Optimal Operation of EVs, EBs and BESS Considering EBs-Charging Piles Matching Problem Using a Novel Pricing Strategy Based on ICDLBPM

Abstract

Electric vehicles (EVs), electric buses (EBs), and battery energy storage system (BESS), as both controllable power sources and load, play a great role in providing flexibility for the power grid, especially with the increased renewable energy penetration. However, there is still a lack of studies on EVs’ pricing strategy as well as the EBs-charging piles matching problem. To address these issues, a multi-objective optimal operation model is presented to achieve the lowest load fluctuation level, minimum electricity cost, and maximum discharging benefit. An improved load boundary prediction method (ICDLBPM) and a novel pricing strategy are proposed. In addition, reduction in the number of EBs charging piles would not only impact normal operation of EBs, but also even lead to load flexibility decline. Thus a handling method of the EBs-charging piles matching problem is presented. Several case studies were conducted on a regional distribution network comprising 100 EVs, 30 EBs, and 20 BESS units. The developed model and methodology demonstrate superior performance, improving load smoothness by 45.78% and reducing electricity costs by 19.73%. Furthermore, its effectiveness is also validated in a large-scale system, where it achieves additional reductions of 39.31% in load fluctuation and 62.45% in total electricity cost.

1. Introduction

1.1. Background

With the promotion of carbon peaking and carbon neutrality, a high penetration of renewable energy sources (RESs) has increasingly become a fundamental characteristic and development trend of modern power grids [1]. Nevertheless, the inherent volatility, intermittency, and randomness of RESs introduce significant operational pressure and challenges to grid stability [2]. In response, tapping into the flexibility potential across different segments of the power system has become crucial for maintaining grid stability and enhancing operational adaptability [3,4,5].

As demand-side resources, flexible loads can be optimally dispatched to shave peak loads, fill valleys, mitigate intermittent energy fluctuations, and provide ancillary services [5,6,7]. Compared with conventional generation units, flexible loads exhibit advantages such as fast response speed and low inertia constants [8]. Therefore, leveraging flexible loads for energy-balance ancillary services represents a key pathway to address supply–demand imbalances in power systems with high RES penetration [9].

1.2. Literature Reviews

Peak load shaving, as a key grid-flexibility measure, is among the most important ancillary services, alongside frequency regulation, spinning reserve, and reactive power support [9,10,11,12]. Extensive research has examined flexible loads and their role in ancillary services. For example, A. Arcos-Vargas et al. [13] analyzed recent BESS trends and proposed an optimal sizing model to reduce consumer electricity costs. R. Tulabing et al. [6] introduced a load-aggregation method that prioritizes flexible loads to dynamically adjust demand in response to price signals. A. Dadkhah et al. [14] utilized electrolyzers in hydrogen refueling stations as flexible loads to provide frequency-regulation services, while X. Hu et al. [15] developed a stochastic algorithm to coordinate distributed generators’ reactive power and improve voltage profiles. In recent years, battery energy storage systems (BESSs) have been the predominant flexible resource studied for peak shaving [16,17,18]. Studies have focused on optimizing BESS capacity, power, and placement to flatten load curves [19], and on controlling individual batteries within a BESS to enhance shaving efficiency [20]. In addition, a novel battery management system (BMS) [21] was designed to mitigate state-of-charge (SOC) imbalances among battery packs, thereby extending battery lifespan. This approach has been successfully extended to grid-connected energy storage systems.

Recently, electric vehicles (EVs) have played an increasingly important role as flexible resources, providing not only peak load shaving but also various ancillary services [22]. For instance, S. Gao et al. [23] explored the potential of EV aggregation in ancillary service markets and proposed a coordinated charging/discharging strategy to enhance frequency regulation and economic benefits. Similarly, a reactive-power optimization model with ordered charging/discharging was developed to improve voltage profiles, reduce losses, and lower costs [24].

With the maturation of vehicle-to-grid (V2G) technology, EVs have become integral to peak shaving [17]. K. L. Zhou et al. [25] introduced a charging-scheduling model based on an urgency indicator to balance user needs and shaving effectiveness, while an improved decision-tree algorithm was applied to coordinate EVs, BESS, and PV units for peak reduction [26]. However, the inherent randomness of EV loads can exacerbate fluctuations and undermine shaving performance. To address this, prediction-based approaches have been widely adopted. Studies show that incorporating forecasting into scheduling can smooth grid operations, improve shaving effects, and reduce costs [27,28,29]. Examples include day-ahead plans based on elastic EV demand [30], load-prediction systems for optimal EV dispatch [31], composite models for short-term charging-load forecasts [32], and boundary-based predictions of schedulable EV charging loads [33]. Yet, most existing prediction methods focus solely on charging; forecasts that jointly consider charging and discharging remain scarce.

