Lightning Protection Design for Small Riverboats Using the FDTD Method

The Amazon region has high lightning occurrence rates, with many lightning hot spots. Different rivers cross the Amazon rainforests and the people that live there use these rivers as roads to go from one place to another. Every year, lightning deaths are reported in the Brazilian Amazon region. In 2018, a lightning discharge struck a small riverboat, which killed 4 people and injured 5 in the North region of Brazil. In this study, we investigate lightning fatalities in Amazon rivers and compute electric field distribution in a small riverboat due to a lightning strike using the finite-difference time-domain (FDTD) method. The results show that the electric field produced inside the boat is large enough to cause air breakdown. Additionally, we propose a cost-effective method to improve lightning safety for people that use small boats in the Amazon region.

The lightning protection of riverboats or small watercrafts in general is a challenging task. Their small dimensions, the different types of materials used in their construction, and the relatively poor electricity conduction of river water are some of the issues faced in the lightning protection design of small riverboats in the Amazon region. According to Thomson [7], about 3% or more of southwest Florida boats have their marine electronics damaged due to direct or indirect lightning strikes.
Unfortunately, there is no statics for lightning related damages on small riverboats for the Brazilian Amazon region. In the U.S, the lightning protections of boats are defined by different agencies including the American Boat and Yacht Council, the Coast Guard, the Florida Sea Grant, and the National Fire Protection Association. However, in Brazil, to the best of our knowledge, there is no guidance for lightning protection for small watercrafts.
Currently, there are significant efforts in lightning protection of electric power systems [8], telecommunication systems [9], [10], and structures [11]. However, there is limited information in the scientific literature on lightning protection of boats or small vessels in general. There are studies on large vessels such as ships. For instance, Nicolopoulou et al. [12] investigated the lightning protection zone in ships through impulse voltage tests on a scaled-down ship model. They found that the international instructions of marine regulations are in most cases insufficient. Nicolopoulou et al. [13] studied lightning-induced overvoltages in the electric network of a ship using the CST Cable Studio environment. They simulated lightning strikes in different parts of the ship and evaluated the induced overvoltage. Their major conclusion concerns improving the immunity testing of marine equipment.
In this work, we extend the analyses of Ferreira et al. [4] regarding the case study and include a survey regarding lightning safety in boats in the Amazon Region rivers (see the Appendix) to understand if the locals are aware of how dangerous lightning can be. Additionally, for the first time, we perform a numerical study of the electric fields inside the watercraft due to a direct lightning strike to assess risks to which passengers are subjected to. Subsequently, we design and optimize the performance of a lightning protection system for a common riverboat in the Amazon region by balancing the produced transient magnetic fields. Results are obtained by solving Maxwell's equations numerically with the FDTD method. Finally, it is shown that the proposed optimized lightning protection systems can improve the lightning safety of the riverboat occupants during thunderstorms.

II. METHODOLOGY
The riverboats are widely used in the rivers of the Amazon region, and they are very important for the locals. These riverboats are used in the economic activities of the region, as well as in the transportation of people from one city/village to another city/village. Hence, we tried to identify how the boat construction process takes place. Most of the riverboats (called "Catraia" by the locals) in the region are small and can support 5 to 25 occupants. Some boats have no roof, consisting only of the structure responsible for floating. The riverboat roofs are mostly composed of a layer of wood and a thin layer of metal. In some cases, an additional layer of plastic is added. In our simulations, we considered the riverboat with the roof made by a single layer of metal, the same in which 4 people died and 5 were injured due to a lightning strike in the state of Acre, Brazil (see Fig. 2).
Positions over the domain are indexed using integer indexes i, j, and k. The physical position of the cell reference node in the domain is given by x = i.Δx, y = j.Δy, and z = k.Δz. The electric field is calculated at time t = n.Δt and H  is computed at t = (n + ½).Δt. Therefore, (3) and (4) can be approximated by and 2) Precision and stability: The FDTD method, like other numerical methods, has the criteria to yield stable and accurate solutions. Precision is understood as the convergence of numerical solution to analytical or experimental solution and stability as finite-amplitude solutions for all t. Thus, the parameters that guarantee these criteria are the spatial and temporal increments. To guarantee stability, the time step must satisfy the Courant condition [15] where vmax is the greatest wave propagation speed in the analysis domain. In order to restrict errors associated with dispersion, ensuring accuracy, spatial increments must satisfy the criteria min , , , 10 x y z l D £ (8) where λmin is the shortest wavelength produced by the excitation source [15].
3) Lightning excitation source: In this work, the lightning return stroke current I(t) is represented by the set of equations given by Tanabe [16], which are a n t a n t f a n t a n t f I e e n t n t In (9), the amplitude was set to Imax = 1 kA, the raise and tail times were set to Tf = 0.22 µs and Tt = 139.8 µs, respectively, and the other constants are given by a1 = 0.693147180/Tf, a2 = 2.5584279 / Tt,  [17]. Uniaxial perfectly matched layer (UPML) [18], [15] was used to truncate the analysis region. To simulate the lightning discharge impinging on the vessel, a FDTD model was conceived with a wooden structure and a metallic roof, based on the description in [4]. The dimensions of the boat modeled in the simulations are given in Fig. 2, in which a graphical representation of the FDTD model is also shown. The electromagnetic properties of the materials used in this work are given in Table I and are found in [19], [20]. Electromagnetic properties of the river were measured in [20]. In order to represent the metal covering, a thin metallic plate was modeled in FDTD, as described in [15]. For representing the electrical discharge and its channel, it is used the method described in [17], in which the leader channel is represented by a thin cylindrical conductor parallel to the z-axis, starting from the point in which the discharge reaches the boat (the metal plate), up to the cell immediately below the FDTD mesh border, thus entering the UPML region.
This way, because of the absorption of the fields around the part of the conductor inside the absorbing region, the conductor can be considered to be virtually infinite in length [21]. At the attachment point, the magnetic field around the conductor is excited, as governed by Ampère's law (2), for implementing the current source.

