Joint optimization of UAVs nest siting and parallel scheduling under solar power supply constraints

1 Introduction

Unmanned aerial vehicle (UAV) technology has shown great potential to replace manual operations in power line inspection and related applications He et al. (2024). Traditional inspection methods, including manual patrols and manned helicopters, face inherent limitations in cost, efficiency, and operational safety Huang and Wang (2024). In recent years, UAVs have become an essential tool for power grid inspection. Their ability to operate at low altitude, perform close-range perception, and maintain safety in hazardous environments offers distinct advantages Mendu and Mbuli (2025). However, the limited endurance of onboard batteries remains a major obstacle for long-duration missions. Frequent recharging increases scheduling complexity and reduces overall inspection efficiency Fendji et al. (2020). Transmission line inspection is characterized by long linear corridors, widely distributed towers, and strong periodicity. Simply enlarging battery capacity or increasing the frequency of returns cannot fundamentally solve the structural contradiction between short endurance and long-range missions Huang and Savkin (2021). Therefore, it is essential to integrate the energy replenishment process into the operational workflow. By deploying accessible charging infrastructure along inspection corridors, a closed-loop operational pattern of inspection–recharging–redeployment can be established, as shown in Figure 1.

Against this background, UAV nest technology has emerged as a practical solution. A UAV nest generally refers to an automated base or charging station that enables UAVs to take off, land, and recharge autonomously. It provides nearby energy replenishment and standby support for UAVs engaged in inspection or surveillance missions (Huang et al., 2025). A well-planned deployment of UAV nests can significantly reduce the travel distance and time required for energy replenishment. This, in turn, enhances the continuity and efficiency of UAV operations Zeng et al. (2025). In recent years, extensive research has been conducted on the site selection and spatial layout of UAV nests. Most studies treat this problem as a facility location optimization task and develop mathematical models and algorithms for its solution. Depending on the primary optimization objectives, existing methods can be broadly classified into two categories: coverage maximization and cost minimization Caballero et al. (2025); Chen et al. (2025); Meng et al. (2024); Yildiz (2024); Abdel-Basset et al. (2021).

The objective of coverage maximization is often formalized as the Maximum Coverage Location Problem (MCLP). Under a limited budget or a predefined number of deployable stations, this model seeks to select a set of UAVs base locations that allow the largest possible number of mission targets (e.g., transmission tower nodes) to fall within their operational service radius. Zeng et alZeng et al. (2025) formulates the UAV nest location problem as a joint optimization model that aims to maximize both the weighted response time satisfaction and the regional coverage rate. The optimization process is constrained by several engineering factors, including budget limitations, minimum inter-base spacing, and cooperative operation requirements. The study employs an improved multi-dimensional ant colony optimization algorithm to identify a feasible set of candidate nest locations and to determine the optimal deployment configuration. The central objective in Zhang et al. (2025) is to maximize tower coverage under the constraints of battery endurance and flight range. This objective is achieved through a weighted Monte Carlo sampling method integrated with Pareto-based multi-objective optimization, which enables an effective balance between coverage performance and operational efficiency. In addition, Bae et al. (2020) introduces the k-GRAPE algorithm, which combines game-theoretic decision-making with k-means clustering. This approach aims to maximize user utility in charging station placement by considering user preferences and congestion effects. Theoretical analyses and empirical experiments further confirm the algorithm’s convergence and demonstrate its superiority over existing methods.

Research on cost minimization aims to achieve the lowest possible construction and operational expenses while maintaining complete coverage and accessibility. The objective function generally includes the fixed construction cost of UAV nests, the configuration cost of individual UAVs or fleets, and additional costs related to energy consumption, maintenance, or operational time. Shavarani et al. (2021) formulates a bi-objective facility location model for time-sensitive UAV delivery tasks. The model minimizes both total cost and demand loss under service constraints. It applies the nondominated sorting genetic algorithm II (NSGA-II) and a reference-point-based NSGA to obtain a Pareto-optimal set of solutions that balance cost efficiency and service quality. Zahedi et al. (2020) models the UAV base station deployment problem as a P-median formulation. The study employs a bisection method to determine the number of facilities and integrates a fuzzy clustering algorithm to generate candidate station sites. This method achieves higher bandwidth coverage with fewer base stations under coverage and capacity constraints, thereby minimizing overall system costs. Park et al. (2022) establishes a robust location–allocation model in which both coverage distance and capacity vary with the number of UAVs. The study proposes an extended mathematical formulation and introduces a branch-and-price (B&P) framework to solve large-scale problems efficiently. Subproblems are handled through MILP reformulation and decomposition-based dynamic programming, while a restricted master heuristic (RMH) is designed to enhance lower-bound quality and computational efficiency. Ribeiro et al. (2019) develops a mixed-integer linear programming (MILP) model that jointly optimizes UAV routing and charging station placement. The objective is to minimize total inspection cost while ensuring full coverage of the conveyor system. The study validates its approach using real-world industrial data, demonstrating strong applicability and scalability. Chung et al. (2020) constructs a location optimization model aimed at minimizing the number of UAVs required for inspection missions. It introduces a heuristic algorithm to efficiently address the NP-hard nature of the problem while accounting for environmental factors such as wind conditions.

It should be noted that most existing studies focusing on coverage maximization or cost minimization assume that UAV nests can access energy on demand. However, this assumption is not always valid in transmission corridors characterized by high voltage, strong electromagnetic interference, and complex terrain Muñoz-Gómez et al. (2023). Direct connection to high-voltage lines involves multiple regulatory and technical challenges. These include compliance with live-line operation standards, maintaining corridor clearance, and obtaining grid connection approval. Moreover, such configurations require multi-layer insulation and protective measures, which significantly increase the cost of electromagnetic compatibility and lightning protection Boukabou and Kaabouch (2024). In addition, the strong coupling between the energy supply and the main grid introduces further vulnerabilities. Scheduled maintenance or unexpected outages can interrupt the charging process, weakening the overall system resilience and increasing the life-cycle cost of operation and maintenance. Although several studies have proposed equipping UAVs with solar panels to extend flight endurance, their performance is often unstable due to shading and seasonal variations in solar irradiance Chu et al. (2021). Therefore, a more practical approach is to deploy off-grid UAV nests along transmission corridors. These nests integrate autonomous takeoff, landing, and local energy replenishment capabilities, while explicitly incorporating solar irradiance availability into the joint optimization of site selection and inspection scheduling.

