In this study, I investigate how large-scale electric vehicle charging and discharging can be coordinated so that power system load fluctuations are reduced while electric vehicle owners obtain meaningful arbitrage revenue. My motivation comes from the rapid growth of electric vehicle adoption and the increasing importance of vehicle-to-grid participation in low-carbon energy systems. I treat each electric vehicle as a flexible distributed energy resource whose charging and discharging behavior can be scheduled over a rolling optimization horizon. To make the problem tractable and practical, I combine a multi-objective evolutionary algorithm based on decomposition with a finite discrete action set and a price-driven auction mechanism. The resulting method, which I refer to as MOEA/D-EV, produces forward-looking charging and discharging plans under different trade-off preferences. I evaluate the method with simulations under multiple time-of-use tariff structures and different electric vehicle population sizes. The results show that my method can smooth the net load curve, reduce peak-valley differences, and improve electric vehicle arbitrage income while respecting battery state-of-charge constraints and discrete charging and discharging capabilities.
Research Context and Motivation
The transition toward cleaner energy systems has made electric vehicle integration a central issue in power system operation. An electric vehicle is not only a transportation asset but also a mobile energy storage unit. When many electric vehicles charge during high-demand periods or discharge without coordination, the aggregate effect can create new load peaks, widen peak-valley differences, and increase stress on distribution networks. Conversely, when electric vehicle charging and discharging are coordinated properly, the same electric vehicle fleet can provide peak shaving, valley filling, demand response, and local balancing services. This dual role makes electric vehicle scheduling a multi-objective problem in which system operators and electric vehicle owners have different but connected interests.
Time-of-use tariffs provide a practical price signal for coordinating electric vehicle behavior. A time-of-use tariff divides the day into peak, flat, and valley periods and assigns different prices to each period. Electric vehicle owners are encouraged to charge when the price is low and discharge when the price is high. However, if all electric vehicle owners respond to the same low price at the same time, a new electric vehicle charging peak may appear. Therefore, price signals alone are not sufficient. I need an intelligent scheduling mechanism that can dynamically allocate charging and discharging tasks among electric vehicles while considering both grid-side load smoothing and user-side arbitrage revenue.
Existing studies often formulate electric vehicle scheduling as a single-objective problem, either minimizing system load variance or minimizing user charging cost. Such formulations cannot explicitly reveal the trade-off between grid stability and electric vehicle owner benefit. Other studies treat electric vehicle charging and discharging power as continuous variables. Although continuous formulations are mathematically elegant, they can create very high computational and communication burdens when thousands of electric vehicle decisions are coordinated over many time intervals. A continuous decision space also makes real-time implementation difficult because the number of possible power combinations grows extremely fast. Therefore, I focus on a discrete scheduling framework that converts continuous power choices into a finite set of charging and discharging actions. This design reduces complexity and supports practical electric vehicle coordination.
My main contributions are as follows. First, I build a multi-objective electric vehicle charging and discharging scheduling model that minimizes grid load fluctuation and maximizes electric vehicle arbitrage revenue. Second, I use a decomposition-based multi-objective evolutionary algorithm to obtain a set of non-dominated scheduling plans with different trade-off preferences. Third, I design a finite discrete action set for electric vehicle charging and discharging so that each electric vehicle selects from a small number of feasible power levels according to its state of charge and the current tariff interval. Fourth, I develop a price-driven auction algorithm that assigns charging and discharging tasks to electric vehicles and adjusts bids iteratively to reduce the gap between the aggregated electric vehicle response and the grid requirement. The combination of these components allows me to generate forward-looking schedules that are both economically attractive and operationally feasible.
Problem Formulation
I consider a rolling optimization horizon of length T. The horizon is divided into equal time intervals, and I use t as the time index. There are n electric vehicles participating in scheduling. For electric vehicle i at time interval t, I define p(i,t) as its charging or discharging power. A positive value represents discharging to the grid, a negative value represents charging from the grid, and zero represents an idle state. The base load at time t is p(L,t). The average net load after electric vehicle integration is p(avr). The first objective is the sum of squared deviations of the net load from its average value. This objective represents grid load fluctuation and is expressed as
$$
F_1=\sum_{t=1}^{T}\left(p_{L,t}-\sum_{i=1}^{n}p_{i,t}-p_{avr}\right)^2
$$
where
$$
p_{avr}=\frac{1}{T}\sum_{t=1}^{T}\left(p_{L,t}-\sum_{i=1}^{n}p_{i,t}\right).
$$
Minimizing F1 smooths the net load curve. A smaller F1 means that the difference between the net load and its average is smaller, which indicates better peak shaving and valley filling. I treat F1 as a minimization objective.
The second objective is electric vehicle arbitrage revenue. In each time interval, the aggregate electric vehicle power determines whether the interval is a charging interval or a discharging interval. If the aggregate power is negative, the electric vehicle fleet is charging and pays the charging price. If the aggregate power is positive, the electric vehicle fleet is discharging and receives the discharging price or subsidy. If the aggregate power is zero, the charging cost and discharging revenue cancel. The arbitrage revenue is written as
$$
F_2=\sum_{t=1}^{T}\left|\sum_{i=1}^{n}p_{i,t}\right|Price_t\beta_t
$$
where Price(t) is the time-of-use price at interval t, and the sign indicator is
$$
\beta_t=
\begin{cases}
-1, & \sum_{i=1}^{n}p_{i,t}\lt 0,\\
0, & \sum_{i=1}^{n}p_{i,t}=0,\\
1, & \sum_{i=1}^{n}p_{i,t}\gt 0.
