Efficient charging scheduling through coordination of electric vehicle platoons and charging stations

Electric vehicles (EVs) play a key role in low-carbon societies, especially when supplied with renewable energy resources. The International Energy Agency (IEA) reported that the number of EVs worldwide could reach 230 million by 2030¹. The widespread adoption of EVs presents numerous opportunities and challenges², e.g., the development of eco-driving designs³, the enhancement of charging networks for households, workplaces, and public charging stations⁴, and the formulation of new adaptive cruise control (ACC) strategies⁵. The integration of the Internet-of-Vehicles can help address traffic-related challenges, e.g., reducing traffic congestion and accidents^6,7.

Related work

Existing research related to this paper can be broadly categorized into four streams: (i) eco-driving and range evaluation, (ii) charging station planning and charging management, (iii) mixed traffic and platoon-related charging, and (iv) V2I-enabled intelligent EV networks.

The development of EVs is subject to the driving range limitation. Energy-efficient driving is a promising approach to address this issue. The accurate evaluation of an EV’s driving range is complex, as it involves the EV’s maneuvers and climatic conditions⁸. In light of a state-space dynamic model⁹, reduced energy consumption by optimizing the EV’s velocity and travel duration. In¹⁰, a model-based method was proposed to predict the current driving range and optimize the speed profile for eco-routing and eco-driving. Considering the joint motion and energy control design¹¹, presented a hierarchical control architecture with a learning-based velocity planner and real-time energy management to improve the hybrid EV’s energy efficiency. In¹², the optimal velocity control of a hybrid EV was achieved by using model predictive control, improving both energy efficiency and computational speed. Recent work¹³ adopted a data-driven approach to optimize the hybrid EV trajectory for eco-driving. A lightweight EV was studied in¹⁴ to minimize the energy consumption by controlling the battery’s maximum current under travel-distance constraints. Considering the inter-vehicle distance and road speed limit, the authors of¹⁵ built a speed advisory system for safe and eco-driving via a model predictive control approach. These studies focus on optimizing driving efficiency but do not address the challenges posed by heterogeneous traffic flows, particularly when manned EVs and EV platoons share the same charging stations. In contrast, our study tackles the combined challenges of optimizing both driving speed and charging allocation for manned EVs and EV platoons, a novel approach that has not been addressed in these works.

While energy-efficient EV driving can alleviate driving range anxiety, the lack of charging stations is a serious setback for the popularity of EVs. Therefore, the appropriate deployment of charging stations and charging management is pivotal in EV networks, which can further prolong the driving range, boost energy utilization, and create various opportunities for greener intelligent transportation systems. In addition, charging guidance and scheduling protocols have been investigated for EV charging services. For example, a guidance model for EV charging services was proposed in¹⁶, and a charging/discharging scheme with a load-management protocol was developed in¹⁷. A tour-based method was investigated in¹⁸ to minimize the travel and charging time without running out of charge. In¹⁹, a decentralized game-theoretical approach was developed to determine the locations of charging stations under congestion and individual preference concerns. The work of²⁰ addressed the problem of charging station placement at certain bus stops and analyzed the impact of battery size on the overall cost. In^21,22, a (dis)charging scheme was proposed based on the queuing model, subject to the EV requirements and smart grid stability. Since ultra-fast charging stations can substantially reduce the charging time²³, studied random charging demand in the EV fast charging system with an aggregator, and provided a bidding strategy to deal with the corresponding operation and market participation problem. While these studies focus on infrastructure and cost optimization, they often assume homogeneous traffic and model each EV as an independent single-pile charging demand, thereby neglecting the capacity coupling induced by platoons occupying multiple charging piles.

Intelligent EVs are capable of not only finding their optimal charging stations but also leveraging cooperative adaptive cruise control (CACC) to avoid collision and mitigate traffic jams. As a key CACC application, vehicle platooning can improve the road throughput and disengage the following vehicles from driving tasks^24,25,26. Thus, EV platooning can significantly save energy and prolong the driving range²⁷. The work of²⁸ proposed a dynamic-flexible model to evaluate the travel duration of an EV platoon. In²⁹, an energy-saving-aware platoon control method was proposed to address the effects of EVs’ drive/brake power deviation and distance deviation. Recent work³⁰ provided a central platform to optimize the EV platoon formation process, where energy consumption can be minimized. In³¹, a method combining charging pricing and scheduling decisions in a platoon charging network was proposed to alleviate the overload of each charging station. In³², an EV platoon control scheme was developed to enable efficient eco-driving in mixed traffic scenarios. In³³, a machine-learning approach was proposed for platooning control in mixed platoons consisting of automated and human-driven vehicles. However, while mixed-traffic research has extensively investigated platooning/CACC control in scenarios involving both automated and human-driven vehicles, it has primarily focused on safety and motion control, rather than on charging scheduling. Furthermore, platoon-related charging studies have not addressed the complexities of a heterogeneous charging network that jointly serves both manned EVs and multiple EV platoons under a unified station-capacity constraint.