Electricity pricing also critically influences shaving outcomes. While fixed time-of-use (TOU) tariffs have been used to guide user behavior [34,35,36], they are not universally suitable. Developing adaptive pricing strategies is therefore essential. L. Zhang et al. [37] redesigned TOU rates according to daily load curves, and L. Zhu et al. [38] redefined peak-valley periods based on load-deviation thresholds. However, these strategies typically operate on given load curves without integrating load forecasting, limiting their responsiveness to real-time conditions.

1.3. Research Gap and Contributions

While existing research has extensively explored the integration of EVs and BESS, a significant gap remains in models that adequately incorporate EBs. As a high-power, schedulable fleet with distinct operational patterns, EBs represent a critical but often neglected flexible load. Moreover, the practical mismatch between EB charging demands and charging pile availability is rarely addressed, limiting the realism of current coordinated dispatch strategies. Moreover, only a few studies have investigated the combination of load forecasting and pricing strategy in ancillary service provided by flexible load. To bridge these gaps, as systematically compared in

Table 1. Comparison of the proposed strategy with different studies.

Reference	Flexible Load Type			Load Prediction	Pricing Strategy/TOU Electricity Price	Handling Method of EBs-Charging Piles Matching Problem
	EVs	EBs	BESS	–	–	–
[25]	√	–	–	–	–	–
[26]	√	–	√	√	–	–
[30]	√	–	–	√	–	–
[31]	√	–	–	√	–	–
[32]	√	–	–	√	–	–
[33]	√	–		√	√	–
[34]	–	–	√	–	√	–
[35]	–	–	√	–	√	–
[36]	–	–	√	–	√	–
[37]	√	–	–	–	√	–
[38]	√	–	–	–	√	–
This study	√	√	√	√	√	√

, this study aims to explore ancillary services provided by mixed flexible load, such as EVs, EBs, and BESS, considering different pricing strategies with load forecasting as well as the charging pile matching problem of EBs.

To summarize, the contributions of this study can be highlighted as follows.

(1)

The presented multi-objective optimization model explicitly incorporates EBs as co-optimizable entities alongside EVs and BESS, effectively coordinating them to minimize load fluctuation and total electricity cost.

(2)

The proposed improved charging/discharging load boundary prediction method (ICDLBPM) characterizes both charging and discharging power as variables within dynamic boundaries, providing a more flexible foundation for dispatch optimization.

(3)

The developed ICDLBPM-based pricing strategy introduces a boundary-driven dynamic TOU pricing mechanism that adjusts electricity tariffs in real-time based on predicted load boundaries and actual EV load, thereby effectively guiding charging/discharging behavior to enhance peak shaving and valley filling while maintaining grid-user alignment.

(4)

The introduced handling method for EBs-charging piles matching problem not only ensures operational feasibility under resource constraints but also strategically utilizes the flexible charging/discharging capacity of matched EBs to support grid stability and renewable integration.

2. Problem Formulation and Modeling

As flexible sources and loads, EVs and BESS have played an increasingly important role in providing auxiliary services to the power grid such as suppressing daily load fluctuation. In addition, due to the significant driving regularity of EBs, their charging and discharging power can be adjusted when they are connected to the charging piles to supply flexible power to the power grid. In this paper, the network structure of charging and discharging resources is established as shown in Figure 1. It can be seen that batteries can directly exchange energy with the power grid, while EBs and EVs can connect to the power grid through charging piles with bidirectional energy flow. It should be noted that each EV has one charging pile; EVs do not need to queue up for charging to achieve bidirectional energy flow with the power grid. However, for EBs, due to the fact that the number of charging piles (m) is less than the number of EBs (n), charging and discharging strategies of EBs are more complex. The conditions above make it feasible for EVs, EBs, and BESS to improve the reliability, flexibility, and economy of power grid operation.

2.1. Optimal Objectives

To achieve more reliable and flexible operation of the power grid as well as lower charging cost and higher discharging benefit, an optimal multi-objective coordinated charging/discharging model of EVs, EBs, and BESS is proposed considering the mismatching problem between EBs and charging piles. The static deviation rate quantifies the overall deviation of a load curve from its peak value. A smaller value indicates a flatter curve with lower volatility, making it a direct and effective metric for assessing load-smoothing performance. Therefore, minimizing this rate as shown in Equation (1) is aligned with the objective of reducing load fluctuation. It simultaneously aims to minimize the charging costs and maximize the discharging revenues of EVs, EBs, and BESS. It should be noted that the positive power of EVs, EBs, and batteries means they are in a charging state and their charging cost should be added in operation cost as shown in Equation (2). Similarly, the negative power of EVs, EBs, and batteries means they are in a discharging state and their discharging benefit is calculated as shown in Equation (3). The established multi-objective optimization model is given in Equations (1)–(3).

$$\min F_1={\displaystyle \sum _{t=1}^{n_{\mathrm{T}}}\frac{\left(P_{\mathrm{L}}^{\max }-P_{\mathrm{L}}\left(t\right)\right)}{P_{\mathrm{L}}^{\max }}}$$