In the domain of multi-UAV cooperative inspection, researchers focus on algorithms for task allocation and path optimization. The primary goal is to complete inspection missions efficiently while satisfying energy constraints. These problems are often formulated similarly to the Vehicle Routing Problem (VRP) or the Multiple Traveling Salesman Problem (MTSP) He et al. (2025); Hu et al. (2023); Rios et al. (2021); Zhang et al. (2022); Cheikhrouhou and Khoufi (2021). Qi et al.Qi et al. (2025) constructs a multi-UAV path planning model that accounts for energy consumption at different flight phases and battery capacity limitations. The study minimizes the total flight time of all UAVs and introduces an Improved Bee Foraging Learning Particle Swarm Optimization (IBFLPSO) algorithm, which significantly outperforms conventional methods in finding near-optimal solutions. Qiao et al. (2019) proposes a cooperative patrol task planning model for multiple UAVs. By integrating Floyd’s initial solution with the Improved Push Forward Insertion Heuristic (IPFIH), the study maximizes patrol rewards under complex constraints and validates the feasibility and effectiveness of the approach in resource-limited scenarios through simulation. Phalapanyakoon and Siripongwutikorn (2023) develops a path planning model for rechargeable UAVs that incorporates multiple round trips and mission time limits. The study integrates flight distance-based charging costs, fleet scale costs, and penalty for time constraint violations into a mixed-integer programming (MIP) framework. Simulation results demonstrate that the model can obtain optimal solutions for small-scale instances and feasible solutions for large-scale problems. Min et al. (2022) establishes an energy consumption model for UAV inspection of transmission towers. It further combines this model with a simulated annealing algorithm to generate energy-optimal inspection paths. Zhang et al. (2025) introduces the Adaptive Nearest Point (ANP) path planning method for multi-UAV inspection. Under battery endurance and maximum flight range constraints, this method significantly improves coverage and on-time task completion. After determining UAV nest locations, Huang et al. (2025) proposes an iterative local search—simulated annealing (ILS-SA) heuristic algorithm. This algorithm is used to optimize nest type selection and task assignment schemes effectively.

Based on these studies, two major discrepancies can still be observed between current multi-UAV inspection path planning research and practical engineering applications. First, most existing approaches assume that UAV nests serve as continuously available recharging nodes. However, they rarely achieve dynamic coordination between UAV battery levels and nest energy management. Key operational parameters—such as nest storage capacity, charging rate, the number of simultaneous charging ports, UAV charging efficiency, and waiting time—are often neglected. This leads to a mismatch between flight scheduling and recharging cycles, which can compromise both mission completion time and overall operational feasibility. Second, in overlapping coverage areas among multiple nests, current methods usually rely on static fixed-boundary or nearest-distance assignments determined during the site selection phase. Such static allocation rules lack adaptive mechanisms that respond to real-time operating conditions. Specifically, they fail to adjust task ownership based on the current energy levels of UAVs and nests or on the workload intensity of individual UAVs. As a result, inspection tasks for towers located in overlapping zones may be assigned inefficiently, leading to globally suboptimal scheduling and reduced coordination efficiency.

From the perspective of solution methodologies, existing studies can generally be categorized into two main approaches: single-stage joint optimization and two-stage decomposed optimization. The single-stage joint optimization approach integrates nest site selection and UAV scheduling within a unified model. This framework captures the coupling relationships comprehensively but often suffers from high computational complexity and limited stability when dealing with large-scale problems or multiple constraints. In contrast, the two-stage decomposed optimization approach first determines the locations and capacities of UAV nests, and then solves the path planning and scheduling problem based on the established layout. This method offers greater scalability and practical feasibility for engineering applications. Considering the problem’s scale and inherent complexity, this study adopts the two-stage optimization framework. Furthermore, to address the limitations discussed above, the key innovative contributions of this work are summarized as follows:

• Under the objective of minimizing the total operational cost, a composite scoring metric is introduced as a key constraint in the site selection process. This score integrates factors such as solar irradiance, tower density, and nest capability parameters through a weighted formulation. By jointly optimizing the locations and capacities of UAV nests, the method ensures a structured alignment between energy replenishment capacity and regional inspection demands.
• Based on the established nest layout, an energy-aware multi-UAV cooperative inspection path planning model is developed. In areas where nest coverage overlaps, a dynamic secondary allocation mechanism is introduced. This mechanism adjusts tower assignments in real time according to the energy reserves of UAVs and their workload intensity, with the overall objective of minimizing the global task completion time.
• For the UAV nest siting problem, we propose the Solar-Aware Siting (SAS) algorithm, which follows a “two-end calibration + ε-constrained optimization” pipeline. Subsequently, for inspection routing and for optimizing tower ownership in cross-covered regions, we develop the Parallel Scheduling with Cross-Tower Reassignment (PS) algorithm.

2 Problem description and assumptions

2.1 Problem description

With the continuous development of UAV technology, UAV inspection systems have gradually become an efficient and flexible monitoring and inspection tool, particularly in inspection tasks within the power, telecommunications, and transportation sectors. This study primarily addresses the problem of collaborative UAV inspections within a specified region, aiming to improve inspection efficiency and reduce overall costs through the rational deployment of nests and optimization of inspection scheduling. The core elements considered in this study are as follows:

• Collaborative operation of multiple UAV types

In UAV inspection tasks, this study considers three types of UAVs: large, medium, and small. Each type of UAV has different performance parameters, such as flight speed, endurance time, and charging requirements. Large UAVs, with longer endurance and greater payload capacity, are suitable for inspecting large areas; medium-sized UAVs are designed for medium-range tasks, while small UAVs are more suited for smaller-scale inspections. Therefore, the construction and service radii of the nests should vary according to the type of UAV.

• Nest siting problem

The siting of nests is crucial for both the economic and operational efficiency of the system. The selection of nest locations must consider not only the distribution of towers but also solar irradiance. Solar irradiance directly impacts the charging efficiency of nests, and solar power charging systems rely on high-irradiance areas to enhance energy efficiency. Furthermore, nest locations should ensure coverage of all inspection tasks while minimizing construction costs. The choice of nest type (large, medium, or small) should also be determined based on the density and distribution of towers within the region.

• Tower assignment in overlapping service areas

In practice, the service radii of nests may overlap, meaning that multiple nests can cover the same tower. Therefore, it is necessary to determine which nest will be responsible for inspecting the towers in these overlapping regions. This issue requires a balance between the siting and scheduling stages. In this study, the assignment of towers in overlapping areas is not rigidly decided during the siting phase but is instead dynamically determined in the scheduling phase, taking into account real-time factors such as charging demand and flight distance to select the most suitable nest.