V2I communication-based intelligent EV networks can further improve energy utilization with the help of edge computing³⁴. In³⁵, the velocity of a hybrid EV was predicted by leveraging a chaining neural network at the road-side units (RSUs). A V2I-based wireless charging highway architecture was designed in³⁶, with the EV energy demands and road density monitored by the RSUs. The real-world experiments in²⁴showed that V2I could bring in more gains in traffic efficiency for vehicle platooning. In the V2I-based vehicle platooning systems³⁷, investigated the eco-driving strategies for a platoon with mixed gasoline vehicles and EVs. Nevertheless, the above V2I/edge-assisted studies mainly emphasize communication architectures, prediction, and motion-control performance, and they do not address charging scheduling jointly with speed optimization for heterogeneous traffic flows; in contrast, our work leverages V2I-enabled information exchange to jointly coordinate speed and charging allocation for manned EVs and EV platoons under unified station-capacity constraints.

In summary, existing studies typically optimize eco-driving, charging management, and platooning control in isolation and often assume homogeneous traffic and single-vehicle charging demands, thereby overlooking a unified station-capacity constraint when manned EVs and multiple EV platoons coexist. The key missing piece is a unified optimization that jointly captures heterogeneous entities, platoon-induced capacity-coupled charging demands, and the coupling between charging allocation and speed decisions. In contrast, this paper formulates a joint charging scheduling and speed optimization problem for V2I-enabled EV networks with heterogeneous traffic flows, where each platoon consumes multiple charging piles according to its size and allocation decisions are tightly coupled with speed variables.

Contribution

In this work, we address the charging scheduling problem in V2I-based EV platoon networks with various traffic flows, including manned EVs and EV platoons, which has never been studied before to the best of our knowledge. Compared with the conventional EV systems^{19,20,21,23,38,39}, EV platoons may lead to severe load imbalance in such a heterogeneous charging network due to the limited charging piles in a charging station.

In contrast to previous studies, our work uniquely tackles the charging scheduling problem in EV networks by considering heterogeneous traffic flows, including both human-driven EVs and EV platoons. This work also incorporates a comprehensive evaluation of factors such as charging station availability, platoon size, driving speed, and road congestion. Furthermore, we introduce a new problem objective focused on minimizing total time cost, which encompasses both travel and charging times under various traffic conditions. To solve this complex non-convex combinatorial problem, we develop a two-stage alternating optimization algorithm that integrates dynamic programming, heuristic methods, and constrained particle swarm optimization. This approach provides an effective solution for managing mixed traffic flows and reducing charging time. To the best of our knowledge, this work is the first to jointly optimize charging allocation and speed for both manned EVs and EV platoons under a shared charging station capacity constraint.

The main contributions of this paper can be summarized as follows:

• For V2I-based heterogeneous EV networks featuring multiple manned EVs and EV platoons, we formulate joint charging scheduling and speed optimization problems. The goal is to minimize the overall time cost. Additionally, the effects of varying platoon sizes and road traffic conditions are considered.

• Since the considered problem is discrete and combinatorial, we propose a two-stage alternating minimization algorithm to solve the problem effectively by alternating between charging scheduling and velocity optimization. For charging scheduling, a dynamic programming-based method is developed to determine the association between a manned EV or EV platoon and a charging station. A greedy strategy with lower complexity is also studied. For velocity optimization, a swarm-inspired method optimizes the velocities of manned EVs and EV platoons. Extensive simulations corroborate that our design can substantially reduce the time cost.

• The important insights and guidelines are established: Given a fixed number of charging piles, deploying additional charging stations with fewer charging piles (multiple charging piles per charging station) can save more charging time than adding more charging piles at fewer charging stations. Moreover, a larger platoon size leads to longer charging time. Our results show that platoon-based charging scheduling significantly affects network efficiency, as platoons consume multiple charging piles, leading to complex interactions with individual EVs.

The rest of this paper is organized as follows. The system model is described in Section “System descriptions”. The design of the algorithm is proposed in Section “Algorithm design”. Section "Experiment and results" provides the numerical results. Finally, the paper is concluded in Section "Conclusion and future work". Here, the main notations in this paper and their definitions are summarized in Table 1.

In a V2I-based EV charging network (illustrated in Fig. 1), the considered hybrid traffic flows consist of \(N_\textrm{H}\) manned EVs (i.e., human-driven, non-automated EVs) and L EV platoons which are collected in \(\mathscr {N}_\textrm{H}\) and \(\mathscr {L}\), respectively. Each EV platoon \(l\in \mathscr {L}\) has \(N_\textrm{P}^l\) connected and automated EVs. Since each platoon has a leader vehicle and at least one following vehicle⁴⁰, we have \(N_\textrm{P}^l \ge 2\) for \(\forall l\). In this study, an EV platoon specifically refers to a fixed-size, fleet-managed convoy (e.g., logistics vans, robo-taxi pods) whose vehicles are dispatched under a single operator; the platoon size \(N_\textrm{P}^l\) remains constant throughout the planning horizon, and operations such as splitting, merging, or re-ordering of vehicles are not considered. This assumption is made for a short scheduling horizon and to maintain tractability of the joint charging allocation and velocity optimization problem under fixed platoon membership. Both manned EVs and EV platoons adhere to the charging allocation strategy of the scheduling center. Privately-owned EVs that may only form spontaneous groups on the road are modeled as manned EVs rather than platoons, because their charging decisions stay fully individual. There are M charging stations as collected in \(\mathscr {M}=\{1,\cdots ,M\}\), where each charging station \(m \in \mathscr {M}\) has \(S_m\) charging piles. Manned EVs and EV platoons send requests for charging along with the current battery and driving state information to the intelligent scheduling center (ISC) via V2I links (For an EV platoon, there are two options for sending the vehicles’ battery and driving state information to the RSU: a) The leader vehicle collects all the platoon vehicles’ states via vehicle-to-vehicle (V2V) communications and sends the information to the ISC; and b) All the platoon vehicles directly send their state information to the ISC by themselves). Although we use the term “ISC”, it can be implemented as a V2I-enabled edge controller with regional partitioning. Regarding communication overhead, vehicles periodically upload only essential battery and driving state information for scheduling. Distance to candidate stations and segment-level traffic conditions are available at the ISC side and do not require per-vehicle reporting. Therefore, the overall uplink traffic increases with the number of vehicles in a region, while the message size remains small. Each ISC can communicate with manned EVs and EV platoons via multiple-input multiple-output (MIMO) radio technology, as widely adopted in 4G and 5G networks^{41,42,43,44,45}.

