In recent years, global energy infrastructure has been stepping towards a more sustainable and low-carbon pathway. This transition has been driven by rising environmental awareness, tightening carbon policies, and the continuous growth of clean energy production. A crucial part of this evolution is the increasing deployment of electric cars, which not only contributes to reducing transport emissions but also creates new possibilities for the power grid through bi-directional energy exchange. Electric cars are no longer viewed merely as transportation devices; they are equipped with sizable battery packs that can function as distributed energy storage units, and this property has given rise to a broad range of vehicle-to-grid (V2G) research topics, including peak shaving, valley filling, demand response, and data-driven scheduling mechanisms.
The acceleration of electric vehicle adoption, however, brings about several challenges. Large-scale, uncoordinated charging of electric cars can place severe stress on distribution networks and lead to higher peak load, deeper valley-to-peak differences, and larger load variations. These effects threaten the stable and safe operation of modern power systems. In addition, the economics of electric vehicle participation is still a major issue. If vehicle owners are not sufficiently rewarded for their flexibility, they may refuse to participate in grid-supporting programs. Consequently, an effective scheduling framework must deal with both grid stability and user profitability at the same time. In practice, minimizing grid load fluctuation and maximizing the arbitrage revenue of electric car owners often conflict, which makes the scheduling problem a typical multi-objective optimization problem.
Time-of-use (TOU) pricing is widely regarded as an efficient mechanism to coordinate large populations of electric cars. By setting different electricity prices in peak, flat, and valley periods, TOU tariffs can naturally guide electric car owners to charge during periods with surplus generation and to discharge during periods with high demand. The concept is often expressed through the price elasticity of electricity demand, which defines how the demand for electricity changes in response to variations in the price of electricity. For example, a self-elasticity coefficient describes the response of the current time period to its own price, while a cross-elasticity coefficient describes the response of demand in one period to price changes in another. These elasticity relations form the foundation of TOU-based charging and discharging strategies and are widely used in demand response studies.
Many existing scheduling models, however, treat the charging and discharging rates of electric cars as continuous variables. Since each vehicle can choose an infinite number of charging levels at every time step, the resulting decision space becomes enormously complex as the number of cars grows. This high-dimensional continuous search problem leads to excessive computational and communication costs. In particular, real-time coordination among thousands of electric vehicles in a distribution system would require considerable information exchange and optimization time, making continuous formulations difficult to apply in practical V2G operations. As a result, recent research has paid increasing attention to the discretization of charging and discharging behavior, for instance, through finite action sets, which compress the solution space while still preserving good scheduling performance.
To further address the conflict between grid-side and user-side objectives, many studies make use of multi-objective optimization algorithms. Popular evolutionary algorithms for multi-objective optimization include NSGA-II and MOEA/D. NSGA-II uses non-dominated sorting to preserve convergence and diversity. However, one practical weakness of NSGA-II is that it does not provide straightforward trade-off preference information for each non-dominated solution. This matters in engineering applications because the decision maker has to select a single implementable schedule from many Pareto-optimal alternatives, and the choice usually depends on whether more importance is assigned to grid-side performance or user-side profitability. MOEA/D, in contrast, decomposes the multi-objective problem into a group of scalar sub-problems. Each sub-problem is associated with a fixed weight vector, and the final solution set naturally reflects different weighting preferences. This makes MOEA/D a more suitable basis for the trade-off analysis in the present work.
Based on the above considerations, this thesis proposes a MOEA/D-based framework for the multi-objective optimal scheduling of electric vehicle charging and discharging. The method coordinates the charging and discharging behavior of electric cars over a 24-hour rolling horizon, using a discrete action set to reduce complexity and a novel auction algorithm to respond dynamically to TOU tariffs. The main contributions of this work are highlighted below:
1. A multi-objective optimization model for electric vehicle charging and discharging is constructed. The first objective minimizes the load fluctuation of the distribution network, while the second objective maximizes the total arbitrage revenue of electric car owners. Both charging/discharging power constraints and battery state constraints are introduced.
2. A finite discrete action set strategy is proposed to coordinate the behavior of many electric cars. Instead of optimizing real-valued charging power, each vehicle is allowed to choose one of a limited number of charging or discharging rates according to its current battery state of charge. This method considerably reduces the computational overhead and communication burden, while still supporting precise and practical charging schedules.
3. A novel price-driven auction algorithm is developed. Electric cars are regarded as bidders in the auction, and the microgrid is considered as the seller that assigns charging and discharging tasks. The auction algorithm acts on the TOU price signal and repeatedly updates bids so that the total charging/discharging power approaches the power imbalance that must be compensated by the electric vehicle fleet.