\end{cases}
$$
I maximize F2 for electric vehicle owners. For consistency with the minimization framework used by MOEA/D, I minimize -F2. Thus, the electric vehicle scheduling problem can be written as
$$
\min \mathbf{F}(\mathbf{x})=\left(F_1(\mathbf{x}),-F_2(\mathbf{x})\right).
$$
The decision vector is defined over all electric vehicles and all time intervals. I write
$$
\mathbf{X}=\left[\vec{P}_1,\vec{P}_2,\ldots,\vec{P}_T\right],
$$
where
$$
\vec{P}_h=\left[p_{1,h},p_{2,h},\ldots,p_{n,h}\right],\qquad h=1,2,\ldots,T.
$$
Each element p(i,h) belongs to the finite discrete action set allowed for electric vehicle i at time h. This encoding makes the decision vector directly interpretable as a charging and discharging schedule for the entire electric vehicle population.
Power and Battery Constraints
Electric vehicle charging and discharging must satisfy power limits. I write
$$
p_i^{\min}\le p_{i,t}\le p_i^{\max},
$$
where p(i,min) is the maximum charging power, expressed as a negative value, and p(i,max) is the maximum discharging power, expressed as a positive value. The battery current limits are also considered. According to typical lithium-ion battery charging and discharging characteristics, I use
$$
0\le I_c\le \frac{c}{15},\qquad 0\le I_d\le \frac{c}{3},
$$
where I(c) is the charging current, I(d) is the discharging current, and c is a battery capacity parameter. The state of charge of an electric vehicle battery is bounded to protect battery health and avoid overcharging or overdischarging. The constraint is
$$
SOC_i^{\min}\le SOC_{i,t}+\frac{p_{i,t}\Delta t}{E_i}\le SOC_i^{\max},
$$
where SOC(i,t) is the state of charge of electric vehicle i at time t, E(i) is the battery capacity, and Delta t is the interval length. In my experiments, I set the lower state-of-charge bound to 0.2 and the upper bound to 0.9. These bounds keep the electric vehicle battery within a safe operating range and reduce degradation risk.
Multi-Objective Optimization Foundations
A multi-objective optimization problem contains several objective functions that may conflict with each other. In general form, I write
$$
\min \mathbf{F}(\mathbf{x})=\left(f_1(\mathbf{x}),f_2(\mathbf{x}),\ldots,f_m(\mathbf{x})\right)
$$
subject to
$$
g_k(\mathbf{x})\le 0,\quad k=1,\ldots,K,\qquad h_j(\mathbf{x})=0,\quad j=1,\ldots,J.
$$
Here x is a decision vector in the feasible region Omega. Because the objectives may conflict, there is usually no single solution that minimizes all objectives at the same time. Instead, I use Pareto dominance. A solution x(a) dominates x(b) if x(a) is no worse than x(b) in every objective and is strictly better in at least one objective. Formally,
$$
\mathbf{x}_a\prec\mathbf{x}_b
\iff
\forall i,\ f_i(\mathbf{x}_a)\le f_i(\mathbf{x}_b)
\ \text{and}\
\exists j,\ f_j(\mathbf{x}_a)\lt f_j(\mathbf{x}_b).
$$
A Pareto optimal solution is not dominated by any feasible solution. The set of all Pareto optimal solutions is the Pareto optimal set, and its image in objective space is the Pareto front. I write
$$
PS=\left\{\mathbf{x}^*\in\Omega\mid \nexists \mathbf{x}\in\Omega,\ \mathbf{x}\prec\mathbf{x}^*\right\},
$$
$$
PF=\left\{\mathbf{F}(\mathbf{x}^*)\mid \mathbf{x}^*\in PS\right\}.
$$
In my electric vehicle scheduling problem, each solution on the Pareto front represents a different balance between grid load fluctuation and electric vehicle arbitrage revenue. This set is useful because a decision maker may prefer a smoother load curve, a higher electric vehicle revenue, or a balanced compromise. I use MOEA/D because it decomposes a multi-objective problem into a set of single-objective subproblems and maintains clear weight preferences. This property supports practical decision making.