The ISC manages the EV charging network and makes decisions on charging allocation for corresponding manned EVs and EV platoons. In the present framework, every vehicle follows a predetermined origin–destination path; hence, the road network is represented by a set of segments characterised by their lengths, real-time congestion levels, and statutory speed limits rather than by a full traffic-flow model. This abstraction suffices to capture how speed control influences energy consumption, which in turn determines the feasibility of the charging-station allocation under capacity constraints. The ISC is aware of (a) the distance from each vehicle to every candidate station, (b) segment-specific traffic conditions, (c) the nonlinear speed–energy map of the powertrain, (d) the user-requested energy increment \(B_{\zeta ,m}^{\textrm{cha}}\), and (e) charger power \(P_{\zeta ,m}^{\textrm{cha}}\). Given the data, it can evaluate both travel time and charging time for every admissible combination of speed and charging-station choice, and ultimately determine a feasible pairing between each manned EV or platoon and a charging station.

Let \(D_{n,m}\) and/or \(D_{l,m}\) denote the travel distances between a manned EV \(n\in \mathscr {N}_\textrm{H}\) and/or EV platoon \(l\in \mathscr {L}\) and charging station m, respectively (In this work, an EV platoon’s travel distance \(D_{l,m}\) depends solely on the EV with the lowest battery level, which takes part in the charging allocation for choosing the targeted charging station. Future work can explore more scalable EV platooning designs with energy awareness, allowing for the flexible disengagement of individual EVs within a platoon). Since the corresponding travel duration needs to be evaluated based on the road traffic conditions, a commonly-used model is expressed as⁴⁶

where \(\zeta \in \mathscr {N}_\textrm{H} \cup \mathscr {L}\); \(\nu _\zeta\) is the velocity of manned EV or platoon; a and b are two coefficients based on specific observations in practice; \(\gamma _\textrm{tf}\) is the current traffic flow; and C is the road capacity. In addition, the energy cost for traveling to the charging station m is given by

where \(g\left( \nu _{\zeta }\right)\) is the energy consumption per unit distance, relying on the velocity \(\nu _{\zeta }\)⁴⁷. Let \(\Xi _{\zeta ,m}\) denote the remaining energy at the EV or platoon \(\zeta\) when requesting service from charging station m, subject to \(\Xi _{\zeta ,m}>\vartheta _{\zeta ,m}\). In practice, an EV platoon’s remaining energy \(\Xi _{l,m}\) depends on the EV with the minimum remaining energy in this platoon.

Without loss of generality, when an EV or platoon arrives at a selected charging station m, additional energy cost \(\vartheta _{\zeta ,m}\) needs to be recharged for traveling. Therefore, its charging time is given by

where \(B_{\zeta ,m}^\textrm{cha}\) is the required amount of energy when the EV (or platoon) \(\zeta\) sends the charging requirement signal to the ISC for the charging station m, here \(B_{\zeta ,m}^\textrm{cha}\) is a user-requested energy increment that may correspond either to a full battery recharge or to a partial top-up; thus the model inherently accommodates both complete and partial charging requests; \(\eta _{\zeta ,m}^\textrm{cha}\) and \(P_{\zeta ,m}^\textrm{cha}\) represent the charging efficiency and charging power at charging station m, respectively. For an EV platoon at charging station m, the required amount of energy \(B_{l,m}^\textrm{cha}\) depends on the average of required energy across its EVs, i.e.,

where \(B_{l,m}^\textrm{cha}(i)\) is the required energy of the i-th vehicle in EV platoon l.

We aim to minimize the overall time cost in the considered EV network. Our charging design seeks to avoid overloading such that all the EVs that arrive at the charging station can be served immediately without waiting time. Assuming the consistent tariffs across charging piles, we neglect the impact of price. Therefore, the problem of interest can be cast as

where \(\textbf{x}=\left[ x_{n,m}\right]\), \(\textbf{y}=\left[ y_{l,m}\right]\) and \(\varvec{\nu }=\left[ \nu _\zeta \right]\).

