diff --git "a/9tFAT4oBgHgl3EQfpx3L/content/tmp_files/load_file.txt" "b/9tFAT4oBgHgl3EQfpx3L/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/9tFAT4oBgHgl3EQfpx3L/content/tmp_files/load_file.txt" @@ -0,0 +1,1221 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf,len=1220 +page_content='Optimal multiple FSO transceiver configuration for using on High-altitude platforms Dieu Linh Truong ∗1 and The Ngoc Dang †2 1School of Information and Communication Technology, Hanoi University of Science and Technology, Vietnam 2Department of Wireless Communications, Posts and Telecommunication Institute of Technology, Vietnam January 23, 2023 Abstract Free-space optical (FSO) communication requires light of sight (LoS) between the transmitter and the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' For long-distance communication, many research projects have been conducted towards using a network composed of high- altitude platforms (HAPs) flying at an elevation of 20 km to carry intermediate FSO transceivers that forward data be- tween ground stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The clear environment at high el- evations prevents terrestrial obstacles from cutting the LoS between the transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' An FSO transceiver on a HAP can communicate with ground stations within a small area owing to its limited beam size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' We suggest using multiple FSO transceivers on a HAP to extend its ground coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' However, the use of too many FSO transceivers may quickly exhaust the onboard energy of the HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' As a result, HAP must be lowered to recharge frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' In this study, we first propose a configuration of multiple FSO transceivers to widen the ground coverage of a HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' We then propose a set of closed-form expressions to calculate the extended coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Finally, to implement a HAP network using multiple FSO transceivers, we seek the optimal config- uration of multiple FSO transceivers that minimizes the to- tal cost of the HAP network, including amortization, energy, and maintenance costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The simulation results show that the proposed multiple FSO transceiver configuration clearly in- creases the ground coverage of a HAP and significantly re- duces the cost of the HAP network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Keywords— Free Space Optics, High-altitude platform, Beam size optimization, HAP based FSO network 1 Introduction Free-space optical (FSO) communication uses light propagation in free space to transmit data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' In recent years, this technology has emerged as a promising choice for short-distance high-speed communication between endpoints with a clear light of sight (LoS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' ∗linhtd@soict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='vn †ngocdt@ptit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='vn Commercial FSO transmitters available in the market at prices of thousands of dollars can operate at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='25 ��� 10 Gbps over 1 − 2 kilometers, for example, the SONABeam series of fSona [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' To reach a long distance, a multi-hop FSO system can be used, where data are transmitted through intermediate FSO transceivers [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' To avoid obstacles that cut the LoS between terrestrial FSO transceivers, researchers from academia and industry have proposed placing intermediate FSO transceivers of the multi-hop FSO system on high-altitude platforms (HAPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' High-altitude platforms are flying objects that operate at altitudes of 17–24 km in the stratosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Several HAP models have been proposed and piloted previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Some projects continue until recently, such as the Loon Project of Google [4], the UAV project of Facebook [5], and the Stratobus project of Thales Alenia Space [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' A multi-hop FSO system using a HAP network is described in [7] and illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' According to this model, FSO transceivers on the ground (so-called ground FSO nodes) are re- grouped into clusters to become the serving zones of HAPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' A HAP has an FSO transceiver looking down to exchange data with the ground FSO nodes of the cluster under it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' This FSO transceiver is called serving FSO transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' A HAP also carries several FSO transceivers pointing towards other HAPs for inter- HAP communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' These FSO transceivers are known as inter- HAP FSO transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Although the ITU recommends a HAP footprint width of ap- proximately 500 km in radius, experimental projects show much smaller coverage areas [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Nevertheless, a network of multiple HAPs can cover a country entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' For example, a constellation of 16 HAPs with multiple radio frequency antennas was considered to cover Japan [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' An end-to-end data-switching scheme for a multi-hop FSO sys- tem using HAP was proposed in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Since the communication between a HAP and the ground is point-to-multipoint, the serv- ing FSO transceiver on the HAP controls multiple accesses from ground FSO nodes under it using the WDM technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Each ground node is assigned a separate wavelength for up and down communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' An IP router on the HAP aggregates IP packets heading toward a common cluster within a single flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The flow will be carried by one or more continuous lightpaths between the source and destination HAPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The number of lightpaths is deter- mined according to the size of the flow and the transport capacity of a wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' A WDM switch is installed on each HAP to route 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='08642v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='NI] 20 Jan 2023 these lightpaths over the HAP network on a wavelength-switched basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' In Figure 1, the blue path HAP1-HAP2-HAP4-HAP5 and the red path HAP1-HAP2-HAP3 are two flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 1 2 2 1 1 1 1 2 3 3 3 3 3 WDM switch IP router IP router WDM switch 1 2 2 p-HAP-2 1 3 3 p-HAP-1 toward HAP-2 of cluster-2 toward HAP-1 of cluster-1 HAP2 Inter-HAP FSO transceiver Ground FSO node Serving FSO 3 1 2 A cluster A cluster transceiver HAP1 HAP3 HAP4 HAP5 Serving zone of HAP 1 Serving zone of HAP 2 inter-HAP link inter-HAP link Figure 1: Multi-hop FSO communication system using HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' In terrestrial FSO communications, the light beams are usually set to be very narrow for low transmission energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' However, for HAP and ground communication, the serving FSO transceiver of the HAP must project a sufficiently wide laser beam for covering distributed ground FSO nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' A single serving FSO transceiver has a relatively small foot- print owing to the low capacity of the current laser source, and the limited sensibility and aperture sizes of ground receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The calculation in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='1 shows that with a laser source of 1 Watt, required received power at receivers of -41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='1dBm, and receiver aperture radius of 2 m, a single serving FSO transceiver at an elevation of 20 km can cover a ground area of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='691 km radius only (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' To extend the coverage of a HAP, we propose using multiple serving FSO transceivers arranged in a bundle, as shown in Fig- ure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Each serving FSO transceiver points in a slightly different direction to cover a particular ground area that overlaps other ar- eas to create a continuous coverage region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Given a ground region to be served, using HAPs with multiple serving FSO transceivers reduces the number of required HAPs compared to using HAPs with a single serving FSO transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' However, the expenditure for serving FSO transceivers increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Therefore, the number of serving FSO transceivers to be used on a HAP should be carefully considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Regarding the communication between ground nodes and a HAP, the multiple serving FSO transceiver model still uses the WDM technique, where each ground node is assigned a unique wavelength within its cluster to communicate with its HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The number of ground nodes to be served by a HAP is restricted by the number of wavelengths offered by the WDM technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' In this study, we focus on identifying the optimal configuration of multiple serving FSO transceivers to achieve a minimal-cost HAP network for serving a set of ground FSO nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The optimal configuration should define the number of serving FSO transceivers to be set up on a HAP and the beam width for each transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The cost of the HAP network includes the investment, energy, and maintenance costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Compared with the previous study in reference [7], the current research differs in two aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' First, the current research pro- Figure 2: A HAP with multiple serving FSO transceivers and its footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' poses the use of multiple serving FSO transceivers on each HAP instead of a single serving FSO transceiver, as in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Second, the current research identifies the optimal beam widths for serving FSO transceivers, whereas in [7], the beam widths are predefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The current study also differs from that in [10], where beam size was optimized for an inter-HAP link, which is a point-to-point link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' First, we analyze the single and multiple serving FSO transceivers configu- rations in Section 2 to determine their ground coverage sizes and constraints on transmitter beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' In Section 3, we state the