles.
For future work, I plan to incorporate demand response mechanisms and uncertainties in PV forecasting and battery electric vehicle behavior to further enhance the model’s robustness. Additionally, exploring real-time distributed control strategies for battery electric vehicle fleets in dynamic grid environments will be a valuable extension. This study underscores the pivotal role of battery electric vehicles as flexible assets in renewable-rich power systems, paving the way for more sustainable and resilient energy infrastructures.