Design and Assessment of Robust Persistent Drone-Based Circular-Trajectory Surveillance Systems

Abstract

We study the use of a homogeneous fleet of drones to design an unattended persistent drone-based patrolling system for vast circular areas. The drones follow flight missions supported by auxiliary on-ground charging stations, whose location and number must be determined. To this end, we first present a mixed integer non-linear programming model for defining cyclic schedules of drone flights considering the selection of the drone model from a set of candidate drone platforms. By imposing a minimum acceptable time between consecutive visits to any perimeter point, the objective consists of minimizing the total surveillance system deployment cost. The solution provides the best platform, the location of base stations, and the number of drones needed to monitor the perimeter, as well as the flight mission for each drone. We test five commercial platforms in six different scenarios whose radios vary between 1196 and 1696 m. In five of them, the MD4-100 Microdrones model achieves the lower cost solution, with values of EUR 66,800 and 83,500 for Scenarios 1 and 2 and EUR 116,900 for Scenarios 3, 4 and 5, improving its rivals in average percentages that vary between 8.46% and 70.40%. In Scenario number 6, the lower cost solution is provided by the TARTOT-500 model, with a total cost of EUR 168,000, improving by 20% the solution provided by the MD4-100. After obtaining the optimal solution, to evaluate the system robustness, we propose a discrete event simulation model incorporating uncertain flight times taking into account the possibility of accelerated depletion of drones’ Lithium-Ion polymer (Li-Po) batteries. Overall, our research investigates how various factors—such as the number of drones in the fleet and the division of the perimeter into sectors—impact the reliability of the system. Using Scenario number 3, our tests demonstrate that under a risk of battery failures of 2.5% and three UAVs per station, the surveillance system reaches a global percentage of punctually patrolled sectors of 92.6% and limits the number of delayed sectors (the relay UAV reaches the perimeter slightly above the required time, but it positively re-establishes the cyclic pattern for patrolling) to only a 5.6%. Our findings provide valuable insights for designing more robust and cost-effective drone patrol systems capable of operating autonomously over large planning horizons.

1. Introduction

Persistent aerial surveillance around unmanned aerial systems (UAVs) is a routine operation in many Intelligence, Surveillance, and Reconnaissance (ISR) missions. Examples of applications include both military and civil [1], such as detecting illegal ground activities [2], maritime surveillance [3], or the provision of coverage and connectivity to ground users and first responders in emergencies [4]. Typically, such applications are aimed at ensuring that the maximum amount of information is gathered, often over a large area and long time.

The criticality of the ISR task sometimes justifies the use of complex systems, such as those proposed for the US–México border [5,6]. Ref. [5] considers the deployment of an electrification line that supports small multi-rotor drones capable of wirelessly recharging their batteries. Ref. [6] reports the use of sophisticated long-range, big-wingspan UAVs (Predator B) to prevent drug interdiction. More often, the novel UAV-based security routines use commercial-off-the-shelf platforms, as it occurs in the protection of railways or transport hubs to early detect and limit incidents that otherwise would adversely impact the integrity and availability of these assets [7]. Instead of deploying security cameras around areas to be covered and security personnel patrols, authorities can plan real-time monitoring of various sites, taking advantage of the mobility and cost-effectiveness of drones.

In this paper, we deal with the practical design of a stand-alone persistent drone-based surveillance system for continually monitoring the boundary of a secured area. Our assumption about the drones to be used does not suppose any complex UAV platform: it is compatible with today’s professional drones with features such as waypoint navigation and the ability to follow trajectories using the Global Positioning System (GPS). However, specific assets where drones land and automatically recharge without any human intervention are desirable in the mission area to ensure full automation of UAV operations for an indefinite time. The option of using battery-swapping platforms that automatically swap the battery from the drone with a fresh one—e.g., [8]—is costly and further requires great maintenance efforts that can be justified in other contexts, e.g., the last-mile delivery [9] or the material handling operations for production [10], but not in the service under study. Instead, we propose on-site drone recharging stations based on wireless replenishment technology—several conceptions appeared in recent years [11,12]—thereby allowing us to design a cheaper and more flexible and robust persistent surveillance system.

Specifically, in this paper, we consider a vast circular perimeter surrounding a critical area to be persistently patrolled by a fleet of homogeneous UAVs working in automatic mode. We aim to conceptualize this patrolling service according to a list of practical considerations. First, we assume that communication transceivers are installed at the same locations wherein the charging stations are installed. Thus, the deployed base stations also act as service stations for transmitting the live video from the UAVs to an on-ground control room located at the centre of the area. There are practical reasons to avoid their placement too close to the perimeter: (i) avoid the need to extend the power line required to feed the charging ground stations too far from the control room and (ii) limit the risk of vandalism when setting the base stations too close to the border. Second, our collection of data is not only serving the purpose of providing the security staff with the real-time visualization of the received images gathered by the guarding UAVs, for instance, using a first-person view system as in [13], but also of benefiting from the possibilities of current photogrammetry techniques on the images captured by the UAVs.

We propose a mixed integer non-linear programming (MINLP) model for selecting the best UAV platform from a list of candidate commercial UAVs and determining the cyclic persistent schedule of flight missions—with non-coincident start (take-off) and end (landing) locations—to minimize the total deployment cost. Several design decisions are involved, including the number of base stations and their placement and the perimeter segment to be patrolled by each UAV.

For each of the candidate drone platforms, the design should specify the perpetual cycle (taking off from a base station, flying, monitoring, landing at a different base station than that from which it took off, recharging, and awaiting to be launched again) for an unknown number of UAVs. For example, Figure 1 depicts a possible cyclic schedule design with eight base stations: the circular perimeter is divided into eight sectors, and sixteen drones (every arrow) are required to fly at the same time, that is, at any time we have sixteen drones moving according to synchronized routes that together satisfy the above-listed practical constraints. In Figure 1, we outline the cyclic scheduling of drone flights showing the planned motion planning of the swarm. The drone $i\in \{1,\dots ,8\}$ starts its route (a) and reaches the perimeter (b). In (c), we present the situation when it has just finished the first sector. In (d), we present the moment in which a second subset of drones $\{9...16\}$ are taking off to replace the drones $i\in \{1,\dots ,8\}$ which are then approaching the end of their second sector. Thus, in (e), this relay is effective, whereas in (f) we present the landing of the exhausted drones at the charging stations. Notice that the second subset is now in charge of the surveillance task. In this example, each route consists of a flight mission servicing two of the eight sections into which the circular perimeter is split.

Crucially, the solution arising from the MINLP resolution must be tested and validated before proceeding to its real-life implementation, mainly due to the uncertainty of drone flight times that frequently appear in reality (as a consequence of the possible rapid depletion of their Li-Po batteries [14]). To this aim, we developed a discrete-event simulation (DES) model that considers the power drained of every UAV as a function of its flight velocity. The DES introduces random occurrences of accelerated depletion of batteries to check the extent to which the service from a certain fleet size moving according to a given cyclic pattern is robust.

The remainder of the paper is as follows. Section 2 reviews the literature to position this research. Section 3 outlines the main contributions of this research. Section 4 presents the description of our problem. The compact formulation of the multiplatform model is presented in Section 5. In Section 6, we propose an algorithm to obtain the optimal solution for a specific platform. Section 7 outlines a DES approach to find the fleet required to achieve a robust performance of our circular perimeter patrolling system. Section 8 contains a set of experiments to assess the validity of the proposed approach. First, we solve the multiplatform MINLP model and perform a sensitivity analysis on several parameters. Then, we apply the algorithm to obtain the optimal solution for every specific platform. Finally, we analyze the robustness of the patrolling schemes around the DES. Section 9 presents the conclusions and further research lines.

2. Literature Review

In this section, we first discuss routing papers relevant to the optimization of cyclic drone schedules and then we analyze several contributions to perimeter surveillance. Later, we review the literature on UAV lifetime of electrical batteries. Finally, we discuss the positioning of our research concerning the revised literature.

The need to periodically monitor selected targets (in our case, points on the circular trajectory) could be seen like something similar to a ‘time window’, which may suggest a relationship with vehicle routing problems with time windows (VRP-TW). Moreover, similar to what is achieved in the Green-VRP and the Electric-VRP when addressing the optimal routing for a fleet of limited driving range vehicles, we must pay attention to scheduling the visits to replenishment stations located within our circular area. However, important differences arise with both of them. (i) We do not consider that travel times are affected by the road infrastructure; that is, direct flights are always possible between any two points. Moreover, every point in the perimeter has to be visited periodically. Also, (ii) we do not consider just the decision of locating or not a recharging station in a set of candidate fixed positions, but consider the chance of locating the recharging station in the continuous space.

The strategy of transiting through intermediate facilities other than the one the vehicle starts has been previously applied to gain flexibility both in classical ground delivery problems ([15,16,17]) and in the context of drone-based delivery problems [18,19,20,21,22,23,24,25]. Ref. [18] proposed a distributed system of autonomous UAVs that self-coordinated and cooperated to ensure spatial and temporal coverage of specific points of interest. Ref. [19] addressed a continuous monitoring problem with inter-depot routes and priorities to account for supporting sites wherein drones can renew their energy. Ref. [20] addressed a last-mile delivery problem in a large-scale urban area (Phoenix, AZ, USA) to maximize the coverage of customers. Ref. [23] proposed first finding a Hamiltonian tour ignoring the recharging needs of drones and then applying a learning-based heuristic for inserting recharging events, thus replanning a new route. Similarly to the work in [20], refs. [9,24] addressed the problem of extending the drone operating range from a network design perspective in which UAVs can recharge their batteries on their journey to their final destinations using suitably located charging stations. Unlike these contributions, in our research, we need the opportunity to freely place the static installation of charging stations based on open contact pads, where every UAV with low-level batteries lands and automatically recharges without any human intervention.

Since the goal in our design approach consists of defining a persistent continuous (non-stopping) monitoring system with recharging events and periodic visits to points, our case study could be viewed as a continuous-location periodic drone scheduling problem, where the period corresponds to the time needed for the full coverage of the set of points. Our surveillance design is based on the synchronous take-off of a set of UAVs from a variety of depots (base stations) so that those UAVs reach the perimeter at the same time, patrol it, and leave it to ultimately land at any other of the depots again at once.

