With the global push for environmental protection and sustainable development, the “dual carbon” goals have become a common objective worldwide. As a major source of carbon emissions, the transportation industry’s shift toward electrification is a crucial step for low-carbon development. The rapid adoption of electric cars, while offering benefits like zero emissions and high energy efficiency, poses significant challenges to power grids due to uncoordinated charging behaviors. This can exacerbate peak-valley load differences, reduce grid equipment utilization, and complicate carbon emission control. Time-of-use (TOU) pricing, a demand-side management tool, uses price signals to guide user behavior, mitigating adverse impacts from electric car charging. From a low-carbon dispatch perspective, multi-objective optimization of TOU pricing for electric cars not only ensures economical grid operation but also encourages charging during low-carbon periods, reducing emissions and promoting cleaner energy structures. This study delves into this optimization problem, providing theoretical and technical support for synergistic development between electric cars and the grid.
In this article, we analyze the electricity consumption characteristics of electric cars and grid low-carbon dispatch requirements, construct a multi-objective optimization system, propose TOU pricing strategies, and validate them through a case study. We emphasize the integration of economic and environmental goals, leveraging advanced algorithms for solution. The proliferation of electric cars necessitates intelligent grid management to balance costs, user satisfaction, and carbon reduction. Our work aims to address these intertwined challenges, offering a framework that aligns with global sustainability targets.
Analysis of Electric Car Electricity Consumption Characteristics
Electric cars exhibit distinct electricity consumption patterns that complicate grid scheduling. Key characteristics include uncertainty in charging time and location, concentration of charging loads, and diversity in charging capabilities. Users’ varying travel routines lead to charging at disparate times and places—such as home chargers, public stations, or commercial spots—making accurate load forecasting difficult. Charging tends to cluster during specific periods, like evening hours after work, intensifying peak loads. Additionally, charging power varies widely: home chargers typically range from 3-7 kW, while fast chargers can exceed 50 kW, imposing higher demands on grid prediction and调度. Understanding these traits is essential for designing effective pricing strategies that accommodate the unique behavior of electric car users.
To summarize, we present the key characteristics in Table 1, highlighting their impacts on the grid.
| Characteristic | Description | Grid Impact |
|---|---|---|
| Uncertainty in Time/Location | Charging occurs at random times and diverse locations based on user habits. | Increases调度 complexity; complicates load prediction. |
| Load Concentration | Peaks during evening hours (e.g., 20:00-22:00) when users return home. | Amplifies peak-valley differences; risks equipment overloading. |
| Charging Power Diversity | Power levels vary from 3-7 kW (slow) to 50+ kW (fast). | Requires flexible grid management; affects stability. |
These characteristics necessitate a robust optimization approach. For instance, the total charging power from electric cars at time t can be expressed as:
$$ P_{ev,t} = \sum_{j=1}^{N_{ev}} P_{ev,j,t} $$
where \( P_{ev,j,t} \) is the charging power of electric car j at time t, and \( N_{ev} \) is the total number of electric cars. The unpredictability of \( P_{ev,j,t} \) underscores the need for incentive-based strategies like TOU pricing to guide behavior.
Grid Low-Carbon Dispatch Requirements
As electric cars become widespread, grids face pressing low-carbon dispatch needs. These include load balancing to smooth demand curves, carbon emission reduction aligned with “dual carbon” goals, and ensuring供电 reliability. Grids must prioritize调度 of wind, solar, and other low-carbon sources, steering electric car charging to periods with high clean energy share. This not only cuts emissions but also enhances grid efficiency by leveraging off-peak resources. Reliability is paramount, as electric cars depend heavily on continuous供电; poor调度 could lead to outages, undermining user trust. Thus, dispatch strategies must integrate economic, environmental, and technical constraints holistically.
