Coordinated Optimization Scheduling of Electric Vehicle Charging and Discharging Considering Wind-Solar Uncertainty

1. Introduction and Research Context

In the contemporary landscape of global energy consumption, fossil fuels continue to hold a dominant position. However, the inherent finiteness of these traditional energy resources, combined with the escalating severity of environmental pollution, has necessitated a fundamental shift toward cleaner and more sustainable energy alternatives. The increasing penetration of renewable energy sources, particularly wind and photovoltaic (PV) power, is a defining feature of the ongoing energy transition. In parallel, propelled by national strategies for carbon peak and carbon neutrality, the electric vehicle industry has experienced explosive growth. Electric vehicles, with their significant potential for energy conservation and emission reduction, are progressively reshaping the conventional transportation system.

The large-scale integration of renewable energy generation into the power grid, alongside the proliferation of electric vehicles, presents unprecedented challenges to the secure, stable, and economical operation of power systems. The output powers of wind turbines and PV panels are characterized by significant randomness, volatility, and intermittency, which pose serious threats to grid security. Simultaneously, the uncoordinated charging behaviors of large numbers of electric vehicles can dramatically amplify peak-valley differences in the load profile, as illustrated in my research using Monte Carlo simulation. The charging demands of a substantial fleet of electric vehicles tend to cluster during specific periods, such as the evening hours after commuting, thereby superimposing a new peak upon the existing residential load peak. This concurrent rise in demand not only intensifies the difficulty of peak load management but also increases the operational pressure on conventional generation units.

Recognizing the urgent need to address these interrelated issues, my research focuses on the development of coordinated optimization scheduling strategies for electric vehicle charging and discharging, with the dual objectives of facilitating the accommodation of wind and solar energy while ensuring the economic and stable operation of the power grid. A comprehensive review of the existing literature reveals several key research gaps. Prior studies often adopted decomposed approaches to model wind and solar uncertainty, without adequately capturing the spatial and temporal correlations between these two renewable sources. Furthermore, many optimization frameworks prioritized single objectives, such as minimizing operational costs, without sufficiently incorporating metrics related to system stability, renewable energy accommodation rate, or multi-stakeholder interests. In addition, some earlier research overly focused on the perspective of the power grid, occasionally overlooking the behavioral response and economic benefits of electric vehicle users. To overcome these limitations, I have developed a multi-pronged approach in this work. First, I present a multi-objective optimization scheduling model that integrates a dynamic time-of-use pricing mechanism to coordinate electric vehicle charging and discharging. This model is designed to simultaneously minimize the equivalent load peak-valley difference and the total operating cost for electric vehicle users. Second, I propose a bi-level optimization scheduling framework based on scenario generation to manage the uncertainty associated with wind and PV output, aiming to harmonize renewable energy accommodation with the secure and economic operation of the distribution network. The overarching goal is to enhance grid operational efficiency and increase the penetration level of clean energy.

2. Mathematical Modeling of Wind Power, PV Generation, and EV Load

Accurate modeling of the fundamental components, including renewable generation units and electric vehicle charging loads, constitutes the bedrock for subsequent optimization efforts. My analysis begins with the physical and probabilistic models governing wind and PV power output.

For wind power, the power output is fundamentally dependent on the wind speed. A two-parameter Weibull distribution is widely accepted for describing the probability density function of the wind speed \(v\), which is given by:

\[
f_p(v) = \frac{k}{c} \left( \frac{v}{c} \right)^{k-1} \exp \left[ – \left( \frac{v}{c} \right)^k \right]
\]

Here, \(k\) represents the shape parameter, and \(c\) is the scale parameter. The actual electrical power output, \(P_p\), from a wind turbine is a piecewise function of the wind speed, determined by parameters such as the cut-in speed \(v_{in}\), rated speed \(v_n\), and cut-out speed \(v_{out}\), as follows:

\[
P_p = \begin{cases} 0 & v < v_{in} \text{ or } v > v_{out} \\ \frac{v^2 – v_{in}^2}{v_n^2 – v_{in}^2} P_{wn} & v_{in} \le v \le v_n \\ P_{wn} & v_n < v \le v_{out} \end{cases}
\]

where \(P_{wn}\) is the rated power. For photovoltaic systems, the stochastic nature of solar irradiance is effectively modeled using a Beta distribution, for which the probability density function of irradiance \(l\) in a specific period is:

\[
f(l) = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)} \left( \frac{l}{l_{max}} \right)^{\alpha – 1} \left( 1 – \frac{l}{l_{max}} \right)^{\beta – 1}
\]

In this equation, \(\alpha\) and \(\beta\) are the shape parameters of the Beta distribution, and \(l_{max}\) is the maximum irradiance. The output power of a PV array, \(P_{PV}\), is subsequently calculated by:

\[
P_{PV} = \begin{cases} P_{km} \cdot \frac{r}{r_k} & 0 \le r < r_k \\ P_{km} & r \ge r_k \end{cases}
\]

where \(r\) is the solar irradiance, \(r_k\) is the rated irradiance, and \(P_{km}\) is the rated power of the PV array.