2.1. Routing Approaches for ISR Missions

Prior ISR work has applied routing approaches to set up a continuous monitoring system on a fleet of UAVs. Ref. [26] recently considered the problem of replacing security UAVs performing a surveillance mission along a perimeter. The authors determined which UAVs to replace and when while minimizing the effect this replacement has on the chances of observing an intruder attempting to penetrate the perimeter. Ref. [27] minimized the number of UAVs to be used when the replacement point was a single depot from which the UAVs were launched, while [28] did it around a chosen number of multiple replacement stations. Differently from the former studies, ref. [29] dealt with a cyclic schedule of missions on a set of points of interest so that every point is visited by some UAV at certain intervals, according to an every-point-dependent deadline. Ref. [30] focused on finding the number of UAVs required and their cyclic routes, assuming they start and end at the same vertex and can visit vertices multiple times. Conversely, our surveillance system takes advantage of scheduling cyclic flight missions with non-coincident start (take-off) and end (landing) points (base stations acting both as charging stations for the replenishment of the UAVs batteries and as parking receptacles where UAVs await to be launched).

Some works have reported the use of mobile units to charge everywhere without taking into account the number of locations that are finally located, which further allows the UAVs to be charged while being transported to another location. Ref. [31] considered mobile charging vehicles to support long-scale long-term ISR military missions within a battlefield environment. Ref. [32] considered multiple battery-swapping mobile stations to support ISR tasks for a team of UAVs whose trajectories are given within a planning horizon, and recharging may occur only within specific time windows. In the ISR context, the location of static stations is typically limited to a short list of specific sites [33,34], which is also common in the Location-Routing Problems (LRPs). While this feature makes this work closely related to our work, the assumption of base station placement limited to a discretized candidate set is not valid in our application, since we need to locate the recharging station at any point within a circle of a certain radius.

2.2. Perimeter Surveillance

We review related work on persistent monitoring, focusing on approaches closely aligned with our scheduling of periodic photogrammetry patrol missions along a large perimeter at a fixed, typically low, UAV velocity. To enforce an inter-vehicle safety time between adjacent patrol segments, we adopt a unidirectional patrolling scheme, aligning our problem with the unidirectional circular patrolling model in [35]. Given a fixed patrolling speed and circular radius, we follow [35] to determine the required number of UAVs by partitioning the perimeter into equal-length segments based on the desired patrol frequency. In contrast, ref. [36] addresses surveillance planning at unitary speed for a fixed number of robots, aiming to minimize maximum weighted idleness, while [37] focuses on minimizing target revisit time in a graph-based environment. The authors consider finite patrolling cycles constrained by battery limits, where each UAV returns to a replenishment station located at one of the targets. Our approach diverges from these works in two key aspects: (i) we do not fix the fleet size a priori, instead examining the trade-off between minimizing latency and fleet size, and (ii) we consider service stations that are remote and not located at target locations—an essential constraint absent in the aforementioned studies.

To some extent, the work in [38] is the most related to our case study. The authors address the optimal placement of UAV refuelling stations for persistently monitoring the boundary of an island modelled as a simple polygon with a discrete number of vertices. The authors propose an algorithm that achieves an optimal solution under the assumption that base stations are placed at points within the intersection of the polygonal boundary and its convex hull. However, despite being able to cope with more general perimeters, they do so under the assumption that the flight route for each UAV consists of a polygonal path between two consecutive refuelling stations, thereby implicitly using a simple replacement policy at them.

2.3. Li-Po Batteries and Flight Time

The effective discharge duration of Li-Po batteries is influenced by various factors, including environmental temperature and the number of charge–discharge cycles. Experimental studies at different constant power rates, such as [39], have shown that the linear energy–power relationship described by Peukert’s law—originally developed for lead–acid batteries—does not hold under high constant power loads that exceed the cells’ rated specifications.

The revised applications of UAVs have neglected this dynamic behavior. The classical power consumption models for electrical UAVs combine the basic equations of flight dynamics and translate them into the electrical domain, based mostly on the specifications given by cell manufacturers along with the precise knowledge of the payload. However, according to [40], ‘…estimates of flight distances and energy consumption are likely to be inaccurate when based on flight times and speeds reported by manufacturers that are either maximum values or from unknown operating conditions’. Ref. [40] identifies key factors that affect drone energy consumption and discuss similarities and differences among various models. For cruising flight, drone power consumption can be modelled as a convex function of the drone’s total weight, e.g., [41,42], while for hovering it is proportional, weight to power, as 1.5 [43]. Notice that in our case study, no delivery is involved (no effect of weight changes needs to be incorporated) and consumptions are mainly due to cruising horizontal flights.

Among previous research considering electrical power consumption as a function of the UAV cruise velocity, we can refer to works  [40,44,45,46]. Ref. [44] approximated the power consumption of such flights as a linear function of velocity once the lift-to-drag ratio and the efficiency of power transfer were fixed. Ref. [45] also analyzed the profile of power as a function of velocity, further claiming that there are two practical velocities of interest: that providing maximal endurance (or time) and that providing maximal range (or distance). Ref. [46] followed the non-linear power consumption model of [45] to decide the speed of the UAV in a drone delivery problem. Ref. [40] studied the propeller power consumption as a function of the UAV speed. In addition, since it is well known that for maximizing Li-Po battery lifetime and efficiency, it is not recommended that the depth of discharge exceeds 70–80%. There are several practical approaches to the energy-aware UAV flight modeling that propose using safety factors when planning real-life UAV missions: ref. [47] assumed 80% of the energy; ref. [42] considered a safety factor of 1.2 to reserve energy for unusual conditions (which is equivalent to using 83.3% of maximum flight distance); ref. [48] assumed that the drone flight range was only 70% of the maximum flight distance to account for unknown factors (e.g., bad weather), thus applying a safety margin to accommodate an eventual increase in energy consumption.

As a first approximation to incorporate a per-metre flight energy consumption into UAV patrol routes, we consider the linear approach in [44]. Then, the proposed MINLP model considers a power consumption that is linear with the flight velocity and further assumes that the depth of discharge of batteries never exceeds an 80% level. As explained in Section 7, we use the more detailed non-linear power consumption model of [45] in a second stage when analyzing the robustness of the system before finally defining the definitive system design.

3. Research Positioning Concerning the Literature

The literature review demonstrates that existing studies, to the best of our knowledge, are, to some extent, different from the problem addressed here.

1.

Our problem can be viewed as a particular case of the Fence Patrolling Problem first introduced by the authors of [35], who studied a circle patrolling problem where agents are allowed to move in both directions, i.e., bidirectional policy or not. The implicit assumption in [35] is that the patrolling tours end at service stations that are co-located with any of the target locations, which is usually considered in persistent monitoring approaches [37,38], except for some recent articles [49,50]. However, while both propose the use of only one service station that is different from the subset of target points that must be visited, in our case, we are forced to deploy a variety of them to guarantee that the spanned time between consecutive visits to every point in the perimeter is below a maximum value.

2.

In our case, several decisions must be made, including selecting the number of base stations, the circular trajectory split, and the standard drone product among a set of commercial UAV platforms according to the specific data that the different manufacturers publish in their data sheets.

3.

Moreover, we must plan the take-off of a replacement UAV to let it reach the perimeter at the time a weary UAV leaves and flies to a base station to obtain fresh energy, acting as a relay UAV to maintain the revisit time to any perimeter point.

4.

Noticeably, the trajectories of leaving and replacing UAVs are different, a policy we apply to ensure inter-UAV safety distances to avoid potential collisions.

Autonomous drone surveillance systems, as the one proposed in this research, capable of perpetual, unattended operation through automated charging stations, offer significant advantages over traditional human-attended drone deployments. These systems enable continuous, high-frequency monitoring without the limitations of operator availability or fatigue, thereby enhancing situational awareness and reducing response time to security incidents. In contrast, human-attended drones require manual operation for launch, navigation, and landing, leading to higher labor costs and less consistent coverage. Furthermore, autonomous systems standardize flight paths and data collection, facilitating reliable long-term monitoring and anomaly detection. While initial setup costs may be higher due to the need for docking infrastructure and advanced control software, the long-term operational savings and scalability make these systems particularly well-suited for critical infrastructure and large-scale perimeter security. Ultimately, the shift toward fully autonomous drone surveillance reflects a broader trend in leveraging automation to improve efficiency, safety, and reliability in security operations. Our research contributes to this field by providing valuable insights for designing more robust and cost-effective patrol systems capable of operating autonomously over large planning horizons.

The main contributions of this research are:

–

First, we develop an original MINLP model for defining the movements of the UAVs as cyclic schedules of flight missions—with non-coincident start (take-off) and end (landing) locations—to cyclically visit an unknown number of sectors in which the perimeter will be split. The objective function aims at minimizing the system deployment cost.

–

Second, due to the non-linear and non-convex nature of the model, state-of-the-art non-linear solvers cannot guarantee global optimality. Thus, we propose a platform-dependent algorithm to obtain the optimal patrolling scheme.

–

We assume the use of vertical take-off and landing (VTOL) drone platforms with brushless electromotors powered by rechargeable Li-Po batteries, which are lighter weight and have higher capacity than their counterparts [51]. It is worth commenting that such vehicles can travel at a constant speed between a pair of locations, even though windy conditions may impact the cruise speed. This aspect introduces uncertainty in the battery lifetime, which is, in fact, a major concern for the robustness of system operation. To deal with this issue, we develop a DES model that includes the drained power as a function of flight velocity (fixed by the MINLP solution) and considers random occurrences of accelerated depletion of batteries. The DES model allows us to measure the robustness of the system against uncertainty.

4. Problem Description

In this section, we first derive the equations that, for a given drone platform and the practical considerations on its communications equipment, velocity range, and endurance constraints, relate (i) the number of sectors in which the circular perimeter must be split, (ii) the placement of the base stations, and (iii) the number of drones required. We then extend this reasoning to the case where the selection of a commercial UAV platform from a set of candidate platforms is considered.

4.1. Definition of the Mobility Pattern on a Given Drone Platform

We consider a circular perimeter of radius R containing the critical assets, mainly concentrated in the centre of the area. Note that, according to Figure 2, a circular trajectory allows for monitoring not only circular areas but also irregular areas that can be covered by a certain circular crown as in (a) and (b) depending on the UAV camera vision angle and the cruise height, or even more irregular shapes if the drone is equipped with rotating lens, as in (c), which allows for more general perimeters monitoring. The UAV flight is set at a constant height h, arising from the specifications of the on-board camera and the required footprint on the terrain. A typical requirement from the photogrammetry procedure concerns the percentage of interleaving needed between successive perceived images that, along with the constant height h, determines a fixed velocity $V^{patrol}$ at which the UAV should follow the perimeter. In our case, this value is set to $V^{patrol}=2$ m/s (note that the patrol velocity depends on several factors regarding the speed of communications and the quantity and quality of data to be transmitted to the control center. Moreover, the usage of photogrammetry techniques to reconstruct objects from images requires a patrol velocity related to the camera shot interval and the UAV altitude. A patrol velocity of 2 m/s ensures the adequate overlap between consecutive photos for a shot interval of 2 s). In contrast, the flight speed on moving to or returning from the perimeter $V^{cruise}$ is a decision variable. Figure 3 illustrates a single circular flight route of radius R for one of the guarding UAVs, which patrols a portion of the perimeter, flying at a constant height h, and continuously sending the video surveillance images to a base station settled at a radius $r\le r_{max}$.