4. The complete MOEA/D-EV algorithm is implemented and tested under two different TOU price structures and with different numbers of electric cars, i.e., 10 cars and 50 cars. The results confirm that the proposed method can effectively balance grid load fluctuation and user profit.
The remainder of the article is organized as follows. The next section describes the basic concepts, including multi-objective optimization definitions, TOU pricing mechanisms, the MOEA/D algorithm, and the auction algorithm. The proposed model and algorithm are then explained in detail, followed by simulation experiments and a discussion of the results. Finally, conclusions and future directions are presented.
Multi-objective optimization is widely encountered in engineering when several conflicting measures must be improved at the same time. A general continuous multi-objective optimization problem can be expressed as
$$ \min \mathbf{F}(\mathbf{x}) = \bigl(f_1(\mathbf{x}), f_2(\mathbf{x}), \dots, f_m(\mathbf{x})\bigr) $$
subject to the inequality and equality constraints
$$ g_j(\mathbf{x}) \le 0,\quad j = 1,\dots,K $$
$$ h_l(\mathbf{x}) = 0,\quad l = 1,\dots,M $$
where \(\mathbf{x} = [x_1, x_2, \dots, x_D]^\mathrm{T}\) is the decision vector in the \(D\)-dimensional search space. In multi-objective optimization problems, the notion of optimum is different from that in single-objective optimization. Instead of a single global best solution, the solution of a multi-objective problem is usually a set of Pareto-optimal solutions. The concept of Pareto dominance can be described as follows: for two decision vectors \(x_a,x_b\), \(x_a\) is said to dominate \(x_b\), denoted \(x_a \prec x_b\), if for every objective \(i\), \(f_i(x_a) \le f_i(x_b)\), and there exists at least one objective such that \(f_i(x_a) < f_i(x_b)\). A solution \(x^* \in \Omega\) is Pareto-optimal if there does not exist any feasible solution that dominates \(x^*\). The collection of all Pareto-optimal solutions is named the Pareto-optimal set, and the images of these solutions in the objective space form the Pareto frontier.
The load variation of a power system is often evaluated with the help of the load variance or the difference between the daily peak and valley loads. A smaller variance indicates a flatter load curve and safer grid operation. The economics of vehicle participation is usually evaluated by comparing the charging cost and the discharge subsidy over the scheduling horizon. In a V2G situation, the vehicle owner earns revenue when discharging during peak-price periods and spends money when charging in low-price periods. The difference between these payments creates an arbitrage opportunity. This profit mechanism is essential for improving the willingness of electric car owners to participate in demand response activities.
Peak-valley TOU pricing is a straightforward strategy that divides the day into several intervals such as peak, flat and valley periods. The price in the peak period is the highest, while the price in the valley period is the lowest. Under the influence of TOU tariffs, rational users naturally move electricity consumption from peak to valley times. This movement is captured by the elasticity matrix \(\mathbf{E}\), in which the diagonal element \(\eta_{ii}\) is the self-elasticity coefficient and the off-diagonal element \(\eta_{ij}\) is the cross-elasticity coefficient. The elasticity relationship is
$$\begin{bmatrix} \Delta D_1 / D_1 \\ \Delta D_2 / D_2 \\ \vdots \\ \Delta D_n / D_n \end{bmatrix} \;=\; \begin{bmatrix} \eta_{11} & \eta_{12} & \cdots & \eta_{1n} \\ \eta_{21} & \eta_{22} & \cdots & \eta_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ \eta_{n1} & \eta_{n2} & \cdots & \eta_{nn} \end{bmatrix} \begin{bmatrix} \Delta \lambda_1 / \lambda_1 \\ \Delta \lambda_2 / \lambda_2 \\ \vdots \\ \Delta \lambda_n / \lambda_n \end{bmatrix}$$
where \(\lambda_i\) is the electricity price in period \(i\), \(D_i\) is the corresponding demand, and the symbol \(\Delta\) denotes the change in these variables. Such a matrix can help the grid operator evaluate how much charging demand can be moved from peak periods to valley periods.
The MOEA/D algorithm has gained great popularity for such problems. It first generates a set of weight vectors \(w^{(1)}, w^{(2)},\dots,w^{(N)}\), equally distributed over the objective space. For each weight vector, a sub-problem is defined as a scalar aggregation function of all objectives. The solution to the entire multi-objective problem is then obtained by evolving a population of \(N\) solutions, one for each sub-problem, in a collaborative manner. The Chebyshev aggregation function is widely used because it can handle non-convex Pareto fronts. The Chebyshev function is written as
$$ g^{te}\bigl(x^{(j)} \mid w^{(j)}, z^*\bigr) = \max_{i=1,\dots,m} \bigl\{ w^{(j)}_i \cdot \bigl| F_i(x^{(j)}) – z^*_i \bigr| \bigr\} $$
where \(z^*\) is the ideal point vector. In each iteration, the ideal point is updated, and neighboring sub-problems exchange information through crossover and mutation operations. The population gradually converges toward a well-distributed approximation of the Pareto frontier. Unlike NSGA-II, MOEA/D has a natural ability to show the trade-off direction of every individual in the final solution set because each individual is associated with its weight vector.