• UAV scheduling problem

Once the nest locations and tower assignments are determined, the next step is the UAV scheduling problem. This involves how to schedule UAV inspection tasks within the service areas of different nests, when to charge, and how to optimize inspection routes. The objective is to minimize the total inspection time. During scheduling, factors such as UAV flight speed, charging rate, and endurance time must be considered to achieve efficient inspections.

2.2 Assumptions

To simplify the modeling process and ensure the solvability of the problem, the following assumptions are made in this study:

• The flight speed and endurance time of the UAVs are fixed and unaffected by external weather conditions;
• Each UAV is associated with a specific nest and can only operate within the service range of the corresponding nest;
• The elevation differences between the towers are not considered;
• The service range of each nest is determined by its type (large, medium, small), corresponding to large, medium, and small UAVs, with different nest types having distinct service radii;
• The solar irradiance at each tower is fixed, determined by its geographical coordinates, and does not vary with time. The average solar irradiance for that location is quantified based on the coordinates.

3 Modeling

To address the multi-UAV inspection and tower siting problem in practical applications, this study decomposes the problem into two hierarchical subproblems: the nest siting problem and the UAV scheduling problem within the service area. This hierarchical approach effectively reduces the computational complexity of each subproblem, thereby improving the overall solution efficiency.

3.1 Model I: solar-Aware Ssting

3.1.1 Construction of the mathematical model

In this phase, the modeling is based on the following static assumptions: only solar energy supplementation is used, and each nest is equipped with a UAV of the same type (large, medium, small, corresponding to each nest type). Dynamic factors such as operation time, path, and charging or discharging queues are not considered at this stage, as these will be addressed in the UAV scheduling tasks in the second phase. Under these static assumptions, the focus of this phase is on the one-time siting and type configuration, with the objective of minimizing the total construction cost while ensuring coverage of all towers. A ′collaborative score lower bound’ is introduced to ensure that the selected sites are in areas with high solar irradiance, high tower density, and good proximity. To retain scheduling flexibility for the second phase, a framework is adopted where ’service circles can overlap, and service areas are defined by actual or potential service objects.’ The service radii may overlap, but the final assignment of towers in the overlapping areas is not fixed in this phase, only requiring that at least one constructed nest covers each tower.

Table 1 lists the parameters used in this stage. The service domain induced by the one-way service radius R τ of a nest–type pair (candidate site i, nest type τ) is defined by d i j ≤ R τ, meaning that the pair ( i , τ ) can serve tower j. Accordingly, for any tower j ∈ J, the set of feasible serviceable candidate pairs is defined in Equation 1:

A j = i , τ ∈ I × T : d i j ≤ R τ (1)

TABLE 1. Notation of model I (solar-aware siting).

where I denotes the set of candidate nest locations, T denotes the set of nest/UAV types. Based on this, two types of tower sets, C and O, are generated from the set structure, as shown in Equations 2, 3.

C = j ∈ J : A j = 1 (2)

O = j ∈ J : A j ≥ 2 (3)

Where C denotes the set of non-overlapping towers, in which each tower is uniquely covered by a single feasible nest–type pair. In contrast, O denotes the set of overlapping towers located within the intersection of at least two service regions. This classification is based solely on reachability and is independent of any assignment decision. The final ownership of overlapping towers is determined in the UAV scheduling stage.

To reflect the on-site energy supply conditions and spatial demand intensity during the siting stage, and to ensure that each objective indicator possesses comparable scales and explicit economic significance, two categories of weighting factors are defined in this stage.

• Static Tower Density: to eliminate the influence of UAV type differences, a unified reference radius R ref, independent of the UAV model, is adopted. The static tower density around candidate nest i is defined as Equation 4:

N i = ∑ i ∈ J 1 d i j ≤ R ref (4)

To eliminate regional disparities in tower quantity, N i is normalized using the Min–Max method. Let min N k and max N k denote, respectively, the minimum and maximum values of N k | k ∈ I over all candidate nest sites. The normalized static tower density is then given by Equation 5:

D ̃ i = N i − min N k max N k − min N k , k ∈ I , D ̃ i ∈ 0,1 (5)

This normalized index quantitatively characterizes the spatial intensity of transmission towers surrounding each candidate nest. The normalization ensures consistency in dimensional scale with solar irradiance metrics while avoiding deviations caused by regional variations. Therefore, D ̃ i is defined as the static tower density. where k i j τ represents the proximity attenuation weight, which decreases linearly with increasing distance d i j. When d i j ≤ R τ, the weight gradually declines from 1 to 0, reflecting the spatial incentive of the ’closer-is-better’ effect. Meanwhile, R τ captures the coverage scale corresponding to different UAV types.

• 2. Proximity Attenuation Weight: To characterize the marginal effect of ’the closer, the better’, a proximity attenuation weight is introduced within the reachable domain. Specifically, as shown in Equation 6, a linearly decaying neighborhood function is adopted according to the single-trip service radius R τ defined for each UAV type τ:

k i j τ = max 0,1 − d i j R τ , k i j τ ∈ 0,1 , i , τ ∈ A j (6)

In the first stage, the siting problem of UAV nests under solar power-supply constraints is formulated using an Epsilon-constraint approach. The primary objective minimizes the total construction cost, while a lower bound is imposed on the integrated quality that jointly reflects illumination, tower density, and service distance. The objective function is expressed as Equation 7:

min C o s t = ∑ i ∈ I ∑ τ ∈ T F i τ + C u , τ y i τ (7)

where F i τ denotes the fixed construction cost of establishing a type-τ nest at candidate site i, and C u , τ denotes the procurement cost of the UAV associated with nest type τ. Therefore, ( F i τ + C u , τ ) represents the one-time capital investment for activating site–type pair ( i , τ ), following the construction-before-operation principle. Since no operational energy consumption occurs at this stage, illumination and tower-density indicators are incorporated into the comprehensive quality to guide candidate-site prioritization. The comprehensive quality function consists of three components:

• Prior-site term: ∑ i ∈ I ∑ τ ∈ T φ τ S i ̃ D ̃ i y i τ, where φ τ denotes the weighting factor for UAV type τ. This term is independent of tower assignment and quantifies the inherent locational advantage of establishing a nest in areas with higher solar irradiance and static tower density.
• Non-overlapping tower assignment term: ∑ j ∈ C ∑ ( i , τ ) ∈ A j φ τ k i j τ x i j τ, where x i j τ ∈ { 0,1 } indicates whether tower j is served by a type τ nest located at candidate site i. The set C represents towers with clearly defined service responsibilities in the first stage, that is, those located within non-overlapping regions.
• Overlapping-coverage term: For towers j ∈ O located in overlapping regions, multiple candidate nest–type pairs ( i , τ ) ∈ A j are capable of providing service. Since routing and energy constraints are not yet incorporated at the siting stage, directly introducing assignment variables would lead to premature commitment under incomplete information, potentially deteriorating the overall solution quality. To address this issue, an auxiliary variable g j is introduced to represent the maximum achievable service quality for tower j under the current siting configuration, which is defined by Equation 8:

g j ≤ φ τ k i j τ y i τ , ∀ j ∈ O , ∀ i , τ ∈ A j , g j ≥ 0 (8)

It restricts g j to be no greater than the service quality corresponding to any feasible nest–type pair. Since g j is maximized in the objective function, it is driven to the largest feasible upper bound at optimality, yielding: g j = max ( i , τ ) ∈ A j : y i τ = 1 ( φ τ k i j τ ). Therefore, g j effectively captures the best achievable service level for tower j under the current siting configuration, without explicitly determining its service assignment at this stage.