The modeling of electric vehicle charging behavior requires a probabilistic approach to capture the inherent randomness of driver behaviors. In my work, modeling is based on daily travel statistics, specifically the arrival time to the grid, departure time from the grid, and daily driving mileage. The travel times and distances follow established probability distributions, such as the normal distribution for arrival and departure times and the lognormal distribution for daily mileage. The probability density functions for arrival time \(t_x\) and departure time \(t_y\) are formulated as:

\[
f_a(t_x) = \begin{cases} \frac{1}{\sigma_a \sqrt{2\pi}} \exp \left[ -\frac{(t_x – \mu_a)^2}{2\sigma_a^2} \right] & 0 \le t_x < 12 \\ \frac{1}{\sigma_a \sqrt{2\pi}} \exp \left[ -\frac{(t_x – \mu_a – 24)^2}{2\sigma_a^2} \right] & 12 \le t_x \le 24 \end{cases}
\]

\[
f_b(t_y) = \begin{cases} \frac{1}{\sigma_b \sqrt{2\pi}} \exp \left[ -\frac{(t_y – \mu_b)^2}{2\sigma_b^2} \right] & 0 \le t_y < 12 \\ \frac{1}{\sigma_b \sqrt{2\pi}} \exp \left[ -\frac{(t_y – \mu_b – 24)^2}{2\sigma_b^2} \right] & 12 \le t_y \le 24 \end{cases}
\]

Similarly, the probability density function for driving distance \(s\) is given by a lognormal distribution. Based on these distributions, my research employed a Monte Carlo simulation framework to forecast the aggregate charging load profile of a large fleet of electric vehicles. The investigation into different fleet sizes, such as 500, 1000, and 2000 electric vehicles, demonstrated that uncoordinated charging results in a significant concentration of load in the evening hours, with a peak-valley difference ratio that can be as high as 3 to 4 times, thereby underscoring the need for coordinated control mechanisms.

3. Multi-Objective Optimization Scheduling Model with Dynamic Time-of-Use Pricing

To steer electric vehicle charging and discharging behaviors, I proposed a multi-objective optimization scheduling model underpinned by a dynamic time-of-use pricing mechanism. Static pricing models often lack the accuracy needed to reflect the real-time fluctuation of renewable energy generation and load demand. Thus, my research formulates a dynamic pricing strategy. Building upon the day-ahead forecast of load, the optimization first determines a static time-of-use tariff, delineating peak, flat, and valley periods, which is then dynamically updated at intervals (for instance, every 15 minutes) to align pricing signals with real-time system conditions, including the availability of wind and solar power.

Within this pricing framework, the response of electric vehicle users is modeled through a price elasticity matrix, \(E_n\). The self-elasticity coefficient \(\varepsilon_{ii}\) and cross-elasticity coefficient \(\varepsilon_{ij}\) capture how a change in electricity price in period \(i\) affects demand in the same period or in a different period \(j\), respectively. The matrix of elasticity is defined as:

\[
E_n = \begin{bmatrix} \varepsilon_{11} & \varepsilon_{12} & \cdots & \varepsilon_{1n} \\ \varepsilon_{21} & \varepsilon_{22} & \cdots & \varepsilon_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ \varepsilon_{n1} & \varepsilon_{n2} & \cdots & \varepsilon_{nn} \end{bmatrix}
\]

The strategic scheduling of electric vehicle charging and discharging is then framed as a multi-objective optimization problem. Two principal objectives are defined. The first objective \(F_1\) aims to minimize the mean square value of the equivalent load peak-valley difference of the microgrid. This objective effectively smooths the overall load curve by incorporating the load from electric vehicles with the base load. The function is:

\[
F_1 = \min \frac{1}{N_{n}} \sum_{n=1}^{N} \left( P_{n-1}^{charge} + P_{al,EV} – P_{al,av}^{n-1} \right)^2
\]

The second objective \(F_2\) focuses on minimizing the total charging and discharging cost for electric vehicle users. This ensures that the scheduling strategy yields economic benefits for the participating vehicle owners. The function is:

\[
F_2 = \min \sum_{t=1}^{24} \left( P_c(t) \cdot r_c(t) – P_d(t) \cdot r_d(t) + P(t) \cdot r_b \right) \Delta t
\]

The model incorporates a suite of constraints, including the State of Charge levels, charging/discharging power limits, and limits on total energy traded, ensuring system safety and user battery protection.