The auction algorithm is a distributed method often used for assigning tasks to agents. In the classical auction algorithm, each agent submits bids for tasks, and the auctioneer repeatedly assigns each task to the highest bidder while updating prices. This approach has good efficiency and can be adapted to resource allocation problems in the electricity market. In this paper, the auction framework is not used to maximize monetary bidding but rather to map the charging and discharging capacity of electric cars into appropriate discrete actions for the power grid.
In the proposed scheduling model, all electric cars in a given region are coordinated during an optimization period of \(T=24\) hours. The total load in the grid is composed of a base load \(p_{L,t}\) and the aggregate charging/discharging power of the electric vehicle fleet. The charging/discharging power of vehicle \(i\) at time \(t\) is represented by \(p_{i,t}\). A positive \(p_{i,t}\) indicates that the vehicle is discharging, a negative \(p_{i,t}\) indicates charging, and zero means that the vehicle is idle.
The first objective function is to minimize the load fluctuation over the optimization period, which is expressed as the load variance:
$$ \min F_1 = \sum_{t=1}^{T} \bigl( p_{L,t} – \sum_{i=1}^{n} p_{i,t} – p_{avr} \bigr)^2 $$
where
$$ p_{avr} = \frac{1}{T} \sum_{t=1}^{T} \bigl( p_{L,t} – \sum_{i=1}^{n} p_{i,t} \bigr) $$
The second objective is to maximize the total arbitrage revenue of electric car owners. Since charging and discharging in the same period are cleared at the same price interval, the revenue is calculated according to the absolute value of the total charging or discharging power and the sign of the aggregate power. The expression is given by
$$ \max F_2 = \sum_{t=1}^{T} \Bigl| \sum_{i=1}^{n} p_{i,t} \Bigr| \cdot Price_t \cdot \beta_t $$
where \(\beta_t\) is defined as
$$ \beta_t = \begin{cases} -1, & \text{if } \sum_{i=1}^n p_{i,t} < 0 \\ 0, & \text{if } \sum_{i=1}^n p_{i,t} = 0 \\ 1, & \text{if } \sum_{i=1}^n p_{i,t} > 0 \end{cases} $$
When the aggregate power is negative, the fleet is charging and the electricity price is paid by the users. When the aggregate power is positive, the fleet is discharging and the income from the participation is positive. For convenient minimization in the multi-objective algorithm, the second objective is multiplied by \(-1\) so that the problem turns into a two-objective minimization problem.
The constraints in the scheduling model are essential for safe operation. First, the charging and discharging power of each electric car must be kept between pre-defined limits:
$$ p_{i,t}^{\min} \le p_{i,t} \le p_{i,t}^{\max} $$
where \(p_{i,t}^{\min}\) represents the maximum charging power (a negative number) and \(p_{i,t}^{\max}\) represents the maximum discharging power.
Second, the battery state of charge must not exceed its safe range. In this study, the lower bound is \(SOC_{\min}=0.2\) and the upper bound is \(SOC_{\max}=0.9\). The SOC evolution is written as
$$ SOC_{i,t} + \frac{p_{i,t}}{E_i} \le SOC_{\max} $$
$$ SOC_{i,t} + \frac{p_{i,t}}{E_i} \ge SOC_{\min} $$
where \(E_i\) is the battery capacity of vehicle \(i\). Overcharging and deep discharging are both prevented under these constraints, which helps extend battery life and retain the economic value of electric cars.
The decision variable in the proposed algorithm is the vector of charging/discharging power for all vehicles over the entire scheduling horizon:
$$ \mathbf{X} = [\mathbf{P}_1, \mathbf{P}_2, \dots, \mathbf{P}_T] $$
where
$$ \mathbf{P}_h = [p_{1,h}, p_{2,h}, \dots, p_{n,h}],\quad h=1,2,\dots,T $$
This decision variable is optimized by the MOEA/D-EV algorithm. The algorithm begins by initializing a population of \(N\) candidate plans according to the current SOC and the finite discrete action set. Then, for every individual in the population, the auction procedure is run for each time step to correct the action choices and to ensure they satisfy the battery constraints. The objective functions are evaluated and the ideal point is updated. In the main loop, crossover and mutation operations generate a new plan, and the auction correction is applied again. The Chebyshev aggregation function determines whether the new plan can replace any plan in the neighborhood. This process continues until the stopping criterion is fulfilled.