MOEA/D Decomposition and Update
MOEA/D uses a set of uniformly distributed weight vectors to define subproblems. For weight vector w and ideal point z*, the Tchebycheff aggregation function is
$$
g^{te}(\mathbf{x}\mid\mathbf{w},\mathbf{z}^*)=
\max_{1\le i\le m} w_i\left|f_i(\mathbf{x})-z_i^*\right|.
$$
For comparison, the weighted sum aggregation function is
$$
g^{ws}(\mathbf{x}\mid\mathbf{w})=\sum_{i=1}^{m}w_i f_i(\mathbf{x}),
$$
and the penalty-based boundary intersection function is
$$
g^{pbi}(\mathbf{x}\mid\mathbf{w},\mathbf{z}^*)=d_1+\theta d_2,
$$
where
$$
d_1=\frac{\left\|(\mathbf{z}^*-\mathbf{F}(\mathbf{x}))^T\mathbf{w}\right\|}{\|\mathbf{w}\|},
\qquad
d_2=\left\|\mathbf{F}(\mathbf{x})-(\mathbf{z}^*-d_1\mathbf{w})\right\|.
$$
In each iteration of MOEA/D, each subproblem has a neighborhood defined by the nearest weight vectors. Two parent solutions are selected from the neighborhood, and crossover and mutation produce an offspring solution. The offspring is repaired by the electric vehicle auction procedure and evaluated. If the offspring improves a neighboring subproblem according to the aggregation function, it replaces the current solution. The ideal point is updated whenever a better objective value is found. This process gradually approximates the Pareto front. In my method, the population size is 200, and the maximum number of iterations is 1000. These settings provide sufficient diversity and convergence for the electric vehicle scheduling experiments.
Proposed MOEA/D-EV Framework
My proposed MOEA/D-EV framework integrates three elements: a multi-objective decomposition framework, a finite discrete action set for electric vehicle charging and discharging, and a price-driven auction algorithm. The framework works in a rolling manner. At the beginning of an optimization cycle, a decision center collects forecast data, including base load, tariff, and electric vehicle state information. The center determines the remaining charging capacity and the potential discharging capacity gap. The auction algorithm maps these capacities to electric vehicle actions, and MOEA/D evaluates and evolves the resulting scheduling plans. The final output is a set of Pareto optimal electric vehicle charging and discharging schedules, each associated with a trade-off preference.
The finite discrete action set is essential for practical electric vehicle coordination. If each electric vehicle can choose any continuous power value at every interval, the number of possible schedules becomes extremely large. This creates high computational complexity and high communication overhead. In my design, each electric vehicle selects from a small set of normalized charging and discharging actions. I define c as the minimum normalized charging power and d as the minimum normalized discharging power. The charging actions are expressed as multiples of c, and the discharging actions are expressed as multiples of d. For example, slow charging uses c, normal charging uses 3c, and fast charging uses 7c. The corresponding discharging actions are d, 3d, and 7d. The action set depends on the state of charge, as shown in the following table.
| State-of-charge range | Allowed discharging actions | Allowed charging actions |
|---|---|---|
| [0.2, 0.3] | {d} | {7c, 3c, c} |
| (0.3, 0.5] | {3d, d} | {7c, 3c, c} |
| (0.5, 0.6] | {3d, d} | {3c, c} |
| (0.6, 0.8] | {7d, 3d, d} | {3c, c} |
| (0.8, 0.9] | {7d, 3d, d} | {c} |
This state-of-charge-dependent design has two purposes. First, it protects the electric vehicle battery by limiting high-power discharging when the battery is nearly empty and limiting high-power charging when the battery is nearly full. Second, it reduces the decision space. An electric vehicle does not need to communicate a continuous power value to the decision center. Instead, it reports or selects a discrete action index. This significantly reduces communication load and makes large-scale electric vehicle coordination more practical.
The price signal further guides action selection. I divide the time-of-use price into six intervals. D1, D2, and D3 represent low, medium, and high discharging price intervals. C1, C2, and C3 represent high, medium, and low charging price intervals. The interval definitions use the average price, the maximum price, the minimum price, and a standard deviation parameter. In a high-price interval, electric vehicles are encouraged to discharge at a higher rate. In a low-price interval, electric vehicles are encouraged to charge at a higher rate. The following table summarizes the price-driven rate selection logic.
| State-of-charge region | Price interval | Charging rate | Discharging rate |
|---|---|---|---|
| High state of charge | D3 | Not preferred | High rate, such as 7d |
| High state of charge | D2 | Not preferred | Normal rate, such as 3d |
| High state of charge | D1 | Not preferred | Slow rate, such as d |
| High state of charge | C3 | Slow rate, such as c | Not preferred |
| High state of charge | C2 | Slow rate, such as c | Not preferred |
| High state of charge | C1 | Slow rate, such as c | Not preferred |
| Middle state of charge | D3 | Not preferred | Normal or high rate |
| Middle state of charge | D2 | Not preferred | Normal rate |
| Middle state of charge | D1 | Not preferred | Slow rate |
| Middle state of charge | C3 | High rate, such as 7c | Not preferred |
| Middle state of charge | C2 | Normal rate, such as 3c | Not preferred |
| Middle state of charge | C1 | Slow rate, such as c | Not preferred |
| Low state of charge | D3 | Not preferred | Not preferred |
| Low state of charge | D2 | Not preferred | Not preferred |
| Low state of charge | D1 | Not preferred | Not preferred |
| Low state of charge | C3 | High rate, such as 7c | Not preferred |
| Low state of charge | C2 | Normal rate, such as 3c | Not preferred |
| Low state of charge | C1 | Slow rate, such as c | Not preferred |
Formally, the electric vehicle power at time t belongs to a finite set determined by its state of charge:
$$
p_{i,t}\in\mathcal{A}\left(SOC_{i,t}\right)
\subset
\left\{-7c,-3c,-c,0,d,3d,7d\right\}.
$$
The aggregate electric vehicle power is
$$
\sum_{i=1}^{n}p_{i,t}
=
\sum_{i=1}^{n}
\begin{cases}
-PC_{i,t}, & \text{charging action},\\
PD_{i,t}, & \text{discharging action},\\
0, & \text{idle action}.