(1)

$$\min F_2={\displaystyle \sum _{t=1}^{n_{\mathrm{T}}}\left(k_1{P}_{\mathrm{unc}}\left(t\right)+k_2\left({\displaystyle \sum _{i=1}^{n_{\mathrm{EV}}}{P}_{\mathrm{EV}}(i,t)C_{\mathrm{EV}}(i,t)+{\displaystyle \sum _{j=1}^{n_{\mathrm{EB}}}{P}_{\mathrm{EB}}(j,t)C_{\mathrm{EB}}(j,t)+{\displaystyle \sum _{k=1}^{n_{\mathrm{B}}}{P}_{\mathrm{B}}(k,t)C_{\mathrm{B}}(k,t)}}}\right)\right)}$$

(2)

$$\max F_3={\displaystyle \sum _{t=1}^{n_{\mathrm{T}}}\left(k_3\left({\displaystyle \sum _{i=1}^{n_{\mathrm{EV}}}{P}_{\mathrm{EV}}(i,t)D_{\mathrm{EV}}(i,t)+{\displaystyle \sum _{j=1}^{n_{\mathrm{EB}}}{P}_{\mathrm{EB}}(j,t)D_{\mathrm{EB}}(j,t)+{\displaystyle \sum _{k=1}^{n_{\mathrm{B}}}{P}_{\mathrm{B}}(k,t)D_{\mathrm{B}}(k,t)}}}\right)\right)}$$

(3)

2.2. Constraints

The operation of EVs, EBs, and BESS is subject to various constraints, such as capacity constraints, SOC-related charging and discharging power constraints, and real-time load balance equations. Specifically, EVs and EBs are both subject to driving time limits and a minimum quantity of electricity required for operation.

2.2.1. Capacity Constraints of EVs, EBs, and BESS

To protect the battery and satisfy the minimum quantity of electricity for operation, in addition to setting the upper capacity limits, the lower capacity limits should also be set. The capacity limits and capacity calculations are shown in Equations (4)–(7). Specifically, the capacity of EVs, EBs, and BESS at time t is obtained based on the capacity of EVs, EBs, and BESS at time t − 1 and the charging and discharging power at time t − 1. And it should be noted that when the EB is connected to the charging pile after completing one operation, the power required for operation should be subtracted.

$$S_{\mathrm{x}}^{\min }(m)\le S_{\mathrm{x}}(m,t)\le S_{\mathrm{x}}^{\max }(m,t), \mathrm{x}\in \left\{\mathrm{EV},\mathrm{EB},\mathrm{B}\right\}$$

(4)

$$S_{\mathrm{x}}\left(m,t\right)=S_{\mathrm{x}}\left(m,t-1\right)+{\eta }_{\mathrm{x}}\left(m\right)P_{\mathrm{x}}\left(m,t\right), \mathrm{x}\in \left\{\mathrm{EV},\mathrm{EB},\mathrm{B}\right\}$$

(5)

$$S_{\mathrm{y}}^{\min }(u)+E{r}_{\mathrm{y}}(u)\le S_{\mathrm{y}}(u,t),T_{\mathrm{y}}(u,t-1)=1\&T_{\mathrm{y}}(u,t)=0, \mathrm{y}\in \left\{\mathrm{EV},\mathrm{EB}\right\}$$

(6)

$$S_{\mathrm{EB}}\left(j,t\right)=S_{\mathrm{EB}}\left(j,t-1\right)-E{r}_{\mathrm{EB}}(j) , T_{\mathrm{EB}}(j,t-1)=0\&T_{\mathrm{EB}}(j,t)=1$$

(7)

2.2.2. SOC-Related Charging/Discharging Power Constraints

In addition to indicating the available state of remaining charge in the battery, SOC also affects the charging/discharging power of batteries. In the actual operation, the maximum charging and discharging power of EVs, EBs, and BESS vary with SOC. For this reason, the charging and discharging power limits should be calculated separately based on the SOC at different times. Charging and discharging power limits are given as shown in Equations (8) and (9), respectively.

$$0\le P_{\mathrm{x}}(m,t)·C_{\mathrm{x}}(m,t)\le {\displaystyle \frac{S_{\mathrm{x}}^{\max }(m)-S_{\mathrm{x}}(m,t)}{S_{\mathrm{x}}^{\max }(m)}}·P_{\mathrm{x}}^{\max }(m), \mathrm{x}\in \left\{\mathrm{EV},\mathrm{EB},\mathrm{B}\right\}$$

(8)

$${\displaystyle \frac{S_{\mathrm{x}}(m,t)-S_{\mathrm{x}}^{\min }(m)}{S_{\mathrm{x}}^{\max }(m)}}·P_{\mathrm{x}}^{\min }(m)\le P_{\mathrm{x}}(m,t)·D_{\mathrm{x}}(m,t)\le 0, \mathrm{x}\in \left\{\mathrm{EV},\mathrm{EB},\mathrm{B}\right\}$$

(9)