prob- lem of designing a minimal-cost HAP-based FSO network, which is the target of the optimization of multiple serving FSO transceiver configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Then, in Section 4, we define a HAP energy con- sumption formula and show that solar energy is necessary for keep- ing the HAP working in space for a long period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' We also present a constraint that a HAP must respect to relying uniquely on so- lar energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' In Section 5, we present the algorithms for identifying the optimal multiple serving FSO transceiver configuration and its footprint radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Section 6 presents the process designing the minimal cost HAP-based FSO network using the optimal multi- ple serving FSO transceiver configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Section 7 presents the simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Finally, Section 8 concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 2 Serving FSO transceiver configu- rations 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='1 Single serving FSO transceiver configu- ration In this section, the allowable beam width and ground coverage of a single serving FSO transceiver are determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The beam size is restricted to ensure that the received power at a ground point within the beam footprint is detectable by receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 2 Figure 3: Surface of the part of sphere blocked by solid angle α is calculated as the sum of the surface of all ribbons around the sphere when the solid angle varies from α to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Assume that the transmitter source radiates within a solid angle α and that the radiation density is uniform in all directions within the solid angle at a distance r from the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The radiation den- sity at distance r is inversely proportional to the surface of the part of the sphere radius r blocked by the solid angle α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' To calculate this surface, we divide the sphere into thin ribbons corresponding to open angles of d(α/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The width of a ribbon is rd(α/2), as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The radius of the ribbon at zenith angle α/2 is r sin(α/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Thus, the ribbon surface is 2πr sin(α/2)rd(α/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The surface of the part of the sphere blocked by the solid angle α is the sum of the surfaces of all ribbons when zenith angle varies from α to 0, as follows: � 0 α 2πr sin (α 2 )rd(α 2 ) = 2πr2(1 − cos (α 2 )) Let Ur be the radiation density at distance r and Ptx be the transmitted power at the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' We deduce: Ur = Ptx 2πr2(1 − cos (α/2)) (1) Let P rx j be the received power at ground FSO node j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The received power is proportional to the radiation density and the received aperture of the ground node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' It is: P rx j = e−σLjULjAR (2) where Lj is the distance between ground FSO node j and its serving HAP Hi (see Figure 4), σ is the attenuation coefficient of the links between the HAP and ground, ULj is radiation density at distance Lj from the source, AR is the aperture area of the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Let Rrx be the receiver aperture radius, then, AR = πR2 rx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' In (2), the first term represents the attenuation of laser power through the atmosphere, which is described by the exponential Beer–Lambert Law [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Figure 4: Received power on border nodes of a coverage area is the smallest amongst all nodes in the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' By substituting ULj from (1) into (2), we obtain the received power at node j as follows: P rx j = e−σLj × Ptx × R2 rx 2L2 j × 1 1 − cos (α/2) (3) The power received at node j must not be less than the required level of the receiver, denoted by ρrx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' It is obvious that point j at the border of the ground coverage area receives the least power because it is the furthest from the source (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Hence, all points in the coverage areas of HAP Hi receive sufficient power if and only if the border points receive at least the required power;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' that is, P rx j = e−σH/ cos ( α 2 ) PtxR2 rx cos2 ( α 2 ) 2H2(1 − cos ( α 2 )) ≥ ρrx (4) where Lj is substituted by H/ cos( α 2 ) for border node j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Solving inequation (4) yields the beam width of the single serv- ing FSO transceiver configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Corresponding to beam width α, the ground coverage radius of the configuration is: Ri = H tan(α 2 ) (5) Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Function P rx j decreases with α ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='.π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Proof of Lemma 1 is given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Figure 5 shows the received power at the border of the cover- age area with different receiver aperture radius Rrx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' This figure confirms that P rx j decrease with an increase in α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Let αmax be the value for α that makes P rx j (αmax) = ρrx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' then according to Lemma 1, P rx j (α) ≥ P rx j (αmax) = ρrx, ∀α ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='.αmax] thus all α ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='.αmax] satisfy constraint (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Calculations using the parameters given in Table 1 show that when Rrx = 2 m, αmax = 37° and the coverage radius is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='691 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' When Rrx = 4 m, αmax = 67° and the coverage radius is 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='237 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 3 Ribbon surface= 2πr sin(α/2) r d(α/2) Kd(a/2) r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='sin(a/2) a/2 SourceHAP H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' α H Received power Prx R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Nodej coverage area of HAP H 0 2 4 6 8 10 0 20 40 60 80 100 120 140 160 180 Received power at coverage border (10-8 W) Beam size α(degree) Rrx=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='125m Rrx=1m Rrx=2m Rrx=4m Required at receiver (Prx) Figure 5: Received power at the coverage border of the single serving FSO transceiver configuration with different receiver apertures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='2 Multiple serving FSO transceiver config- uration The ground coverage of a HAP can be widened by combining sev- eral serving FSO transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Different combinations are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' In this research, we study a straightforward configuration in which a principal serving FSO transceiver is in the center projecting light perpendicular to the ground, and several identical supplemen- tary serving FSO transceivers are set evenly around the principal one (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Each supplementary transceiver projects slanted beams to extend the coverage in one direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' This arrangement is referred to as mFSO configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Usually, the transmitters in a bundle are considered to project signals in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' However, because of the large principal beam, the supplementary serving FSO transceiver projection directions are far from being perpen- dicular to the ground, and their footprints are ellipses instead of circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' To create a continuous coverage region, the footprint of the principal serving FSO transceiver and those of the supplemen- tary serving FSO transceivers should overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Therefore, there should be a sufficiently large number of supplementary serving FSO transceivers to cover entirely the contour of the principal foot- print.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The extended coverage area is defined as the largest circle covered by these footprints (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The principal transceiver is responsible for the region defined by its footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' A supplemen- tary serving FSO transceiver is responsible for the part limited by its footprint, principal coverage circle, and extended coverage circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Let α be always the beam width of the principal serving FSO transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' To ensure that ground nodes under principal coverage receive sufficient power, α should still respect constraint (4), as in the single serving FSO transceiver configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Let the beam width of a supplementary serving FSO transceiver be β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' In the responsible area of the supplementary transceiver, the points on the extended coverage circle are the farthest from the supplementary transceiver;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' thus, they receive the least power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' If these points receive at least ρrx, all other points receive sufficient power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' It is easy to note that the footprints of the neighboring supple- Figure 6: footprint of multiple FSO transceiver (mFSO) con- figuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' mentary serving FSO transceivers join each other on the extended coverage circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Let J be such a joint point, the power J receives from the supplementary FSO transceiver is defined similar to (3) but with beam width β, which is P rx J = e−σ×LJ × Ptx × R2 rx 4L2 J × 2 1 − cos (β/2) (6) Thus, β is constrained by the condition P rx J ≥ ρrx, which gives: e−σ×LJ PtxR2 rx 2L2 J(1 − cos (β/2)) ≥ ρrx (7) Let us denote the extended coverage radius by Rext then LJ = � H2 + R2 ext (8) Appendix B presents detailed calculations of LJ and Rext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The calculations yielded the following results Rext = H2 tan( ξ+α 2 ) − tan( α 2 )(1 − tan2( ξ+α 2 )) 1 − tan2( ξ+α 2 ) + 2 tan( ξ+α 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' tan( α 2 ) (9) where tan(ξ + α 2 ) = tan(γ) + tan(θ) 1 − tan(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' tan(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' cos( π m) tan(γ) = tan(α 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' cos( π m) tan(θ) = � sin2( β 2 ) − sin2( α 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' sin2( π m) cos( β 2 ) (10) (11) (12) and m is the number of supplementary FSO transceivers set around the principal one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' We can remark that Rext and thus LJ depend on α, β and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Hereafter, Rext is sometimes denoted by Rext(α, m, β) and LJ by LJ(α, m, β) to express these dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=" 4 Principle coverage circle 0 K' K 2T m Extended coverage circle3 Problem of designing minimal cost HAP network There are several costs in a HAP network, such as investment, en- ergy, and maintenance costs." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Based on the expected life duration and maintenance cycle of a HAP, these costs can be distributed by day as 1) daily amortization cost representing investment cost, 2) average daily maintenance cost, and 3) daily energy cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Con- sequently, the problem of minimizing network cost becomes min- imizing the daily network cost, which comprises these three com- ponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Following variables are introduced for formulating mathemati- cally the daily network cost: K: Number of HAPs in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The HAPs are indexed by i ∈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='.K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' niF i : Number of FSO transceivers used on HAPi for inter- HAP communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' nsF i : Number of serving FSO transceiver of HAPi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Let ζday H and ζday F be constants that express the daily amortiza- tion costs of a HAP and an FSO transceiver, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' These costs are defined as the ratio of the prices of the HAP or FSO transceiver to their expected lifetime duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Then, the overall daily amortization cost of the HAP network is: Kζday H + ( K � i=1 nsF i + K � i=1 niF i )ζday F (13) To evaluate the daily maintenance and energy costs, we need to consider the HAP design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' HAPs are classified into two categories based on the underlying physical principle that provides the lifting force for the HAPs: aerodynamic (the HAP is heavier than air) and aerostatic (the HAP is lighter than air).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' While aerostatic platforms use buoyancy to float in the air, aerodynamic platforms use dynamic forces created by movement through the air [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' In general, both aerostatic and aerodynamic systems require a “flying energy” to keep the HAP relatively stable for maintaining FSO communication between HAPs and that between HAPs and FSO ground nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' An aerodynamic system requires a large propulsion power to move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Aerostatic systems typically consume less energy than aerodynamic systems do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' To be able to operate for a long duration in space, HAPs are mainly unmanned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' HAPs are equipped with different energy resources such as on- site production (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=', solar energy harvested by solar panels) or rechargeable energy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=', batteries or fuel cells brought from the ground).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Solar energy-based HAPs can operate continuously in space until they are lowered for maintenance purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Recharge- able energy-based HAPs are lowered once the reserved energy is depleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' In brief, the continuous in-space working duration of a HAP is limited by its available energy, which is relatively fixed by the HAP design, its energy consumption level, which varies de- pending on the payload weight and communication of the HAP, and its maintenance cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' We define the maintenance cost of a HAP as the expense of low- ering the HAP to perform technical maintenance, energy recharge on the ground, and then reinstall it in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Let di be the number of days on which HAPi can operate con- tinuously in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Let ζmtn be constant expressing the cost of one time lowering a HAP, maintaining it, recharging it, and then reinstalling it in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The daily maintenance cost of the HAP network is K � i=1 ζmtn di (14) Regarding the daily energy cost, we consider solar energy to be free, whereas the solar panel cost is counted in the cost of the HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The cost of rechargeable energy is part the maintenance cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' As a result, the energy cost does not explicitly represent the total cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Nonetheless, the energy consumption level of a HAP affects its in-space working duration di;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' therefore, we analyze this in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Combining (13) and (14), we obtain the following overall daily cost of the HAP network: Cost = Kζday H + ( K � i=1 nsF i + K � i=1 niF i )ζday F + K � i=1 ζmtn di (15) The problem of minimizing daily cost of the HAP network is stated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Given input parameters including – NFSO: Set of ground FSO nodes and their coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The number of nodes in the set is denoted as |NFSO|, – M: Data traffic to be carried between ground FSO nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' This is the list of bandwidth demands between the ground nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Outputs to seek are – A HAP network with HAP locations and inter-HAP links, – Beam width to set to each serving FSO transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Optimization objective is – Minimizing the daily cost expressed in (15) of the HAP network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The following two remarks drive us to conduct further analyses in subsequent sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' First, if a HAP has self-sufficient solar energy, its in-space working duration di is not limited by its energy consumption but depends uniquely on the maintenance cycle of the HAP, which is usually constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' In Section 4, we show the daily energy consumption of a HAP and the constraint that a HAP needs to respect to rely solely on solar energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Second, the cost of the HAP network increases with an increase in the number of FSO transceivers and HAPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The number of HAPs can be reduced by increasing ground coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' To increase ground coverage, more serving FSO transceivers can be used on each HAP, but this introduces greater energy consumption and ex- tra amortization cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Section 5 focuses on identifying the optimal configuration for serving FSO transceivers on a HAP to achieve a minimal HAP network cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 4 Daily energy consumption of a HAP with payload Several parameters affect the power consumption of a HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The descriptions and notations of these parameters are listed in section Energy parameters of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Most parameters were set based on industrial experimental projects such as the Loon project [4], Stra- tobus project [6], and other studies listed in the reference column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='1 presents the choice of parameter values in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 5 Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' nota- tions Descriptions Values References Cost related parameters ζday H Daily amortization cost of a HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 100 ζday F Daily amortization cost of an FSO transceiver on HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 10 ζmtn Cost of one-time maintenance of a HAP including lowing it down, 1000 maintenance, charging and reinstall it in the stratosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Dm Maintenance cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 365 days [6] Energy parameters Esolar Minimum daily harvested solar energy by a HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 42 - 290 kWh [12] ρavion Power consumed by the avionic part of a HAP to carry an unit of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 2 W/kg ρHCM F Power for heating, cooling, and management for each FSO on HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 20 W [4] ρPAT Power consumed by a PAT system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 15W [13] ρinter F Power consumed by inter-HAP FSO transceivers for laser source (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='1 W), heating/cooling/management (20 W) and PAT (15 W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='1 W [4] Inter-HAP FSO link parameters C2 n Atmosphere structure parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='0 × 10−18m−2/3 Attenuation coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='5 × 10−6m−1 [4] Coupling loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 45 dBm Transmitted power of an inter-HAP FSO transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='1 W [4] Receiver aperture diameter of an inter-HAP FSO transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='037 m [4] Beam width of an inter-HAP FSO transmitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 280 µrad [4] HAP-ground link parameters and variables σ Attenuation coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='5 × 10−6m−1 ρFSO tx Transmitted power of the laser source of a serving FSO transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 1 Watt Rrx Receiver aperture radius of a ground FSO transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='05 m SONABeam [1] ρrx Required received power at a ground FSO transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='10−8 W 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='1 dBm in [4] Other parameters H Elevation of HAPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 20 km LHH Maximum length of an inter-HAP link so that its BER is under δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 88 km δ BER threshold for inter-HAP links and lightpaths between HAPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' W The number of wavelengths in WDM technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 40;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 80 µH Platform mass excluding FSO transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='5 kg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 500 kg [4] µF FSO transceiver mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='3 kg [4] Table 1: Parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Greek characters are used for denoting constant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Let us consider the power