\end{cases}
$$
Here PC(i,t) belongs to the charging action set and PD(i,t) belongs to the discharging action set. The sign convention keeps charging negative and discharging positive, which is consistent with the net load and arbitrage formulation. This discrete formulation is one of the key reasons why my electric vehicle scheduling method remains computationally manageable.
Price-Driven Auction Algorithm
The auction algorithm coordinates electric vehicle actions in response to the time-of-use tariff and the grid requirement. I model the microgrid or decision center as the seller of charging and discharging tasks, and I model electric vehicles as buyers of these tasks. A discharging task corresponds to peak shaving, and a charging task corresponds to valley filling. Electric vehicles submit bids based on their current state of charge, their available discrete actions, and the tariff interval. The decision center allocates tasks to electric vehicles and updates bids until the aggregate response is close to the required demand.
The auction procedure has three stages. In the first stage, the decision center collects the grid demand and the electric vehicle bids. In the second stage, the center allocates charging and discharging tasks according to price priority and state-of-charge feasibility. In the third stage, the center sums the accepted bids and computes the gap between the aggregate electric vehicle response and the grid demand. If the gap is not minimal, the center updates the bids and repeats the allocation. The gap is
$$
Gap_t=\left|\sum_{i=1}^{n}b_{i,t}-D_t\right|,
$$
where b(i,t) is the accepted bid of electric vehicle i at time t and D(t) is the grid demand or target response at time t. The bids are sorted in descending order for discharging tasks and in ascending order for charging tasks, depending on the price interval. This ranking allows the auction to select the most competitive electric vehicle actions first.
I use the following procedure inside MOEA/D-EV. For each candidate schedule, the auction algorithm verifies feasibility with respect to the state of charge and the finite action set. It then adjusts the bid distribution according to the tariff interval. The adjusted bids are returned to the evolutionary algorithm. This interaction ensures that the candidate solution is not only economically meaningful but also physically feasible for the electric vehicle fleet. The auction algorithm also eliminates the need for a central controller to specify continuous power values for every electric vehicle. Instead, the center only needs to coordinate discrete bids, which reduces communication complexity.
The price-driven auction logic can be summarized as follows. When the tariff is in the highest discharging interval D3, electric vehicles with sufficient state of charge are encouraged to choose the highest feasible discharging action. When the tariff is in the highest charging interval C3, electric vehicles with sufficient available capacity are encouraged to choose the highest feasible charging action. In intermediate intervals, the auction selects normal or slow actions according to the gap and the state of charge. If an electric vehicle is near its lower state-of-charge bound, discharging bids are restricted. If an electric vehicle is near its upper state-of-charge bound, charging bids are restricted. This design maintains battery safety while supporting grid flexibility.
Experimental Design
I evaluate MOEA/D-EV under two time-of-use tariff scenarios and two electric vehicle population sizes. The first scenario uses 10 electric vehicles and the first tariff structure. The second scenario uses 50 electric vehicles and the second tariff structure. Each electric vehicle has a battery capacity of 43 kWh and a maximum charging or discharging power of 7 kW. The rolling horizon is 24 hours. The initial state of charge of each electric vehicle is generated randomly between 0.2 and 0.9. The population size is 200, and the maximum number of iterations is 1000. The base load profile represents a commercial building load with a daily peak-valley difference of 46 kW. The base load values are listed in the following table.
| Hour | Base load (kW) | Hour | Base load (kW) |
|---|---|---|---|
| 1 | 65 | 13 | 99 |
| 2 | 64 | 14 | 81 |
| 3 | 64 | 15 | 82 |
| 4 | 62 | 16 | 74 |
| 5 | 64 | 17 | 78 |
| 6 | 64 | 18 | 93 |
| 7 | 80 | 19 | 100 |
| 8 | 77 | 20 | 92 |
| 9 | 81 | 21 | 87 |
| 10 | 90 | 22 | 74 |
| 11 | 101 | 23 | 66 |
| 12 | 101 | 24 | 55 |
I use two time-of-use tariff structures to test the adaptability of my method. The first tariff has a peak price of 1.070, a flat price of 0.687, and a valley price of 0.360. The peak periods are 8:00 to 14:00 and 18:00 to 23:00. The flat periods are 7:00 to 8:00 and 14:00 to 18:00. The valley period is 23:00 to 7:00. The second tariff has a higher peak price and a lower valley price. It includes two peak levels, a flat period, and a valley period. The price data are summarized in the following table.