To balance the preference for high-quality sites with the goal of broader tower coverage, a fusion parameter α ∈ [ 0,1 ] is introduced. The integrated scoring function is defined as Equation 9:

Q = α ∑ i ∈ I ∑ τ ∈ T φ τ S ̃ i D ̃ i y i τ + 1 − α ∑ j ∈ C ∑ i , τ ∈ A j φ τ k i j τ x i j τ + ∑ j ∈ O g j , α ∈ 0,1 (9)

where α is a policy preference parameter controlling the trade-off between strategic site quality and immediate service coverage. Larger α values place more emphasis on solar-rich and demand-dense candidate sites, whereas smaller α values prioritize broader effective tower accessibility. Meanwhile, Q is incorporated as a constraint term to regulate the cost-minimization objective. The first term encourages the establishment of nests in favorable locations (high irradiance and dense tower nest); the second term ensures explicit service allocation for non-overlapping towers; and the third term captures potential coverage contribution for overlapping towers whose final assignments remain undetermined.

3.1.2 Constraints

3.1.2.1 Consistency and coverage feasibility

• Single-type construction constraint: as defined in Equation 10, each candidate site can accommodate at most one type of UAV nest:

∑ τ ∈ T y i τ ≤ 1 , ∀ i ∈ I (10)

• Uniqueness of Assignment and Construction Consistency for Non-overlapping Towers: each non-overlapping tower must be assigned to its sole reachable candidate nest, and that nest must be constructed. This constraint is defined as Equation 11:

∑ i , τ ∈ A j x i j τ = 1 , ∀ j ∈ C ; x i j τ ≤ y i τ , ∀ j ∈ C , i , τ ∈ A j (11)

• Overlapping-tower coverage constraint: as defined in Equation 12, each tower located within overlapping service regions must be covered by at least one constructed UAV nest:

∑ i , τ ∈ A j y i τ ≥ 1 , ∀ j ∈ O (12)

3.1.2.2 Integrated quality threshold

To ensure coordination between the energy supply and spatial coverage in the siting scheme, as defined in Equation 13, an Epsilon-constraint is employed to impose a lower bound on the integrated quality:

Q ≥ Q ε (13)

It should be clarified that Cost and Q play different roles in the first-stage model. Specifically, Cost in Equation 7 represents the objective cost function of the siting scheme, which evaluates economic efficiency from the perspectives of nest construction and UAV configuration expenses, and is minimized as the primary optimization objective. In contrast, Q is introduced as an optimization constraint to measure the integrated siting quality, thereby ensuring that the final deployment scheme achieves a proper balance between economic efficiency and overall planning rationality.

Although Equation 9 has already incorporated solar irradiance, tower density, and coverage effectiveness into the integrated quality score, an explicit threshold is still required in the optimization process. This is because Equation 9 only defines how quality is evaluated, whereas Equation 13 determines the minimum acceptable quality level when minimizing deployment cost. Without such a constraint, the optimization may excessively favor low-cost solutions at the expense of service capability and locational suitability.

The threshold Q ε is determined according to a baseline–upper interpolation principle. First, by solving the baseline model of objective Equation 7 without the quality constraint, the lowest-cost UAV nest configuration is obtained. The integrated quality score of this solution, evaluated using Equation 9, is recorded as Q min. This value represents the quality level naturally achieved when cost is prioritized.

Then, the maximum attainable quality value Q max is identified by solving a quality-maximization model. This solution reflects the highest siting quality that can be achieved when cost is not the dominant concern. Therefore, Q min and Q max jointly define the feasible quality interval.

Based on these two benchmark values, the quality threshold is defined as Equation 14:

Q ε = Q min + λ Q max − Q min , λ ∈ 0,1 (14)

where λ is the quality intensity control coefficient that establishes a linear interpolation relationship between the minimum-cost and maximum-quality solutions. A larger λ imposes a stricter quality requirement, while a smaller λ places greater emphasis on cost efficiency.

It should be emphasized that quality is not measured based on cost itself. Instead, cost-optimal and quality-optimal solutions are only used as benchmark reference points, while the actual quality value is always calculated by Equation 9. By adjusting λ, the model continuously regulates the quality level under cost constraints, guiding UAV nest deployment toward regions with higher solar irradiance and tower density, thereby achieving coordinated optimization of economic efficiency and spatial coverage.

3.2 Model II:UAV Parallel Ssheduling

3.2.1 Construction of the mathematical model

The UAV scheduling task is a typical vehicle routing problem (VRP). The VRP is a class of combinatorial optimization problems in which a set of vehicles is required to serve a set of customers from one or multiple depots under various operational constraints, such as route length, capacity, or time. Although VRP has been extensively studied in the context of freight distribution, its modeling framework is broadly applicable to routing and scheduling problems involving multiple agents and spatially distributed tasks.

In this study, the UAV scheduling problem can be naturally cast within the vehicle routing problem (VRP) framework. Specifically, UAVs correspond to vehicles, transmission towers correspond to service nodes (customers), and UAV nests serve as depots. Each UAV departs from its associated nest, visits a sequence of assigned towers, and returns to the same nest after completing a sortie. Moreover, due to energy limitations, a UAV may need to perform multiple sorties within one inspection cycle, which introduces a multi-trip characteristic. Accordingly, the problem is formulated as a multi-depot, multi-trip VRP with energy (recharging) constraints, with the objective of minimizing the overall system completion time.