A central component of the proposed method is the finite discrete action set. In practice, charging at arbitrary power levels is not necessary for the coordination of electric cars. Instead, the charging and discharging power can be selected among a small number of standardized rates. This is similar to the manner in which a central bank issues cash of fixed denominations so that daily transactions can be carried out conveniently. In this paper, the minimum standardized charging and discharging powers are denoted by \(c\) and \(d\), respectively. The available rates are classified as slow, normal and fast levels, such as \(\{c,3c,7c\}\) for charging and \(\{d,3d,7d\}\) for discharging.
The action set is selected based on the state of charge. When the SOC is high, the vehicle is allowed to discharge at a fast rate but its charging rate is restricted; when the SOC is low, the vehicle can charge at a high rate but discharging is carefully limited. The corresponding action set is presented in the following table.
Table 1: Finite discrete action set for different battery state intervals
| SOC range | [0.2,0.3] | (0.3,0.5] | (0.5,0.6] | (0.6,0.8] | (0.8,0.9] |
|---|---|---|---|---|---|
| Discharging actions | {d} | {3d,d} | {3d,d} | {7d,3d,d} | {7d,3d,d} |
| Charging actions | {7c,3c,c} | {7c,3c,c} | {3c,c} | {3c,c} | {c} |
During each scheduling period, the choice within this discrete action set depends on the TOU price zone. The price interval in a typical TOU tariff can be classified into several levels, such as D1, D2, D3 for discharging and C1, C2, C3 for charging. At high prices in zone D3, the vehicle is encouraged to discharge quickly. At low prices in zone C3, the vehicle is expected to charge rapidly. In this way, the action-set selection follows a “buy low, sell high” strategy, which is one of the key price-driven mechanisms in V2G scheduling.
Table 2: Charging and discharging strategy under TOU price levels
| Price level | Price condition | Charging action preference | Discharging action preference |
|---|---|---|---|
| D3 | \(P_{max}-0.5\sigma_T \le P_t \le P_{max}\) | restricted | fast discharging |
| D2 | \(P_{max}-\sigma_T \le P_t < P_{max}-0.5\sigma_T\) | restricted | normal discharging |
| D1 | \(P_{avg} \le P_t < P_{max}-\sigma_T\) | restricted | slow discharging |
| C1 | \(P_{min}+\sigma_T \le P_t < P_{avg}\) | slow charging | restricted |
| C2 | \(P_{min}+0.5\sigma_T \le P_t < P_{min}+\sigma_T\) | normal charging | restricted |
| C3 | \(P_{min} \le P_t < P_{min}+0.5\sigma_T\) | fast charging | restricted |
Here, \(\sigma_T\) represents the spread of the price forecast over the optimization horizon, \(P_{\max}\) and \(P_{\min}\) are the highest and lowest expected prices, and \(P_{avg}\) is the average price. The use of several price zones provides a flexible interface between the electricity price and the charging/discharging decision.
To implement the combination process, this study designs a new auction algorithm. In the auction model, the microgrid is treated as the seller that announces its need for peak shaving and valley filling, while the electric cars are treated as buyers. In every time slot, each electric car chooses a charging or discharging action from its finite discrete action set, and this action is transformed into a bid. The total bid is the sum of all vehicle powers. The gap between the total power and the microgrid requirement is called the demand gap. The auction algorithm repeatedly updates bids, that is, the discrete power rates, until the demand gap is minimized. This procedure generates a corrected plan that is suitable for later objective evaluation in MOEA/D-EV.
To validate the proposed algorithm, simulation experiments are performed under two commercial TOU pricing scenarios. The first TOU price plan and the second TOU price plan are listed in the tables below. The daily base load curve of the commercial building is also shown.
Table 3: Daily base load of a commercial building
| Hour | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Power/kW | 65 | 64 | 64 | 62 | 64 | 64 | 80 | 77 | 81 | 90 | 101 | 101 |
| Hour | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |
| Power/kW | 99 | 81 | 82 | 74 | 78 | 93 | 100 | 92 | 87 | 74 | 66 | 55 |
Table 4: Commercial TOU tariff plan 1
| Period type | Time intervals | Price / CNY·kWh\(^{-1}\) |
|---|---|---|
| Peak | 8:00-14:00, 18:00-23:00 | 1.070 |
| Flat | 7:00-8:00, 14:00-18:00 | 0.687 |
| Valley | 23:00-7:00 | 0.360 |
Table 5: Commercial TOU tariff plan 2
| Period type | Time intervals | Price / CNY·kWh\(^{-1}\) |
|---|---|---|
| Peak 1 | 11:00-12:00, 19:00-23:00 | 1.210 |
| Peak 2 | 8:00-11:00, 12:00-14:00, 18:00-19:00 | 1.120 |
| Flat | 7:00-8:00, 14:00-18:00 | 0.712 |
| Valley | 23:00-7:00 | 0.300 |
In the simulations, the number of electric cars is set to \(n=10\) for TOU plan 1 and \(n=50\) for TOU plan 2. Each car has a battery capacity of \(43\,\text{kWh}\) and a maximum charging/discharging power of \(7\,\text{kW}\). The minimum standardized power is equal to \(1\,\text{kW}\). The state of charge is randomly initialized between \(20\%\) and \(90\%\) to represent different operating states. The population size of the MOEA/D-EV algorithm is 200, and the maximum number of iterations is set to 1000.