We formalize these requirements through objectives and constraints in subsequent sections. For example, carbon emissions from generation at time t are:
$$ E_{t} = \sum_{i=1}^{N_g} e_i \cdot P_{i,t} $$
where \( e_i \) is the carbon intensity of generator i (e.g., 0.8 kg/kWh for coal, 0 for renewables), and \( P_{i,t} \) is its power output. Minimizing \( E_{t} \) over time is a core low-carbon goal, directly influenced by when electric cars charge.
Construction of Multi-Objective Optimization System
We build a multi-objective optimization system that balances grid operation cost, user electricity cost, and carbon emissions. This system incorporates various constraints to ensure feasibility and safety.
Objective Functions
Three key objectives are considered:
- Grid Operation Cost (\( C_{grid} \)): Includes generation, transmission, and maintenance costs. Generation cost depends on power output and fuel type, while transmission and maintenance cover infrastructure expenses. Minimizing this cost improves grid economy.
- User Electricity Cost (\( C_{user} \)): The total expense for electric car users based on charging energy and TOU prices. Reducing this cost enhances user satisfaction and promotes electric car adoption.
- Carbon Emissions (\( E_{carbon} \)): Total emissions from electricity generation, tied to the energy mix. Lower emissions support environmental sustainability.
Mathematically, the objectives are defined as:
$$ C_{grid} = \sum_{t=1}^{T} \left( \sum_{i=1}^{N_g} c_i(P_{i,t}) + C_{trans,t} + C_{main,t} \right) $$
where \( c_i(P_{i,t}) \) is the cost function of generator i at time t, \( C_{trans,t} \) is transmission cost, \( C_{main,t} \) is maintenance cost, T is the调度 horizon (e.g., 24 hours), and \( N_g \) is the number of generators. For simplicity, we often linearize \( c_i(P_{i,t}) \) as \( a_i \cdot P_{i,t} + b_i \), with coefficients \( a_i, b_i \).
$$ C_{user} = \sum_{t=1}^{T} \sum_{j=1}^{N_{ev}} p_t \cdot E_{ev,j,t} $$
where \( p_t \) is the TOU price at time t, and \( E_{ev,j,t} = P_{ev,j,t} \cdot \Delta t \) is the charging energy of electric car j at time t, with \( \Delta t \) as the time interval.
$$ E_{carbon} = \sum_{t=1}^{T} \sum_{i=1}^{N_g} e_i \cdot P_{i,t} $$
with \( e_i \) as the carbon intensity.
The multi-objective problem is to minimize \( \mathbf{F} = [C_{grid}, C_{user}, E_{carbon}] \). We employ a weighted sum approach for solution:
$$ \min \left( w_1 \cdot C_{grid} + w_2 \cdot C_{user} + w_3 \cdot E_{carbon} \right) $$
where weights \( w_1, w_2, w_3 \geq 0 \) and \( w_1 + w_2 + w_3 = 1 \). These weights reflect priority trade-offs, e.g., higher \( w_3 \) emphasizes carbon reduction.
Constraints
The optimization is subject to physical and operational limits:
- Power Balance Constraint: At each time t, generation must meet total load, including base demand and electric car charging:
$$ \sum_{i=1}^{N_g} P_{i,t} = L_t + \sum_{j=1}^{N_{ev}} P_{ev,j,t}, \quad \forall t $$
where \( L_t \) is the non-electric car base load. - Generator Power Limits: Each generator has minimum and maximum output bounds:
$$ P_{i,min} \leq P_{i,t} \leq P_{i,max}, \quad \forall i,t $$ - Electric Car Charging Limits: Charging power per electric car is capped by its charger rating, and total energy must satisfy daily needs:
$$ 0 \leq P_{ev,j,t} \leq P_{ev,j,max}, \quad \forall j,t $$
$$ \sum_{t=1}^{T} E_{ev,j,t} = E_{j,req}, \quad \forall j $$
where \( E_{j,req} \) is the required daily energy for electric car j, ensuring users’ travel needs. - Carbon Emission Limit: Total emissions must not exceed a cap, aligning with low-carbon goals:
$$ E_{carbon} \leq E_{max} $$ - TOU Price Bounds: Prices must stay within reasonable ranges to avoid user backlash or financial infeasibility:
$$ p_{min} \leq p_t \leq p_{max}, \quad \forall t $$
These constraints ensure the solution is practical and grid-compliant. For instance, the charging limits protect electric car batteries and grid stability.