In this stage, the VRP is modeled based on the solar-aware siting results obtained in the first stage. Specifically, the deployment of UAV nest locations has already been determined, and the set of established nest–type pairs is defined as I ′ = { ( i , τ ) : y i τ = 1 }. Table 2 summarizes the parameters used in the second-stage model. The objective of this stage is to complete the inspection of all transmission towers in set J within a single inspection cycle while minimizing the overall operation completion time, i.e., the time at which the last UAV nest finishes its assigned inspection tasks, under the parallel execution of all ( i , τ ) ∈ I ′.

TABLE 2. Notation of Model II (UAV parallel scheduling).

Accordingly, a parallel UAV inspection scheduling model is established, in which each UAV operates only within its serviceable area and performs multiple round trips with intermediate recharging. Each pair ( i , τ ) corresponds to a type-τ UAV stationed at nest i, which executes several closed routes of the form “nest–multi-tower–nest”. After returning to the nest, the UAV must first replenish the energy consumed in the previous sortie before taking off for the next mission. To capture the influence of solar irradiance S i obtained in the first stage on the charging performance of each nest, it is converted into a time-advantage parameter, and the effective charging rate is defined as r i τ > 0. This parameter characterizes the charging efficiency per unit time.

Given the inter-tower distance d j k ( j , k ∈ J ), the cruising speed v τ > 0, and the unit-distance flight energy consumption e τ fly > 0, the flight time and energy consumption between towers are computed as Equation 15:

t j k = d j k v τ , E j k = e τ fly d j k (15)

Similarly, for nest–tower routes, the corresponding flight time and energy consumption are defined as Equation 16:

t i j τ = d i j v τ , E i j τ = e τ fly d i j , (16)

where i ∈ I denotes a UAV nest node and j ∈ J denotes a transmission tower. The term d i j represents the Euclidean distance between nest i and tower j. For notational simplicity, the route from nest i to tower j is denoted as ( i , j ), and the return route is denoted as ( j , i ). To ensure feasibility, the maximum number of sorties associated with each ( i , τ ) is conservatively bounded by S i τ max = | J |, which corresponds to the theoretical upper bound where each tower is visited at most once.

The objective is to minimize the overall system completion time. By introducing the auxiliary variable C max, the min–max objective can be linearized as Equation 17:

min C max = max q i τ , s , C max ≥ q i τ , s , ∀ i , τ ∈ I ′ , s = 1 , … , S i τ max (17)

where q i τ , s denotes the completion time of the s-th sortie executed by UAV ( i , τ ). In this way, minimizing C max is equivalent to minimizing the maximum sortie completion time over all ( i , τ ) ∈ I ′ and all feasible s, thereby yielding the minimum overall system completion time.

3.2.2 Constraints

Next, we establish the corresponding constraints for the scheduling model.

3.2.2.1 Consistency and coverage feasibility

• To ensure that each tower is assigned to a feasible nest pair and is visited exactly once by its assigned UAV, the constraints shown in Equation 18 are imposed.

∑ i , τ ∈ A j a i j τ = 1 , ∀ j ∈ J , a i j τ = x i j τ j ∈ C , ∑ s = 1 S i τ max z i τ , j , s = a i j τ , ∀ i , τ , ∀ j (18)

The first relation ensures the unique ownership of each tower, the second fixes the assignment of non-overlapping towers according to the first-stage result, and the third links the ownership decision with the rule that each assigned tower must be visited exactly once.

• Since each sortie is associated with a specific established nest–type pair ( i , τ ), the depot of this sortie is represented by the actual nest node i, rather than by an additional dummy node. Each sortie must start from its assigned nest and return to the same nest. For every tower visited in the sortie, the in-degree and out-degree must both equal one, as defined in Equation 19:

∑ k ∈ J w i k i τ , s = h i τ , s , ∑ k ∈ J w k i i τ , s = h i τ , s , ∑ k ∈ i ∪ J \ j w k j i τ , s = z i τ , j , s , ∑ k ∈ i ∪ J \ j w j k i τ , s = z i τ , j , s , ∀ j ∈ J (19)

3.2.2.2 Path initiation and connectivity constraints

To prevent cross-region linkages and the activation of infeasible sorties, coupling constraints between arc variables and visit variables are introduced, as shown in Equation 20:

w i j i τ , s ≤ a i j τ , w j i i τ , s ≤ a i j τ , w j k i τ , s ≤ a i j τ , w j k i τ , s ≤ a i k τ , w j k i τ , s ≤ h i τ , s , z i τ , j , s ≤ h i τ , s , ∀ j , k ∈ J , j ≠ k (20)

The above degree, initiation, and coupling constraints collectively establish a single-route structure analogous to the Traveling Salesman Problem (TSP). Nevertheless, the overall formulation remains a Vehicle Routing Problem (VRP), because each route is restricted to the set of towers assigned to ( i , τ ).

3.2.2.3 MTZ subtour elimination constraints

To eliminate subtours other than the UAV nest, the Miller–Tucker–Zemlin (MTZ) linearization is adopted. First, the visiting-order index of an unvisited tower is set to zero: 0 ≤ μ i τ , j , s ≤ | J | ⋅ z i τ , j , s ( ∀ j ). If tower j is not visited in sortie s ( z = 0 ), then μ = 0; otherwise, when z = 1, μ ∈ [ 0 , | J | ] denotes its position in the visiting sequence. The sequential consistency constraint is given in Equation 21:

μ i τ , j , s − μ i τ , k , s + | J | ⋅ w j k i τ , s ≤ | J | − 1 + | J | 2 − z i τ , j , s − z i τ , k , s , ∀ j ≠ k (21)

When both j and k are visited in sortie s and the arc j → k ( w = 1 ) is selected, the above constraint reduces to μ i τ , k , s ≥ μ i τ , j , s + 1. Thus, the sequence indices strictly increase along any active route, thereby preventing the formation of inner subtours.

3.2.2.4 Time and energy constraints

The total flight time and energy consumption of sortie s are defined in Equation 22:

T i τ , s fly = ∑ j , k ∈ J j ≠ k t j k w j k i τ , s + ∑ j ∈ J t i j τ w i j i τ , s + w j i i τ , s , E i τ , s fly = ∑ j , k ∈ J j ≠ k E j k w j k i τ , s + ∑ j ∈ J E i j τ w i j i τ , s + w j i i τ , s (22)

To ensure full-charge takeoff and avoid energy depletion, the total energy consumption of each active sortie must satisfy E i τ , s fly ≤ E τ use h i τ , s.