Table 6 reports the simulation results under TOU plan 1 for different weight combinations \([w_1,w_2]\), where \(w_1\) is the weight assigned to the load fluctuation objective and \(w_2\) is the weight assigned to the arbitrage revenue objective. The row \(F_1\) gives the load variance. The row \(|F_2|\) shows the absolute value of the electricity cost, i.e., the arbitrage revenue magnitude, in CNY. The table also contains the average daily load, the daily peak load, the valley load, the peak-valley difference, the total charging energy, and the total discharging energy.
Table 6: Simulation results under TOU plan 1
| \([w_1,w_2]\) | [0,1] | [0.1,0.9] | [0.2,0.8] | [0.3,0.7] | [0.4,0.6] | [0.5,0.5] | [0.6,0.4] | [0.7,0.3] | [0.8,0.2] | [0.9,0.1] | [1,0] |
|---|---|---|---|---|---|---|---|---|---|---|---|
| \(F_1(kW^2)\) | 242.000 | 48.000 | 31.333 | 22.958 | 14.958 | 12.500 | 8.500 | 5.958 | 4.625 | 2.000 | 0.950 |
| \(\|F_2\|\) (CNY) | 196.278 | 191.248 | 189.098 | 187.318 | 185.515 | 184.445 | 183.735 | 182.665 | 181.945 | 178.735 | 177.665 |
| Average load/kW | 73.000 | 73.333 | 73.500 | 73.583 | 73.667 | 73.708 | 73.708 | 73.625 | 73.708 | 73.750 | 73.750 |
| Load peak/kW | 76 | 75 | 75 | 76 | 74 | 75 | 75 | 75 | 75 | 75 | 75 |
| Load valley/kW | 60 | 69 | 70 | 72 | 69 | 71 | 72 | 73 | 74 | 74 | 74 |
| Peak-valley gap/kW | 16 | 6 | 5 | 4 | 5 | 4 | 3 | 2 | 1 | 1 | 1 |
| Total charging/kW | 113 | 119 | 119 | 124 | 123 | 123 | 123 | 124 | 126 | 128 | 128 |
| Total discharging/kW | 255 | 249 | 249 | 249 | 248 | 247 | 247 | 247 | 247 | 246 | 245 |
The results in Table 6 clearly show that the two objectives are contradictory. When the weight \(w_1\) is equal to one, the algorithm completely focuses on minimizing the load fluctuation. The load variance decreases to 0.950, and the peak-valley gap is only 1 kW. Under this extreme preference, the total charging energy is high but the arbitrage revenue \(|F_2|\) is the lowest in the table. Conversely, when \(w_2=1\), all weight is placed on the arbitrage revenue. The revenue reaches 196.278 CNY but the load variance rises to 242.000. The Pareto front obtained by the MOEA/D-EV algorithm provides a full set of trade-off alternatives for the decision maker.
Table 7: Simulation results under TOU plan 2
| \([w_1,w_2]\) | [0,1] | [0.1,0.9] | [0.2,0.8] | [0.3,0.7] | [0.4,0.6] | [0.5,0.5] | [0.6,0.4] | [0.7,0.3] | [0.8,0.2] | [0.9,0.1] | [1,0] |
|---|---|---|---|---|---|---|---|---|---|---|---|
| \(F_1(kW^2)\) | 498.625 | 254.500 | 112.958 | 92.958 | 72.958 | 59.958 | 45.333 | 34.500 | 24.625 | 17.625 | 10.958 |
| \(\|F_2\|\) (CNY) | 273.836 | 269.118 | 264.236 | 260.188 | 253.822 | 245.898 | 239.114 | 237.102 | 233.652 | 219.986 | 153.026 |
| Average load/kW | 71.875 | 71.750 | 71.708 | 71.702 | 72.041 | 72.458 | 72.667 | 72.750 | 72.875 | 73.375 | 76.958 |
| Load peak/kW | 82 | 79 | 76 | 75 | 75 | 76 | 76 | 75 | 75 | 75 | 79 |
| Load valley/kW | 62 | 65 | 68 | 68 | 69 | 70 | 71 | 71 | 71 | 72 | 76 |
| Peak-valley gap/kW | 20 | 14 | 8 | 7 | 6 | 6 | 5 | 4 | 4 | 3 | 3 |
| Total charging/kW | 1249 | 1259 | 1250 | 1119 | 1250 | 1251 | 1254 | 1256 | 1258 | 1256 | 1199 |
| Total discharging/kW | 1418 | 1431 | 1423 | 1421 | 1415 | 1406 | 1404 | 1404 | 1403 | 1389 | 1246 |
Under TOU plan 2, 50 electric cars participate in the scheduling. Similar trade-off behavior can be observed. When the algorithm is formulated with weight \([0,1]\), the absolute revenue \(|F_2|\) is equal to 273.836 CNY, the largest arbitrage income in the table, yet the load variance is as high as 498.625. When the weight becomes \([1,0]\), the load variance decreases to 10.958, but the revenue falls to 153.026. Compared with the original peak-valley difference of 46 kW in the base load, even the extreme revenue-oriented schedule in Table 7 reduces the peak-valley gap to 20 kW. The schedule with the lowest load variance achieves a peak-valley difference of only 3 kW. This strongly confirms the significant potential of the proposed approach for peak shaving and valley filling.