2.2.3. Real-Time Load Balance Equation

Apart from the capacity limits and power limits of EVs, EBs, and BESS, the real-time power balance constraints should also be satisfied for the whole system. In particular, the total power demand is equal to the total power supply at each time. EVs, EBs, and BESS can exchange power with the power grid as mobile and flexible power load or power sources. In addition, the time interval is selected as 15 min. The real-time power balance equations are described as shown in Equations (10) and (11).

$$\begin{array}{c}{P}_{\mathrm{unc}}(t)+{\displaystyle \sum {\displaystyle \sum _{m=1}^{n_{\mathrm{x}}}{P}_{\mathrm{x}}(m,t)C_{\mathrm{x}}(m,t)}}\\ =P_{\mathrm{s}}(t)+{\displaystyle \sum {\displaystyle \sum _{m=1}^{n_{\mathrm{x}}}{P}_{\mathrm{x}}(m,t)}}{D}_{\mathrm{x}}(m,t), \mathrm{x}\in \left\{\mathrm{EV},\mathrm{EB},\mathrm{B}\right\}\end{array}$$

(10)

$$C_{\mathrm{y}}(u,t)D_{\mathrm{y}}(u,t)=0, \mathrm{y}\in \left\{\mathrm{EV},\mathrm{EB}\right\}$$

(11)

2.2.4. Driving Time Constraints of EVs and EBs

EVs cannot be charged or discharged during driving as shown in Equation (12). The maximum charging and discharging power of EVs will be set to zero during driving. However, due to the shortage of EBs’ charging piles, even if EBs stay in charging stations, charging and discharging operations can only be carried out when EBs are connected to charging piles as shown in Equation (13). The maximum charging and discharging power of EVs will be set to zero during driving or they are not connected to charging piles.

$$\left\{\begin{array}{l}{P}_{\mathrm{EV}}^{\max }(i)=P_{\mathrm{EV}}^{\max }\left(i\right)T_{\mathrm{EV}}(i,t) \\ P_{\mathrm{EV}}^{\min }(i)=P_{\mathrm{EV}}^{\min }\left(i\right)T_{\mathrm{EV}}(i,t)\end{array}\right.$$

(12)

$$\left\{\begin{array}{l}{P}_{\mathrm{EB}}^{\max }(i)=P_{\mathrm{EB}}^{\max }\left(i\right)T_{\mathrm{EB}}(j,t)Z_{\mathrm{EB}}(j,t)\\ P_{\mathrm{EB}}^{\min }(j)=P_{\mathrm{EB}}^{\min }\left(j\right)T_{\mathrm{EB}}(j,t)Z_{\mathrm{EB}}(j,t) \end{array}\right.$$

(13)

3. Methodology and Strategy

3.1. Improved Charging/Discharging Load Boundary Prediction Method (ICDLBPM)

A load boundary prediction method that forecasts the maximum and minimum EV load for the next time period based on the actual EV load at the current moment was proposed by J. S. Yang et al. [1]. However, this method focuses solely on charging load prediction and does not account for potential discharging load. To address this limitation, an improved charging and discharging load boundary prediction model (ICDLBPM) is proposed. With ICDLBPM, the predicted charging and discharging power of EVs are no longer treated as fixed values but as variables within defined upper and lower boundaries, thereby providing a more comprehensive and flexible input for dispatch optimization. The procedure of the proposed ICDLBPM, as outlined in Figure 2, consists of three sequential steps.

Step 1: The actual charging/discharging power and adjustable boundaries of each EV are initialized. The maximum adjustable power margin for each EV at the next time t + 1 is then calculated using Equation (14).

Step 2: The adjustable power boundaries are dynamically updated according to the real-time grid-connection status of each EV. Specifically, the load boundaries for any EV that has just stopped charging or is scheduled to leave the grid by time t + 1 are set to zero.

Step 3: For the fleet of EVs that remain connected to the grid from time t to t + 1, the aggregate upper and lower load boundaries for the system at t + 1 are computed via Equation (15).

$$\left\{\begin{array}{l}\Delta P_{\mathrm{EV}}^{\max }(i,t+1)=P_{\mathrm{EV}}^{\max }(i,t)-P_{\mathrm{EV}}(i,t)\\ \Delta P_{\mathrm{EV}}^{\min }(i,t+1)=P_{\mathrm{EV}}^{\min }(i,t)-P_{\mathrm{EV}}(i,t)\end{array}\right.$$

(14)

$$\begin{array}{l}{P}_{\mathrm{EV}}^{\mathrm{up}}(i,t)=P_{\mathrm{EV}}^{\mathrm{down}}(i,t)=P_{\mathrm{EV}}(i,t), t=1\\ \left\{\begin{array}{l}{P}_{\mathrm{EV}}^{\mathrm{up}}(i,t+1)=P_{\mathrm{EV}}^{\mathrm{up}}(i,t)+\Delta P_{\mathrm{EV}}^{\max }(i,t)\\ P_{\mathrm{EV}}^{\mathrm{down}}(i,t+1)=P_{\mathrm{EV}}^{\mathrm{down}}(i,t)+\Delta P_{\mathrm{EV}}^{\min }(i,t)\end{array}\right., \mathit{else}\end{array}$$

(15)

3.2. Pricing Strategy Based on ICDLBPM

To reduce electricity costs, TOU pricing and similar strategies are widely adopted. For instance, L. Zhang et al. [37] classified daily periods into peak, flat, and valley based on power load and adjusted TOU prices accordingly. Building on this, L. Zhu et al. [38] further refined the period classification by incorporating the k-fold peak–valley difference and the deviation between daily average load and real-time load, followed by corresponding tariff adjustments.