consumption of a single HAP Hi that has m serving FSO transceiver and niF i inter-HAP FSO transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The power consumption includes: P avion Hi : Power draw of avionic part for maintaining Hi with payload in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' P down Hi : Power draw of all serving FSO transceivers on HAP Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' This power includes the heating/cooling/management power, laser transmitted power of all serving FSO transceivers on the HAP, and the power consumed by the Pointing Acqui- sition and Tracking (PAT) system of the HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' P inter Hi : Power draw of all inter-HAP FSO transceivers on HAP Hi for inter-HAP communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The power includes the heating/cooling/management, and PAT power for each inter-HAP FSO transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Inter-HAP FSO transceivers are oriented towards different remote HAPs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' therefore, each transceiver must have a PAT system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The total daily energy consumption (by 24 hours) of Hi is Econsum = (P avion Hi + P down Hi + P inter Hi ) × 24 (16) To breakdown further P avion Hi , P down Hi , and P inter Hi , we introduce following parameters: ρavion: Power consumed by the avionic part of the HAP to carry a unit of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' ρFSO tx : Transmitted power of each serving FSO transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Because the current power of laser source is limited to 1 W, which is very small in comparison with the power consumed by other factors on the HAP, we consider that ρFSO tx = 1 W, regardless of the beam width of the serving FSO transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' ρHCM F : Power draw for heating, cooling, and management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' It is also considered constant for each serving FSO transceiver and is set to ρHCM F = 20 W, according to reference [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' ρPAT : Power draw for Pointing, Acquisition and Tracking activity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' it is another constant and is set to ρPAT = 15 W [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' A HAP system uses a single PAT for its set of serving FSO transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' ρinter F : Power draw of a single inter-HAP FSO transceiver including communication, heating, cooling, management, and 6 PAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' According to [4], 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='1 W laser power is sufficient for an inter-HAP communication of 100 km distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' In this study, we limited the inter-HAP link length to less than 100 km and considered the laser power constantly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='1 W regardless of the distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Therefore, ρinter F = ρHCM F + ρPAT + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' µH: Mass of the HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' µF: Mass of an FSO on the HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Assuming that P avion Hi is linearly proportional to the weight of the HAP by ρavion, P avion Hi = [µH + (nsF i + niF i )µF]ρavion (17) P down Hi is the sum of the power consumed by serving FSO transceivers and PAT activity of the HAP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' thus, P down Hi = nsF i (ρFSO tx + ρHCM F ) + ρPAT (18) P inter Hi is the sum of the power consumed by inter-HAP FSO transceivers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' thus, P inter Hi = ρinter F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='niF i (19) Substituting (17), (18), and (19) into (16), we obtain the daily power consumption of a HAP as Econsum = {[µH + (nsF i + niF i )µF]ρavion + nsF i (ρFSO tx + ρHCM F ) + ρPAT + ρinter F niF i } × 24 (20) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='1 Necessity of solar energy and utilization constraint Current HAPs mainly use energy from solar panels mounted on HAP wings and/or energy from batteries or hydrogen fuel cells (HFC) onboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Solar energy can be harvested and charged into batteries during the day for nighttime use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Harvested solar energy varies with year time and location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' According to the experiments in [12], in York, UK, the harvested solar power is 42–80 kWh/day, and in Enugu, Nigeria, it is 290–545 kWh/day, depending on the size of the solar panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Figure 7 depicts the total daily energy consumption of a HAP, calculated from (20), versus the number of serving FSO transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Parameters were ρavion = 2 W/kg, ρPAT = 15 W, HAP weights µH = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='5 kg or 500 kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The HAP carried 10 inter-HAP FSO transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The referenced daily solar en- ergy levels were the minimum daily solar energy levels in York and Enugu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' From a certain number of serving FSO transceivers, a HAP consumes more energy than the harvested solar energy in York, and an HFC would be necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Owing to the limited payload capacity of a HAP, its HFC capacity is also very limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' According to [8], the current state-of-the-art fuel-cell density is approximately 1600 Wh/kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' A lightweight HAP, such as a Google balloon weights 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='5 kg, cannot carry heavy long-lasting fuel cells on board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The larger HAP Stratobus can carry up to 450 kg, but it weights already 7 tons leading to high energy consumption for flying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Even if the Stratobus payload capacity is reserved for the HFC, its energy would quickly run out within a few days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Based on this observation, we believe that long-duration flights should consider solar energy as the principal energy source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' In this case, the power consumption of a HAP with payload must not 0 50000 100000 150000 200000 250000 300000 350000 0 20 40 60 80 100 Total daily energy consumption (W-hr) Number of serving FSO transceivers µΗ=28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='5 kg µΗ=500 kg Min solar energy at York Min solar energy at Enugu Figure 7: Energy consumption by a HAP with different num- ber of serving FSO transceivers in comparison with the min- imum harvested solar energy at York and Enugu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' ρavion = 2/kg W and ρPAT = 15 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' exceed the daily harvested solar energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Let the daily harvested solar energy be Esolar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' then, � [µH + (nsF i + niF i )µF]ρavion + ρPAT + nsF i (ρFSO tx + ρHCM F ) + ρinter F niF i � ≤ Esolar 24 (21) According to Figure 7, solar energy provision does not need to be very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' A solar energy level between the minimum harvested in York and Enugu allows a 500 kg HAP to carry at least 6 serving FSO transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' A HAP can carry hundreds FSO transceivers with more than 125 kWh solar energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Therefore, it is realistic to rely on the solar energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Hereafter, we consider that HAPs solely use solar energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Despite self-sufficient solar energy, HAPs still need to be lowered periodically for maintenance, for example, after one year in the case of Stratobus [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Let us denote the maintenance cycle as a constant Dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Then di = Dm, ∀i ∈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='.K (22) 5 Optimal mFSO configuration Using multiple serving FSO transceivers increases the expense of FSO transceivers, although it can reduce the expense of HAPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' This section aims to determine the mFSO configuration that min- imizes the HAP network cost defined in (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' We assume that all HAPs use identical mFSO configurations, that is, identical prin- cipal beam width α, supplementary beam width β and number of supplementary serving FSO transceivers m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Let us now consider the dependence of the HAP network cost on mFSO configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' As each HAP has m supplementary serving FSO transceivers and uses only solar energy, the cost (15) becomes Cost = Kζday H + (Km + K � i=1 niF i )ζday F + Kζmtn Dm Cost is a function of K, m and niF i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' K depends on the coverage radius Rext(α, m, β) of the mFSO configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' niF i , as the num- ber of inter-HAP links of HAP i, depends on the traffic demand 7 set M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Hence, Cost depends on mFSO configuration and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' It is difficult to determine the optimal mFSO configuration without considering M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' To relax the dependance on M, we estimate Cost by a function that depends solely on mFSO configuration, that is, tuple (α, m, β);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' then try to find an instance (α, m, β) minimizing the estimated cost in expecting that the instance also drives the real cost to a minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='1 Cost estimation Figure 8: A ground area is divided into grid of square cells;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' each cell is circumscribed by a circle representing a serving zone of a HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' First, we estimate the number of HAPs K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Samples of the estimation are datasets with uniformly distributed ground nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Let S be the surface of the ground area containing those nodes, and W the number of wavelengths in the WDM technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' We divide the ground zone S into a grid of square cells of size ℓ × ℓ, each one will be covered by a HAP (see Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' To be served by a HAP, a cell must satisfy the following two conditions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' A cell can contain at most W ground nodes because a HAP can use at most W wavelengths to serve ground nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Owing to the uniform distribution of ground nodes, we