The dynamic behavior of the scheduling method is further revealed by examining the total charging and discharging powers in each hour. Under TOU plan 1, the hourly aggregate powers of 10 electric cars are shown in the next table.
Table 8: Hourly aggregate charging/discharging power of 10 electric cars under TOU plan 1
| Hour | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Discharging/kW | 2 | 3 | 3 | 1 | 1 | 0 | 8 | 6 | 11 | 18 | 28 | 26 |
| Charging/kW | -11 | -13 | -13 | -13 | -11 | -10 | -2 | -3 | -3 | -2 | -1 | 0 |
| Hour | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |
| Discharging/kW | 25 | 10 | 10 | 6 | 7 | 20 | 26 | 18 | 13 | 5 | 0 | 0 |
| Charging/kW | 0 | -2 | -2 | -6 | -3 | 0 | 0 | 0 | 0 | -2 | -8 | -18 |
Under TOU plan 1, the scheduling results clearly follow the “discharge at high price, charge at low price” principle. During the valley period from hour 1 to hour 6, most electric cars are charging, and the aggregate charging power is uniformly distributed. In the peak period from hour 9 to hour 14, the aggregate discharging power reaches high values, and the electricity price is correspondingly high. This behavior verifies that the finite discrete action set and the auction algorithm respond effectively to the TOU price signal. At the same time, the absolute value \(F_2\) increases when the market price difference is large, which provides an economic incentive for electric car owners to participate in V2G.
For TOU plan 2, the hourly schedule of 50 electric cars is presented below. Because more cars participate in the scheduling, the aggregate power values are larger than those under plan 1.
Table 9: Hourly aggregate charging/discharging power of 50 electric cars under TOU plan 2
| Hour | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Discharging/kW | 78 | 53 | 49 | 47 | 44 | 45 | 55 | 30 | 67 | 81 | 81 | 44 |
| Charging/kW | -88 | -65 | -58 | -59 | -54 | -56 | -47 | -25 | -57 | -63 | -51 | -14 |
| Hour | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |
| Discharging/kW | 76 | 69 | 32 | 30 | 31 | 70 | 83 | 67 | 74 | 72 | 70 | 58 |
| Charging/kW | -50 | -59 | -21 | -28 | -25 | -50 | -53 | -48 | -57 | -69 | -78 | -76 |
Figure 4-3 and Figure 4-4 in the original visualization show the same values in the form of bar charts and line charts. Even without directly presenting the figures, the data table itself confirms that the proposed algorithm can adjust the aggregate charging and discharging powers to match the TOU price levels. During high-price intervals, discharging reaches its peak; during low-price intervals, charging becomes dominant. The integration of the auction mechanism helps the algorithm identify whether an individual electric car should charge or discharge according to its current battery SOC and the current price zone. It also improves the robustness of the scheduling result against different price structures.
The ability of the framework to manage individual battery state evolution is also important. Under the middle weight combination \([0.5,0.5]\), the state of charge trajectories of selected vehicles can be examined. At the beginning of the optimization period, all electric cars receive an initial SOC that is randomly distributed in the interval \([0.2,0.9]\). The simulation results show that the SOC is maintained in the safe range at every time step. In the high price period, the SOC tends to decrease because the cars discharge to earn revenue. In the low price period, the SOC recovers as the cars are charged. This is exactly the desired behavior for a V2G program: the grid does not need to directly ask each user to charge or discharge because the price signal provides sufficient guidance. The state constraint also protects the battery from overcharge and deep discharge. This dynamic regulation ability is a major advantage of the proposed method.