Constraint \(\textrm{C1}\) describes the binary indicators, i.e., \(x_{n,m}\) equals 1 if the EV n is recharged by the charging station m and 0 otherwise, and \(y_{l,m}\) equals 1 if the platoon l is recharged by the charging station m and 0 otherwise; \(\textrm{C2}\) ensures that the number of arrival EVs in a charging station cannot exceed its capacity \(S_m\) to avoid overloading, i.e., the waiting time for charging at the charging station is mitigated; \(\textrm{C3}\) and \(\textrm{C4}\) guarantee that only manned EVs or EV platoons with enough corresponding energy can reach the charging station and be recharged, which also enable that there always exist feasible solutions of the problem (5); \(\textrm{C5}\) and \(\textrm{C6}\) guarantee that an EV cannot be recharged concurrently by multiple charging stations; and \(\textrm{C7}\) bounds the velocity with the minimum \(\nu _\textrm{min}\) and the maximum \(\nu _\textrm{max}\).

As shown in the objective function (5) and its constraint \(\textrm{C2}\), EV platoons are more likely to incur high charging times and overloading at charging stations. Therefore, appropriate charging allocation is key to charging time reduction and load balance. In (5), the discrete allocation variables \(\{x_{n,m},y_{l,m}\}\) and continuous velocities \(\{\nu _\zeta \}\) are coupled through constraints \(\mathrm {C3-C4}\), where the energy-consumption term \(g(\nu _\zeta )\) depends on \(\nu _\zeta\) and is multiplied by the allocation indicators. To facilitate online updates under time-varying traffic, we decouple the optimization into two stages: (i) charging allocation under fixed velocities, which enforces the combinatorial capacity and exclusivity constraints \(\mathrm {C1-C2}\) and \(\mathrm {C5-C6}\); and (ii) velocity optimization under fixed allocation, which adjusts \(\{\nu _\zeta \}\) to satisfy \(\mathrm {C3-C4}\) and \(\textrm{C7}\) while reducing the total time cost. This decomposition enables fast re-optimization when traffic conditions change, without repeatedly searching over the entire mixed discrete–continuous decision space.

Problem (5) is challenging for three reasons: i) it is combinatorial and non-convex; ii) the association indicators \(\{x_{n,m},y_{l,m}\}\) and velocity variables \(\{\nu _\zeta \}\) are coupled through constraints \(\mathrm {C3-C4}\); and iii) the traveling-cost term in(2) involves a generic and implicit function⁴⁸. We note that dynamic programming (DP), heuristic methods, and swarm-inspired approaches are classical tools. The novelty of this work comes from the heterogeneous charging setting where each platoon occupies multiple charging piles (i.e., \(N_\textrm{P}^l\) piles in \(\textrm{C2}\)), and from the coupled mixed discrete-continuous optimization via the speed-dependent energy constraints \(\mathrm {C3-C4}\). Algorithmically, we tailor and integrate these classical tools into a two-stage alternating minimization framework, where charging allocation is handled by DP/greedy methods and velocity update is handled by a constrained PSO scheme. Therefore, we first solve the charging-allocation subproblem with respect to \(\{x_{n,m},y_{l,m}\}\) under fixed velocities using a DP approach. Specifically, under the fixed velocity values \(\{\nu _\zeta \}\), for \(\gamma =\left| \Upsilon \right| ,\cdots ,1\), the Bellman optimality condition for problem (5) is expressed as

where \(s_m\in \{0,\cdots , S_m\}\) represents the slack capacity for accommodating extra EVs at charging station m,

with \(l=\left\lfloor {\frac{{\left| \Upsilon \right| - 1}}{M}} \right\rfloor - N_\textrm{H} + 1\), and \(n=\left\lfloor {\frac{{\left| \Upsilon \right| - 1}}{M}} \right\rfloor - L + 1\) is the charging allocation indicator which is searched under \(\textrm{C1}\)–\(\textrm{C6}\) constraints provided that \(\Upsilon =\left\{ x_{1,1},\cdots ,x_{1,M},\cdots ,x_{N_\textrm{H},M},y_{1,1},\cdots ,y_{L,M}\right\}\) is the set of binary indicators with the cardinality \(\left| \Upsilon \right| =N_\textrm{H}M+LM\), \(N_\gamma = \left\{ \begin{array}{l} N_P^l ,\quad \gamma = \left| \Upsilon \right| , \cdots ,\left| \Upsilon \right| - LM + 1, \\ 1,\quad \;\gamma = \left| \Upsilon \right| - LM, \cdots ,1, \\ \end{array} \right.\), and \(\mathscr {T}_{0}\left( \cdot \right) =0\).

Based on (6), we seek the optimal charging configuration to solve (5) given \(\{\nu _\zeta \}\). In Algorithm 1, \(\{\left( \mathscr {R},\mathscr {T}_{\gamma }\right) \} \in \mathscr {Q}_{\gamma }\) contains the subsets \(\mathscr {R}\) of the first \(\gamma\) elements, to maximize the value \(\mathscr {T}_{\gamma }\), and only the optimal paths in the tree data structure \(\mathscr {Q}_{\gamma }\) capture the optimal value function \(\mathscr {T}_{\gamma }\left( \cdot \right)\); besides keeping only the optimal paths after branch expansion, lines \(12-16\) are also executed to guarantee that when there are two user association configurations \(\mathscr {R}^{'}\) and \(\mathscr {R}^{''}\) that have the same objective value \(\mathscr {T}_{\gamma }\), a larger platoon is prioritized for charging under the same charging time; the optimal charging configuration \(\mathscr {R}^*=\{x_{n,m}^*,y_{l,m}^*\}\) is chosen in \(\mathscr {Q}_\gamma\) at \(\gamma =\left| \Upsilon \right|\).