Individual UAVs have a restricted communication range and a maximal flight time, both limiting their effectiveness. According to the illustrative example in Figure 1, a swarm of drones flies at the same time, so major attention is needed to precisely coordinate their behavior. In Figure 4, we present one such flight mission, where a UAV (a) takes off from a base station (point $Q_o$), (b) approaches the fence in an oblique flight (segment $Q_o-P_1$), as a practical way of avoiding collisions with a returning UAV following the shorter radial trajectory, (c) patrols the perimeter ($P_1-P_2$), and (d) returns to land at a different base station before its maximal flight time or endurance E ($P_2-Q_2$). As depicted in Figure 4, along a typical drone flight, communication control is handed over successive base stations—e.g., in the example in Figure 4, the base station at $Q_0$ passes the control to the one at $Q_1$ once the UAV is at point $P_1$. On the assumption that the base stations are equally distributed under the inner circle of radius r, we need to ensure that the maximal range for the in-use communications $d_{max}$ is not exceeded. We next reduce our analysis on the achievement of this communication constraint to a 2D study, since we are interested in relatively large scenarios where the circle radius is of the order of thousands of meters, while the value of the height h is quite small (practical values of $h=30$ m). To determine the patrolling pattern of a given platform, it is necessary to specify the number of sectors in which the circle area must be divided. As shown in Figure 4, the width of each sector is given by the angle $\delta$, which is related to three main design characteristics: (i) the maximum length of the communication segment and the radius of the circumference where the base stations are located, (ii) the required time between revisits at any perimeter point, and (iii) the UAVs’ battery endurance. The next two sections devise the equations that relate these variables.

4.2. Range of Communications and Sectorization

First, we can relate the range of communications and the angle of each sector in which we split the circle, $\delta$, using the following trigonometry analysis. In Figure 4, a full drone’s flight comprising the segments $Q_0-P_1$, $P_1-P_2$ and $P_2-Q_2$ serves the purpose of relating the length of the communication segment, denoted as L, and the angle $\beta$ where the handover from a base station to the next one occurs.

We can solve the triangle $O-Q_0-P_1$ to obtain:

$$r^2=L^2+R^2-2LRcos\beta .$$

(1)

Since the length of the segment $Q_1-P_2$ is similar to that of the segment $Q_0-P_1$, we can apply again the cosine theorem to resolve the triangle $Q_0-Q_1-P_1$ to express that length as

$${\left(S^{Q_0{Q}_1}\right)}^2={(R-r)}^2+L^2-2(R-r)Lcos\beta .$$

(2)

We can substitute the term $2Lcos\beta$ deduced from Equation (1) into Equation (2) to obtain

$${\left(S^{Q_0{Q}_1}\right)}^2={(R-r)}^2+L^2-(R-r)[(L^2+R^2-r^2)/R].$$

(3)

and from it, we can simplify to obtain the following expression for the distance till the hand-over point projection on the inner circle:

$${\left(S^{Q_0{Q}_1}\right)}^2=r/R[L^2-{(R-r)}^2].$$

(3’)

However, we can compute the same Euclidean distance according to the angle $\delta$ as $S^{Q_0{Q}_1}=2rsin\left({\displaystyle \frac{\delta }{2}}\right)$ which, once substituted in the former, leads to

$$L^2=4Rrsi{n}^2\left({\displaystyle \frac{\delta }{2}}\right)+{(R-r)}^2.$$

(4)

We finally can obtain the length of this communications link at the maximal range of the access link from the base stations, $d_{max}$, as follows:

$$L\le d_{max}.$$

(5)

Equation (4) and Constraint (5) are used later in the MINLP formulation to relate the angle defining a sector, the maximum communication length, and the radius of the circumference where base stations must be located. For practical reasons, a bound on the value of the latter is also included.

4.3. Revisit Time and Sectorization

As mentioned above, the width of each sector is also related to the maximum allowed revisit time. Since the goal is to patrol the circular perimeter so that the period between the revisits is relatively short, the revisit time $T^r$ must be lower than a maximum value, denoted as $MRT$.

$$T^r\le MRT.$$

(6)

For the sake of usability, the whole perimeter is split into an integer number of sectors, S, as follows:

$$S\delta =2\pi .$$

(7)

The whole perimeter comprises $2\pi R$ meters to be patrolled at a fixed velocity $V^{patrol}$, from which the time $T^r$ and the angle of each sector $\delta$ is related as follows:

$$T^r={\displaystyle \frac{R\delta }{V^{patrol}}}.$$

(8)

Constraint (6) along with Equations (7) and (8) adjusts the angle according to an integer splitting of the circle while setting the revisit time at a feasible value. Notice that both decision variables are platform specific, and hence Equations (6)–(8) need to be incorporated separately for each considered platform into the MINLP model proposed in Section 5.

4.4. Endurance and Number of Sectors Covered by a Drone’s Flight

In addition to deciding the number of base stations (one for every sector, i.e., S), we must establish the number of sectors n that are patrolled by every drone’s flight. For example, in Figure 5, we illustrate a case in which $n=2$ sectors are served by any single UAV mission flight. Deciding on the value of n is highly dependent on the commercial UAV platforms that we consider as candidates to compose our fleet. To begin with, the manufacturers publish an endurance or maximal flight time, E, that we can relate to the number of sectors served by a single UAV flight, n, and the revisit time, $T^r$, so that it allows the realization of a complete flight, starting at a base station, completing n sectors and ending in a different base station:

$${\displaystyle \frac{(R-r)}{V^{cruise}}}+n{T}^r+{\displaystyle \frac{L}{V^{cruise}}}\le E.$$

(9)

Similarly, the manufacturer typically indicates the range of valid speeds:

$$MinSpeed\le V^{cruise}\le MaxSpeed.$$

(10)

These two constraints refer to data commonly spread by the platform manufacturer and are later incorporated separately for each one of the platforms into the multiplatform MINLP model. Note that the number of sectors covered by each drone is closely related to the angle of each sector.

4.5. Safety Margin on the UAV Endurance and the Number of Sectors Covered by a Drone’s Flight

In addition to considering the duration provided by the manufacturer, a more realistic analysis advises modeling the drone battery discharge process in more detail. The overwhelming majority of commercial drones available today employ rechargeable Li-Po batteries composed of cells with a nominal voltage of $2.4$ V, which are typically connected in series to attain a higher voltage. Importantly, as mentioned in Section 4.4, when the depth of discharge exceeds a threshold, continuing to use the Li-Po battery not only leads to its rapid depletion but also compromises the battery’s lifetime. Next, we detail the use of a drone energy consumption model to determine the value of n where, applying this conservative approach, a safety threshold of 80% is considered.

Notice that in Equation (9) we do not consider either take-off or landing times, since the energy consumption involved in both manoeuvres is negligible compared to that of the long runs of horizontal flight. We follow [44] to calculate the energy per meter in steady-level flight considering that the fundamental forces are thrust to move forward, the weight of gravity action on the UAV mass, and lift and drag. Therefore, when assuming a flight at a constant altitude and constant speed, the energy per meter can be expressed in terms of the lift-to-drag (a constant for steady flight), $LtD$, the mass of the UAV (summing up the frame mass $m^{frame}$ and the payload mass $m^{payload}$, the later corresponding to the weight of the camera), the percentage of battery energy that we are effectively transferring to motors and propellers, and the flight speed.

There are two modalities of steady-level flight in our surveillance application: one at flight speed $V^{cruise}$ and another at flight speed $V^{patrol}$. The power consumption at each one can be obtained by multiplying the energy per meter by the corresponding constant velocity and adding the power consumption of avionics ($PO{W}^{Avio}$):

$$PO{W}^{cruise}=3.6{\displaystyle \frac{m^{frame}+m^{payload}}{370\Omega LtD}}{V}^{cruise}+PO{W}^{Avio},$$

(11)

$$PO{W}^{patrol}=3.6{\displaystyle \frac{m^{frame}+m^{payload}}{370\Omega LtD}}{V}^{patrol}+PO{W}^{Avio}.$$

(12)

Observe that the number $3.6$ corresponds to the conversion factor applied to our velocity in m/s to convert it into km/h as considered in [44]. We recall here that the units in use are as follows: kg for masses ($m^{frame}$, $m^{payload}$), $m/s$ for velocities ($V^{cruise}$, $V^{patrol}$), and $kW$ for the powers ($PO{W}^{cruise}$, $PO{W}^{Avio}$, $PO{W}^{patrol}$).

As for the total battery energy, we start from the usual data on capacity and voltage that the manufacturer provides to establish it as $BatteryEnergy\left(inkJ\right)=3.6·Cap\left(inAh\right)·Voltage\left(inV\right)$. To avoid the work of batteries within their accelerated depletion range, the number of kJ to plan on every flight is limited to a battery energy bound, denoted as $BEB$, that includes a safety threshold of 80%. Then,

$$BEB=0.8·3.6·Cap·Voltage.$$

(13)

This energy ($BEB$) should be enough to cover the total energy consumed along the mission. Then, we obtain a new constraint connecting the number of patrolled sectors and the battery load, which, together with Constraints (11), (12), and (13), is incorporated into the multiplatform MINLP model proposed in Section 5.

$${\displaystyle \frac{LPO{W}^{cruise}}{V^{cruise}}}+n{T}^{r}PO{W}^{patrol}+{\displaystyle \frac{(R-r)PO{W}^{cruise}}{V^{cruise}}}\le BEB.$$

(14)

4.6. Fleet Size on Assuming a Constant Number of UAVs per Base Station

From a practical perspective, perhaps the most important expected result of the patrolling system is the number of drones required to perform the surveillance mission. As we explain, the whole number of drones w can be expressed as a function of the distance from the take-off point to the perimeter (L), the period between two successive visits to the same point at the perimeter ($T^r$), the number of sectors which each UAV guards on every single flight (n), the recharging time at the ground base stations ($ChT$), and the number of sectors (S). We explain the reasoning based on the example depicted in Figure 6 as follows. Let us inspect what happens in the interval a drone arrives at the perimeter at point $P_{(i+1)}$, and it is back to revisit this point: it patrols n segments (out of the S in which we split the perimeter) in a $n·T^r$ unit of time, then flies a radial distance $(R-r)$ to reach a ground recharging point, then stays charging during $ChT$ seconds, and later takes off and travels along L meters to reach again the perimeter at point $P_{(i+n+2)}$. Therefore, the number of UAVs necessary to persistently guard each of these n sectors can be computed as follows.