have ℓ2 S |NFSO| ≤ W 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' A cell must be contained inside by a circle radius equivalent to the extended radius Rext of a HAP ℓ ≤ √ 2Rext The maximum number of HAPs required to cover region S is the number of cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Let this number be ˆK, then, ˆK = S ℓ2 = ⌈max {|NFSO| W , S 2R2 ext }⌉ (23) Hence, ˆK is an overestimation of the number of HAPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Next, we estimate the value of niF i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Let V be the maximum number of inter-HAP links that a HAP may have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Then niF i ≤ V, ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Finally, Cost can be overestimated as: � Cost = ˆK � ζday H + (m + V + 1)ζday F + ζmtn Dm � (24) � Cost is a function of Rext(α, m, β) and m while V is a parameter of the estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The estimation is more precise when V is set close to the actual number of inter-HAP links of a HAP, and coarser otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='2 Algorithms finding optimal configura- tion Given α and m, a larger β results in a larger Rext, and thus a smaller ˆK and � Cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Therefore, β should be set to the largest value according to (7) for a given α and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' It is worth noting that the value of β does not affect the solar energy consumption because the laser power ρFSO tx is small and is considered constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Determining the optimal configuration becomes finding the optimal values of α and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Algorithm 1 Find the optimal mFSO configuration 1: function Find-optimal-mFSO 2: niF i ← V 3: cMin ← ∞ ▷ cost min 4: αMax ← maximum α by (4) 5: for α = αMax .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 0 do 6: mMax ← calculated by (25) ▷ max m 7: mOpt ← 0 ▷ optimal m 8: for m = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' mMax do 9: β ← Beta-max(α, m) ▷ max β 10: Calculate Rext(α, m, β) using (9),(10),(11) (12) 11: Calculate � Cost(α, m, β) using (24) 12: if � Cost < cmin then 13: cmin ← � Cost 14: αOpt ← α ▷ optimal α 15: mOpt ← m ▷ optimal m 16: βOpt ← β ▷ optimal β 17: end if 18: end for 19: end for 20: return αOpt, mOpt, βOpt 21: end function Algorithm 2 Find the maximum β given α, m 1: function Beta-max(α, m) 2: for β = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 180 do 3: Calculate Rext(α, m, β) using (9),(10),(11) (12) 4: Calculate LJ using (8) 5: Calculate P rx J using (6) 6: if P rx J ¡ρrx then ▷ Looking for the first β violate constraint (7) 7: return β-1 ▷ the previous trial β was the maximum 8: end if 9: end for 10: end function Following an exhaustive search approach, we examine all possi- ble values of α and m to seek for the pair that minimizes � Cost in (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The search range of α is from 0° to the maximum 8 Serving zone of a HAPvalue set by constraint (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The number of supplementary serving FSO transceivers m is also limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Indeed, since the number of inter-HAP links of a HAP can go up to V as set in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='1, and nsF i = m + 1, ∀i, then from the energy constraint (21), we deduce the upper bound for m: m ≤ Esolar 24 − (VµFρavion + Vρinter F + µHρavion + ρPAT) µFρavion + ρHCM F + ρFSO tx − 1 (25) Algorithm 1 implements the exhaustive search idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' First, two nested loops scan all possible values of α satisfying constraint (4) and all possible values of m satisfying (25) to find the pair that minimizes � Cost in (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' For each pair (α, m), the largest value of β according to constraint (7) is selected using Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The optimal mFSO configuration is reported by the algorithms as (αOpt, mOpt, βOpt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Algorithm 2 finds the maximum β that satisfies constraint (7) for a given pair of (α, m) by testing the possible values of β in- creasingly from 0 until the received power P rx J at the border of the extended coverage area reaches the required received power ρrx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The received power P rx J is calculated using the set of equa- tions (6), (8), (9),(10),(11), and (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' In the implementation of both algorithms, α and β step by 1° after each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Finer stepping allows obtaining more accu- rate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' However, even with 1° stepping, the variation in the optimal Rext is only a few hundred meters, which is negligible in comparison to the absolute value of Rext which is in the range of 6-30 kilometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The complexity of Algorithm 1 is O(m) because α ≤ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The complexity of Algorithm 2 is constant because β ≤ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 6 Design HAP network topology This section presents the HAP network design using the optimal configuration identified above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Let denote Linter as the number of inter-HAP links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Since �K i=1 niF i is the total number of inter- HAP FSO transceivers, it is equal to 2Linter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The network cost becomes: Cost = Kζday H + (K(m + 1) + 2Linter)ζday F + K ζmtn Dm and is equivalent to Cost = K � ζday H + (m + 1)ζday F + ζmtn Dm � + 2Linterζday F (26) The cost is proportional to the number of HAPs K and the number of inter-HAP links Linter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' We consider that the daily amortization cost of a HAP is much greater than that of an FSO transceiver;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' thus, the coefficient of K is much greater than the coefficient of Linter in Cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Consequently, K should be prioritized to minimize over Linter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Therefore, the topology design is broken into following two steps: i) ground nodes are clustered into equal radius circles that will become serving zones of HAPs in such a way that the number of clusters is the smallest for minimizing K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' ii) corresponding HAPs are located at the centers of clusters but at an elevation of 20 km and are interconnected by the fewest number of inter-HAP links, Linter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' A HAP network topology design algorithm was proposed in [7] following these two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' In this algorithm, the clustering radius was not determined but was left as an input of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' In the current study, we set the clustering radius as the extended coverage radius Rext of the optimal mFSO configuration to drive towards a HAP network with minimal Cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The main steps of the Figure 9: HAP network design flowchart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' HAP network design process are presented in Figure 9, where the steps taken from [7] are shown in color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The process is explained as follows: Initialize V, the maximum number of inter-HAP links of a HAP, by a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Calculate the optimal mFSO configuration using Algorithm 1, and set the clustering radius as its Rext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Apply the clustering algorithm proposed in [7] to distribute ground nodes into clusters of radius Rext while keeping the number of ground nodes in each cluster under W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Each cluster becomes a serving zone of a HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The HAP is located at the center of the cluster but at an elevation of 20 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Bandwidth demands between ground nodes belonging to dif- ferent serving zones are bundled into lightpaths between corresponding HAPs, creating the inter-HAP traffic matrix MHAP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Apply HAP topology design algorithm proposed in [7] to build the HAP topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The algorithm begins with an empty topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' It finds a route for each lightpath demand of MHAP 9 Init V Find optimal conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' (α, m, β) and Rext Clustering ground nodes with Rext radius [7] V=V+1 Calculate inter-HAP Design HAP topo [7] M HAF No is routed entirely Report HAP topofrom a full-mesh graph linking all HAPs within communica- tion distance limit LHH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Each time a lightpath uses an inter- HAP link that has not yet been included in the current HAP topology, the link is incorporated into the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The link in the topology is prioritized for use in building routes for the next lightpath demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Once all lightpath demands in MHAP are routed, the final topology is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Otherwise, routing may fail due to the low connectivity between HAPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' In this case, V is increased by one, and the process is repeated until all lightpath de- mands in MHAP are routed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 7 Simulation results The algorithms for finding the optimal mFSO configuration were implemented and integrated with the topology designed algorithm described in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' We performed simulations with practical parameters and evaluated the efficiency of mFSO configuration compared to the single serving FSO transceiver configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='1 Parameter values The simulation parameters are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The values of these parameters were chosen according to experiments reported in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' This subsection explains the choices of the pa- rameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Cost-related parameters: The cost-related parameters are set such that the daily amortization cost of a HAP is significantly greater than that of an FSO transceiver, and the one-time mainte- nance cost is significantly higher than the daily amortization cost of a HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The maintenance cycle of a HAP is set as Dm = 1 year according to published information on Stratobus [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Energy-related parameters: Esolar - daily harvested solar energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' We considered daily solar energy levels between the minimum daily solar energy values in York and Enugu reported in [12], which were 42 kWh and 290 kWh, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' ρavion - power