| Tariff item | Period | Price (currency per kWh) |
|---|---|---|
| Tariff 1 peak | 8:00 to 14:00 and 18:00 to 23:00 | 1.070 |
| Tariff 1 flat | 7:00 to 8:00 and 14:00 to 18:00 | 0.687 |
| Tariff 1 valley | 23:00 to 7:00 | 0.360 |
| Tariff 2 peak level 1 | 11:00 to 12:00 and 19:00 to 23:00 | 1.210 |
| Tariff 2 peak level 2 | 8:00 to 11:00, 12:00 to 14:00, and 18:00 to 19:00 | 1.120 |
| Tariff 2 flat | 7:00 to 8:00 and 14:00 to 18:00 | 0.712 |
| Tariff 2 valley | 23:00 to 7:00 | 0.300 |
I use the following performance indicators: load variance, peak-valley difference, daily average load, peak load, valley load, total charging power, total discharging power, and electric vehicle arbitrage revenue. I also examine the state-of-charge trajectories of selected electric vehicles to verify that the discrete actions and auction bids respect battery constraints. The experiments are designed to answer four questions. First, can the method generate a well-distributed Pareto front for electric vehicle scheduling? Second, does the selected trade-off preference produce the expected change in load fluctuation and electric vehicle revenue? Third, does the auction mechanism dynamically adjust electric vehicle behavior according to the time-of-use tariff? Fourth, does the finite discrete action set reduce computational and communication complexity without sacrificing feasibility?
Results and Analysis
Multi-Objective Trade-Off
I first analyze the trade-off between grid load fluctuation and electric vehicle arbitrage revenue. I select representative weight combinations to illustrate different decision preferences. A weight combination of [1, 0] emphasizes load fluctuation minimization. A weight combination of [0, 1] emphasizes electric vehicle arbitrage revenue maximization. A weight combination of [0.5, 0.5] represents a balanced preference. The results confirm that the Pareto front is continuous and informative. When the weight shifts toward load fluctuation minimization, the net load curve becomes smoother and the peak-valley difference decreases. When the weight shifts toward electric vehicle revenue maximization, the electric vehicle fleet discharges more aggressively during high-price periods and charges more during low-price periods, which increases arbitrage revenue but may increase load fluctuation.
The following table reports representative solutions for the first tariff scenario with 10 electric vehicles. The table includes the two objective values, the daily average load, the peak load, the valley load, the peak-valley difference, the total charging power, and the total discharging power. The results show that F1 can be reduced from 242.000 to 0.950 as the weight preference moves from [0, 1] to [1, 0]. At the same time, the absolute value of F2 changes from 196.278 to 177.665. This is a clear trade-off: a smoother load curve corresponds to lower electric vehicle arbitrage revenue, while higher electric vehicle revenue corresponds to larger load fluctuation.
| Preference | F1 (kW²) | Absolute F2 | Average load (kW) | Peak load (kW) | Valley load (kW) | Peak-valley difference (kW) | Total charging (kW) | Total discharging (kW) |
|---|---|---|---|---|---|---|---|---|
| [0, 1] | 242.000 | 196.278 | 73.000 | 76 | 60 | 16 | 113 | 255 |
| [0.1, 0.9] | 48.000 | 191.248 | 73.333 | 75 | 69 | 6 | 119 | 249 |
| [0.2, 0.8] | 31.333 | 189.098 | 73.500 | 75 | 70 | 5 | 119 | 249 |
| [0.3, 0.7] | 22.958 | 187.318 | 73.583 | 76 | 72 | 4 | 124 | 249 |
| [0.4, 0.6] | 14.958 | 185.515 | 73.667 | 74 | 69 | 5 | 123 | 248 |
| [0.5, 0.5] | 12.500 | 184.445 | 73.708 | 75 | 71 | 4 | 123 | 247 |
| [0.6, 0.4] | 8.500 | 183.735 | 73.708 | 75 | 72 | 3 | 123 | 247 |
| [0.7, 0.3] | 5.958 | 182.665 | 73.625 | 75 | 73 | 2 | 124 | 247 |
| [0.8, 0.2] | 4.625 | 181.945 | 73.708 | 75 | 74 | 1 | 126 | 247 |
| [0.9, 0.1] | 2.000 | 178.735 | 73.750 | 75 | 74 | 1 | 128 | 246 |
| [1, 0] | 0.950 | 177.665 | 73.750 | 75 | 74 | 1 | 128 | 245 |
I also repeat the analysis for the second tariff scenario with 50 electric vehicles. The larger electric vehicle population provides more flexibility, but it also increases the number of possible action combinations. The auction mechanism and the finite discrete action set keep the problem manageable. The following table shows that the load variance ranges from 498.625 to 10.958, and the absolute electric vehicle arbitrage revenue ranges from 273.836 to 153.026. The second tariff has a stronger price difference between valley and peak periods, which creates a higher incentive for electric vehicle discharging during high-price intervals. As a result, the electric vehicle arbitrage revenue is larger than in the first scenario, but the load variance can also become larger when the preference heavily favors electric vehicle revenue.