A. FDTD numerical results
By analyzing the model of Fig. 2(b), which has no lightning protection and is often used in the Amazonian rivers, it is noted that the wooden beams are not good conductors and, thus, it is expected that electric fields are intense inside the watercraft during direct lightning strikes. This is illustrated by In order to transport charges from the roof to the water, the electrically conductive river water can be exploited by using metal rods connecting the metallic roof to a point beneath the water surface. As a result, the current conducted by a rod produces a magnetic field as given by Ampère's law. Using more than one rod, arranged vertically, parallelly to each other, their respective magnetic fields tend to be produced in opposite directions, consequently decreasing the z-component of the electric field inside the boat (see Fig. 4(b) and Fig. 6). It is difficult in the boat to approximate Ez to zero since there are lots of field reflections occurring during the electrodynamic processes described by Maxwell's equations (1) and (2) over time. Therefore, simulations were generated considering several different cases, as seen in the models of Fig. 7: increasing the number of rods per wooden beam and the number of beams with metallic rods. Observing Ez evolution over time in Fig. 8(a), obtained at the point P1 (0.8 m above the watercraft floor -see Fig. 7(a)), we notice that increasing the number of vertical metallic rods connected to the roof and in contact with water is, in fact, a method that drastically reduces the field intensity inside the vessel by a factor of about one hundredth, concerning the case shown in Fig. 5 (non-protected watercraft). The field can be reduced more efficiently by distributing the rods over the perimeter of the watercraft. Therefore, adding more rods per beam reduces the electric field inside the geometric center of the watercraft and near the beams due to magnetic field balancing. However, it is impracticable to add so many rods in this watercraft. We found the most cost-effective case the one with two rods on each of the six beams ( Fig. 7(d)), for which we estimate an additional weight of about 8.5 kg, which is negligible concerning the total weight of the riverboat. Moreover, adding more metallic parts increases cost, producing minor effects on field reduction as is seen in Fig. 8(a). Figure 9 shows the three-dimensional design of the watercraft with such lightning protection scheme. The local (Belém, PA, Brazil) cost for including this lightning protection was estimated to be between 10 and 20 U.S dollars in February 2020.  Have you ever been involved in a lightning accident in a watercraft? Q4 Have you ever heard of a lightning accident in a watercraft? Q5 Q6 Have you ever heard of lightning struck any watercraft in your region? Have you ever heard about lightning protection systems for watercraft?
The questionnaire survey was applied to a group of thirty-eight people, in which all of them were men. The interviews were conducted close to the river, in the ports of Belem and Ponta de Pedras. All interviewed people work in some way with activities related to small riverboats. Some of them were the owners of the boats and others worked transporting goods among the cities. The age ranges and the education levels of respondents (who are quantified in percentages) are given in Table A2. According to Table A2, the majority of respondents do not have a high educational level. Most of the people did not complete high school and they are adults. All the questions in the survey were Yes/No questions and they are shown in Table A1. The questionnaire started with a basic question about lightning. No deep knowledge regarding lightning physics was asked. Whether the interviewed knew how to recognize a lightning flash or thunder it was enough. Then, it was asked about selfprotection against lightning in watercraft. Whether the respondent claimed to remain inside the watercraft during thunderstorms, it was considered that the interviewer knew how to be protected.
According to Table A3, most of the respondents are aware of what lightning is. However, they do not know how to protect themselves during thunderstorms. Specifically, 65% of people who know what lightning is do not know how to be protected against lightning in watercraft. None of the respondents has ever suffered an accident with lightning on boats.
Regarding question 4 (Q4), only two respondents answered yes. For question 5 (Q5), twelve of them stated that they had heard about lightning accidents in watercraft. For question 6 (Q6), some of the respondents claimed to know about lightning protection for buildings, however, 32 of them have never heard about lightning protection for watercraft. Hence, there is a lack of knowledge on lightning protection available to watercraft and a lack of guidelines for local communities in the Amazon region.