consumed by the avionic part of a HAP to carry a unit of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Although the power-to-mass ratio can be estimated as 6 W/kg according to [12], the published power rates of real systems are smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' For aerodynamic systems such as Zephir-S, Zephir-T [14], and Phasa-35 [15], ρavion varies from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='68 -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='04 W/kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Indeed, Zephir-S weighs 80 kg (75 kg platform and 5 kg payload) and consumes 243 W, Zephir-T weighs 160 kg (140 kg platform and 20 kg payload) and consumes 429 W, and Phasa-35 weighs 165 kg (150 kg platform and 15 kg payload) and consumes 459 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Aerostatic systems consume even less power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The Stratobus weighs 7000 kg and consumes 5 kW when it carries a 250 kg payload and 8 kW when it carries 450 kg [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Thus, the power-to- mass ratio of Stratobus is between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='69 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='07 W/kg only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Therefore, in this simulation ρavion was set to 2 W/kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' ρPAT - power consumed by a PAT system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' it was set to 15 W according to [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' ρHCM F power consumed for heating, cooling, and manage- ment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' it was set to 20 W according to [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' ρinter F power consumed by an inter-HAP FSO transceiver;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' it was set to 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='1 W including laser power, ρHCM F and ρPAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Inter-HAP link parameters: These parameters were set to values similar to those provided in the Loon project [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' HAP-ground FSO link parameters: The attenuation coefficient of an FSO link between a HAP and a ground node is set identical to that of inter-HAP links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The required received power ρrx at a ground node was set according to [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The aperture radius Rrx of a ground FSO receiver was set according to the commercial FSO transceiver SONABeam [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Other parameters: δ - BER threshold for inter-HAP links and lightpaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' We set δ = 10−3 because errors with that BER can be corrected using current Forward Error Correction (FEC) techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' LHH - the maximum allowable distance between two HAPs such that the BER of an inter-HAP link is less than δ = 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Using the inter-HAP FSO link parameters listed in Table 1, the calculation yielded LHH = 88 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' µH - platform mass;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' it varies significantly from one design to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The Loon balloon weighs just 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='5 kg while the Stratobus weighs 7000 kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' With ρavion = 2 W/kg, a HAP weighing more than 7000 kg already consumes 326 kWh/day to carry itself, which is more than the maximum harvested solar energy, leading to no remaining energy to carry FSO transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Therefore, µH = 500 kg was used in the simula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' µF - mass of an FSO transceiver on HAPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' It was set accord- ing to the FSO transceiver used in the Loon project, which weighs 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='3 kg [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' This value is consistent with the weights between 8 and 10 kg of commercial terrestrial SONABeam FSO transceivers [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' W - the number of wavelengths per FSO link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' It was set to 40 or 80 according to the current WDM technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The test dataset contained 19 test cases, each with 400 – 2800 ground nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The ground FSO node locations were randomly generated on a square surface of 100 × 100 km, which is the size of a large metropolis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The test cases had different numbers of ground nodes, reflecting different ground node densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The traf- fic requirement M contained demands randomly generated between ground FSO nodes such that the total incoming or outgoing traffic of a ground FSO node did not exceed 1 Gbps, which is the capacity of a single wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Initially, V was set to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The optimal multiple serving FSO transceiver configuration (α, m, β) was calculated using Algo- rithms 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The extended radius Rext of the optimal configura- tions was calculated using (9) and was then used as the clustering radius in the HAP topology design step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' With Esolar = 42 kWh and W = 40, V must be increased to 12 to get all demands in MHAP routed successfully for all test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' With all other Esolar and W values, the topology design algorithm successfully routed all demands in MHAP for all test cases right with initial V = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Figure 10 illustrates the HAP locations and their footprints calculated using the proposed algorithms for a test case of 1005 ground FSO nodes, Esolar = 75 kWh, and W = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='2 mFSO configuration versus single serv- ing FSO transceiver configuration Table 3 lists the maximum beam width αmax according to (4) and the maximum ground coverage radius of the single serving FSO transceiver configuration when the receiver aperture radius 10 Esolar = 42 kWh Esolar = 50 ∼ 290 kWh W = 40,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' V = 12 W = 80,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' V = 10 W = 40,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' V = 10 W = 80,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' V = 10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='|NFSO| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='Rext ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='Cost ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='37 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='0 6691 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='23346 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='37 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='0 6691 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='19926 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='37 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='8946 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='20284 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='37 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='11929 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='11178 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='Table 2: Optimal configurations and costs of all test cases with Rrx = 2 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Receiver aperture Maximum beam Maximum radius Rrx (m) width αmax coverage radius (m) 2 37 ° 6691 4 67 ° 13237 Table 3: Maximum beam width and coverage radius of single serving FSO transceiver configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' was varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Table 4 lists the extended coverage radius of the max- imum mFSO configuration for different solar energy levels and receiver aperture radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The maximum mFSO configuration was obtained using the largest principal beam αmax, largest m accord- ing to (25), and largest β according to (7), given αmax and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The coverage radius of the maximum mFSO configuration was ex- tended approximately twice in comparison with that of single FSO transceiver configuration, except for Esolar = 42kWh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' When solar energy level increased, the maximum m increased;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' thus, the ex- tended coverage radius increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' However, when m was already large, the extention increased slowly with m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Additionally, the maximum extended coverage was much larger when Rrx = 4 than Rrx = 2m because a receiver can accept weaker signals with larger apertures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' To compare the network costs incurred by the two configura- tions, we examined the detailed results in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The table lists the optimal mFSO configurations and network costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' When Esolar = 42kWh, the optimal number of supplementary serving FSO transceivers is m = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' thus, the configuration uses a single serving FSO transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Therefore, these cases were used as ref- erences for single serving FSO transceiver configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' When Esolar > 50kWh, all optimal configurations were truly mFSO, and the results were identical for all solar energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The numbers indicate that mFSO configuration offered significantly Esolar Max Rext (m) (kWh) Max m Rrx = 2 (m) Rrx = 4 (m) 42 6 6691 13237 50 16 12174 25582 75 47 13559 28403 100 78 13678 28845 125 109 13711 28969 150 140 13724 29020 175 171 13731 29047 200 202 13735 29062 225 233 13738 29071 250 264 13739 29077 275 295 13740 29082 290 314 13741 29084 Table 4: Maximum extended coverage radius of mFSO con- figuration when V = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' lower costs (listed in columns 16th and 21th) than those of single serving FSO transceiver configuration (listed in columns 6th and 11th) for the same test cases and number of wavelengths W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The costs resulting from mFSO configuration were as low as 54–87% of those resulting from single serving FSO transceiver configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' These numbers confirm that when there is sufficient solar energy, mFSO configuration is definitively a better choice than single serv- ing FSO configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 11 0 20 40 60 80 100 120 0 20 40 60 80 100 y-axis x-axis Figure 10: Footprints of HAPs with mFSO configuration ob- tained from the topology design for a test case of 1005 ground FSO nodes when