Since in tracking a particular UAV the time between two consecutive take-offs is $n{T}^r+(R-r)/V^{cruise}+ChT+L/V^{cruise}$, we determine how many drones are needed to guard the n sectors involved—see Figure 6—if we divide this into the revisiting time established for any sector $T^r$. However, notice that typically the flight mission of this particular UAV is only able to cover a part of the whole circular perimeter ($n<S$), and therefore, we have to multiply by factor $S/n$ and round up to find an integer fleet size capable of persistently guarding the whole perimeter:

$$w=⌈{\displaystyle \frac{S}{n}}{\displaystyle \frac{\frac{L}{V^{cruise}}+\frac{(R-r)}{V^{cruise}}+n{T}^r+ChT}{T^r}}⌉.$$

(15)

Finally, if we assume that to produce a symmetrical solution there is a preference for having the same number of drones assigned to each base station, we obtain a slightly different fleet size $w^s$ as the one to be ultimately used in our system:

$$w^s=S⌈1+{\displaystyle \frac{\frac{L}{V^{cruise}}+\frac{(R-r)}{V^{cruise}}+n{T}^r+ChT}{T^r}}⌉.$$

(17)

For example, the illustration shown in Figure 5 considers that $ChT=5600$ s and $E=3000$ s, from which Equation (9) along with Equation (8) leads to a fleet of $w^s=36$ (different to the minimal required fleet, which was $w=32$). Therefore, our symmetric pattern for this persistent surveillance consists of splitting the circular area into $S=9$ sectors, served with nine base stations, in each of which we place 4 UAVs to continuously monitor the perimeter with a successive pass for any point of $T^r=1182$ s, each UAV’s flight covering $n=2$ sectors. Equation (16) is considered in the next section as part of the multiplatform MINLP model.

5. Compact Formulation Extended to a Variety of Drone Platforms

To decide among a variety of commercial drone platforms $p\in P=\{1,\cdots ,|P\left|\right\}$, a binary selection variable ${\alpha }_p$ is introduced for selecting the most appropriate one, along with the rest of the variables defining the surveillance system. Therefore, the decision variables previously described for a generic platform are now transformed to account for each of the considered platforms by incorporating the subindex p, ($w_p^s$, $r_p$, $T_p^r$, $L_p$, $S_p$, $V_p^{cruise}$, $PO{W}_p^{cruise}$, $PO{W}_p^{patrol}$); see

Table 1. Notation.

Sets
P	The Set of Commercial Drone Platforms Under Consideration.
Parameters
R	The radius of the circular perimeter, measured in meters.
$MRT$	Maximal span time between revisiting every point in the perimeter, measured in seconds.
$V^{patrol}$	The patrolling velocity, fixed at 2 m/s.
$r_{max}$	The upper bound to the radius for placement of the base stations, expressed in meters.
$v_{max}$	The maximal range for communications for a flight mission, expressed in meters.
$ChT$	The charging time required to warranty the proper replenishment of Li-Po batteries, expressed in seconds.
c	Purchasing cost for the charging pad, fixed at EUR 8000.
$c_p$	Purchasing cost for the platform $p\in P$, measured in EUR$/unit$.
$PO{W}^{Avio}$	Power consumption due to the camera on board, no matter which platform $p\in P$, measured in $kW$.
$E_p$	Maximal flight time or endurance for the platform $p\in P$, measured in seconds.
$m_p^{frame}$	The mass of the frame of the commercial platform $p\in P$, measured in $kg$.
$m^{payload}$	The mass of the payload (other than the common camera) installed on the commercial platform $p\in P$, measured in $kg$.
${\Omega }_p$	Efficiency in the transference of electrical energy to motors and propeller, according to the specifications of platform $p\in P$. It stands for ratio ($0\le {\Omega }_p\le 1)$ of the power drawn at batteries that are effectively converted into power used by the propeller.
$Lt{D}_p$	The lift-to-drag ratio combines in a single parameter the aerodynamic and drone design aspects for each commercial platform $p\in P$.
$BE{B}_p$	The limit to the battery energy drained on every flight mission, depending on the features of the platform $p\in P$. The bounded number of $kJ$ to consider is limited to a value $BE{B}_p=80\%3.6Ca{p}_{p}Voltag{e}_p$, where $Ca{p}_p$ is the capacity in $Ah$ and $Voltag{e}_p$ is the voltage according to the specifications for the Li-Po batteries on board the commercial platform p.
Variables
${\alpha }_p$	Binary variable. Takes value 1 if the platform $p\in P$ is selected.
$S_p$	Integer number of segments in which we divide the circular perimeter, if the platform $p\in P$ is selected.
$w_p^s$	Integer number of drones in the fleet for a symmetric pattern for persistent surveillance—i.e., with an equal number of drones at every base station—if the platform $p\in P$ is selected.
$n_p$	Integer number of segments that a platform patrols, if the platform $p\in P$ is selected. For each platform $n_p$ is selected. For each platform $[1,S_p]$.
${\delta }_p$	The angle of each sector (measured in radians) in which we split the circle, if the platform $p\in P$ is selected.
$r_p$	The radius of the inner circle where we place the $S_p$ equally distributed base stations, expressed in meters.
$L_p$	The distance from the source base station to the points where the control of the flight mission of platform is transferred to the next base station (measured in meters) if the platform $p\in P$ is selected.
$T_p^r$	The revisit time (measured in seconds), namely the time elapsed for patrolling every sector on platforms of type $p\in P$, if the platform p is selected.
$V_p^{cruise}$	The velocity (measured in meters per second) that platform $p\in P$ uses on moving to or returning from the perimeter, if the platform p is selected.
$PO{W}_p^{cruise}$	The power consumption (measured in kW) considered from steady level flight at velocity $p\in P$, if the platform p is selected.

for notation.

The MINLP model to cope with such decisions is as follows:

$$\begin{array}{ccc}\hfill & \left(FO\right)\hfill & \hfill Min\sum _{p\in P}{\alpha }_p\left(c_p{w}_p^s+c{S}_p\right)\end{array}$$

(17)

Subject to

$$\begin{array}{cccccc}\hfill & \hfill & \hfill r_p\le r_{max},& & \hfill p\in P& \hfill \end{array}$$

(18)

$$\begin{array}{cccccc}\hfill & \hfill & \hfill L_p^2=4R{r}_{p}si{n}^2\left({\delta }_p/2\right)+{(R-r_p)}^2,& \hfill & \hfill p\in P& \hfill \end{array}$$

(19)

$$\begin{array}{cccccc}\hfill & \hfill & \hfill L_p\le d_{max},& \hfill & \hfill p\in P& \hfill \end{array}$$

(20)

$$\begin{array}{cccccc}\hfill & \hfill & \hfill T_p^r\le MRT,& \hfill & \hfill p\in P& \hfill \end{array}$$

(21)

$$\begin{array}{cccccc}\hfill & \hfill & \hfill S_p{\delta }_p=2\pi ,& \hfill & \hfill p\in P& \hfill \end{array}$$

(22)

$$\begin{array}{cccccc}\hfill & \hfill & \hfill T_p^r={\delta }_{p}R/V^{patrol},& \hfill & \hfill p\in P& \hfill \end{array}$$

(23)

$$\begin{array}{cccccc}\hfill & \hfill & \hfill L_p/V_p^{cruise}+n_p{T}_p^r+(R-r_p)/V_p^{cruise}\le E_p,& \hfill & \hfill p\in P& \hfill \end{array}$$

(24)

$$\begin{array}{cccccc}\hfill & \hfill & \hfill PO{W}_p^{cruise}=3.6{\displaystyle \frac{\left(m_p^{frame}+m^{payload}\right)}{\left(370{\Omega }_{p}Lt{D}_p\right)}}{V}_p^{cruise}+PO{W}^{Avio},& \hfill & \hfill p\in P& \hfill \end{array}$$

(25)

$$\begin{array}{cccccc}\hfill & \hfill & \hfill PO{W}_p^{patrol}=3.6{\displaystyle \frac{\left(m_p^{frame}+m^{payload}\right)}{\left(370{\Omega }_{p}Lt{D}_p\right)}}{V}^{patrol}+PO{W}^{Avio},& \hfill & \hfill p\in P& \hfill \end{array}$$

(26)

$$\begin{array}{cccccc}\hfill & \hfill & \hfill PO{W}_p^{patrol}\left({\displaystyle \frac{L_p}{V_p^{cruise}}}+n_p{T}_p^r+{\displaystyle \frac{R-r_p}{V_p^{cruise}}}\right)\le BEB,& \hfill & \hfill p\in P& \hfill \end{array}$$

(27)

$$\begin{array}{cccccc}\hfill & \hfill & \hfill w_p^s=S_p⌈1+{\displaystyle \frac{\frac{L}{V_p^{cruise}}+\frac{(R-r_p)}{V_p^{cruise}}+n_p{T}_p^r+Ch{T}_p}{T_p^r}}⌉,& \hfill & \hfill p\in P& \hfill \end{array}$$

(28)

$$\begin{array}{cccccc}\hfill & \hfill & \hfill \sum _{p\in P}{\alpha }_p=1,& \hfill & & \hfill \end{array}$$

(29)

$$\begin{array}{cccccc}\hfill & \hfill & \hfill SpeedMi{n}_p\le V_p^{cruise}\le SpeedMa{x}_p,& \hfill & \hfill p\in P& \hfill \end{array}$$

(30)

$$\begin{array}{cccccc}\hfill & \hfill & \hfill r_p,T_p^r,L_p,S_p,V_p^{cruise},PO{W}_p^{cruise},PO{W}_p^{patrol}\ge 0,& \hfill & \hfill p\in P& \hfill \end{array}$$

(31)

$$\begin{array}{cccccc}\hfill & \hfill & \hfill {\alpha }_p\in \{0,1\},n_p,w_p^s\in {\mathcal{N}}^{+},& \hfill & \hfill p\in P.& \hfill \end{array}$$

(32)

Note that from a resolution point of view, the above formulation allows us to decompose the problem into a set of sub-problems, each one corresponding to a specific platform, i.e., the model could be solved one by one for each one of the drone platforms. However, since the number of variables and constraints of the complete formulation is small, it is not strictly necessary to perform this decomposition. It should be noted that along with the best design (that from the selected platform), the compact model resolution also determines a feasible design for each of the considered platforms. Thinking of the convenience of performing continuous surveillance of the perimeter, one might think on an objective function focused exclusively on minimizing the revisit time. However, minimizing solely the revisit time would give rise to dividing the area into a very high number of sectors, and later assigning a large number of very small perimeter portions to each UAV. This naïve solution does not consider the economic implications of installing as many charging pads as sectors nor the cable installation itself from the control centre to recharging stations. Then, a more reasonable objective function (F$O_2$) aimed at minimizing the cost of the fleet size of UAVs, as well as the cost of charging pads (which is equivalent to the number of sectors), is proposed. The parameter $c_p$ accounts for the cost of each drone platform, while c represents the cost of a charging pad. Note that minimizing the total deployment cost of the system probably implies revisit times near the maximum allowed value ($MRT$). Beyond costs, there is a final check to be addressed before validating the design of the surveillance system: we need to ensure its robustness. For this purpose, the cyclic patrol scheduling that arises from solving the above optimization model needs to be further analyzed to assess the robustness performance for unexpected battery depletion.