3.2.2.5 Cumulative completion time and solar-charging coupling

To compute the cumulative completion time, the sequence “fly first, recharge later” is adopted. For the first sortie, only flight time is considered, with no charging delay, i.e., q i τ , 1 ≥ T i τ , 1 fly. For each subsequent sortie ( s ≥ 2 ), the UAV must recharge the consumed energy E i τ , s − 1 fly at the effective charging rate r i τ before the next takeoff, resulting in a waiting time of E i τ , s − 1 fly / r i τ. Therefore, the cumulative completion time is updated as Equation 23:

q i τ , s ≥ q i τ , s − 1 + E i τ , s − 1 fly r i τ + T i τ , s fly , s ≥ 2 (23)

The charging rate is determined by Equation 24:

r i τ = r τ min + r τ max − r τ min S ̃ i (24)

where S ̃ i is the normalized solar irradiance index obtained from the first-stage optimization, thereby ensuring consistency between the charging rate expression and the solar indicator used in the first-stage quality evaluation. This equation establishes the relationship between charging efficiency and solar exposure, and further realizes the coupling between cumulative completion time and route execution sequence within a linear framework.

4 Model solution and algorithm design

The algorithmic framework of this study, as shown in Figure 2, follows a two-level coupled optimization structure integrating siting and scheduling. The input components include equivalent task mileage (covering takeoff, landing, and operational mileage), inspection speed, spatial topology, solar irradiance distribution, nest service radii, and construction costs.

At the first level, the Solar-Aware Siting module formulates a mixed-integer linear program (ILP) on the coverage graph. This module identifies the cost-minimal baseline configuration and further incorporates a cooperative quality score derived from solar-irradiance zones, core towers, and boundary-crossing towers. Based on this score, a quality–cost weighted effective-cost objective is constructed, and a quality threshold is imposed. The module then selects the optimal set of nest locations and their radius tiers (S/M/L) from the feasible solution space.

At the second level, the Parallel Scheduling module performs multi-nest inspection scheduling under the fixed siting configuration. Towers are first allocated to nests using a nearest-feasible assignment within the corresponding coverage radii. The assigned towers are then organized into multiple sorties through an energy-cap-constrained packing process that satisfies the equivalent-mileage limits, while flight, charging, and turnaround durations are explicitly accounted for.

Based on this initial assignment, a cross-tower reassignment procedure is applied at coverage boundaries to reduce the makespan of bottleneck nests. The final timeline is then constructed to obtain per-nest durations and the global makespan (Gantt chart). This two-layer workflow provides the basis for evaluating the effectiveness of the scheduling strategy.

4.1 Algorithm solar-aware siting (SAS)

SAS follows a “two-end calibration + ε-constrained optimization” pipeline. Given the coverage graph and solar-weighted costs, we first solve a cost-only ILP to obtain a baseline siting and aggregate its synergy quality to form Q min. We then maximize the quality (under the same feasibility constraints) to obtain Q max. With a user-set quality intensity λ ∈ [ 0,1 ], we define the quality threshold in Equation 14 and solve a second ILP that minimizes construction cost subject to Q ≥ Q ε. The output is the selected site–type set N, the calibration triplet Q = ( Q min , Q max , Q ε ), and the total cost C, thereby yielding an interpretable cost–quality trade-off supported by data-driven endpoint calibration. The algorithm pseudocode is summarized in Table 3.

TABLE 3. Pseudocode of Solar-Aware Siting (SAS). The routine returns the selected site–type set N, the calibration triplet Q = ( Q min , Q max , Q ε ), and the total cost C.

4.2 Parallel scheduling with cross-tower reassignment

Given the siting result N and type radii { R τ }, we build per-nest coverage sets and construct an initial “nearest feasible” assignment of towers, then pack energy-capped routes (sorties) for each nest to obtain a baseline schedule and its makespan M.

To reduce the bottleneck without violating endurance/charging limits, we identify cross-covered towers and iteratively reassign one tower at a time from the bottleneck nest to an alternative feasible nest; a move is accepted only if it does not increase the global makespan and strictly shortens the bottleneck’s total time. After each accepted move, routes are repacked under the same energy model and the schedule is updated until convergence or a move budget is reached. This stage preserves spatial feasibility via coverage sets and radii while enforcing temporal feasibility through endurance/charging caps (“caps”) in route packing and timeline construction. The algorithm pseudocode is summarized in Table 4.

TABLE 4. Pseudocode of the parallel scheduling stage with cross-tower reassignment. Coverage feasibility is enforced by { R τ } and CoverSets; temporal feasibility is enforced by endurance/charging caps in the route-packing and timeline construction.

4.3 Parameter calibration and implementation details

To improve the reproducibility of the proposed two-stage framework, this subsection further specifies the calibration principles, implementation settings, and algorithmic details for both the Solar-Aware Siting (SAS) stage and the Parallel Scheduling with Cross-Tower Reassignment (PS) stage.

4.3.1 Calibration of model I and SAS

In the siting stage, the candidate nest locations are co-located with transmission towers, and the set of candidate site–type pairs is generated according to the service radius of each nest type. The three nest classes (small, medium, and large) follow a strict hierarchical setting in service radius and relative construction cost, as summarized in Table 5. These radius parameters determine the feasible coverage relationship between candidate sites and towers and are therefore used to construct the coverage graph { A j } j ∈ J.

TABLE 5. Parameter settings of nest types.

C is a dimensionless relative cost coefficient.

To quantify local demand concentration in a manner independent of nest type, a unified reference radius R ref is introduced for the calculation of static tower density. This parameter is used only to evaluate the surrounding tower concentration of a candidate site and does not affect the actual service feasibility determined by R τ. The resulting density indicator is normalized by the min–max rule so that it can be combined with the normalized solar irradiance index on a comparable dimensionless scale.

The type-weight coefficient φ τ is used to reflect the relative strategic importance of different nest types in the integrated quality evaluation. In this study, these coefficients are calibrated in accordance with the service capability hierarchy of the three nest classes, so that larger nest types receive higher quality contributions when they provide broader and more flexible coverage. The parameter α ∈ [ 0,1 ] is introduced as a policy preference coefficient to balance two aspects of siting quality: intrinsic locational suitability, represented by the prior-site term, and explicit service effectiveness, represented by the assignment-related terms. A larger α gives more weight to solar-rich and demand-dense candidate sites, whereas a smaller α places greater emphasis on direct tower accessibility and proximity efficiency.

The quality-intensity coefficient λ ∈ ( 0,1 ) is calibrated through the ε-constraint mechanism. Specifically, three mixed-integer linear programs are solved under the same feasibility constraints. First, the cost-only model is solved to obtain the minimum-cost siting solution, whose integrated quality value is evaluated as Q min. Second, the quality-maximization model is solved to obtain the upper benchmark Q max. Third, a quality threshold is constructed as Q ε = Q min + λ ( Q max − Q min ), and the final siting model minimizes the total construction cost subject to the additional constraint Q ≥ Q ε. In this way, λ does not directly alter the quality function itself; rather, it determines how strict the minimum acceptable quality requirement is between the two benchmark endpoints.