Let \(|\Gamma |\) denote the number of candidate allocation states actually evaluated by the DP procedure in Algorithm 1 after feasibility screening and pruning. The computational complexity is \(\mathscr {O}(|\Gamma |)\), where \(|\Gamma |\) is instance-dependent. In particular, the full search space is exponential in \(|\Upsilon |\) with \(|\Upsilon |=N_\textrm{H}M+LM=M(N_\textrm{H}+L)\), whereas in Algorithm 1 many branches are discarded early due to the station-capacity and energy-feasibility constraints. Consequently, \(|\Gamma |\) is typically much smaller than the exhaustive enumeration space, which is also reflected by the millisecond-level runtime in Table 2. A heuristic method can be developed in a greedy-based manner with much lower complexity, as summarized below:

Stage 1: Given velocity values \(\{\nu _\zeta \}\), calculate the total amount of time \(T_{\zeta ,m}^\textrm{tra}+T_{\zeta ,m}^\textrm{cha}\) for charging a manned EV or platoon.

Stage 2: Repeat

1. a)

   Select the manned EV or platoon with the minimum value of \(T_{\zeta ,m}^\textrm{tra}+T_{\zeta ,m}^\textrm{cha}\) subject to \(z_\gamma D_\gamma g\left( \nu _\gamma \right) <\Xi _\gamma\) and \(s_m \ge N_\gamma\). Note that when a platoon and a manned EV have the same minimum value of \(T_{\zeta ,m}^\textrm{tra}+T_{\zeta ,m}^\textrm{cha}\) subject to \(s_m \ge N_\gamma\), the platoon takes priority since more EVs in a platoon can be served. Then, the selected manned EV or platoon is allocated to the corresponding charging station, and excluded from charging allocation in the following iterations;

2. b)

   Update \(T_{\zeta ,m}^\textrm{tra}+T_{\zeta ,m}^\textrm{cha}\) of the remaining manned EVs or platoons.

Upon convergence (all manned EVs or platoons are selected), we obtain the corresponding greedy-based solution \(\{x_{n,m}^*,y_{l,m}^*\}\). The greedy-based method has a computational complexity on the order of \(\mathscr {O}\left( N_\textrm{H}+L\right)\).

After optimizing the charging allocation under fixed velocities \(\{\nu _\zeta \}\), we update the optimal velocities by solving the following subproblem:

The energy consumption per unit distance \(g\left( \nu _{\zeta }\right)\) is a function of \(\nu _{\zeta }\) and may not be explicitly computed. Existing work⁴⁷ attempted to adopt approximation methods. However, only certain types of EVs are studied. Instead, we adopt and tailor a constrained particle swarm optimization method to update \({\nu _\zeta }\) under constraints \(\mathrm {C3-C4}\) and \(\textrm{C7}\), which remains applicable even when the energy-consumption function \(g(\cdot )\) is arbitrary or implicitly given. Particle swarm optimization is a popular swarm-inspired approach to effectively solve the continuous problem with only a few hyperparameters and dispenses with gradients^49,50. In the particle swarm optimization process, a particle’s (swarm member) optimal movement oscillates between its locally optimal and globally optimal solutions, as obtained through interactions with other particles.

Due to the energy consumption constraints \(\mathrm {C3-C4}\), we adopt the constrained particle swarm optimization method. According to (8), the fitness function can be written as

where \(e_{n,m}^\textrm{H}=\max \left\{ 0, x_{n,m} D_{n,m} g\left( \nu _{n}\right) - \Xi _{n,m}\right\}\), \(e_{l,m}^\textrm{P}=\max \left\{ 0, y_{l,m} D_{l,m} g\left( \nu _{l}\right) - \Xi _{l,m}\right\}\), \(\lambda _{n,m}^\textrm{H}\) and \(\lambda _{l,m}^\textrm{P}\) are penalty weights. In (9), the penalty terms are employed to meet the energy cost constraints \(\mathrm {C3-C4}\).

The initial positions of the particles are randomly generated within the feasible region of the search space. If any particle’s position violates the constraints, a correction procedure is applied to adjust the position within the feasible region, ensuring that the optimization process begins with valid solutions. Therefore, given charging allocation \(\{x_{n,m},y_{l,m}\}\), the velocity values \(\{\nu _\zeta \}\) are obtained by using particle swarm optimization; see Algorithm 2. Let k be the iteration index and \(K_{\max }\) the maximum number of iterations. Let J denote the swarm size. Each iteration evaluates \(\widetilde{F}\!\left( \varvec{\nu }\right)\) for all J particles; denote the per-particle fitness-evaluation cost by \(C_F\). This yields a complexity of \(\mathscr {O}\!\left( K_{\max } J C_F\right)\).