6. An Algorithm to Devise the Optimal Solution for a Specific Platform

We present an algorithm to obtain the minimal cost design achievable for a specific drone platform. By applying this procedure platform by platform and comparing the solutions, we obtain the optimal solution of the multiplatform MINLP formulation described in Section 5. On considering a specific platform, we can avoid using the label ‘p’ in the variables listed in

Table 2. Input parameters for the Critical Scenarios to be addressed.

			Scn#1	Scn#2	Scn#3	Scn#4	Scn#5	Scn#6
Input Parameters	Radius	R (m)	1196	1496	1696	1696	1696	1696
	Maximal comms link	$d_{max}$ (m)	1444	1444	1444	1444	1444	900
	Far placement of base stations	$r_{max}$ (m)	900	1333	1333	1333	900	1333
	Patrolling velocity	$V^{patrol}$ (m/s)	2	2	2	2	2	2
	Patrolling recurrence	$MRT$ (s)	1222	1222	1222	1222	1222	1222
	Recharging time	$ChT$ (s)	4000	3600	4000	5600	5600	3600

. According to Expression (16), the required fleet size increases as the number of sectors increases. Moreover, the number of base stations also increases, since in our surveillance pattern there is a base station per sector.

The number of sectors can be obtained by dividing the circle by the angle $\delta$; the smaller the angle, the higher the number of sectors. As a consequence, we are interested in solutions with the least possible number of sectors, or equivalently with the highest possible value of $\delta$. After obtaining the minimum number of sectors and assigning sectors to drones, in certain circumstances, as we explain later, it is possible that an increment in the number of sectors in which the surveillance area is divided allows us to assign a different number of sectors to drones, taking better advantage of the endurance of each drone and thus reducing the value of the objective function. Then, by enlarging iteratively the number of sectors one by one, we can compute the objective function and stop the procedure when its value increases, thus obtaining the optimal surveillance pattern.

6.1. Obtaining the Maximum Value of $\delta$

The value of the angle of a sector is a consequence of the maximum allowed revisit time, $MRT$, and the maximum length of the communication segment, L. The shorter the revisit time, the higher the number of sectors dividing the circular area. Then, to achieve a sectorization with the minimum number of sectors, one of the two next conditions must hold.

1.

On one hand, if possible, the revisit time must be equal to the maximum allowed value, represented by $MRT$, which gives rise to a maximum angle ${\delta }_1=V^{patrol}MRT/R$.

2.

On the other hand, the angle is conditioned by $L=d_{max}$, which leads us to a different value ${\delta }_2$. In this case,

(a)

if $R-r_{max}\le d_{max}\le R$, the maximum angle ${\delta }_2$ is obtained when the base station is located at a distance $r_{max}$ from the circle centre—see Figure 7. Hence, the maximum value of $\delta$ is given by the minimum of ${\delta }_1$ and ${\delta }_2$.

(b)

However, for values of $d_{max}\ge R$, larger angles are obtained if the base station is located near the centre. Notice that if ${\delta }_2$ is lower than ${\delta }_1$, it is possible to keep the communications link when enlarging ${\delta }_2$ by moving the base station towards the circle centre until ${\delta }_2 = {\delta }_1$; see Figure 8.

Lemma 1.

In the case $R-r_{max}\le d_{max}\le R$, there is a natural bound on the minimal number of sectors derived from placing the base stations at a radius $r_{max}$ considering the ultimate angle as the minimum of ${\delta }_1$ and ${\delta }_2$, conveniently decremented, if needed, to ensure an integer number of sectors.

Lemma 2.

In the case $d_{max}\ge R$, there is a natural bound on the minimal number of sectors derived from considering ${\delta }_1=V^{patrol}MRT/R$. However, the ultimate angle could be different and smaller from accounting for the communications link condition when ${\delta }_2\le {\delta }_1$. Then, it is possible to enlarge ${\delta }_2$ by moving the base station towards the circle centre until ${\delta }_2$ is near to ${\delta }_1$. Such a tighter bound is denoted ${\widehat{\delta }}_2$ and leads to the obtention of a tightened value of the fleet size.

Summarizing, when $R-r_{max}\le d_{ma}x\le R$, (case 2.a), the maximum value of $\delta$ is given by the minimum of ${\delta }_1$ and ${\delta }_2$. Nevertheless, in the case $d_{max}\ge R$, (case 2.b), if ${\delta }_2$ is lower than ${\delta }_1$, it is possible to enlarge ${\delta }_2$ by moving the base station towards the circle centre until ${\delta }_2$=${\delta }_1$, so, in this case, somehow it is the $MRT$ condition which guides the sector partitioning. From fixing the angle $\delta$ we can obtain the number of sectors (S), the revisit time ($T^r$), the distance of the base stations to the centre of the area (r), and the effective value of the communication distance (L).

6.2. Determining the Number of Sectors Surveilled by Each Drone

After obtaining the number of sectors, the endurance and battery usage conditions—Constraints (9) to (14)—allow us to determine the number of sectors surveilled by each drone. Specifically, Constraint (9) provides a value $n_1$ and Constraint (14) a different value $n_2$. The minimum of these two values is the maximum number of sectors that can be surveilled by each drone, $n=min\{n_1,n_2\}$.

At this stage, we can compute the upper bound on the fleet size using Expression (16) and the value of the objective function $F{O}_2$.

6.3. Analyzing the Convenience of Increasing the Number of Sectors

After obtaining the patrolling scheme, to determine its optimality, it is necessary to essay the solution with a number of sectors equal to $S+1$, thus decreasing the value of $\delta$ from $2\pi /S$ to obtain ${\delta }^N=2\pi /(S+1)$. We repeat the calculation of the patrolling pattern, determining new values $T^{rN}=\delta R/V^{patrol}$, $r^N$ and $L^N$ as well as $n_1^N$ (9), $n_2^N$ (14), $n^N=min\{n_1,n_2\}$, and compute a new value for the objective function $F{O}_2^N$. If $F{O}_2^N\ge F{O}_2$, the pattern previously obtained $(\delta ,S,T^r,r,L)$ was optimal, with objective value $F{O}_2$.

Otherwise, we can set $S=S^N$ and repeat the procedure by increasing the number of sectors and recomputing the patrolling shape till obtaining $F{O}_2^N>F{O}_2$. The algorithm is summarized in Algorithm 1. The algorithm is illustrated for the case of $Scn\#1$ using the MD4-100 Microdrones platform in Appendix A.

Algorithm 1 Algorithm to obtain the optimal solution for a given UAV platform

1: Set iteration counter k = 0; 2: ${\delta }_1=V^{patrol}MRT/R$ 3: if  $R-r_{max}\le d_{max}\le R$  then 4:    ${\delta }_2=arccos((R^2+r_{max}^2-d_{max}^2)/\left(2R{r}_{max}\right)$ 5:    $\delta =min\{{\delta }_1,{\delta }_2\}$ 6: else 7:     if $d_{max}\ge R$ then 8:        $\delta ={\delta }_1$ 9:     end if 10: end if 11: Compute $S^k=⌈2\pi /\delta ⌉$ 12: Adjust ${\delta }^k=2\pi /S$ 13: Compute $T^{rk},r^k,L^k$ 14: Compute $n_1^k\mathrm{using}\left(9\right)$ 15: Compute $n_2^k\mathrm{using}\left(14\right)$ 16: $n^k=min\{n_1^k,n_2^k\}$ 17: Compute $F{O}^k$ 18: Set Condition = True 19: while Condition do 20:     $k=k+1$ 21:     $S^k=S^{(k-1)}+1$ 22:     ${\delta }^k=2\pi /S$ 23:     Compute $T^{rk},r^k,L^k$ 24:     Compute $n_1^k\mathrm{using}\left(9\right)$ 25:     Compute $n_2^k\mathrm{using}\left(14\right)$ 26:     $n^k=min\{n_1^k,n_2^k\}$ 27:     Compute $F{O}^k$ 28:     if $F{O}_2^k\ge F{O}^{(k-1)}$ then 29:        Set Condition = False 30:     else 31:        $\mathbf{Solution}\leftarrow ({\delta }^k,S^k,T^{rk},r^k,L^k,n^k,F{O}^k)$ 32:     end if 33: end while 34: $\mathbf{Optimal}\mathbf{solution}\leftarrow \mathbf{Solution}$

7. Simulation Approach to Test the Robustness of the Cyclic Surveillance Service

In this section, we present a DES model to check the extent to which a cyclic patrol design exhibits appropriate protection against the accelerated depletion of batteries as an uncertainty source limiting the flight of UAVs.

On solely considering the specifications from cell manufacturers, many reported UAV applications have neglected the dynamic behavior of their energy storage. Recently, ref. [52] contributed to a novel battery-aware model that allows us to better estimate the point of accelerated depletion of batteries. While the authors of [52] claim that energy consumption is a function of weight, distance, and flight speed, we are in a case study where the latter is the only control variable since the considered drones always have the same mass and the distances to cover in the ISR missions are fixed. Hence, in studying the uncertain flight times (endurance), we consider the electrical power required as a function of the cruise velocity [40,45,46]. We apply a similar shape to that in [45] for the propulsion power consumption for rotary-wing UAVs (steady-level flight at $V^{cruise}$ or $V^{patrol}$ speed)—see Figure 6—since in all the cases the drones to be selected in our practical scenario are multi-rotor UAVs. Nevertheless, we recall here that we do not attempt to use the shape to select the flight velocity but solely to calculate the per-meter flight energy consumption. In Figure 6, each UAV travel two segments when the perimeter is divided into eight sectors.