Esolar = 75 kwh, W = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' A circle repre- sents an extended coverage area of a HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Small points in- side the circle are ground nodes and the dot at the center of the circle is the projected location of its serving HAP on the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='3 Factors impact optimal mFSO configu- ration Comparing the values of the optimal extended coverage radius in Table 2 and the maximum extended coverage radius in Table 4, we can see that the optimal extended coverage radius was gener- ally not the maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' This is reasonable because the maximum configuration uses an excessive number of supplementary serving FSO transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Low solar energy may render mFSO configuration impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Indeed, Esolar = 42 kWh could afford maximally 6 supplemen- tary serving FSO transceivers (see Table 4), which was too few to entirely cover the contour of the principal coverage area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Thus, single FSO transceiver configuration was the unique choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' When the solar energy level exceeds 50 kWh, its exact value does not affect the optimal configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The simulation showed that the optimal configurations were identical for all solar energy levels from 50 kWh/day and above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' This is explained by the fact that a greater solar energy level allows to accept configurations with large coverage but may be more expensive because of using more supplementary serving FSO transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' As a result, large con- figurations were not selected as optimal configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' In other words, increasing solar energy does not necessarily improve the HAP network cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Since the optimal multiple serving FSO transceiver configura- tions were identical for all Esolar ≥ 50 kWh, all other numerical results related to topology design and routing with these solar en- ergy levels were identical and are presented as single results in subsequent figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The coverage of the optimal configurations decreased when the ground nodes became denser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Indeed, test cases with large num- bers of ground nodes had greater ground node densities, and columns 13th and 16th of Table 2 shows that the optimal m and Rext decreased when the density increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The reason is that, with a greater ground node density, is a small ground region al- ready contains W ground nodes, which is the maximum serving capacity of a HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Therefore, a HAP could serve only a small zone and required only a few supplementary FSO transceivers to cover the zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='Numbers of HAPs and inter-HAP links ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='140 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='160 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='Number of HAPs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='Number of ground FSO nodes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='ˆK for Esolar=42 kWh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='K for Esolar=42 kWh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='ˆK for Esolar ≥ 50 kWh kWh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='K for Esolar ≥ 50 kWh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='Lower bound ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='(a) W=40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='140 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='160 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='Number of HAPs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='Number of ground FSO nodes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='ˆK for Esolar=42 kWh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='K for Esolar=42 kWh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='ˆK for Esolar ≥ 50 kWh kWh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='K for Esolar ≥ 50 kWh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='Lower bound ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='(b) W=80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='Figure 11: Number of HAPs and lower bound with (a) W = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='40 and (b) W = 80 in different solar energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Since each HAP can serve at most W ground FSO nodes, a lower bound for the number of HAPs is: nLB HAP = |NFSO| W (27) Figure 11 shows the number of HAPs, the estimated number of HAPs ˆK and lower bound nLB HAP when (a) W = 40 and (b) W = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' With Esolar ≥ 50 kWh, the actual number of HAPs was almost identical to ˆK in both subfigures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Furthermore, when W = 40 12 0 100 200 300 400 500 600 0 500 1000 1500 2000 2500 3000 Number of inter-HAP links Number of ground FSO nodes Esolar=42 kWh Esolar ≥ 50 kWh (a) W=40 0 100 200 300 400 500 600 0 500 1000 1500 2000 2500 3000 Number of inter-HAP links Number of ground FSO nodes Esolar=42 kWh Esolar ≥ 50 kWh (b) W=80 Figure 12: Number of inter-HAP links when (a) W = 40 and (b) W = 80 for different solar energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' and Esolar ≥ 50 kWh, the number of HAPs approached the lower bound starting from test cases with 1000 ground nodes or above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' This implies that the number of HAPs was almost optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Figure 12 presents the absolute numbers of inter-HAP links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The number of inter-HAP links increased with the number of ground nodes, because the network size and traffic demand in- creased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The number of inter-HAP links clearly decreased when the wavelength density increased from W = 40 to W = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' In other words, denser WDM technique helps reduce the number of inter-HAP FSO transceivers and consequently the network cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' mFSO configuration allows reducing significantly both the num- bers of HAPs and inter-HAP links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Indeed, according to Figure 11, the number of HAPs was much smaller with Esolar ≥ 50 kWh where mFSO configuration was used, in comparison with Esolar = 42 kWh, where single serving FSO configuration was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' A similar phenomenon is observed in Figure 12 for the num- ber of inter-HAP links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='10000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='15000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='20000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='25000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='30000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='35000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='40000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='Cost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='Number of ground FSO nodes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='Cost of Esolar = 42 kWh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='Cost of Esolar = 42 kWh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='Cost of Esolar ≥ 50 kWh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='Cost of Esolar ≥ 50 kWh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='(a) W=40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='10000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='15000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='20000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='25000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='30000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='35000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='40000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='Cost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='Number of ground FSO nodes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='Cost of Esolar = 42 kWh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='Cost of Esolar = 42 kWh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='Cost of Esolar ≥ 50 kWh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='Cost of Esolar ≥ 50 kWh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='(b) W=80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='Figure 13: Real costs and overestimated costs with W = 40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='and W = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='5 Quality of cost estimation Figure 13 presents the estimated and actual costs for different solar energy levels and wavelength densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The estimated cost was very close to the actual cost, mostly for Esolar ≥ 50kWh and W = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Parameter V, the threshold of the number of inter-HAP links of a HAP, affects the quality of the cost estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' To evaluate the choice of V, we compared it with the number of inter-HAP links that a HAP finally has.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Figure 14 shows the average number of inter-HAP links per HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' When there were 40 wavelengths per link, the average number of inter-HAP links per HAP varied between 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='7 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='3 for Esolar ≥ 50 kWh and V = 10, and between 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='8 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='8 for Esolar = 42 kWh while V raised up to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Hence, the value of V was close to the actual number of inter-HAP links required by a HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' However, when there were 80 wavelengths per link, the average number of Inter-HAP links per HAP was reduced to between 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='4 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='4, which is slightly far from the threshold V = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' A smaller V may help better estimate of the optimal cost in these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 13 0 5 10 15 20 0 500 1000 1500 2000 2500 3000 Average number of inter-HAP links per HAP Number of ground FSO nodes Esolar=42 kWh Esolar ≥ 50 kWh (a) W=40 0 5 10 15 20 0 500 1000 1500 2000 2500 3000 Average number of inter-HAP links per HAP Number of ground FSO nodes Esolar=42 kWh Esolar ≥ 50 kWh (b) W=80 Figure 14: Number of inter-HAP links per HAP when (a) W = 40 and (b) W = 80 for different solar energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 8 Conclusions Using mFSO configuration widens a HAP footprint, however, its application is constrained by the available solar energy of the HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Moreover, mFSO configuration may imply an extra investment cost due to additional serving FSO transceivers in comparison with single FSO transceiver configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' This study focused on de- termining the optimal mFSO configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' First, we proposed a set of closed-form expressions for computing the coverage of an mFSO configuration in terms of beam widths of the princi- pal and supplementary transceivers and number of supplementary FSO transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Second, we proposed an algorithm to determine the optimal mFSO configuration that minimizes the total HAP network cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Third, we designed a HAP network topology using the optimal configuration to achieve a minimal final cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The simulation results showed that mFSO significantly ex- tended the HAP footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' With the testing dataset, the extended footprint radii were generally two times larger than the single FSO transceiver footprint radii, leading to a four-fold larger coverage surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The network cost with the optimal mFSO configuration was as low as 54% of the network cost when using a single serving FSO transceiver on a HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Acknowledgements This research was funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='02-2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' References 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Horwath, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Leeb, “Optical Communications for High-Altitude Platforms,” IEEE Journal of Selected Topics in Quantum Electronics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 16, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 1058– 1070, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' [14] Airbus, “Zephir: Persistance and flexibility.” https://lf5422.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='com/wp- content/uploads/2018/08/0296 18 2 zephyr datasheet e horizontal a4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='pdf, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Accessed Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' [15] BAE Systems, “Phasa-35.” http://prismaticltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='uk/products/phasa- 35/, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Accessed Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' A Proof of Lemma 1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Let x = cos(α/2), a = σH, and b = PtxR2 rx 2H2 then P rx j (x) = e−a/x bx2 (1 − x) (28) Calculate the derivative of P rx j (x) we get P ′rx j (x) = e−a/x � a 1 − x + 2x − x2 (1 − x)2 � b (29) Thus, the derivative of P rx j (α) is P ′rx j (α) = P ′rx j (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' (− sin(α)) (30) Beam α is limited between [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='.π] because it orients to the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Thus, x ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='.1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Consequently, 1 − x > 0 and 2x − x2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' In addition, a, b > 0, then P ′rx j (x) > 0 for all x ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='.1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Be- cause − sin(α) < 0, ∀α ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='.π], thus, P ′rx j (α) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Consequently, P rx j (α) decreases with α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' B Calculation of extended coverage radius of mFSO configuration This section identifies formulas that calculate the extended cover- age radius of an mFSO configuration characterized by the princi- pal beam width α, supplementary beam width β and number of supplementary beams m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Conventionally, the coverage provided by a bundle of transmit- ters is calculated as if the transmitters project perpendicular to the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' In mFSO configuration, the principal beam in the center is large, and it pushes the supplementary serving FSO transceiver projection directions far from perpendicular to the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' These supplementary beams form oblique cones that intersect with the ground plane in ellipses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Considering of the elliptical form adds more complexity to the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' In Figure 15, H denotes the position of a HAP, and its projec- tion on the ground plane is O, thus HO = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The principal beam forms a right circular cone whose axis is HO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The cone intersects the ground plane by a circle of radius Rα, which defines the prin- cipal footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The beam of a supplementary FSO transceiver is an oblique cone intersecting the ground plane by an ellipse that defines the corresponding supplementary footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The cone of the supplementary beam intersects with the cone of the principal beam by two lines: HK and HK′ where K and K′ are the two intersection points of the principal and supplementary footprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Thus, OK = OK′ = Rα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' m supplementary FSO transceivers are arranged evenly around the principal transceiver, each of which is responsible for extending the coverage within an angle of 2π/m from the center O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' The responsible angle of the supplementary FSO transceiver in Figure 15 is defined by rays −−→ OK and −−→ OK′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Thus, � KOK′ = 2π/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Ray −−→ OK intersects with the supplementary beam cone at J, then OJ is the radius of the extended coverage region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Readers refer to Figure 6 for a complete view of the extended coverage circle and the positions of K, K′ and J on the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Figure 15: Computation of the distance from supplementary FSO transceivers and the border of extended coverage area LJ in function of Beta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Since the principal beam width is α, then � OHK = α/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Let the base plane containing K and K′ of the supplementary beam cone cuts the cone axis at T, the primary cone axis HO at P, and HJ at J1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Then � THK = β/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' In addition, the supplementary cone intersects with this base plane by a circle containing K, K′ with center T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Let Rβ be the radius of the circle, then TK = TK′ = Rβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Let M be the midpoint of KK′ then H, O, T, M belong to the same plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Let ξ = � KHJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=" The extended coverage radius is Rext = OJ = 15 H β/2 LY a/2 Supplementarycone base plane K' R Pilm a Supplementary foot print Principal foot print GroundHO tan(� OHJ) = H tan( α 2 + ξ)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Thus, Rext = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' tan � 2(ξ + α 2 ) − α 2 � Rext = H2 tan( ξ+α 2 ) − tan( α 2 )(1 − tan2( ξ+α 2 )) 1 − tan2( ξ+α 2 ) + 2 tan( ξ+α 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' tan( α 2 ) (31) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='1 Calculation of tan( ξ+α 2 ) Let N be the midpoint of KJ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' As K and J1 are at the intersection of the supplementary cone and its base plane, HK = HJ1, HN ⊥ KJ1, and HN is the angle bisector of � KHJ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Therefore, � NHK = ξ/2, thus � NHP = ξ+α 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' In addition, since KO is on the base plane of the principal cone, HO ⊥ KO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Thus, △PNH and △POK are similar right triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Consequently, � OKP = � NHP = ξ+α 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Furthermore, tan(ξ + α 2 ) = OP OK = OP Rα (32) Let � OHM = γ and � THM = θ Then � OHT = θ + γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Because MO is on the base plan of the principal cone, MO ⊥ HO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' In addition, as PT is on the base plane of the supplementary cone whose axis is HT then HT ⊥ PT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Consequently, △PTH and △POM are similar right triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' We can deduce that � PMO = � PHT = θ + γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Therefore, tan(θ + γ) = OP OM = OP Rα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' cos( π m) Combining with (32) we deduce : tan(ξ + α 2 ) = tan(θ + γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' cos( π m) (33) Thus tan(ξ + α 2 ) = tan(γ) + tan(θ) 1 − tan(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' tan(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' cos( π m) (34) Since γ = � OHM then, tan(γ) = MO HO .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' From right triangle △OMK we have MO = OK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' cos( π m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' From right triangle △HOK we have HO = OK/ tan( α 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Thus tan(γ) = tan(α 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' cos( π m) (35) It remains to calculate tan (θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content='2 Calculation of tan (θ) Look at the right triangle △HTM, we can see that: tan(θ) = TM TH (36) Since K and K′ are on a circle centered at T, and M is the midpoint of KK′ then △TMK is a right triangle, then TM = � TK2 − KM 2 = � R2 β − R2α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' sin2( π m) (37) Easy to find that △THK is another right triangle then TH = TK/ tan(β 2 ) = Rβ/ tan(β 2 ) (38) Replacing (37) and (38) in to (36) we get tan(θ) = � R2 β − R2α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' sin2( π m) Rβ/ tan( β 2 ) = tan(β 2 ) � 1 − (Rα Rβ )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' sin2( π m) (39) From right triangle △HTK we obtain Rβ = HK sin( β 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' From right triangle △HOK we obtain Rα = HK sin( α 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Replacing these values to (39), we obtain: tan(θ) = � sin2( β 2 ) − sin2( α 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' sin2( π m) cos( β 2 ) (40) Substituting the values of tan(γ) in (35) and tan(θ) in (40) into (34), we obtain tan( ξ+α 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' Subsequently, replacing the obtained tan( ξ+α 2 ) to (31) we get Rext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'} +page_content=' 16' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}