However, the method proposed by L. Zhang et al. may inadvertently create new load peaks, and its predefined periods may misalign with user consumption habits, potentially compromising user satisfaction. Although the strategy introduced by L. Zhu et al. offers greater flexibility and helps avoid secondary peaks, its practical application is hindered by the challenge of determining an optimal k-factor.

$$\left\{\begin{array}{l}{d}_{\mathrm{C}}(t)={\displaystyle \sum _{i=1}^{n_{\mathrm{EV}}}{P}_{\mathrm{EV}}^{\mathrm{up}}}(i,t)-{\displaystyle \sum _{i=1}^{n_{\mathrm{EV}}}{P}_{\mathrm{EV}}}(i,t)\\ d_{\mathrm{DC}}(t)={\displaystyle \sum _{i=1}^{n_{\mathrm{EV}}}{P}_{\mathrm{EV}}}(i,t)-{\displaystyle \sum _{i=1}^{n_{\mathrm{EV}}}{P}_{\mathrm{EV}}^{\mathrm{down}}}(i,t)\end{array}\right.$$

(16)

$$\left\{\begin{array}{l}{q}^{\prime }\left(t\right)=q(t)\times {\displaystyle \frac{P_{\mathrm{EV}}(t)}{P_{\mathrm{EV}}^{\max \_\mathrm{peak}}}} t\in t_{\mathrm{peak}}\\ q^{\prime }\left(t\right)=q(t)\times {\displaystyle \frac{P_{\mathrm{EV}}(t)}{P_{\mathrm{EV}}^{\max \_\mathrm{flat}}}} t\in t_{\mathrm{flat}}\\ q^{\prime }\left(t\right)=q(t)\times {\displaystyle \frac{P_{\mathrm{EV}}(t)}{P_{\mathrm{EV}}^{\max \_\mathrm{valley}}}} t\in t_{\mathrm{valley}}\end{array}\right.$$

(17)

To address these limitations, a ICDLBPM-based pricing strategy is proposed. This approach adjusts TOU prices in response to predicted load boundaries and actual EV load in real-time. Specifically, the differences between the predicted load boundaries and the actual EV load are first computed using Equation (16) and ranked in descending order. Based on this ranking, time intervals are sequentially categorized into peak, flat, and valley periods, with the top, middle, and bottom 32 intervals assigned to peak, flat, and valley periods, respectively. Subsequently, charging and discharging TOU prices are determined via Equation (17), which leverages the ratio of actual EV load to the maximum load in each corresponding period. As illustrated in Figure 3, this boundary-driven pricing mechanism enables more effective guidance of EV charging and discharging behavior, thereby enhancing peak-shaving and valley-filling performance while maintaining better alignment with real-time grid conditions and user patterns.

3.3. Handling Method of EBs-Charging Piles Matching Problem

In actual operation, fewer EB charging piles are deployed than the number of EBs due to site constraints, high costs, and non-standard charging interfaces. Effectively matching EBs to available piles is therefore a key operational problem. Solving this problem does more than ensure reliable EB service. It also creates grid-service value. EBs with spare battery capacity can discharge to the grid for revenue, while those able to charge can absorb surplus renewable power. Both actions help smooth load fluctuations and increase renewable energy utilization.

Accordingly, a handling method of the EBs-charging piles matching problem considering operational electricity demand of EBs is proposed. Its implementation procedure is detailed below and illustrated in the flowchart of Figure 4.

Step 1: Determine the working status of each charging pile and mark it as charging/discharging or idle.

Step 2: Calculate the remaining electricity of each EB in the station, and mark EBs with insufficient electricity according to Equation (6).

Step 3: Find idle charging piles for EBs with insufficient electricity. If all the charging piles are in charging or discharging mode, and if there are charging piles connected to EBs with sufficient electricity, then connect EBs with insufficient electricity to these charging piles.

Step 4: Calculate the capacity change of each EB connected to the charging pile according to Equation (8).

3.4. Flow Description of the Multi-Objective Optimal Operation Model

The overall flow chart of the model incorporating the novel pricing strategy and EB-charging piles matching method is illustrated in Figure 5.

First, the multi-objective optimal operation model for the integrated EV-EB-BESS system accepts key inputs, including actual charging/discharging power, power capacity limits, initial TOU electricity prices, and the EB operation schedule. Based on these parameters, load boundary prediction is conducted via the ICDLBPM to achieve the modification of TOU price, while the remaining battery capacity of EBs is determined through the EB-charging piles matching handling method.