One of the settings within our DES model is the likelihood of failure, namely aborting the mission of a UAV before time because it is too close to the battery discharging time, as a function of the depth of discharge ($DOD$). Importantly, we are concerned with long distances travelled and neglect both the take-off and landing consumptions, so we first evaluate the $DOD$ from the power consumption on the flight mission computed by entering Figure 9 with $V^{cruise}$ or $V^{patrol}$, then multiply by the respective known distances, and then sum up. Second, we use the obtained $DOD$ as an input to sample the likelihood of running out of battery in a certain simulation of the DES model. Particularly, we propose a piecewise linear function—see Figure 10—wherein a probability of failure in the flight mission appears beyond a threshold point at a $DO{D}^{th}=80\%$ of the nominal Li-Po battery energy whose depletion is let to be up to 5% deeper (up to 82.4% which leads to a risk of failure of $12\%$).

In the simulation, each UAV is managed by a different process running in parallel. A centralized module maintains a list with the changes in the system states. The standard flight mission for every UAV within our simulated surveillance system is modelled according to the flow chart in Figure 11. There are three steps within every drone mission: (i) Take-off and transfer from the base station to the perimeter, (ii) Surveillance of several assigned sectors, and (iii) Return to another base station. Once a UAV returns to a base station, a battery replenishment process takes place. After the recharge time ($ChT$), this UAV becomes active again. The DES model checks in advance the $DOD$ that each UAV will have at the end of the next sector before starting to patrol that sector. If the level of the battery is >$DO{D}^{th}$, a failure signal is sent to the system, since a risk of running out of battery is identified. Note that, as mentioned, the likelihood of running out of battery follows the piecewise linear function illustrated in Figure 10. Once a failure is present, the UAV involved is marked as ’to be replaced’ and cannot patrol the next sector as planned, being commanded to return to a different base station than the one originally planned. In parallel, a relay UAV is then invited to take off with sufficient time to start surveying the non-covered sector on time.

In Figure 12, we illustrate how the system could recover from a single battery failure providing on-time service. Observe how starting from a balanced distribution of idle drones among the base stations, this symmetry is broken once the relay procedure occurs. In Figure 12, (a) shows the take-off from source base stations, (b) is the start of the first segment patrol, (c) is the take-off of a second lot of drones at each base station and the start of second patrol, except for the relay which serves only its first segment, (d) is the end of second segment patrol, and (f) is the return of the first set at a different base station.

The priority among the bases from which to pick up the replacement drone is as follows. First, a request for an idle drone is directed to the base wherein the failing drone will land, that is, a sort of swap between the failing and the idle drone since, in case there is a ready drone there, this provides a quicker response to the disruptive event. If not, an attempt is made to choose the relay drone from the previous base station, in case there is a ready drone on it. If this is not possible either, an attempt is made to launch the relay drone from the rear base station, following an oblique flight, this time backwards, in case there is a ready drone on it. Finally, if the latter is not yet possible, the only option is to wait for a drone to arrive at the base where the failed drone lands.

Please note that the described relay mechanism could lead to monitoring the involved sector with a certain delay (we mark it as delayed when the lag time is >$0.05T^r$). Furthermore, provided that the hypothetical starting time for the relay drone exceeds $T^r$, the emergency surveillance of the affected sector is indeed cancelled. In that case, since the next drone in the cyclic patrolling schedule arrival is expected in a time $T^r$, the relay itself has nonsense.

Noticeably, the drone fleet is shared over time by all base stations and every battery failure produces a dynamic imbalance between the drones available in the different bases derived from the relay procedure that it triggers. Finally, the number of sectors patrolled with delay and the number of cancelled/unattended sectors are the output indicators measured from the long runs of execution of the DES model.

8. Computational Experiments

In this section, we first test the proposed MINLP formulation using the solver DICOPT [53]. Specifically, we consider the alternative drone platforms shown in

Table 3. Drone platforms considered.

	Input Data Concerning the Alternative Drone Platforms
	High-Level			Aerodynamical			Surveillance	Avionics Power	Price
	Specifications			Electrical Settings			Flight Speed	Comsumption
UAV platform	$m_p^{frame}+$$m^{payload}$ (kg)	Range for $V^{cruise}$ (m/s)	$E_p$ (s)	${\Omega }_p$	$LtD$	$Ca{p}_p$ ($Ah$) & $Voltag{e}_p$ (V)	$V^{patrol}$ (m/s)	$PO{W}^{Avio}$ (kW)	$c_p$ (€)
MARVIN-5	6.40 + 0.35	2.50–10	1600	0.5	1.8	21 & 22.8	2	0.1	2500
DJI-M210	6.00 + 0.35	2.22–10	3000	0.65	1.5	15.33 & 22.8	2	0.1	5000
TAROT-500	2.15 + 0.35	2.50–12.50	1580	0.65	1.6	5.3 & 22.2	2	0.1	1500
MD4-100	3.80 + 0.35	2.78–12.22	3450	0.65	1.6	13 & 22.2	2	0.1	2900
Matternet-M2	9.50 + 0.35	2.78–13.88	5400	0.5	3	18 & 22.2	2	0.1	8000

. The five platforms are tested in six different scenarios, which are summarized in Table 2. Our computational experiments are performed using a 64-bit Intel Core i7-4712MQ 2.8 Ghz processor and 16 GB of RAM- and the model is implemented using GAMS v 30.1.0 [54].

8.1. Determining the Best Patrolling Scheme: Results from Solving the MINLP

While the resolution of the multiplatform model (17)–(32) provides us with the optimal patrolling scheme for the best drone platform in each scenario, we can also find which are the more competitive designs for the non-selected platforms. The latter can be obtained either by solving the optimization model by fixing the corresponding platform (enforcing ${\alpha }_p=1$ for each $p\in P$) or by running the algorithm that we propose in Section 6 to obtain the optimal solution for each platform in an isolated way.

On solving the set of scenarios for minimizing the cost of the patrolling system ($F{O}_2$), see

Table 4. Results from solving the scenarios.

	DESIGNS FOR SCENARIO #1					DESIGNS FOR SCENARIO #2
	MARVIN-5	DJI-M210	TAROT-500	MD4-100	Matternet	MARVIN-5	DJI-M210	TAROT-500	MD4-100	Matternet
${\delta }_p$	1.571	1.571	1.571	1.571	1.257	1.257	0.898	1.257	1.257	1.047
$S_p$	4	4	4	4	5	5	5	5	5	6
$R_p$	809.148	809.148	809.148	809.148	800.000	390.999	485.427	708.930	708.930	1.333.000
$L_p$	1444.000	1444.000	1444.000	1444.000	1255.055	1424.566	1422.969	1444.000	1424.569	14.215.263
$T_p^r$	939.336	939.336	939.336	939.336	751.468	939.965	939.965	939.965	939.965	78330.377
Flight time	1122.422	2061.758	1085.804	2967.806	4620.489	1.192.921	2.123.283	1118.450	3002.436	40,306.047
$V^{cruise}$	10	10	12.500	12.222	13.888	10	10	12.500	12.222	13.888
$w_p^s$	24	16	24	12	10	30	20	30	15	12
$n_p$	1	2	1	3	6	1	2	1	3	5
$PO{W}^{cruise}$	0.830	0.734	0.392	0.575	0.987	0.830	0.734	0.392	0.575	0.987
$PO{W}^{patrol}$	0.246	0.227	0.147	0.178	0.228	0.246	0.227	0.147	0.178	0.228
$F{O}_2$	92,000	112,000	68,000	66,800	110,000	115,000	140,000	85,000	83,500	132,000
	DESIGNS FOR SCENARIO #3					DESIGNS FOR SCENARIO #4
	MARVIN-5	DJI-M210	TAROT-500	MD4-100	Matternet	MARVIN-5	DJI-M210	TAROT-500	MD4-100	Matternet
${\delta }_p$	0.785	0.898	0.785	0.898	0.898	0.785	0.898	0.785	0.898	0.898
$S_p$	8	7	8	7	7	8	7	8	7	7
$R_p$	1333.000	1333.000	1333.000	1333.000	1333.000	1333.000	1.333.000	1333.000	1333.000	1333.000
$L_p$	1206.688	1354.317	1206.688	1354.316	13,543.159	1206.688	1354.317	1206.688	1354.316	13,543.159
$T_p^r$	666.018	761.163	666.018	761.163	76,116.302	666.018	761.163	666.018	761.163	76,116.302
Flight time	1489.004	2455.221	1457.610	3195.162	39,294.618	1489.004	2455.221	1457.610	3185.160	39,294.619
$V^{cruise}$	10	10	12.500	12.222	13.888	10	10	12.500	12.222	13.888
$w_p^s$	40	21	40	21	21	48	28	48	21	21
$n_p$	2	3	2	4	55	2	3	2	4	5
$PO{W}^{cruise}$	0.830	0.734	0.392	0.575	0.987	0.830	0.734	0.392	0.575	0.987
$PO{W}^{patrol}$	0.246	0.227	0.147	0.178	0.228	0.246	0.227	0.147	0.178	0.228
$F{O}_2$	164,000	161,000	124,000	116,900	198,000	184,000	196,000	136,000	116,900	203,000
	DESIGNS FOR SCENARIO #5					DESIGNS FOR SCENARIO #6
	MARVIN-5	DJI-M210	TAROT-500	MD4-100	Matternet	MARVIN-5	DJI-M210	TAROT-500	MD4-100	Matternet
${\delta }_p$	0.785	0.898	0.785	0.898	0.898	0.524	0.524	0.524	0.524	0.524
$S_p$	8	7	8	7	7	12	12	12	12	12
$R_p$	900.000	900.000	900.000	900.000	900.000	1333.000	1333.000	1333.000	1333.000	1333.000
$L_p$	1236.026	1335.300	1236.026	1335.300	13,353.001	858.801	858.801	858.801	858.801	8588.015
$T_p^r$	666.018	761.163	666.018	761.163	76,116.302	444.012	444.012	444.012	444.012	444.012
Flight time	1535.238	2496.619	1494.597	3219.031	39,592.687	1454.215	2786.251	1429.779	3208.048	4528.087
$V^{cruise}$	10	10	12.500	12.222	13.888	10	10	12.500	12.222	13.888
$w_p^s$	48	28	48	21	21	48	36	48	36	24
$n_p$	2	3	2	4	5	3	6	3	7	10
$PO{W}^{cruise}$	0.830	0.734	0.392	0.575	0.987	0.830	0.734	0.392	0.575	0.987
$PO{W}^{patrol}$	0.246	0.227	0.147	0.178	0.228	0.246	0.227	0.147	0.178	0.228
$F{O}_2$	184,000	196,000	136,000	116,900	203,000	216,000	276,000	168,000	200,400	264,000

The column of the best platform for each scenario is shaded and the lowest total cost is shown in bold face.