In the implementation, the coefficient α is selected through a grid-based sensitivity analysis, and the corresponding siting layouts are compared in terms of nest number, total cost, and integrated quality. The final parameter combination is chosen by jointly considering budget consistency, spatial regularity of the selected nests, and downstream scheduling feasibility. The same solver and feasibility settings are used for all three optimization runs in the SAS procedure to ensure that Q min, Q max, and the final cost-constrained solution remain directly comparable.

4.3.2 Calibration of model II and PS

In the second stage, the scheduling model is built on the fixed siting result returned by SAS. The established nest–type pairs form the active service set, and each tower is assigned to a feasible nest within the corresponding coverage radius. For towers located in non-overlapping regions, the ownership determined in the first stage is retained. For towers located in overlapping regions, an initial nearest-feasible assignment is adopted to construct the baseline schedule.

To unify heterogeneous operational factors on a common scale, this study adopts an equivalent-distance representation for sortie construction. The equivalent sortie length includes inter-tower flight distance, nest–tower travel distance, takeoff/landing conversion, and per-tower service conversion. Based on the nominal capability of each UAV class, an effective distance limit is further defined by applying a fixed safety margin. This treatment reserves sufficient operational redundancy for return flight and scheduling uncertainty while maintaining a linear and implementation-friendly route-packing procedure. The corresponding UAV capability and energy-related parameters are listed in Table 7.

The sortie construction procedure follows an energy-capped packing strategy. Under a fixed tower ownership pattern, towers assigned to the same nest are grouped into multiple closed routes subject to the effective equivalent-distance limit of the corresponding UAV type. After route packing, the completion time of each nest is computed on a unified time axis that explicitly accounts for flight time, solar-dependent charging time, and fixed turnaround time. The global makespan is then defined as the maximum completion time among all active nests.

To further reduce the system makespan, the PS algorithm introduces a cross-tower reassignment procedure over the overlapping coverage region. In each iteration, only towers that can be feasibly served by more than one established nest are considered as movable candidates. A candidate move is accepted only when two conditions are simultaneously satisfied: the global makespan does not increase, and the total workload of the current bottleneck nest is strictly reduced. After each accepted move, the affected routes are repacked under the same energy constraints, and the schedule is re-evaluated. The iterative search terminates when no improving reassignment exists or when the move budget is exhausted.

The calibration of the scheduling parameters follows engineering interpretability and consistency principles. Cruise speed is fixed by UAV class assumptions; charging time is linked to the effective charging rate of each nest; turnaround time is treated as a fixed operational overhead; and the safety margin is uniformly imposed across all UAV types to preserve comparability in route feasibility. Under this setting, the comparison between the Baseline and Optimized policies isolates the actual contribution of cross-tower reassignment, because the siting configuration, UAV capabilities, and all timing parameters remain unchanged.

Overall, the above calibration strategy ensures that the two-stage framework is not only mathematically well-defined, but also operationally interpretable and reproducible. The first stage uses calibrated quality thresholds to control the cost–quality trade-off of nest deployment, while the second stage uses uniformly calibrated route and charging parameters to evaluate the scheduling benefit brought by boundary-task reassignment.

5 Experimental verification

5.1 Model I experiment result

5.1.1 Experimental setup consistency

To ensure consistency between the empirical results of Model I and the theoretical formulation in Section 3, we keep the same optimization structure and constraints, and unify the settings of nest types, cost coefficients, solar-aware adjustment, and solver strategy. Candidate nest sites are co-located with transmission towers. Considering the heterogeneous coverage capability and construction/operation burden across nest types, we define three nest types; their basic parameters are summarized in Table 5.

5.1.2 Geographical distribution of transmission towers and lighting conditions

This study uses a transmission-corridor region as the experimental testbed. The area covers approximately 18 × 18 k m and includes several primary lines together with multiple branch lines. Within this region, a total of 986 transmission towers are deployed. Their spatial distribution pattern is shown in Figure 3.

In terms of regional lighting characteristics, this study adopts the FAO-56 model (Food and Agriculture Organization-Penman-Monteith framework) to calculate the local daily radiation intensity S i, and corrects it using actual regional meteorological data, obtaining the annual average light distribution. And normalize it to obtain S ̃ i. The normalized spatial distribution of regional lighting is shown in Figure 4.

A clear spatial pattern can be observed in the study area. Solar irradiance is generally higher in the northern and northwestern sectors, while the southern and low-lying zones exhibit comparatively weaker irradiance. The region therefore presents both a non-uniform tower distribution and a pronounced irradiance gradient, offering a representative setting for model validation. The tower distribution captures the geographical constraints of inspection routes. In contrast, the irradiance pattern introduces an energy-resource dimension for optimizing nest locations. The combination of these two factors forms the foundational conditions for the multi-factor coupled optimization examined in this study.

5.1.3 Siting layouts and result analysis

We conduct a two–parameter sweep where α controls the strength of the solar-aware cost shift and λ specifies the quality-intensity level between the cost-only baseline and the theoretical quality upper bound. The quantitative outcomes are reported in Table 6, and the corresponding layouts are visualized in Figure 5.

TABLE 6. Model I: ( α , λ ) synergistic optimization results.

Within the regime that preserves the number of nests ( N = 7 ), increasing α from 0 to about 0.6 yields a consistent yet moderate improvement in the synergy score Q while the total cost stays fixed at C = 17.00. In particular, ( α , λ ) = ( 0.45 , 0.06 ) achieves the best quality among all N = 7 candidates (Q = 789.9 versus the baseline Q = 784.4) without incurring extra cost. When α is further enlarged (e.g., α ≥ . 75 with λ ≈ 0.21), the solution enters an N = 6 regime: larger service circles emerge and trade the nest count for coverage, pushing Q up to ∼ 805.5–805.8 but also increasing cost to C = 17.20. Considering budget adherence, geometric regularity, and downstream scheduling load, we select ( α , λ ) = ( 0.45 , 0.06 ) as the final configuration. The selected layout is highlighted separately in Figure 6.