Subsequently, the aforementioned outputs are integrated into a unified framework and subjected to constraint conditions, including capacity and load balance constraints. This integrated model is formulated under the dual objective of minimizing both the static deviation rate and the total electricity cost.

Finally, the multi-objective random black-hole particle swarm optimization (MORBHPSO) [34] algorithm is adopted to solve the model. In this method, particles are assigned a probability to enter or escape a “black-hole” region within the search space, which enhances exploration and maintains population diversity. Through this optimization, the multi-objective optimal scheduling that incorporates the novel pricing strategy and EB-charging piles matching handling method is effectively realized.

4. Simulation Results and Analysis

4.1. Parameters

To indicate the effectiveness and feasibility of the proposed pricing strategy and the handling method of the EBs-charging piles matching problem, five case studies have been carried out on a regional distribution network using MATLAB2024a language programming and MORBHPSO [34] has been adopted. The network includes 100 EVs, 30 EBs, 25 EB charging piles, 20 BESSs, and uncontrollable load. The key parameters for the EVs, EBs, and BESS are adopted from Reference [34] and are listed in

Table 2. Specifications of capacity, power, and driving time constraints of EVs, EBs, and BESS.

Type	EVs	EBs	BESS	Piles
Number	100	30	20	25
Maximum capacity (kWh)	60	300	400	/
Minimum capacity (kWh)	10	30	0	/
Maximum charging power (kW)	30	100	150	100
Minimum charging power (kW)	0	0	0	0
Maximum discharging power (kW)	30	100	150	100
Minimum discharging power (kW)	0	0	0	0
Driving time	07:00–18:00	06:00–22:00	/	/

. In Cases 1–5, the charging and discharging efficiencies of EVs, EBs, and BESS are uniformly set to 90%. To evaluate the impact of efficiency variations on the system outcomes, a corresponding sensitivity analysis on efficiency is also conducted. The original TOU electricity price is the same as shown in [34]. The parameters of MORBHPSO are shown in

Table 3. Parameters of MORBHPSO.

Algorithm	Number of Iterations	Number of Particles in Each Iteration	Maximum Inertia Weight ωmax	Radius of Black-Hole R	Probability Threshold α0	Learning Factor c1	Learning Factor c2
MORBHPSO	100	100	0.9	0.001	0.1	1	1

. The predicted uncontrollable load can be found in Figure 6. The departure timetable of EBs in Case 5 is shown in

Table 4. Departure time of EBs in Case 5.

Time of Departures
EB No.	1	2	3	4	5
1	06:00	09:15	12:30	13:45	17:00
2	06:15	09:30	12:45	14:00	17:15
3	06:30	09:45	13:00	14:15	17:30
4	06:45	10:00	13:15	14:30	17:45
5	07:00	10:15	13:30	14:45	18:00
6	07:15	10:30	13:45	15:00	18:15
7	07:30	10:45	14:00	15:15	18:30
8	07:45	11:00	14:15	15:30	18:45
9	08:00	11:15	14:30	15:45	19:00
10	08:15	11:30	14:45	16:00	19:15
11	08:30	11:45	15:00	16:15	19:30
12	08:45	12:00	15:15	16:30	19:45
13	09:00	12:15	15:30	16:45	20:00
14	06:00	09:15	12:30	13:45	17:00
15	06:15	09:30	12:45	14:00	17:15
16	06:30	09:45	13:00	14:15	17:30
17	06:45	10:00	13:15	14:30	17:45
18	07:00	10:15	13:30	14:45	18:00
19	07:15	10:30	13:45	15:00	18:15
20	07:30	10:45	14:00	15:15	18:30
21	07:45	11:00	14:15	15:30	18:45
22	08:00	11:15	14:30	15:45	19:00
23	08:15	11:30	14:45	16:00	19:15
24	08:30	11:45	15:00	16:15	19:30
25	08:45	12:00	15:15	16:30	19:45
26	09:00	12:15	15:30	16:45	20:00
27	06:00	09:15	12:30	13:45	17:00
28	06:15	09:30	12:45	14:00	17:15
29	06:30	09:45	13:00	14:15	17:30
30	06:45	10:00	13:15	14:30	17:45

and it should be noted that in this paper, the required round-trip time of EBs is set as one hour. It should be noted that these two objectives described in Equations (2) and (3) are combined together in these case studies in order to analyze cost clearly. Five case studies are described as follows.

Case 1: Neither pricing strategy nor the handling method of the EBs-charging piles matching problem is considered in the multi-objective optimization.

Case 2: Pricing strategy A proposed in [37] is considered in the multi-objective optimization.

Case 3: Pricing strategy B proposed in [38] is considered in the multi-objective optimization.

Case 4: The proposed pricing strategy is considered in the multi-objective optimization.

Case 5: Both the proposed pricing strategy and handling method of EBs-charging piles matching problem are considered in the multi-objective optimization.

Table 5. The load fluctuation index, total electricity cost, and their respective percentage decreases against disordered charging in Cases 1–5.