, the MD4-100 platform achieves the best solution in five of the six scenarios, followed by the TAROT-500 platform, which obtains the best score in the $Scn\#5$ (in which the MD4-100 platform is second). On the whole, the results obtained after the optimization suggest that the MD4-100 is the best candidate among the platforms analyzed. Figure 13 summarizes the deployment cost for each platform.

Finally, since DICOPT can provide the best-found solution but without any judgement on its global optimal condition, we devise a straightforward mechanism for solving the multiplatform model consisting of decomposing the problem for each one of the considered UAV platforms, then solving independently as many problems as platforms using an algorithm presented in the next section that provides the optimal solution for each platform, and finally selecting the one with the better objective function. In any case, for each scenario and platform p, each design defines the value of the angle in radians (${\delta }_p$), the number of sectors ($S_p$), the radius at which base stations are located from the centre of area $(r_p$), measured in meters, the length of the maximum communication distance to the base station ($L_p$), measured in meters, the revisit time ($T_p^r$), measured in seconds, the flight time for each drone, measured in seconds, the speed at which each drone must fly from the base station to the perimeter and return from the perimeter to recharge its battery at a different base station ($V^{cruise}$), measured in m/s, the number of drones required by the patrolling scheme ($w_p^s$), the number of sectors assigned to each drone flight ($n_p$), the individual power consumption in cruising and patrolling operations, measured in $kW$, and the value of the $F{O}_2$ objective function.

8.2. Using the Algorithm to Study the Effect of Reducing the Maximum Allowed Revisit Time

On the assumption that we have already selected the MD4-100 platform as the best platform, we propose here studying the effect that a reduction in the maximum allowed revisit time has on the deployment cost. Again, $Scn\#1$ is selected to apply the above-described algorithm to measure the effect not only in costs but also in the flight times. According to Figure 14, whereas the progressive reduction of the allowed revisit time leads to higher deployment cost, an abrupt evolution in the flight time is observed when moving the bound $MRT$ from 450 to 400 s. The $MRT=450$ s setting is featured by an efficient use of batteries (flight time = 3352 s, close to the endurance for this platform, $E=3450$ s), in a pattern comprising 27 drones and 9 charging pads, valued, in total, at EUR $150,300$.

Concerning the $MRT=400$ s setting, our resolution algorithm indicates that the cheaper deployment cost is EUR $167,000$ achieved around a design needing 10 charging pads and a fleet of 30 drones. In the latter, each sector is patrolled in $375.734$ s (<400 s) whereas in the former each sector is patrolled in $417.483$ s (<450 s). Thus, aside from giving rise to a more expensive deployment, lowering $MRT$ also results in a poorer use of the MD4-100 batteries (flight time $=3087.877$ s). Since the attained revisit time is shortlisted to discrete values resulting from the number of sectors (namely $T^r=2\pi R/\left(S{V}^{patrol}\right)$, we next present the performance of the successive patrolling schemes resulting from applying the algorithm for addressing $Scn\#1$ when we vary the number of sectors. Figure 15 and Figure 16 sketch different indicators for the best solutions with the horizontal axis representing the number of sectors, which now acts as a limitation.

8.3. Analyzing the System Robustness. Results of the DES

In this section, we report the results of the DES model designed according to the guidelines presented in Section 7. The DES is coded as a Python version 3.11 application using the SimPy module. As explained before, we assume a certain number of relay drones in excess over the fleet resulting from solving the MINLP and focusing on attaining a robust surveillance system. Otherwise, this design should not be validated as our operational design choice for the surveillance system. We conduct two independent robustness studies: (a) Study of the fleet size influence on the robustness of the performance and (b) Study of the influence of design decisions under a similar fleet size.

8.3.1. Study (a): Effect of the Fleet Size on the Performance of a Cyclic Patrol Scheduling

The study on how increasing the size of the fleet can modify the performance of the surveillance service is conducted around the preliminary cyclic patrol scheduling presented in Table 4—for the scenario $Scn\#3$ and the MD4-100 platform, see

Table 5. Addressing $Scn\#3$: The preliminary surveillance system design & simulation around the DES model.

Input parameters	Parameters of the persistent surveillance service	The radius of the perimeter circumference	$R=1696$ m
		Maximal comms link	$d_{max}=1444$ m
		Far placement of base stations	$r_{max}=1333$ m
		Patrolling velocity	$V^{patrol}=2$ m/s
		Patrolling recurrence	$MRT=1222$ m
	Li-Po Battery parameters	Recharging time	$ChT=4000$ s
Operational design of the cycling patrol schedule	Selected commercial platform		p * = MD4-100 Microdrones
	Placement of base stations	Radial distance to the center	$r=1333$ m
	Quantity of base stations	Number of sectors	$S=7$
	Number of Sectors patrolled by each single UAV flight mission	Number of sectors to monitor in a standard flight	$n=4sectors$
	Initial fleet size	Number of UAVs per base station	$w^s=21[\#3$ UAVs per Base Station]
	Flight time vs. endurance	$E_p=3450$ s	Flight Time. = $E_{p^{*}}$ $\left[SlackofEquation20\right]=3185.16$ s
	Maximal vs. decided recurrence	$MRT=1222$ m	$T^r=761$ s
	Transfer speed (to the perimeter or back from the perimeter)		$V^{cruise}=12.22$ m/s
DES setting
Operational characteristics of a standard flight	Perimetral velocity (over-monitored sectors)		2 m/s
	Transfer speed (to the perimeter or returning from the perimeter)		$12.222$ m/s
	Minimum threshold before considering a delayed patrol of the sector		5% of the flight time over a sector
Simulation parameters	Warm-up period		$50,000$ s
	Simulated period in a replication		Time elapsed to complete 100 laps around the perimieter
	Number of replications		100 replications

. One hundred replications are considered, the simulation of each one extending along a horizon of one hundred complete cycles of surveillance, and a warm-up period is considered before registering the performance indicators. The simulations are carried out around scenarios generated according to a full factorial design of experiments followed by an ANOVA analysis to ensure the validity of our insights. The design of the experiments contains two factors, as presented in

Table 6. Design of Experiments for the analysis (a).

Factors	Factor 1: $RL$, Risk level considered on a standard flight	$2.5\%,5\%,10\%\mathrm{and}12\%$
	Factor 2: $FS$, Fleet size	Drones-per-base: 3, 4, 5, 6
Performance indicators	Punctually monitored sectors (%)
	Delayed monitored sectors (%)
	Unattended sectors (%)

. The first factor includes four different risk levels ($RL$s) that affect every standard flight. Specifically, $RL$ is set according to the likelihood of interrupting a standard flight, with four different likelihood values: $2.5\%$, $5\%$, $10\%$ and $12\%$. Thus, the lifecycle of batteries is considered till the point at which they exhibit an RL of $12\%$. By doing so, we are indeed including a practical feature: the more we use the Li-Po batteries, the more likely they are to fail and therefore the riskier the flight missions and the lower the expected system performance. While completing a standard flight involves surveying all assigned sectors and landing at the assigned base, the consequences of a battery failure can be three-fold: (i) an on-time service from a relay drone, which forces a readjustment of the cyclic schedules, (ii) a delayed service, owing to a feasible degree of delay on the service from a relay drone, and (iii) absence of service on the involved sector when no relay drone can reach the involved sector, at least not in a feasible delay. The second factor is the fleet size ($FS$), which has varied ranging from 3 drones per base (namely the dimensioning resulting from the optimization model) to 6 drones per base. In

Table 7. Results from the Two-Way ANOVA for study (a).

	Punctually (%)		Delayed (%)		Unattended (%)
Factors	F	p-Value	F	p-Value	F	p-Value
$FS$	31,242	0	30,723.02644	0	28,278.8525	0
$RL$	2972.907099	0	2956.954193	0	2597.794509	0
$FS\times RL$	1473.811071	0	1422.225953	0	1412.368775	0

, we show the F-Statistic and the P-value obtained in the ANOVA analysis on the three performance indicators measured in this study for each independent factor ($RL$ and $FS$) and the interaction between these factors. According to the results listed, the three proposed indicators are affected by each of the two independent factors and also by the interaction between them. Next, we present the performance reports on our DES model simulations in terms of (a) the percentage of punctually patrolled sectors, i.e., according to the mission plan, (b) the percentage of sectors delayed in their surveillance, although only suffering a small lag, and (c) the percentage of unattended (cancelled) sectors. The mean values along the simulation run within the first study (a) are plotted in Figure 17.

The insights are as follows. First, the probability of battery failure in an individual standard flight under the more optimistic battery assumption $RL=2.5\%$ is not null: the global percentage of punctually patrolled sectors for the original fleet size (3 UAVs) reaches only a $92.6\%$. Hence, there is an unneglectable probability of non-compliance with the planned surveillance task even in the more positive hypothesis on battery resiliency. This is due to the dynamic imbalance between the drones available at the different base stations that temporarily appear when a battery failure triggers the relay procedures. This drawback is stressed in the case of the more pessimistic selection of $RL=12\%$, which results in $77.8\%$ of unattended sectors.

Conversely, as the fleet size increases, the robustness of the global mission plan is expected to improve, and more and more punctual surveillance of the sectors is attained. With the sole allocation of an additional drone per base (4 UAVs instead of 3, a $33\%$ increase in the fleet size), the new punctual surveillance scores range from $99.3\%$ at $RL=2.5\%$ to $96.7\%$ at $RL=12\%$. While further increasing the fleet size (5 and 6 UAVs) improves the robustness regarding the attendance of sectors, we conclude that 4 UAVs gives up a satisfactory proportion of above $96.7\%$ on the punctuality of patrolling each sector.

Second, regarding the number of delayed sectors, the surveillance service provides a delay probability of $16.7\%$ under an $RL=12\%$, while this score is reduced to $5.6\%$ when $RL=2.5\%$. Again, the addition of drones should reduce delays. With the sole allocation of an additional drone per base (4, a $33\%$ increase in the fleet size), the variation in the percentage of lagged sectors ranges from $0.6\%$ at $RL=2.5\%$ to $2.6\%$ at $RL=12\%$.