Solution Approach
We utilize the Particle Swarm Optimization (PSO) algorithm to solve this multi-objective problem. PSO is a population-based metaheuristic inspired by bird flocking, known for fast convergence and ease of implementation. It treats each candidate TOU price scheme as a particle in a high-dimensional space, iteratively updating positions and velocities to explore optimal solutions. The weighted sum objective guides the search, with particles evaluated based on combined cost and emission metrics. PSO parameters, such as inertia weight and acceleration coefficients, are tuned to balance exploration and exploitation. This approach efficiently handles the non-linear, constrained nature of electric car调度 problems, providing Pareto-optimal solutions that trade off conflicting goals effectively.
The PSO update equations for particle k are:
$$ \mathbf{v}_k^{(n+1)} = \omega \mathbf{v}_k^{(n)} + c_1 r_1 (\mathbf{pbest}_k – \mathbf{x}_k^{(n)}) + c_2 r_2 (\mathbf{gbest} – \mathbf{x}_k^{(n)}) $$
$$ \mathbf{x}_k^{(n+1)} = \mathbf{x}_k^{(n)} + \mathbf{v}_k^{(n+1)} $$
where \( \mathbf{x}_k \) represents the TOU price vector, \( \mathbf{v}_k \) is velocity, \( \omega \) is inertia weight, \( c_1, c_2 \) are acceleration constants, \( r_1, r_2 \) are random numbers, \( \mathbf{pbest}_k \) is the particle’s best position, and \( \mathbf{gbest} \) is the global best. The fitness function is the weighted sum objective, with penalties for constraint violations.
Time-of-Use Pricing Optimization Strategies
Based on the optimization framework, we propose TOU pricing strategies that include time period division, pricing methods, and dynamic adjustment mechanisms.
Time Period Division
Scientific division of time periods is crucial for guiding electric car charging behavior. We categorize periods based on grid load curves and low-carbon energy availability:
- Peak Periods: High demand times with low clean energy share, e.g., 8:00-11:00 and 18:00-22:00. These see elevated prices to discourage electric car charging, reducing strain and emissions.
- Valley Periods: Low demand times with high renewable output, e.g., 0:00-6:00. Lower prices attract electric car charging, utilizing surplus clean energy and smoothing loads.
- Normal Periods: Moderate demand times, e.g., 11:00-18:00 and 22:00-24:00. Prices are set at baseline levels.
Table 2 summarizes a sample division, adaptable to local grid conditions.
| Period Type | Time Intervals | Characteristics | Pricing Strategy |
|---|---|---|---|
| Peak | 8:00-11:00, 18:00-22:00 | High load, high carbon intensity | High price (e.g., 1.2 times baseline) |
| Normal | 11:00-18:00, 22:00-24:00 | Moderate load, mixed energy | Baseline price |
| Valley | 0:00-6:00 | Low load, high renewable share | Low price (e.g., 0.5 times baseline) |
This division leverages price signals to shift electric car charging from peak to valley periods, achieving load shifting and carbon reduction. The impact can be quantified by the load shifting ratio:
$$ R_{shift} = \frac{\sum_{t \in \text{Valley}} P_{ev,t} – \sum_{t \in \text{Valley}} P_{ev,t}^{base}}{\sum_{t} P_{ev,t}^{base}} $$
where \( P_{ev,t}^{base} \) is the charging power without TOU pricing. A positive \( R_{shift} \) indicates successful load转移.