In addition, the finite discrete action set strategy significantly reduces the computational and communication complexities. Since the charging and discharging power can only be selected from a discrete action set, the decision space is no longer an infinite continuous space. For 50 electric cars over a 24-hour horizon, the optimization dimension is 1200, which is still high but becomes manageable. In terms of communication, a continuous power value would normally require four bytes of floating-point data, while a discrete action index can be represented by one byte, or in some cases only two bits. The communication load can therefore be reduced by about seventy-five percent compared with continuous formulations. This property supports real-time interaction between the central controller and large numbers of electric cars.
Some representative scheduling plans are now presented to demonstrate the practical feasibility. Table 10 gives the charging/discharging powers of 10 vehicles at all hours under TOU plan 1, obtained with the proposed MOEA/D-EV method. In this table, positive values denote discharging, negative values denote charging, and zero means that the vehicle is idle.
Table 10: Charging/discharging schedule of 10 electric cars under TOU plan 1
| Vehicle | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 0 | 3 | 3 | 3 | 3 | 3 | 0 | -1 | 0 | 1 | 1 | 3 | 3 | 1 | 1 | -1 | -1 |
| 2 | -1 | 1 | -1 | 0 | -1 | -1 | 1 | 1 | 3 | 3 | 7 | 3 | 3 | 1 | -1 | -1 | -1 | 1 | 3 | 1 | 3 | 1 | 0 | -1 |
| 3 | -1 | 0 | -3 | -1 | -3 | -1 | 1 | 1 | -1 | 3 | 3 | 7 | 3 | 1 | 3 | 3 | 1 | 3 | 3 | 1 | 0 | 0 | -3 | -1 |
| 4 | -1 | -3 | -1 | 0 | -1 | -1 | -1 | -1 | -1 | 1 | 0 | 3 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | -1 | -1 |
| 5 | 0 | 0 | 1 | -3 | 0 | -1 | 1 | 1 | -1 | 1 | 3 | 1 | 7 | 1 | 1 | -1 | 1 | 7 | 3 | 1 | 3 | 1 | -1 | -3 |
| 6 | 1 | 1 | 1 | -1 | 0 | -1 | 3 | 1 | 1 | 3 | -1 | 3 | 3 | 0 | 3 | 1 | 1 | 0 | 3 | 7 | 3 | 1 | 0 | -3 |
| 7 | -3 | -1 | -3 | -3 | -1 | -1 | 1 | -1 | 1 | 3 | 3 | 1 | 1 | -1 | 0 | 1 | 1 | 3 | 3 | 0 | 1 | -1 | -1 | -3 |
| 8 | -1 | -7 | -3 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 3 | 1 | 1 | 3 | 1 | -1 | -1 | 0 | 1 | 1 | 0 | 0 | 0 | -3 |
| 9 | -3 | -1 | 1 | -3 | -1 | -1 | 0 | 1 | 1 | -1 | 3 | 3 | 3 | 3 | 0 | -3 | -1 | 1 | 3 | 0 | 1 | -1 | -1 | -1 |
| 10 | -1 | -1 | -1 | -1 | -3 | -1 | 0 | 1 | 1 | 1 | 3 | 1 | 1 | -1 | 1 | 0 | 1 | 0 | 3 | 3 | 1 | 1 | 0 | -1 |
The schedule in Table 10 clearly demonstrates that the proposed algorithm yields implementable discrete actions. During the high-price daytime period, fast discharge rates such as 3 kW or 7 kW appear frequently. During nighttime valley hours, the cars switch to charging actions. The table also shows that no vehicle is forced to charge or discharge outside its battery capability. The state of charge constraints are not violated at any point. This observation is particularly important because practical drivers may not always be willing to follow an arbitrary continuous power curve. Discrete schedules are much easier to execute in charging stations because they can be mapped directly to standard charger modes.
Compared with a conventional single-objective approach, the multi-objective framework developed here provides richer information. The Pareto front obtained by the algorithm enables the grid operator to choose a solution that better matches its actual requirement on a particular day. For instance, if the weather forecast predicts a large power imbalance, the operator may select a solution closer to \([1,0]\). If the local power market offers a large price spread and the operator wants to reward electric car users, a solution closer to \([0,1]\) can be selected. The intermediate solutions, such as \([0.5,0.5]\), offer a balanced policy that is often preferred in practical operations because it yields both smooth grid load and respectable user revenue.
Another important feature is data-driven rolling scheduling. In the proposed topology, the decision center repeatedly obtains base load forecasts and price forecasts, constructs a roll-forward optimization window, and then sends the selected schedule to electric car charging piles. This rolling mechanism permits the schedule to be updated when the base load prediction changes or when new vehicles enter the system. The resulting schedule is called forward-looking because it takes into account the future state of the system rather than simply responding to the current interval. The auction algorithm is executed at every time interval to maintain feasibility and to adapt the fleet actions to the most recent TOU signal. This makes the overall framework suitable for online V2G management.