Finally, the more stringent effect of the uncertain failures on batteries is the absence of a surveillance service in a certain sector. The percentage of unattended sectors for the original fleet size (3 UAVs) varies from $45.51\%$ when $RL=12\%$ to $1.73\%$ when $RL=2.5\%$. The effect of adding a drone per base station becomes an improvement in this indicator, reaching $0.19\%4$ at $RL=2.5\%$ and $0.73\%$ at $RL=12\%$. We must highlight also that a certain saturation effect appears when the fleet becomes too populated. For instance, the benefit of moving from 5 to 6 UAVs is unclear, even though a bigger fleet means a greater investment (both purchasing and maintenance).

8.3.2. Study (b): Effect of Sectorization on the Persistent Surveillance Performance

Generally, there exists a variety of feasible sectorization patterns that could be the basis for building the cyclical schedule. Next, we study the influence that the choice of sectors partitioning the area exerts on the robustness of the surveillance service provided by a certain fleet size. Specifically, let us consider that by solving our proposed MINLP for the scenario $Scn\#3$ and the MD4-100 platform, we obtain the two alternative design settings presented in

Table 8. The alternative sector partitioning settings of experiments for study (b).

			Sectorization	Sectorization
			SP6	SP8
	Selected Commercial Drone		$p^{*}$ = MD4-100 MICRODONES
	Li-Po Battery	Recharging time	$ChT=4000$ s
	parameters	Patrolling velocity	$V^{patrol}=2$ m/s
MINLP design	Sectorization pattern	Radius of the perimeter circumference	1696 m
		Number of sectors	6	8
		Span between revisits	888 s	666 s
		Placement of base stations	249 m	389.66 m
		Transfer speed (to the perimeter or back from the perimeter)	$8.48$ m/s	9 m/s
		Number of sectors to monitor in a standard flight	3	4
DES setting	Perimetral velocity (over monitored sectors)		2 m/s
	Transfer speed (to the perimeter or returning from the perimeter)		$8.48$ m/s	9 m/s
	Minimum threshold associated with a delayed surveillance		5% of the flight time over a sector
	Warm-up period		50,000 s
	Simulated period in a replication		Time elapsed to complete 100 laps around the perimeter
	Number of replications		100 replications

. They are based on two different sector partitioning, denoted as $SP6$ and $SP8$, wherein the surveillance missions of each flight cover exactly half of the perimeter in both cases. Note that in both partitions, for the selected platform, the values of $V^{cruise}$ are set to intermediate values (corresponding to the zone of low power consumption in Figure 9), thus reducing the probability of a battery failure as a function of the depth of discharge according to Figure 10. Moreover, the values of $V^{cruise}$ in $SP6$ and $SP8$ guarantee that the UAVs’ mission time is the same in both experiments. Simulation studies have been carried out according to a complete factorial design of experiments followed by an ANOVA analysis.

The design of the experiments contains three factors, as presented in

Table 9. Design of experiments for study (b).

Factors	Factor 1: $SP$, Sectorization	3 sectors are surveilled per flight from a total of 6 sectors compounding the surveilled perimeter.
		4 sectors are surveilled per flight from a total of 8 sectors compounding the surveilled perimeter.
	Factor 2: $RL$, Risk levels considered on a standard flight	$2.5\%$
		$5\%$
		$10\%$
		$12\%$
	Factor 3: $FSS$, Similar fleet size scale	Size A: 24 drones (4 drones per base in $SP6$, or 3 drones per base in $SP8$).
		Size B: 30 drones (5 drones per base in $SP6$, or 3 drones per base in $SP8$).
Performance indicators	Punctually surveilled sectors (%)
	Delayed surveilled sectors (%)
	Unattended sectors (%)

. The first factor comprises two different sector partitioning ($SP$) for surveillance missions that cover exactly half of the perimeter: $SP6$—design composed of 6 base stations and 6 sectors—with $n=3$ sectors patrolled per drone flight, and $SP8$—design composed of 8 base stations and 8 sectors—with $n=4$ sectors patrolled per drone flight.

The second factor again includes the four different risk levels affecting every standard flight. Specifically, RL is set according to the likelihood of interrupting a standard flight, with four different likelihood values: $2.5\%$, $5\%$, $10\%$, and $12\%$. Finally, the third factor evaluates two different fleet size scales ($FSS$). We first evaluate a fleet size of $FSS=24$ drones (namely 4 drones per base in the $SP6$ scenario and 3 drones per base in the $SP8$ design), and then move to evaluate larger fleet size scales: 5 drones per base station in the $SP6$ and 4 UAVs per base station in the $SP8$ ($FS=30$ and $FS=32$, respectively).

Table 10. Results from the Three-Way ANOVA for study (b).

	Punctually (%)		Delayed (%)		Unattended (%)
Factors	F	p-Value	F	p-Value	F	p-Value
$SP$	10,035.99	0	8452.37	0	11,513.64	0
$RL$	395.18	0	488.91	0	294.99	0
$FSS$	8970.4	0	8708.39	0	8954.63	0
$S\times RL$	133.2	0	85.46	0	190.49	0
$S\times FSS$	4644.02	0	4076.1	0	5126.94	0
$RL\times FSS$	103.52	0	109.61	0	93.99	0
$S\times RL\times FSS$	93.72	0	73.27	0	114.95	0

shows the F-statistic and the P-value obtained in the ANOVA analysis on the three robustness indicators, for each independent factor ($SP$, $FSS$ and $RL$) and the double and triple interaction between these factors. According to the results listed, the three proposed indicators are affected by each of the three independent factors and also by the interactions between them.

Next, we present the performance reports on the simulations in terms of (a) the percentage of punctually patrolled sectors, (b) the percentage of sectors delayed in their surveillance, although only suffering a small lag, and (c) the percentage of unattended (cancelled) sectors. In addition, and for the sake of clarity, the performance report on this second run of execution also represents in (d) the proportion of disrupted patrolled sectors (whether delayed or cancelled/unattended) that were finally cancelled sectors. The mean values for these four indicators are plotted in Figure 18. The insights are as follows.

The robustness of the persistent surveillance system decreases when $RL$ increases, as can be seen in (a) for each one of the four simulated scenarios that arise from the crossing of the $FSS$ and $SP$ factors. Furthermore, at a fixed $RL$, the increase in $FSS$ provides a better performance for $SP6$ (squares) and $SP8$ (triangles). More interesting insights arise from comparing the different performance of $SP6$ and $SP8$ with a similar value of $FSS$. A dashed line is used for the results of a value of $FSS=24$, which clearly shows that $SP6$ outperforms $SP8$ for each $RL$ in all the proposed indicators. Although $SP6$ provides punctual surveillance in a range of 95.1–96.9%, the figure decreases to 76.3–86.9% for the $SP8$ case. Similarly, the delayed sectors range from 7.1 to 12% and the cancelled sectors range from 7.0 to 11.9% for the $SP8$, outperformed by those from $SP6$ (the ranges are, respectively, 1.6–3.1% and 1.5–1.8%). The reason is that the $SP6$ solution considers a reserve drone per base station, whereas the $SP8$ does not.

To analyze the effect of adding a reserve drone also to the $SP8$, we refer to the continuous lines used in Figure 18 to assess its results compared to a similar fleet size scale on the $SP6$ case. Noticeably, high robustness is also obtained in the case of $SP8$ (punctual surveillance in a range of 94.3–96.1%); indeed, quite similar to the $SP6$ with 4 UAVs. In short, the addition of a reserve drone is positive in both partitions, providing a persistent surveillance service wherein the span between revisits is indeed slightly better in the $SP8$ ($666s$). However, the advantage of the $SP6$ needs to be stressed: it requires a fleet of 24 drones, whereas the $SP8$ requires a fleet of 32 drones to obtain the same degree of robustness.

With the defined relay procedure, the system can recover from battery failures but at the cost of breaking the balanced distribution of the idle drones among the different base stations. This effect has been observed in the conducted experimentation on the DES model, wherein at every replicated scenario, the asymmetries moved from one base station to another as long as the running simulation evolved. According to our experiments, the appearing imbalance leads to a probability of non-compliance with the planned surveillance task of hardly $2.5\%$, even in the more positive hypothesis on battery resiliency.

The improvements derived from the addition of idle drones exhibit a certain saturation effect when the fleet becomes too populated. In our methodology to plan the operational deployment of the completely automated UAV swarm system, we propose a $95\%$ punctuality in patrolling each sector as a sufficient robustness score. The DES model serves the purpose of identifying the level at which using a larger fleet no longer improves expected performance.

9. Conclusions

In this study, we provide a methodology for planning the operational deployment of a standalone drone-based patrolling service on a vast circular perimeter surrounding a critical area. For attaining regular patrolling above a minimum frequency, we develop a procedure for the definition of the flight missions for a fleet of homogeneous camera-equipped UAVs that are supported by a set of auxiliary on-ground charging stations, wherein the UAVs land and automatically recharge without human intervention, to ultimately specify a completely automated UAV swarm system able to persistently guard the perimeter.

Aside from the revisit of every perimeter point in a limited period, we incorporate operational constraints such as conditions on the placement of charging stations, e.g., not too close to the perimeter, communication range limits, the safety margin on the Li-Po battery’ energy and their recharging times.

We first propose a MINLP model aimed at minimizing the total deployment cost of the surveillance system. On solving the model, we pick the specific drone platform up from a candidate list of commercial platforms, deciding the quantity and placements of base stations from which the already replenished UAVs take off, their location, the number of sectors in which the perimeter must be divided, the revisit time, and the cruise velocity on moving to or returning from the perimeter. Since the solver used (DICOPT) does not provide any judgement on the global optimality condition of the obtained designs, we propose an algorithm to obtain the optimal design for a given platform. This allows us to decompose the problem and solve it independently for each platform, which is useful not only to specify the design around the platform but also to specify the best design for the not-selected drone platforms.

However, as in any realistic approach to a real-life drone-based system, we need to plan a recovery mechanism to react to uncertain battery failures. Our proposal is two-fold. First, we place idle drones at the base stations ready to be incorporated into the system in the event of an earlier-than-planned battery depletion. Second, we propose a ‘relay procedure’ as a strategy that (with a sufficient likelihood) should be successful in patrolling the sector(s) that the battery-drained drone could not. To assess to what extent adding relay drones results in a sufficient robustness performance, we develop a DES model embedding the relay procedure to analyze the robustness of any tentative persistent surveillance system design against uncertain battery discharging time. We can use the two-step procedure to evaluate alternative designs considering variations on both the fleet size and the sector partitioning pattern at once.

The presented methodology could be extended by future research to better estimate the energy drainage due to the exact payload in use and to the evolution of the recovery mechanism to incorporate the reaction to the eventual catastrophic loss of a member of the fleet. Concerning the resilience of the service after a catastrophic failure on a drone occurs, we have to further investigate how best to address the request for a backup drone and which policies to appropriate to decide on the recovery procedures.