In light of the aforementioned analyses, we propose a two-stage alternating minimization approach to tackle problem (5), where the alternating minimization makes iterations between the charging allocation stage and velocity optimization stage; see Algorithm 3. In the charging allocation stage, the solution of charging allocation is calculated by using Algorithm 1 or the greedy method. In the velocity optimization stage, the solution of optimal velocity values is obtained by using Algorithm 2. Since the overall charging time cost is a decreasing function of the iteration index in Algorithm 3 (in lines 3–5), convergence is guaranteed. Let q be the iteration index in Algorithm 3 and \(Q_{\max }\) the prescribed maximum number of alternating iterations. Each iteration performs one charging allocation (DP: \(\mathscr {O}(|\Gamma |)\), or greedy: \(\mathscr {O}(N_\textrm{H}+L)\)) and one velocity optimization (PSO: \(\mathscr {O}(K_{\max }JC_F)\)). Hence, the overall complexity is \(\mathscr {O}\!\left( Q_{\max }\big (|\Gamma |+K_{\max }JC_F\big )\right)\) for DP-based allocation, or \(\mathscr {O}\!\left( Q_{\max }\big (N_\textrm{H}+L+K_{\max }JC_F\big )\right)\) for the greedy allocation. In implementation, \(Q_{\max }\) and \(K_{\max }\) are set to small values and the algorithm terminates once the improvement falls below \(\varepsilon\); together with bounds \(N_\textrm{H}\) and L, this makes the proposed method suitable for online scheduling.

In this section, we provide numerical results to confirm the efficiency of the proposed method. In the simulations, the ratio of the current traffic flow to the road capacity \(\frac{\gamma _\textrm{tf}}{C}=0.6\), \(a=0.8\), and \(b=0.8\) for the road condition; each manned EV or EV platoon is randomly located on a 2D plane, its position on the x-axis and y-axis (x-coordinate and y-coordinate) follow the Gaussian distribution with the mean 40 km and variance 20 km\(^2\); the position of each charging station on the x-axis and y-axis follow the uniform distribution with the interval [0, 50] km; the energy consumption per unit distance \(g\left( \nu _{\zeta }\right) =479.1-18.93\nu _\zeta +0.7876\nu _\zeta ^2\)⁴⁷.The minimum and maximum velocities are \(\nu _\textrm{min}=30\) km/h and \(\nu _\textrm{max}=120\) km/h, respectively. All the EVs need the same amount of energy, i.e, \(B_{\zeta ,m}^\textrm{cha}=30\) kWh, \(\forall \zeta ,m\), and all charging stations have the same charging efficiency value \(\eta =0.9\). This fixed value is adopted solely as a common benchmark; the proposed framework can just as well accommodate heterogeneous \(B_{\zeta ,m}^\textrm{cha}\) values, such as those arising from partial-charging requests, without any modification to the underlying optimization model. The random 2D deployment is adopted to emulate diverse regional charging topologies and to avoid bias toward a specific city layout. Moreover, the considered DC fast-charging regime (40–80 kW, the benchmark request \(B_{\zeta ,m}^\textrm{cha}=30\) kWh, and the charging efficiency \(\eta =0.9\) are consistent with reported public DC fast-charging port power (24–350 kW), empirical paid-session energy (average 22 kWh), and typical charging efficiency (88%–95%), respectively^51,52,53. The simulation results are obtained by averaging over \(5\times 10^3\) trials.

Unless otherwise stated, we evaluate each method using the total time cost (in hours), which is consistent with the objective in (5). For quantitative comparisons, we also report the relative reduction of the proposed method relative to a baseline. In addition, when discussing algorithm efficiency, we report the execution time per scheduling update under the same workstation setting (Table 2). For comparisons, we consider classical baselines including the nearest-neighbor method and the greedy heuristic method, and we further include two recently reported strategies in a dedicated benchmark study in Section "Effects of charging power". We first investigate the effects of the key system parameters on the charging allocation to explicitly show the importance of appropriate charging allocation. The interplay between the number of charging piles per charging station and the number of charging stations is first illustrated in Figs. 2 and 3. Then, Fig. 4 shows the charging power effects from the perspective of different numbers of charging piles per charging station. The interplay between the number of EV platoons and platoon size is presented in Figs. 5,6,7. The performance for joint charging allocation and velocity optimization is shown in the corresponding figures from the perspective of EV platoon features and road conditions.

Runtime and scalability

We report the execution time of the charging-allocation stage under different problem sizes on a standard workstation; see Table 2.

Values are reported as mean ± standard deviation over multiple independent random instances for each \((N_\textrm{H},L,M)\) setting. The results provide numerical evidence of feasible problem sizes for scheduling updates and show that the greedy heuristic maintains a low runtime as the problem size increases. The DP runtime exhibits a larger variance because the number of expanded states after pruning is instance-dependent, whereas the greedy heuristic has a more predictable cost dominated by evaluating vehicle-station pairs.

Effects of number of charging piles per charging station

Consider that there are \(N_\textrm{H}=4\) manned EVs and \(L=6\) EV platoons with their size vector \([N_\textrm{P}^l]=[3,2,2,2,3,2]\), and the number of charging stations is \(M=4\). The nearest neighbor method is based on the travel distance, i.e., each manned EV or EV platoon selects its nearest charging station with idle charging piles; if manned EVs or EV platoons choose the same charging station, the EV platoon with the larger size has priority.

Figure 2 shows that the proposed method achieves the best performance, and the greedy-based method is better than the nearest neighbor one. The overall charging time at a charging station decreases as the number of charging piles increases. This is because a higher number of charging piles allows each charging station to accommodate more EVs simultaneously, incurring lower time costs for charging. Consequently, stations with more charging piles can provide more efficient and faster service to EVs.