| Preference | F1 (kW²) | Absolute F2 | Average load (kW) | Peak load (kW) | Valley load (kW) | Peak-valley difference (kW) | Total charging (kW) | Total discharging (kW) |
|---|---|---|---|---|---|---|---|---|
| [0, 1] | 498.625 | 273.836 | 71.875 | 82 | 62 | 20 | 1249 | 1418 |
| [0.1, 0.9] | 254.500 | 269.118 | 71.750 | 79 | 65 | 14 | 1259 | 1431 |
| [0.2, 0.8] | 112.958 | 264.236 | 71.708 | 76 | 68 | 8 | 1250 | 1423 |
| [0.3, 0.7] | 92.958 | 260.188 | 71.702 | 75 | 68 | 7 | 1119 | 1421 |
| [0.4, 0.6] | 72.958 | 253.822 | 72.041 | 75 | 69 | 6 | 1250 | 1415 |
| [0.5, 0.5] | 59.958 | 245.898 | 72.458 | 76 | 70 | 6 | 1251 | 1406 |
| [0.6, 0.4] | 45.333 | 239.114 | 72.667 | 76 | 71 | 5 | 1254 | 1404 |
| [0.7, 0.3] | 34.500 | 237.102 | 72.750 | 75 | 71 | 4 | 1256 | 1404 |
| [0.8, 0.2] | 24.625 | 233.652 | 72.875 | 75 | 71 | 4 | 1258 | 1403 |
| [0.9, 0.1] | 17.625 | 219.986 | 73.375 | 75 | 72 | 3 | 1256 | 1389 |
| [1, 0] | 10.958 | 153.026 | 76.958 | 79 | 76 | 3 | 1199 | 1246 |
These results confirm that my method can generate a set of electric vehicle charging and discharging schedules with different trade-off preferences. The decision maker can choose a plan that emphasizes system load smoothing, a plan that emphasizes electric vehicle owner revenue, or a balanced plan. This is a practical advantage because electric vehicle scheduling involves multiple stakeholders. The Pareto front provides transparent information about the cost of improving one objective at the expense of another.
Dynamic Regulation and State-of-Charge Evolution
I next examine the dynamic regulation capability of the auction mechanism. I use the balanced preference [0.5, 0.5] and analyze the aggregate charging and discharging power in each interval. The results show clear price responsiveness. During high-price intervals, the aggregate electric vehicle power is often positive, which indicates net discharging. During low-price intervals, the aggregate electric vehicle power is often negative, which indicates net charging. During flat-price intervals, the behavior is more diverse. Some electric vehicles charge, some discharge, and some remain idle. This diversity arises from the auction allocation and the state-of-charge constraints. It is not necessary or desirable for every electric vehicle to take the same action at the same time.
The following table shows the aggregate charging and discharging power for 10 electric vehicles under the first tariff and the balanced preference. The values are expressed in kW. Positive values indicate discharging, and negative values indicate charging. The table shows that electric vehicle discharging is concentrated in high-price periods, while charging is concentrated in low-price periods. The valley period from late night to early morning has substantial charging activity, and the peak period around midday and evening has substantial discharging activity.
| Hour | Discharging power (kW) | Charging power (kW) | Hour | Discharging power (kW) | Charging power (kW) |
|---|---|---|---|---|---|
| 1 | 2 | -11 | 13 | 25 | 0 |
| 2 | 3 | -13 | 14 | 10 | -2 |
| 3 | 3 | -13 | 15 | 10 | -2 |
| 4 | 1 | -13 | 16 | 6 | -6 |
| 5 | 1 | -11 | 17 | 7 | -3 |
| 6 | 0 | -10 | 18 | 20 | 0 |
| 7 | 8 | -2 | 19 | 26 | 0 |
| 8 | 6 | -3 | 20 | 18 | 0 |
| 9 | 11 | -3 | 21 | 13 | 0 |
| 10 | 18 | -2 | 22 | 5 | -2 |
| 11 | 28 | -1 | 23 | 0 | -8 |
| 12 | 26 | 0 | 24 | 0 | -18 |
For the second tariff scenario with 50 electric vehicles, the aggregate power values are larger, but the same price-responsive pattern appears. The following table presents the aggregate charging and discharging power for selected intervals. The electric vehicle fleet discharges more during the high-price evening period and charges more during the low-price valley period. The auction mechanism adjusts the bids so that the aggregate response does not simply follow the price without considering the state of charge. Some electric vehicles remain idle when their state of charge is too high for charging or too low for discharging. This behavior is consistent with battery protection requirements.
| Hour | Discharging power (kW) | Charging power (kW) | Hour | Discharging power (kW) | Charging power (kW) |
|---|---|---|---|---|---|
| 1 | 78 | -88 | 13 | 76 | -50 |
| 2 | 53 | -65 | 14 | 69 | -59 |
| 3 | 49 | -58 | 15 | 32 | -21 |
| 4 | 47 | -59 | 16 | 30 | -28 |
| 5 | 44 | -54 | 17 | 31 | -25 |
| 6 | 45 | -56 | 18 | 70 | -50 |
| 7 | 55 | -47 | 19 | 83 | -53 |
| 8 | 30 | -25 | 20 | 67 | -48 |
| 9 | 67 | -57 | 21 | 74 | -57 |
| 10 | 81 | -63 | 22 | 72 | -69 |
| 11 | 81 | -51 | 23 | 70 | -78 |
| 12 | 44 | -14 | 24 | 58 | -76 |
The state-of-charge trajectories provide additional evidence of dynamic regulation. For the first tariff scenario, I select five electric vehicles with random initial state of charge. During high-price periods, their state of charge generally decreases because the auction assigns discharging tasks to electric vehicles with available energy. During low-price periods, their state of charge generally increases because the auction assigns charging tasks. Some intervals show a flat state-of-charge trajectory, which indicates that the electric vehicle is idle. This idle behavior is important because it prevents unnecessary battery cycling when the marginal benefit is low. For the second tariff scenario with 50 electric vehicles, the state-of-charge trajectories show stronger coordination. When an electric vehicle approaches the lower bound, the auction restricts further discharging and may assign a charging task instead. When an electric vehicle approaches the upper bound, the auction restricts further charging. This behavior keeps the electric vehicle battery within the safe interval and supports long-term participation.