Several additional findings can be observed from the simulation data. First, the total discharging energy is larger than the total charging energy in all tested cases. This is because electric cars discharge at high price and charge at low price, and the aggregate discharge energy reflects the economic incentive under the TOU mechanism. Moreover, the average load after optimization remains close to the average load of the original base load curve, which suggests that the electric vehicle fleet mainly shifts energy consumption instead of adding a huge net load to the grid. The improvement in the load shape is therefore achieved without sacrificing the energy balance. The peak-valley gap of the original base load is 46 kW. Under TOU plan 2, the optimized peak-valley gap is at most 20 kW even in the fully revenue-oriented case, and the gap is reduced to just 3 kW in the fully grid-oriented case. This result illustrates the substantial capability of electric cars to act as flexible resources for the distribution grid.
The proposed method also contributes to the promotion of electric cars as a practical form of distributed energy storage. While many previous studies assume perfectly controllable charging power and ignore the resulting computational burden, this paper directly addresses implementability. By treating charging/discharging rates as selected from the discrete action set, the model becomes more realistic and easier to embed in charging station controllers. Through the action selection procedure, every electric car receives a clear operation command: charge, discharge, or stay idle. The command is chosen according to the battery SOC and the TOU price, so it is both safe and economically beneficial for the vehicle owner. The auction algorithm, meanwhile, supplies the system-level coordination needed so that the individual commands, when aggregated, satisfy the grid-level objective.
Several technical details of the proposed auction algorithm deserve further explanation. Initially, the microgrid sends the current price \(Price_t\) and the current demand signal \(Demand_t\) to the controller. Each electric car then generates an initial bid based on its SOC and the discrete action set. The algorithm verifies whether the initial plan is feasible. If it is not feasible, the invalid actions are repaired. After this repair step, the bids are sorted in descending order. The total bid is aggregated and compared with the demand signal. The gap between them is calculated. If the gap is not sufficiently small, the bids are updated by changing the discrete action levels, and the process iterates. The final plan is then returned to the MOEA/D-EV algorithm, where it is evaluated with respect to both objectives. This bidirectional combination effectively closes the loop between individual vehicle actions and system-level requirements.
From a computational perspective, the finite discrete action set also benefits the aggregation phase of the optimization. Since each vehicle action belongs to a set such as \(\{-7,-3,-1,0,1,3,7\}\), the aggregate load in each time step has a much simpler structure than in the continuous case. The evaluation of the load variance and the arbitrage revenue can be carried out efficiently even for hundreds or thousands of electric cars. This benefit becomes increasingly important as the penetration level of electric cars grows and the number of participating vehicles continues to rise. For a real-world city with tens of thousands of electric cars, the communication and control architecture must be light enough to allow frequent schedule updates. The finite action approach is therefore highly relevant to future large-scale V2G implementations.
There is also room for further research. First, the present work assumes that all electric cars are available for the full 24-hour scheduling horizon and that they do not have mandatory departure times. In reality, trip chains, charging station occupancy, and user behavior will affect availability. Future models can incorporate stochastic driving patterns and departure constraints. Second, the combination of electric cars with renewable energy generation, such as wind and solar power, can be explored. Electric car charging can absorb excess renewable generation during high wind or high irradiance periods, and discharging can provide backup during periods of low renewable output. Third, the degradation cost of the battery is not included in the current objective function. In practice, rapid discharging may reduce the calendar life of the battery pack. Adding a battery degradation cost term would make the scheduling more realistic. The use of machine learning models for predicting base load, TOU prices, and user behavior could also improve the scheduling quality. Finally, the blockchain technique could provide a secure and flexible mechanism for clearing and settling V2G transactions, encouraging more electric car owners to participate in demand response programs.
In summary, this paper has proposed and validated a MOEA/D-based method for solving the multi-objective optimal scheduling of electric vehicle charging and discharging. The method relies on three main components: a multi-objective model, a finite discrete action set strategy, and a new auction algorithm. The model minimizes the load fluctuation and maximizes the arbitrage revenue of electric car owners under TOU tariffs. The finite action set compresses the high-dimensional continuous action space and greatly reduces the computational complexity. The auction algorithm dynamically assigns charging and discharging missions to electric cars in every time interval. Extensive simulation results under two TOU pricing plans demonstrate that the proposed method can effectively shave peaks and fill valleys, suppress load variations, and increase the profitability of electric vehicle owners. The obtained Pareto front provides a rich set of trade-off solutions that can guide practical engineering decisions. Therefore, the proposed framework is both effective and practically feasible for real-world V2G scheduling.