As quantitatively demonstrated in Table 3, the proposed method consistently outperforms benchmark approaches across all capacity configurations. When increasing the number of charging piles from 5 to 9, the proposed method reduces total time consumption by 13.6% (from 35.77 h to 30.90 h), showing superior scalability. In this paper, the terms “time cost” and “time consumption” are used interchangeably and both refer to the same metric.

Effects of number of charging stations

Consider that there is a manned EV and \(L=6\) EV platoons with their size vector \([N_\textrm{P}^l]=[4,2,2,3,2,4]\).

Figure 3 shows that the proposed method achieves a lower time cost compared to the greedy-based method, highlighting its efficiency. Deploying additional charging stations can significantly reduce the overall time cost for charging EVs. Furthermore, increasing the number of charging piles at each charging station further enhances performance by simultaneously charging more vehicles. As both the numbers of charging stations and charging piles per station are expanded, the performance gap between the proposed method and the greedy-based method becomes narrower. This suggests that while the proposed method maintains superior performance, the relative advantage diminishes as the infrastructure becomes more extensive and robust.

Effects of charging power

In the simulations, there are \(N_\textrm{H}=3\) manned EVs and \(L=7\) EV platoons with their size vector \([N_\textrm{P}^l]=[4,3,4,3,2,2,4]\), all the charging piles at each charging station have the same level of charging power.

Figure 4 shows that the higher charging power can significantly reduce the charging time, as indicated by (3). The proposed charging scheduling method consistently outperforms the greedy-based method across various charging infrastructure placements. Under a fixed number of charging piles, the strategy with more charging stations, each with fewer charging piles, can further reduce the overall charging time cost. This is because EVs are more likely to detect nearby charging stations, thereby minimizing travel distances to the stations. This reduction in travel time, combined with the efficient allocation of charging resources, results in a more effective and time-efficient charging network. Consequently, the distribution of more charging stations with fewer piles each provides a significant advantage in reducing the total charging time for EVs.

Table 4 compares the performance of our proposed scheduling method against the greedy method across two infrastructure configurations: 3 stations \(\times\) 4 piles and 6 stations \(\times\) 2 piles. The results demonstrate three significant findings: First, our proposed method exhibits consistent performance superiority across all tested conditions, achieving particularly substantial efficiency gains in the distributed configuration compared to the greedy method. Second, while both methods exhibit the expected inverse relationship between charging power and time consumption, our approach displays significantly steeper improvement gradients, suggesting more effective utilization of increased power capacity. Third, the comparative analysis reveals that the distributed 6\(\times\)2 configuration consistently outperforms the clustered 3\(\times\)4 topology for both scheduling methods. These findings collectively underscore the dual importance of both algorithmic optimization and strategic infrastructure deployment in EV charging network design.

To ensure consistent comparisons across different subsections, the greedy-based method is used as the common classical baseline throughout the parameter-study subsections in Section "Experiment and results", and the nearest neighbor method is additionally included where appropriate (e.g., Section "Effects of number of charging piles per charging station") as a representative distance-based baseline. To further strengthen the evaluation while avoiding redundancy, Section "Effects of charging power" serves as a dedicated benchmark subsection and additionally includes two recently reported baselines from the literature under the same system model and feasibility constraints. Accordingly, we further benchmark Algorithm 3 against two recently reported strategies in the EV charging literature. The first baseline, referred to as the occupancy-aware routing (OAR) baseline⁵⁴, incorporates charging-station occupancy information in the station-selection/routing stage. The second baseline, referred to as the shortest-path guidance (SPG) baseline⁵⁵, adopts a shortest-path guidance strategy and focuses on path optimization between each vehicle and candidate charging stations.

For fairness, all methods are implemented under the same system model and feasibility constraints. For platoons, we adapt these baselines by treating each platoon as a single request that occupies \(N_\textrm{P}^l\) piles, consistent with constraint \(\textrm{C2}\). We perform the benchmark under the charging-power sweep, as charging power directly affects the charging time and is a common evaluation axis in the above references. The results are averaged over \(5\times 10^{3}\) independent random instances. As shown in Table 5, our proposed method consistently achieves the lowest total time cost across all charging powers under the \(6\times 2\) configuration, outperforming both the OAR and SPG baselines. The OAR baseline performs slightly better than the SPG baseline because incorporating charging-station occupancy awareness alleviates capacity contention, whereas the SPG baseline primarily follows shortest-path guidance and is therefore more myopic under shared capacity constraints. Overall, these results confirm that our proposed method remains beneficial even when compared with recent state-of-the-art strategies.

Effects of number of EV platoons

In the simulations, each platoon has 5 EVs, 4 charging stations, and 5 charging piles with the same charging power of 80 kW.

Figure 5 also shows the merit of the proposed method over the greedy method. The performance gap widens as the number of platoons increases. Specifically, the proposed method is capable of selecting the right EV platoons with lower time costs, particularly in the scenario where the total number of charging piles (e.g., \(5\times 4=20\) in this case) is insufficient to accommodate all the platoon EVs simultaneously. This efficiency is achieved by optimizing the allocation of limited charging resources. As a result, the proposed method demonstrates superior scalability and effectiveness in managing larger numbers of EV platoons under constrained charging infrastructure conditions, highlighting its potential for real-world applications, provided limited charging resources.

Joint charging allocation and velocity optimization

In Figs. 6 and 7, there are \(L=5\) platoons and 5 charging stations, each charging station has 10 charging piles with the same charging power 80 kW.