Peak Shaving and Valley Filling
I compare the net load curve before and after electric vehicle scheduling. The base load curve has a clear peak-valley pattern. After scheduling, the electric vehicle fleet charges during low-load periods and discharges during high-load periods. This reduces the peak load and raises the valley load. The peak-valley difference decreases substantially. In the second tariff scenario, the initial peak-valley difference is 46 kW. Under the load-smoothing preference, the optimized peak-valley difference can be reduced to 3 kW. Under the revenue-oriented preference, the peak-valley difference is 20 kW. Both values are lower than the initial difference, but the load-smoothing preference produces a much flatter net load curve. The reduction percentages demonstrate the effectiveness of electric vehicle coordination for peak shaving and valley filling.
The improvement can be expressed using the peak-valley difference and the load variance. Let P(max) be the maximum net load and P(min) be the minimum net load. The peak-valley difference is
$$
\Delta P=P_{\max}-P_{\min}.
$$
The reduction ratio is
$$
R_{\Delta P}=\frac{\Delta P_{\text{before}}-\Delta P_{\text{after}}}{\Delta P_{\text{before}}}\times 100\%.
$$
For the second tariff scenario, the load-smoothing preference reduces the peak-valley difference from 46 kW to 3 kW, which corresponds to a reduction ratio of approximately 93.47 percent. The peak load and valley load also move closer to the average load. This result indicates that electric vehicle charging and discharging can provide significant flexibility when coordinated by the auction mechanism. The finite discrete action set does not prevent effective peak shaving because the aggregate effect of many small discrete actions is sufficient to reshape the load curve.
Computational and Communication Complexity
I analyze the complexity of my method in comparison with a continuous-variable formulation. In a continuous formulation, each electric vehicle can choose any power value within its limits at each interval. For 50 electric vehicles and 24 intervals, the decision space has 1200 continuous dimensions. If the problem is discretized into three power levels per electric vehicle per interval, the theoretical number of combinations is very large, but the finite action set and auction mechanism prune the search space substantially. More importantly, the communication requirement is much lower. A continuous power value is typically represented by a 4-byte floating-point number, while a discrete action index can be represented by 1 byte or even fewer bits. The following table summarizes the comparison.
| Aspect | Continuous formulation | My discrete action formulation |
|---|---|---|
| Decision type | Continuous power values | Finite action indices |
| Decision space per electric vehicle per interval | Infinite or very large | Small finite set |
| Communication per electric vehicle per day | 96 bytes with 4-byte values for 24 intervals | 24 bytes with 1-byte indices, or less with compression |
| Potential compression ratio | Reference value | Up to about 93.75 percent reduction |
| Battery protection | Requires additional post-processing | Built into the state-of-charge-dependent action set |
| Real-time suitability | Limited for large electric vehicle fleets | Improved by discrete bids and auction coordination |
The reduction in communication complexity is particularly important for electric vehicle fleets. In a real system, the decision center must exchange information with many electric vehicles or aggregators. If each electric vehicle sends a continuous power value for each interval, the communication load grows quickly. If each electric vehicle sends a discrete action index, the message size becomes much smaller. The auction mechanism further reduces the number of iterations required to reach a feasible aggregate response because it prioritizes the most competitive bids and updates only the remaining gap. This design supports real-time or near-real-time electric vehicle scheduling.
Charging and Discharging Schedule Example
I also examine an individual electric vehicle schedule to verify the interpretability of the method. The following table shows an example schedule for 10 electric vehicles under the first tariff scenario. Each entry is the charging or discharging power in kW. Positive values indicate discharging, negative values indicate charging, and zero indicates idle. The schedule reflects the price signal and the state-of-charge constraints. During peak price intervals, many electric vehicles discharge. During valley price intervals, many electric vehicles charge. During flat price intervals, the actions are mixed, which helps the auction satisfy the aggregate demand without forcing all electric vehicles to follow the same pattern.
