Fault-Tolerant Relay Node Placement in Wireless Sensor Networks for Surveillance of Overhead Transmission Lines

Wireless sensor networks provide a promising technology to surveil overhead transmission lines in smart grid communication. However, a great challenge for researchers is proposed by adverse outdoor environments and essential requirements of strong as well as flexible smart grid communication. Specifically, nearly linear topology, limited energy, and tolerance for faults are entangled to make the surveillance system complex. Because of the need for deep understanding of system, significant efforts have been made in the past few years.We have proposed a fault-tolerance framework for surveillance of overhead transmission lines. In this paper, we follow the framework and further explore deployment problems systematically. Firstly, we present a fault-tolerance placement model. After analyzing the model, we identify the optimal placement for fault tolerance and then propose a placement algorithmwith optimal deployment. An extensive experiment highlights some useful observations for design of the fault-tolerance system and demonstrates efficient fault tolerance for relay nodes. Numerical results show that the fault-tolerance ratio is improved by at least 6% compared to that of previous algorithms.


Introduction
Smart grid, integrated with widespread green energy (e.g., solar, wind, and thermal energy), represents a revolution for traditional power utilities, because it can exploit advanced information and communication technologies to facilitate the infrastructure to improve the efficiency, sustainability, and flexibility of power grids [1]. Smart grid communication, responsible for two-way, cybersecurity communication, and cutting-edge information technologies, has substantially enhanced the intelligence of current energy systems and has attracted huge attention from many researchers [2]. Human inspection, currently the main maintenance of transmission lines in a distributed power network, suffers from timeconsuming, exhausting, and inefficient inspection and is further likely to subject workers involved in power line repairs to serious hazards [3]. Wireless sensor networks (WSNs) have been widely considered in the efficient communication on monitoring application because of their improving performance, relatively low cost, and flexible nature [4]. Consequently, they are instinctively applied in smart grid communications [5]. For labor-replacing purposes, the authors have initiated a scheme applying WSNs to monitor overhead transmission lines in the power distribution [6]. In this scheme, surveillance involves collecting sensing data from a few sensors scattered around a pole, transmitting data to a relay node installed in the pole, and relaying data from different poles to a control center, which is usually situated in a town or a city. e scheme focuses on relay nodes and ignores ordinary sensors around poles since these sensors are relatively simple in terms of function, but the relay nodes are responsible for a large amount of work, such as collecting data, transmitting data, and communication with a control center [4]. In WSNs [7], some prerequisites (e.g., time delay, energy consumption) are possible to be problems [8]. But how to deal with these problems in the open air? As the poles supporting power transmission lines are generally extend through tens of kilometers in rural areas, relay nodes installed in these poles have a linear topology. e linear topology represents a challenge for a network, because it is vulnerable in light of topology tolerance [9], and makes the monitoring application more complex. e smart grid is a self-healing, strong, and flexible system, in which fault tolerance is indispensable. Most of the existing works, however, emphasize the external characteristics of networks, such as reliability, survivability, and vulnerability [10,11]. ese characteristics are constrained to statistical properties over a long time period and cannot respond to the network in real time. Many works neglect the significance of internal faults so that communication networks are difficult to be understood from the fault perspective. Additionally, N-1 contingency is of critical importance in a power system, because it can provide high reliability and make a system functional in the event of one random component failure [12]. erefore, this N-1 contingency is an essential principle as well as design requirements in the practical power network.
Green energy has aroused wide concern of researchers since it can dramatically reduce their carbon footprint and hopefully facilitate the smart grid [13]. For the scheme, although the overhead transmission lines carry electricity, it is too powerful to supply for relay nodes directly [14]. As faults are inevitable [7], we aim to provide a fault-tolerance system for surveillance of transmission lines by WSNs considering green energy envisioned for the smart grid [15].
In our previous work [16], a fault-tolerance framework has been proposed to address the fault tolerance of relay nodes in WSNs for surveillance. In the framework, three sections were implemented to conduct fault tolerance in different stages: planning, deployment, and operation. In this study, we follow our fault-tolerance framework, explore the deployment of relay nodes, and concentrate on the backup placement of relay nodes to determine the optimal placement for fault tolerance. Specifically, in view of linear topology as well as N-1 contingency, we emphasize the communication of relay nodes for the monitoring application by WSNs. From the fault tolerance perspective, we propose a fault-tolerance placement model (FaTPM) and clarify the deployment question to determine the optimal fault-tolerance placement. Based on the FaTPM, we present an initial backup deployment problem (IBDP) by energy harvesting out of doors. en, we analyze the tolerance requirements for compliance with N-1 contingency. Finally, we propose a dynamic optimal deployment placement (DODP) algorithm to solve the fault-tolerance placement problem.

Contributions
e main contributions of this paper are as follows: (1) On the basis of our fault-tolerance framework, we propose a fault-tolerance placement model for relay nodes to supervise overhead transmission lines. (2) Based on the fault-tolerance placement model, we highlight the placement of relay nodes and propose an initial backup deployment problem in consideration of linear topology.
(3) Complying with the significant N-1 contingency in a power system, we analyze the requirements of fault tolerance and develop a dynamic optimal deployment placement algorithm for fault tolerance. (4) We perform experiments to verify the model and algorithm. Numerical results show that the tolerance ratio improves at least 6% compared to that of previous algorithms.
To the best of our knowledge, this study is the first to address fault tolerance for relay nodes by considering N-1 requirements to build a strong, flexible communication system.