Figure 6 shows that the proposed two-stage alternating minimization method (shown in Algorithm 3) obtains the best performance under the same road condition \(a=0.8\) and \(b=0.8\), and it has much lower travel time than the charging allocation only. Note that when the platoon size is not large, each charging station can accommodate more platoons (more than one platoon can be in a charging station, as the platoon size is lower than 5 in this figure); in this case, the performance of the greedy method is almost identical to the proposed one. A larger platoon size can dramatically increase the travel time since the nearby charging stations are more susceptible to overload. This is because larger platoons require more charging resources simultaneously, which can lead to congestion and longer times at charging stations. As a result, the efficiency of the charging process decreases, thereby causing the overall travel time for the platoon to increase significantly. The proposed method maintains high efficiency regardless of platoon size, and its advantages become more pronounced as platoon sizes grow, highlighting its robustness in managing larger and more platoon charging scenarios.

Figure 7 further shows that the proposed method consistently outperforms the greedy-based method across various platoon sizes and road conditions. In scenarios with dense traffic flows, reaching charging stations can often result in severe travel delays due to congestion and limited availability of charging piles. However, our proposed solution effectively mitigates these adverse effects by optimizing the allocation and scheduling of charging resources. This leads to a more efficient distribution of EVs among the available charging stations, reducing overall travel time even in high-traffic conditions. Consequently, the proposed method demonstrates superior adaptability, ensuring better performance and more reliable service for EV platoons under diverse and challenging traffic environments.

In Fig. 8, each platoon has 5 EVs, there are 3 charging stations, and each charging station has 5 charging piles with the same charging power 80 kW.

Figure 8 shows that the proposed method outperforms the greedy-based approach in terms of the overall time cost under various numbers of EV platoons. Compared to other methods, the joint charging allocation and velocity optimization not only reduces the overall time cost but also achieves superior performance. This can be attributed to the synergistic effect of combining optimal charging resource allocation with velocity optimization, ensuring that EVs reach charging stations in a more timely and efficient manner. By strategically scheduling charging and optimizing travel speeds, the proposed method minimizes both charging times at stations and travel times, leading to a significant reduction in the total time cost. Integrating these two aspects ensures that EVs spend less time on the road and at charging stations, resulting in a streamlined and efficient charging experience.

Table 6 quantitatively compares four strategy implementations: i) Our full optimization integrating charging allocation and velocity optimization, ii) Charging allocation optimization without velocity coordination, iii) Greedy charging with PSO-based velocity adjustment, and iv) Pure greedy charging as baseline. The results demonstrate that the integrated approach achieves 1.4%–15.2% time reduction over alternatives, with two technical insights: First, velocity optimization contributes 1.3%–4.4% additional improvement when combined with our charging allocation algorithm, versus only 0.9%–8.8% when paired with greedy allocation. Second, the performance gap widens nonlinearly with platoon count, suggesting our method’s superior scalability in dense scenarios.

We developed a framework for charging allocation in the V2I-based EV networks with hybrid traffic flows, where each charging station has several charging piles, and multiple manned EVs and EV platoons are served for charging. The overloading issue caused by EV platoons was addressed, and a joint charging allocation and velocity optimization problem was formulated to minimize the overall charging time cost in the considered network. To solve this problem, a two-stage alternating minimization method consisting of the charging allocation stage and velocity optimization stage was proposed. In the charging allocation stage, a dynamic programming-based algorithm was developed, and a low-complexity heuristic method was also provided for optimal charging configuration. In the velocity optimization stage, a swarm-inspired approach was adopted. The results confirmed that the proposed method can achieve better charging allocation than the greedy-based and nearest-neighbor schemes in terms of overall charging time cost.

In future work, data-driven methods may help predict real-time road conditions and manage hybrid traffic flows more efficiently to reduce travel duration further. Moreover, scalable and energy-aware platooning designs need to be investigated, where platoons may be dynamically reconfigured via selective disengagement or split/merge operations. In such cases, new charging allocation solutions are recommended to account for time-varying platoon membership. In addition, charging prices and user behavior are not modeled in this paper and will be incorporated in future work by extending the objective (e.g., time–money trade-off) and introducing incentive-compatible mechanisms under time-varying prices.

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Authors and Affiliations

1. Institute of New Energy, College of Physics and Electromechanical Engineering, Hexi University, Zhangye, 734000, China

   Liwan Qi & Shoubo Li

2. Fudan University, Shanghai, 200433, China

   Liwan Qi, Bochun Wu, Shoubo Li & Yi Gong

3. School of Engineering, Edith Cowan University, Perth, 6027, Australia

   Wei Ni

Contributions

L. Q., B. W., S. L., Y. G., and W. Ni wrote the main manuscript text and L. Q. and S. L. prepared the figures. All authors reviewed the manuscript.

Corresponding author

Correspondence to Bochun Wu.

Competing interests

The authors declare no competing interests.

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Qi, L., Wu, B., Li, S. et al. Efficient charging scheduling through coordination of electric vehicle platoons and charging stations. Sci Rep 16, 8773 (2026). https://doi.org/10.1038/s41598-026-39376-9

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• Received: 04 December 2025

• Accepted: 04 February 2026

• Published: 13 February 2026

• Version of record: 12 March 2026

• DOI: https://doi.org/10.1038/s41598-026-39376-9