		Static Deviation Rate of Daily Load	Total Electricity Cost (104 CNY)	Percentage Reductions in Static Deviation Rate	Percentage Reductions in Cost Savings
Case 1	Disordered charging	64.8624	5.94	/	/
	MORBHPSO	38.3317	5.85	40.90%	1.52%
Case 2	Disordered charging	65.5812/	5.91	/	/
	MORBHPSO	38.1666	5.66	41.8%	4.23%
Case 3	Disordered charging	66.2947	5.91	/	/
	MORBHPSO	37.1221	5.19	44.00%	12.18%
Case 4	Disordered charging	65.5841	5.83	/	/
	MORBHPSO	35.5585	4.68	45.78%	19.73%
Case 5	Disordered charging	47.7825	6.86	/	/
	MORBHPSO	32.2089	5.35	32.59%	22.01%

lists the load fluctuation indices and total costs for all five cases before and after optimization, along with their respective percentage improvements relative to Case 1. It should be noted that the presented results are the best-found solutions obtained from multiple independent runs.

4.2. Result Analysis of the Proposed Multi-Objective Optimization Model

It can be seen from Table 5 that the implementation of the proposed multi-objective optimization leads to a marked decrease in both load fluctuation and cost for all cases, with maximum reductions reaching 45.78% and 22.01%, respectively. This outcome unequivocally demonstrates the effectiveness and superiority of the multi-objective optimization framework. In addition, comparison of the load fluctuation indices with and without optimization in Case 1 is presented in Figure 7. The static deviation rate of daily load and total electricity cost obtained from Case 1 are compared with those obtained from the case of disordered charging/discharging.

Figure 8 shows the charging/discharging power of EVs, EBs, and BESS in Case 1 and Figure 9, Figure 10 and Figure 11 present the charging and discharging power of each EV, each EB, and each BESS. It can be seen that due to the influence of TOU electricity price, electricity cost can be reduced by flexible load charging during periods with lower charging electricity price and discharging during periods with higher discharging electricity price. It should be noted that positive power means charging and negative power means discharging. The POF of Cases 1–5 can be found in Figure 12.

4.3. Effect Analysis of the Proposed Pricing Strategies

Cases 2–4 investigate the performance of the existing pricing strategies (Cases 2–3) and the proposed ICDLBPM-based pricing strategy (Case 4). Figure 12 shows that the proposed ICDLBPM-based pricing strategy yields a superior POF and the best compromise solution. Furthermore, as seen in Figure 13, while existing methods achieve a maximum reduction of 3.16% in load fluctuation and 11.28% in cost relative to the baseline (Case 1), the proposed method delivers a marked improvement, reaching 7.23% and 20%, respectively. These values are roughly twice those attained by the existing methods. Therefore, the proposed ICDLBPM-based pricing strategy demonstrates comprehensive superiority over existing methods in terms of both smoothing load fluctuation and total electricity cost reduction.

This is mainly due to the fact that the TOU electricity price obtained from the proposed pricing strategy has higher flexibility in the regional distribution network compared to the other two pricing strategies. The obtained TOU electricity price by the proposed pricing strategy is shown in Figure 14. It can be obviously seen that the optimized TOU electricity price has higher flexibility.

For pricing strategy B, the k factor has a significant impact on the optimization results. A series of simulations with different k values were conducted to clarify the impact of the k factor on the optimization results as shown in Figure 15. It can be seen that the optimization effect on total electricity cost decreases with the increase in the k value, while the optimization effect on load fluctuation first increases and then decreases with the increase in the k value, and reaches its optimal value at a k value of 0.2. In summary, in the scenario of this paper, the optimization effects are best when the k value is 0.2 for pricing strategy B.

4.4. Effect Analysis of the Handling Method of the EBs-Charging Piles Matching Problem

Obtained daily load profiles and capacity profiles of EBs along with charging and discharging power of EB piles can be found in Figure 16, Figure 17, Figure 18 and Figure 19. It can be seen that the charging and discharging power from the available charging piles suffices to meet the electricity demand for every trip, despite the number of piles being fewer than the number of EBs. This confirms the effectiveness of the proposed EB-charging pile matching method. Furthermore, as shown in Figure 13, owing to this EBs-charging piles matching method, the load fluctuation index decreases by an additional 9.42% on the basis of Case 4, demonstrating its effectiveness in peak shaving and valley filling. A slight cost increase is also observed, which stems from the reduced scheduling flexibility caused by having fewer piles. This limitation is also reflected in Figure 10 and Figure 18, where the charging power per pile rises as the number of piles decreases.

4.5. Effect Analysis of the Proposed Method in Large-Scale Systems

To address the concern regarding scalability and real-world applicability, we have conducted an additional large-scale case study, as detailed in the new Section 4.5 of the revised manuscript. The system scale was significantly expanded to 500 EVs, 150 EBs, 125 EB charging piles, and 100 BESS units. Simulation results as shown in

Table 6. The load fluctuation index, total electricity cost, and their respective percentage decreases against disordered charging in a large-scale system.