Pricing Methods
We explore three pricing methods, each with distinct merits:
- Cost-Plus Pricing: Sets prices based on grid costs plus a profit margin. The baseline price \( p_0 \) is derived from generation, transmission, and maintenance costs. TOU prices are then scaled:
$$ p_t = \alpha_t \cdot p_0 $$
where \( \alpha_t \) is a period-specific coefficient: >1 for peak, <1 for valley, and 1 for normal periods. This ensures cost recovery while incentivizing off-peak charging for electric cars. - User-Response-Based Pricing: Incorporates user price sensitivity to maximize behavior adjustment. Via surveys or data analytics, we estimate demand elasticity \( \epsilon_t \), representing how charging demand changes with price. Prices are optimized to balance grid and user objectives:
$$ \min \sum_t \left( (p_t – p_0)^2 + \lambda \cdot (D_t(p_t) – D_t^{target})^2 \right) $$
where \( D_t(p_t) \) is the charging demand function for electric cars at price \( p_t \), \( D_t^{target} \) is the desired demand (e.g., lower in peak), and \( \lambda \) is a weight. This method enhances user acceptance by aligning prices with willingness-to-pay. - Carbon-Market-Integrated Pricing: Embeds carbon prices into electricity pricing to reflect emission costs. If \( q_{carbon} \) is the carbon market price per ton, the adjusted price includes an emission component:
$$ p_t = p_0 + \beta \cdot q_{carbon} \cdot \bar{e}_t $$
where \( \bar{e}_t \) is the average carbon intensity of generation at time t, and \( \beta \) is a scaling factor. During high-renewable periods, \( \bar{e}_t \) is low, reducing prices to encourage electric car charging; conversely, high-carbon periods see price hikes. This directly ties electric car usage to environmental externalities.
These methods can be combined in practice. For instance, a hybrid approach might use cost-plus as a base, adjusted by user response and carbon factors. Table 3 compares their key aspects.
| Method | Basis | Advantages | Challenges |
|---|---|---|---|
| Cost-Plus | Grid operational costs | Simple, ensures profitability | May ignore user behavior |
| User-Response-Based | Price elasticity and demand patterns | Enhances user engagement and satisfaction | Requires extensive data on electric car users |
| Carbon-Market-Integrated | Carbon emission costs and market prices | Promotes low-carbon charging; aligns with sustainability | Depends on carbon market stability |
Dynamic Pricing Adjustment Mechanism
To adapt to real-time grid conditions, we propose a dynamic pricing adjustment mechanism. This updates TOU prices based on factors like load fluctuations, renewable generation forecasts, and carbon intensity updates. The mechanism uses a feedback loop: monitor grid data, compute optimal prices via the optimization model, and implement adjustments. For example, if wind power generation suddenly increases during a valley period, prices can be lowered further to attract more electric car charging, reducing curtailment and emissions.
Mathematically, the dynamic adjustment can be formulated as a model predictive control (MPC) problem. At each control interval, we solve a shortened optimization horizon to determine price adjustments \( \Delta p_t \):
$$ \min_{\Delta p_t} \left( w_1 \Delta C_{grid} + w_2 \Delta C_{user} + w_3 \Delta E_{carbon} \right) $$
subject to updated constraints based on real-time data. The solution ensures that electric car charging responds promptly to grid needs, enhancing flexibility. This mechanism overcomes the rigidity of static TOU pricing, fostering deeper synergy between electric cars and the grid.
Case Study
We apply our framework to a practical scenario to demonstrate its effectiveness.
Case Background
Consider a urban grid area with 1,000 electric cars, served by 600 home chargers (slow, 7 kW max) and 5 public stations with 50 fast chargers (50 kW max) and 100 slow chargers. The grid includes coal-fired plants (carbon intensity 0.8 kg/kWh), wind farms, and solar PV (zero carbon). The调度 horizon is 24 hours, divided into peak (8:00-11:00, 18:00-22:00), normal (11:00-18:00, 22:00-24:00), and valley (0:00-8:00) periods. Base load data and renewable forecasts are assumed known. Each electric car requires 10 kWh daily on average, with charging flexibility between 18:00 and 8:00 next day.