This model was solved using a Niche-based Multi-Objective Particle Swarm Optimization algorithm, which enhances the standard Particle Swarm Optimization by incorporating a niche technique to preserve population diversity and avoid premature convergence, allowing for a more comprehensive exploration of the Pareto frontier. The simulation results for a microgrid model with 2000 EVs demonstrate the effectiveness of my proposed strategy. The resulting optimal schedule of electric vehicle charging and discharging was sensitive to the dynamic pricing signals.

The results indicated that the strategy successfully guided electric vehicles to charge preferentially during periods of high renewable energy generation and low electricity prices, such as the midday solar peak and the early morning wind peak, and to discharge during peak load periods. This behavior significantly enhanced the accommodation of wind power and PV energy and reduced the fluctuations in the net load profile. The solar energy and wind power output curves and the consumption curves were observed to be well-aligned after optimization, confirming the capacity of electric vehicles to act as flexible resources.

4. Scenario-Based Bi-Level Optimization for Grid-Connected EVs

Given the pronounced uncertainty of large-scale wind and solar power integration, I extended my research to formulate a scenario-based bi-level optimization scheduling model. This framework is designed to systematically manage the randomness of renewable generation while optimizing the operation of a distribution network with numerous connected electric vehicles.

The first stage involves the generation and reduction of representative scenarios to provide a robust input for the optimization model. To accurately depict the correlation between wind and PV power outputs, I employed a Frank-Copula function to establish their joint probability distribution, based on historical data of wind speed and solar irradiance. This approach is suitable since Frank-Copula adeptly models both positive and negative correlations, which cannot be achieved by all types of Copula functions. The dependence structure is constructed via its generating function, and the generic form of a Frank-Copula is:

\[
C(u_1, u_2) = -\frac{1}{\theta} \ln \left( 1 + \frac{(e^{-\theta u_1} – 1)(e^{-\theta u_2} – 1)}{e^{-\theta} – 1} \right)
\]

Using a kernel density estimation method for the marginal distributions of wind and solar power across different time periods, combined with the Frank-Copula function, a large set of 1000 correlated wind-solar output scenarios was generated via Monte Carlo sampling. This scenario set comprehensively reflects the diverse possible states of renewable generation and its stochastic dependency.

To manage the computational complexity in the scheduling problem, this large set of scenarios is reduced to a smaller set of representative scenarios using a modified clustering algorithm. I employed a Rough k-means algorithm that integrates principles of rough set theory to handle overlapping clusters and improve the robustness of the clustering result. The ultimate result was the selection of 6 typical scenarios from the original 1000 that encompass the spectra of both high and low wind-solar output conditions.

The bi-level scheduling model is constructed to handle the distinct yet interrelated objectives of different stakeholders. Upper-level model focuses on the orderly scheduling of electric vehicle charging in a charging station. Here, the objective is to maximize the profitability of the charging station operator, which is equivalent to minimizing the negative of its profit. This is mathematically expressed as:

\[
F_{EV} = \min \sum_{t=1}^{T} \sum_{i=1}^{N_{EV}} S_{i,t} \cdot ( \lambda_{EV,t} – g_{EV,t} ) \cdot P_{EV} \cdot \Delta t
\]

where \(S_{i,t}\) represents the charging state of vehicle \(i\) at time \(t\), \(g_{EV,t}\) is the electricity price charged to the users, and \(\lambda_{EV,t}\) is the price paid to procuring power from the upstream grid. This formulation includes constraints on the user’s daily energy requirement to ensure the vehicle’s battery reaches a desired state of charge before departure, and a limit on the maximum system load to prevent overload. This scheduling problem is efficiently solved using Lagrangian relaxation. The upper-level optimization determines a charging schedule that shifts EV load to lower price periods, minimizing peak load.