	Static Deviation Rate of Daily Load	Total Electricity Cost (104 CNY)	Percentage Reductions in Static Deviation Rate	Percentage Reductions in Cost Savings
Disordered charging	77.3697	11.73	/	/
MORBHPSO	46.9541	4.40	39.31%	62.45%

demonstrate that the proposed strategy not only maintains but enhances its optimization performance in this large-scale scenario, achieving a further 39.31% reduction in the static deviation rate and a 62.45% reduction in total cost compared to the baseline (disordered charging). This confirms the strategy’s robustness, scalability, and significant economic potential in practical, large-scale applications.

4.6. Sensitivity Analysis of Charging Efficiency on System Performance

To investigate the influence of charging efficiency on system performance, a sensitivity analysis was performed based on Case 5. In this analysis, charging efficiency was treated as two independent parameters: one common to both EVs and EBs, and another specific to BESS. By evaluating three efficiency levels (85%, 90%, and 95%) for each parameter in all pairwise combinations, a total of nine simulation scenarios were generated and examined.

As can be seen from Figure 20, both the load fluctuation and the electricity cost exhibit low sensitivity to efficiency variations. Specifically, the static deviation rate remained stable between 33 and 34 across all efficiency scenarios, indicating low sensitivity and a limited impact of efficiency-induced energy loss on load fluctuation. Meanwhile, the overall electricity cost fluctuation is minimal and generally decreased as efficiency improved, since higher efficiency reduces charging losses and lowers overall electricity purchases.

4.7. Reliability Verification of the Proposed Method

A supplementary statistical analysis was conducted for the representative Case 5, which integrates all proposed strategies, based on 10 independent optimization runs. The results, presented in

Table 7. Reliability verification results in Case 5.

Total Electricity Cost (104 CNY)			Static Deviation Rate of Daily Load
Mean Value	Optimal Value	Standard Deviation	Mean Value	Optimal Value	Standard Deviation
5.45	5.35	0.0659	33.3599	32.2089	0.6625

, report the mean, optimum, and standard deviation for both the static deviation rate and total operating cost. For instance, the static deviation rate yielded a mean of 33.3599, an optimum of 32.2089, and a low standard deviation of 0.6625. The total cost likewise exhibited minimal variation, with a standard deviation of 0.0659.

The consistently low standard deviations confirm the stability of the solution process. The reported optimum values represent the performance upper limit achievable by the proposed method, thereby reinforcing the credibility of the findings and fully demonstrating the potential of the integrated scheduling strategy.

5. Conclusions

A multi-objective optimal dispatch model is constructed that incorporates EVs, BESS, and EBs. Moreover, ICDLBPM and a novel ICDLBPM-based pricing strategy are presented to flatten load fluctuation and lower electricity cost. To address the practical constraint of insufficient charging piles for electric buses (EBs), a matching method is also proposed to coordinate EB demand with available pile resources. Case studies have been validated on both a conventional-scale system and a large-scale system. Simulation results confirm the effectiveness and feasibility of the model, the proposed pricing strategy, and the EB-pile matching method. The main findings are summarized as follows.

(1)

The proposed multi-objective optimization model achieves significant peak load shaving and cost reduction compared to disordered charging. For instance, reductions of 45.78% in the daily load fluctuation and 19.72% in the total electricity cost are observed in Case 4.

(2)

The effectiveness of pricing strategies in reducing electricity cost of electric power customers was demonstrated in Cases 2–5. Specifically, the proposed ICDLBPM-based pricing strategy outperforms the benchmark strategy A&B from [37,38]. Specifically, it improves daily load smoothness by 6.83% and 4.21% over Strategy A and Strategy B, respectively. In terms of total electricity cost, the reductions are even more pronounced, achieving a 17.31% decrease compared to Strategy A and a 9.83% decrease compared to Strategy B.

(3)

The proposed EB-charging pile matching method satisfies daily EB driving demand without increasing load fluctuations. Furthermore, the presented ICDLBPM-based pricing strategy remains effective even when charging piles are insufficient, demonstrating its adaptability across diverse operational scenarios.

(4)

The case study conducted on a large-scale system yielded superior outcomes, notably substantial decreases in both load fluctuation and total cost. These results validate the robustness of the proposed method and its exceptional capability in coordinating massive flexible resources.

The presented optimal multi-objective model with the novel pricing strategy and hEBs-charging piles matching strategy is expected to be applied to a distribution network where flexible load has a high proportion. The main limitations of this work lie in its simplified assumptions: load forecasting does not account for the uncertainty introduced by random driving patterns; the adopted optimization algorithm lacks parametric sensitivity analysis; and the model omits detailed battery aging dynamics, grid line constraints, and a real-time pricing mechanism coupled with EV operational limits. Future research should therefore focus on integrating uncertainty modeling, conducting algorithm sensitivity studies, refining the battery and network models, and developing real-time pricing strategies. These directions aim to improve the practical robustness, economic efficiency, and engineering applicability of the proposed scheduling framework.