Pre-Optimization Scenario
Without TOU pricing, electric car users charge freely, predominantly during evening peak hours (18:00-22:00). This leads to high grid costs, elevated user expenses, and significant emissions. Quantitatively, we observe:
- Daily grid operation cost: $50,000
- Daily user electricity cost: $30 per electric car on average
- Daily carbon emissions: 12 tons
The load curve shows sharp peaks, straining coal plants and increasing carbon output.
Post-Optimization Scenario
Implementing our multi-objective optimization with PSO (weights: \( w_1=0.4, w_2=0.3, w_3=0.3 \)) yields an optimal TOU scheme: peak price = $0.12/kWh, normal price = $0.08/kWh, valley price = $0.04/kWh. Electric car users respond by shifting charging to valley periods, aided by smart charging incentives. Results after optimization:
- Daily grid operation cost: $42,000 (16% reduction)
- Daily user electricity cost: $25 per electric car (17% reduction)
- Daily carbon emissions: 9 tons (25% reduction)
Table 4 summarizes the comparison, highlighting the benefits for both grid and electric car users.
| Metric | Pre-Optimization | Post-Optimization | Improvement |
|---|---|---|---|
| Grid Operation Cost ($) | 50,000 | 42,000 | -16% |
| User Cost per Electric Car ($) | 30 | 25 | -17% |
| Carbon Emissions (tons) | 12 | 9 | -25% |
| Peak Load (MW) | 85 | 72 | -15% |
| Valley Load (MW) | 40 | 58 | +45% (load filling) |
The optimization demonstrates that TOU pricing can effectively guide electric car charging, reducing costs and emissions while maintaining user satisfaction. The load curve becomes smoother, as shown by the reduced peak-to-valley ratio:
$$ R_{pv} = \frac{P_{peak}}{P_{valley}} $$
which decreases from 2.125 pre-optimization to 1.241 post-optimization, indicating better grid utilization.
Results Analysis
The case study validates our multi-objective approach. Key insights include:
- Electric car users are price-sensitive; even modest price differentials can shift behavior significantly.
- Low-carbon dispatch is achievable by aligning electric car charging with renewable abundance, cutting emissions substantially.
- The weighted optimization balances stakeholder interests; different weight sets can tailor outcomes to local priorities (e.g., higher \( w_3 \) for greener grids).
- PSO proved effective in solving the complex, non-linear problem, converging to satisfactory solutions within reasonable time.
This underscores the practicality of our framework for real-world grids integrating electric cars.
Conclusion and Future Work
In this article, we developed a multi-objective optimization model for TOU pricing of electric cars from a low-carbon dispatch perspective. The model minimizes grid operation cost, user electricity cost, and carbon emissions, subject to power balance, generator limits, charging constraints, and emission caps. Using PSO, we derived optimal pricing schemes that incentivize off-peak and low-carbon charging. Strategies like time period division, cost-plus, user-response, and carbon-integrated pricing were explored, with a dynamic adjustment mechanism for adaptability. A case study confirmed the model’s efficacy, showing notable improvements in cost, user expense, and emissions after optimization.
Future work can extend this research in several directions. First, incorporate more precise predictions of electric car travel patterns and charging demands using machine learning, enhancing model accuracy. Second, include battery degradation costs in the objective to account for electric car battery health under frequent charging. Third, integrate grid reliability metrics, such as voltage stability or frequency regulation, into the optimization to ensure technical robustness. Fourth, explore advanced multi-objective algorithms like NSGA-II or MOPSO for better Pareto front exploration. Fifth, investigate vehicle-to-grid (V2G) capabilities, allowing electric cars to discharge and support the grid, adding another layer of flexibility. Finally, expand the framework to regional or national scales, considering transmission constraints and inter-grid interactions. These advancements will further solidify the role of electric cars in sustainable energy ecosystems, driving progress toward global carbon neutrality.