Lower-level model receives the optimized charging plan from the upper level and further optimizes the operation of the distribution network. The primary goal of the lower-level model is to minimize the overall operational cost of the distribution system, which is a quadratic problem encompassing multiple components. It includes the cost of network losses (\(C_1\)), penalty costs for violating safety constraints like node voltage limits and branch line capacity (\(C_2\)), the cost associated with adjusting Distributed Generation outputs (\(C_3\)), the cost of purchasing power from the main grid (\(C_4\)), and the revenue from selling electricity to the electric vehicle charging station (\(C_5\)). The objective function is:

\[
F = \min \left( C_1 + C_2 + C_3 + C_4 – C_5 \right)
\]

The lower layer involves extensive operational constraints, including power flow equations, node voltage limits, line capacity constraints, and distributed generation output limits. This layer ensures that the distribution network maintains its technical viability, voltage stability, and economic efficiency. To solve this complex, multi-constraint problem, I employed an enhanced genetic algorithm that integrates the simulated annealing technique and its probabilistic jumping property to evade local optima. The algorithm has been applied to a modified IEEE 33-node test system, where electric vehicles are integrated at specified nodes with wind and PV plants.

5. Results and Case Studies from the Bi-Level Scheduling Model

In my simulation, a load scenario was analyzed, and the outcomes were compared between uncoordinated and coordinated charging modes. The results clearly illustrate that when no coordination strategy is implemented, the uncoordinated charging behavior leads to substantial peak load; the load characteristic can be improved by adopting the proposed control. The charging profile of the orderly scheduling is moved to flat or valley periods where the energy price is lower. Meanwhile, the implementation of a time-of-use pricing mechanism acted as a powerful lever for load shifting. The orderly charging strategy contributed to a flattening of the system load curve, and the load characteristic metrics are shown below.

The peak-valley ratio was reduced by 5 percentage points in the simulated operation. This confirms that effective load management of EV charging can significantly improve the stability and operational capacity for renewable accommodation.

When the bi-level optimization for the distribution grid is applied, the results for this specific IEEE 33-bus case study demonstrate clear improvements across all the evaluation metrics. A statistically significant reduction in power loss was observed in the various scenarios after the coordination of distributed generation and electric vehicle loads, and the operational risks, such as voltage limit violations, were able to be maintained within the prescribed safety limits.

Across the six typical scenarios, the cooperative scheduling model was able to achieve a total operating cost reduction of over 10%. The utilization of flexible charging from electric vehicles not only reduced the dependency on peak generation units, but also reduced total network losses by approximately 15%-20%. In scenarios with excess Distributed Generation, the model was effective in curbing renewable curtailment by instructing electric vehicles to increase their consumption. This confirms the pivotal role that electric vehicles play in a renewable-rich distribution network.

6. Conclusions and Future Perspectives

In summary, my research addresses the complex interactions between high-penetration renewable energy sources and the rising uptake of electric vehicles through coordinated optimization strategies. The presented work demonstrates that:

– The proposed orderly and disorderly EV charging models with Monte Carlo-based load forecasting can effectively predict load and reveal that uncoordinated charging amplifies grid peaks.
– The dynamic time-of-use pricing mechanism guided EV users to shift their charging and discharging schedules. The niche-based Multi-Objective Particle Swarm Optimization algorithm effectively minimized the load peak-valley difference by over 17% while also yielding substantial economic benefits for EV users. This coordinative scheduling proves beneficial for enhancing the consumption of wind and solar power resources.
– The adoption of a scenario-based bi-level optimization, leveraging Copula theory for dependency modeling and Rough k-means clustering for scenario reduction, enables effective management of uncertainties in a comprehensive manner. My framework, using a multi-layer optimization approach and an enhanced genetic-simulated annealing solution, has proven to be an efficient strategy.
– Through the cooperative control of EV loads and distributed generation, my methodology improves the operational economics of the distribution network, decreasing system operational costs by 5% to 7%. More importantly, it enhances grid resilience by decreasing network losses and mitigating voltage deviation risks across multiple typical wind/solar fluctuation scenarios.

Looking toward future research, several avenues present themselves to expand upon the current study. The focus of existing models is often on a limited set of objectives; therefore, a more holistic multi-criteria framework that evaluates social convenience, user equity, and battery life degradation could be more relevant in the field. Moreover, robust handling of “black swan” events, such as sudden and extreme weather conditions or massive, correlated user behavior shifts, needs further investigation. Integrating advanced machine learning techniques for forecasting may bolster predictive capabilities and further enhance the responsiveness of electric vehicles. Overall, this research provides a foundation that will empower the transition towards a more advanced, resilient power grid that heavily relies on renewable sources, with electric cars playing a crucial role in its optimization.