| Electric vehicle | Hour 1 | Hour 2 | Hour 3 | Hour 4 | Hour 5 | Hour 6 | Hour 7 | Hour 8 | Hour 9 | Hour 10 | Hour 11 | Hour 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 0 | 3 | 3 | 3 | 3 |
| 2 | -1 | 1 | -1 | 0 | -1 | -1 | 1 | 1 | 3 | 3 | 7 | 3 |
| 3 | -1 | 0 | -3 | -1 | -3 | -1 | 1 | 1 | -1 | 3 | 3 | 7 |
| 4 | -1 | -3 | -1 | 0 | -1 | -1 | -1 | -1 | -1 | 1 | 0 | 3 |
| 5 | 0 | 0 | 1 | -3 | 0 | -1 | 1 | 1 | -1 | 1 | 3 | 1 |
| 6 | 1 | 1 | 1 | -1 | 0 | -1 | 3 | 1 | 1 | 3 | -1 | 3 |
| 7 | -3 | -1 | -3 | -3 | -1 | -1 | 1 | -1 | 1 | 3 | 3 | 1 |
| 8 | -1 | -7 | -3 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 3 | 1 |
| 9 | -3 | -1 | 1 | -3 | -1 | -1 | 0 | 1 | 1 | -1 | 3 | 3 |
| 10 | -1 | -1 | -1 | -1 | -3 | -1 | 0 | 1 | 1 | 1 | 3 | 1 |
The schedule continues into the later hours in the same manner. The key observation is that the electric vehicle actions are not arbitrary. They emerge from the interaction between the time-of-use tariff, the state of charge, the finite action set, and the auction bid. The resulting schedule respects the battery limits and avoids simultaneous charging and discharging by the same electric vehicle in the same interval. This is important for practical implementation because a single electric vehicle cannot charge and discharge at the same time. The discrete action set enforces this condition by design.
Discussion
My results show that the combination of MOEA/D, finite discrete actions, and price-driven auctioning is effective for electric vehicle charging and discharging scheduling. The multi-objective formulation exposes the trade-off between grid load fluctuation and electric vehicle arbitrage revenue. The finite discrete action set reduces the decision space and communication burden. The auction algorithm provides a dynamic allocation mechanism that responds to time-of-use prices and state-of-charge constraints. The simulation results under two tariff structures and two electric vehicle population sizes confirm that the method is adaptable and practical.
One important observation is that the best schedule for the grid is not necessarily the best schedule for electric vehicle owners. A load-smoothing schedule may require electric vehicles to charge during moderately low-price periods and discharge during moderately high-price periods. This can reduce arbitrage revenue compared with a more aggressive price-following strategy. Conversely, a revenue-oriented schedule may concentrate discharging in the highest-price intervals and charging in the lowest-price intervals, which can create new load peaks or valleys. The Pareto front makes this trade-off explicit. A decision maker can choose a preference that reflects the desired balance between system stability and user participation. This is more informative than a single-objective solution.
Another observation is that the finite discrete action set does not significantly reduce the quality of the scheduling solution. The aggregate effect of many electric vehicles selecting from a small set of actions is sufficient to reshape the load curve. In fact, the discrete action set improves practicality because it matches the way electric vehicle chargers often operate. Many chargers have fixed power levels or a small number of power settings. Representing electric vehicle behavior as discrete actions is therefore realistic. The action set also makes it easier to enforce battery constraints because the allowed actions depend on the state of charge.
The auction algorithm plays a central role in coordinating electric vehicle behavior. Without the auction, the evolutionary algorithm would need to search a much larger space and might produce schedules that are economically attractive but operationally infeasible. The auction converts the grid requirement into a set of bids and allocates those bids to electric vehicles. The iterative gap update ensures that the aggregate response approaches the target. The price-driven bidding rule encourages electric vehicles to charge when the price is low and discharge when the price is high. The state-of-charge constraints prevent the auction from assigning actions that would violate battery limits.
Conclusion and Future Work
I have presented a multi-objective electric vehicle charging and discharging scheduling method based on MOEA/D, a finite discrete action set, and a price-driven auction algorithm. The method minimizes grid load fluctuation and maximizes electric vehicle arbitrage revenue. It produces a Pareto front of scheduling plans with different trade-off preferences. The finite discrete action set reduces computational and communication complexity while respecting battery state-of-charge constraints. The auction algorithm dynamically coordinates electric vehicle behavior in response to time-of-use tariffs. The simulation results under different tariff structures and electric vehicle population sizes show that the method can reduce peak-valley differences, smooth the net load curve, and improve electric vehicle arbitrage revenue. The method also provides decision makers with transparent trade-off information, which is valuable for practical electric vehicle scheduling.
In future work, I plan to extend the method in several directions. First, I will incorporate more detailed user behavior models. The current study assumes that all participating electric vehicles are available for scheduling, but in practice, electric vehicle availability depends on travel patterns and user preferences. I will use data-driven models to predict electric vehicle availability and state of charge at the start of each scheduling cycle. Second, I will integrate renewable generation and storage systems into the scheduling framework. Electric vehicle charging can absorb surplus renewable energy, and electric vehicle discharging can support the grid when renewable output is low. Third, I will consider transportation constraints such as traffic conditions and charging station locations. The mobility of electric vehicles affects when and where they can charge or discharge. Fourth, I will explore decentralized and trustworthy transaction mechanisms for electric vehicle participation. A secure and transparent coordination platform could encourage more electric vehicle owners to participate in charging and discharging scheduling. These extensions will make the method more suitable for complex real-world electric vehicle scheduling scenarios.
Overall, I conclude that MOEA/D-EV provides a practical and effective approach for electric vehicle charging and discharging scheduling. It balances grid-side load smoothing and user-side arbitrage revenue, respects battery constraints, and reduces complexity through discrete actions. The method is well suited to time-of-use tariff environments and can support the integration of large-scale electric vehicle fleets into future power systems.