5.2 Model II experiments result

Under fixed siting, the three nest classes (S/M/L) follow a strict hierarchy (S<M<L) in both coverage radius and scheduling capability: the radius defines each nest’s serviceable tower set, while capability parameters bound the executable length of a sortie and its recharge cadence. We place flight, charging, and turnaround on a common time axis via an equivalent distance measure (out–and–back travel + takeoff/landing conversion + per–tower service conversion), converted to flight time by cruise speed; a fixed safety margin yields the effective distance limit, charging is proportional to energy used, and a fixed turnaround is added (Table 7). Given these unified settings, each nest forms sorties by nearest–neighbor batching capped by the effective equivalent distance, operates in parallel, and the system makespan is the maximum nest completion time. We compare two policies under identical constraints: Baseline assigns each overlap–covered tower to the geometrically closest nest once, whereas Optimized performs local one–tower reassignments among overlapping nests—without altering siting or capabilities—and accepts a move only if it reduces the makespan, iterating to convergence.

TABLE 7. UAV capability and energy parameters used by the scheduler.

5.2.1 Comparison analysis between Baseline and Optimized

We report results for the fixed siting/typed layout selected in Model I ( α = 0.45 , λ = 0.06 ). To make cause–effect relationships explicit, we analyze (i) spatial route geometry, (ii) parallel execution cadence, and (iii) load distribution and sortie granularity, each tied to the corresponding figure.

Figure 7 contrasts the multi-nest routes produced by the Baseline (nearest static assignment) and the Optimized (cross-tower feasible reassignment) policies. The layouts exhibit near-identical trunk corridors and turning patterns; only a small subset of towers in overlap regions switches ownership after reassignment. This indicates that, once siting and service radii are fixed, the remaining spatial degrees of freedom are confined to boundary exchanges rather than global route reshaping. Consequently, potential time savings from geometry alone are bounded.

Given the common equivalent-distance accounting and identical energy/turnaround parameters, the schedule on each nest follows the same flight → charging → turnaround rhythm. The Gantt charts in Figure 8 show that the bottleneck nest remains the same across policies and that the system makespan decreases only slightly from ∼ 803.1 to ∼ 800.3 minutes under the Optimized policy. This behavior confirms that the cadence is energy-paced: once sorties saturate their effective distance caps, charging segments dominate the timeline, leaving little slack for reassignment to compress.

Figure 9 reports per-nest total workload (minutes). Bars for the two policies are almost indistinguishable, implying that reassignment mainly reroutes short boundary links without redistributing the aggregate charging burden that drives each nest’s completion time. Complementarily, Figure 10 plots the histogram of sortie equivalent distances. The distributions nearly overlap and cluster close to type-dependent caps, revealing a “cap-touching” steady state: sorties are formed to the limit, so the number and length of sorties are effectively locked by energy constraints rather than by path geometry. Together, these two plots explain why feasible cross-tower exchanges improve the makespan only marginally: they smooth a few local imbalances but cannot loosen the global charging cadence.

5.2.2 Supplemental validation of model II under economic siting

In the synergy–optimal layout of the previous subsection, only a tiny portion of towers lay on exchangeable boundaries; consequently the scheduling layer had little room to improve and the makespan reduction was marginal. To further substantiate the effectiveness of cross–tower reassignment—the core mechanism of Model II—we repeat the comparison under a purely economic siting while keeping all capability and timing parameters unchanged.

Figure 11 contrasts the spatial routes before and after reassignment on this economic layout. Relative to the Baseline (left), the Optimized policy (right) reassigns a subset of overlap towers from the boundary regions to neighboring nests where they are geometrically and energetically cheaper to serve. These one-tower moves smooth spokes across the M / S interfaces and cut several back-and-forth detours near the large L nest’s fringe, without altering coverage radii or sortie caps.

The impact on execution rhythm is shown by the Gantt timelines in Figure 12. The Baseline schedule is dominated by a single heavy nest whose alternating flight–charge–turnaround cycles set the critical path. By shifting selected overlap towers away from this bottleneck nest, cross–tower reassignment reduces its cumulative workload and compresses charge-dominated plateaus, shortening the system makespan from about 803.1 min to 792.7 min (a decrease of ∼ 10.4 min, i.e., 1.3%). All hard constraints (coverage feasibility, effective distance caps, charger capacity) remain satisfied, and the other nests show only minor timing adjustments.

Across both settings, the two-layer design—Solar-Aware Siting (layout/radii fixed first) followed by Parallel Scheduling with cross-tower reassignment—behaves as intended. When the layout is already synergy-optimal, exchangeable boundary mass is scarce, so reassignment yields only marginal makespan relief (0.4%), revealing a regime dominated by energy/charging cadence. Under the purely economic layout, the same reassignment mechanism redistributes overlap towers away from the bottleneck nest and shortens the makespan by 10.4 min (1.3%) without altering siting, radii, or any feasibility constraints. Hence, cross-tower reassignment is a safe, low-overhead lever for schedule smoothing; larger gains require additional freedoms at the siting or energy (range/charging) layers.

Overall, the evidence demonstrates that the cross–tower reassignment step in Model II provides a safe, low-overhead lever to rebalance boundary tasks and relax charge-constrained bottlenecks when siting leaves more exchangeable mass, yielding a measurable makespan improvement under identical resource limits.

6 Conclusion

We developed and validated a two–stage optimization framework for solar–powered UAV inspection: (1) a Solar–Aware Siting MILP that minimizes construction cost under a calibrated quality threshold coupling irradiance, tower density, and proximity; and (2) a Parallel Scheduling stage that builds energy–capped sorties and performs cross–tower reassignment at service boundaries to compress the makespan. On a realistic corridor with O ( 1 0 3 ) towers, the siting stage identified a N = 7 layout with C = 17.00 and improved quality ( Q = 789.9 ), selected at ( α , λ ) = ( 0.45 , 0.06 ). Under this synergy–optimal layout, reassignment yielded only marginal timeline relief ( ≈ 0.4 % ), revealing an energy–paced regime where charging cadence, not route geometry, sets the critical path. Under a purely economic layout, the same reassignment reduced the makespan by ∼10.4 min ( ≈ 1.3 % ) by shifting overlap towers off the bottleneck nest, with all feasibility constraints preserved.

These findings provide actionable guidance: invest first in synergy–aware siting to align nests with bright corridors and dense demand; deploy cross–tower reassignment to harvest additional (but bounded) gains where boundary mass remains. The framework is scalable and interpretable, making it suitable for planning studies and rolling operational use. Future work will integrate seasonal irradiance and weather uncertainty, charger–port sizing and battery swapping, multi–UAV resource sharing, and field validation, aiming to close the loop between solar–aware infrastructure planning and energy–paced scheduling for large–area, solar–driven UAV inspection.

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Footnotes

Edited by: Qiang Bai, Chang’an University, China

Reviewed by: Muhammad Javed, Donghua University, China

Lin Chen, Chang’an University, China