Organization and Notations
e remainder of this research is organized as follows: Section 2 reviews related works on surveillance by WSNs and fault tolerance. Section 3 introduces the FaTPM and proposes the IBDP. Section 4 presents N-1 implications, analyzes tolerance requirements, and gives the DODP algorithm. e numerical results are provided in Section 5. Finally, Section 6 provides concluding remarks and directions for future research. e frequently used notations are summarized in Table 1. We also use bold to represent vectors.

Related Works
e surveillance of network had been studied in the literature since it was first presented in [6]. A hierarchical network was designed to meet the requirements of smart grid applications by combining wired and wireless technologies [4]. In order to reduce the time delay, P.W.T. Pong et al. introduced a novel configurable network using cellular network on account of the linear network topology [9]. e authors in [17] investigated the transmission delay, selected a "representative node" scientifically, and presented a grouping mechanism to minimize transmission delay for efficient communications. In view of linear topology, these two studies used the cellular network to directly transmit data [9,17]. In addition, for the gateway of the optical fiber composite overhead ground wire alongside the transmission line, Yi Qian et al. studied the power allocation under the assumption that sensors are powered by green energy (e.g., solar energy). en, they proposed centralized and distributed power schemes [18]. For the energy consumption of relay nodes, the authors utilized a magnetoelectric composite to supply sensor nodes and built a self-powered sensor network by scavenging energy from an alternating current power line [19]. Despite the remarkable research works, a network without fault tolerance could hardly work well for practical surveillance systems.
A few previous works also studied fault tolerance. e authors in [20] surveyed technologies for monitoring power lines and proposed future research directions. Broadly speaking, the methods of fault tolerance in communication networks could be categorized into two classes: topology reconfiguration and backup. e topology reconfiguration was extensively studied in general WSNs by estimating cost of reconfiguration [21]. It was an NP-hard problem to determine the minimum numbers of relay nodes to achieve fault tolerance in a fully connected network [22]. To solve the problem, the authors developed heuristic algorithms, such as moth flame optimization algorithm and bat algorithm. Because of the linear network architecture, the fault tolerance by reconfiguring topology was not fit for the surveillance scheme.
Some studies used the backup method to obtain the fault tolerance. A reverse transmission approach was presented by [23].
is approach aimed to ensure intact data and to transmit information in the reverse direction when a fault occurred. In [24], the authors employed several power lines in parallel and backed up data in a neighboring power line to tolerate faults. To enhance the robustness [10], the authors provided multiple node-disjoint paths between a communication node and a control center to achieve fault tolerance.
Although previous studies have produced important results, the fault-tolerance surveillance system is still to be fully understood. e fault-tolerance framework has been proposed [16]. In the framework, we use backup method to obtain the fault tolerance considering the linear topology. e numbers of backup, placement of backup, and adaptive switching algorithm are included in this framework. Although the placement of backup nodes is already referred to, in this study, we follow this framework and discuss the faulttolerance placement of relay nodes comprehensively.

Network Model.
In this section, we take a systematic look at the network model, introduce the FaTPM, and elaborate factors. en, we propose the initial backup deployment problem.
In the power distribution network, the monitoring system by WSNs can be modeled as shown in Figure 1. Between two adjacent power substations (i.e., Sta 1 and Sta 2 ), the relay nodes (RNs) are installed in each pole to relay data to a control center (CC).
e CC is responsible for aggregation of data and operation of algorithm. Data are transmitted in a hop-by-hop style. Owing to the constraint of time delay in linear topology, a cellular-enabled module (CM) has been introduced [10]; then a group is formed by CM, in which several relay nodes share a common CM. e short-distance wireless communication technology (e.g., Zigbee) is used for relay nodes as well as sensors scattered around the pole. e long-distance wireless communication (e.g., 4 G/GSM) is adapted to the CM. Generally, substations are connected with CC by optical fiber. us, we model the monitoring system as an undirected graph G � (V, E) due to two-way communication. V represents vertices, which stand for n poles, two adjacent power substations (sometimes we use RN 0 and RN n+1 refer to Sta 1 and Sta 2 ), and the control center (CC).
at is, which include short-distance wireless communication, longdistance wireless communication, and optical connections.
For the FaTPM, we reinforce this model with fault tolerance. ere are n relay nodes between substations with linear topology, RN 1 . . . RN n . ese relay nodes have real positions, and there is different spacing between poles according to geographical locations.
e CMs (i.e., CM 1 . . . CM m ) are responsible for directly transmitting data. us, relay nodes are grouped by CMs to Gp 1 . . . Gp m . In each group Gp k , k ∈ [1..m], several relay nodes are included, for instance, RN k1 . . . RN kq ∈ Gp k . As the fault is unavoidable, backup for the RN and CM is indispensable to achieve fault tolerance since a network with linear topology can hardly tolerate any failures via reconfiguration. In addition, backup relay nodes should have the same functions with the original ones from the fault-tolerance view. In contrast to [9], we determine that the last relay node is a sink in each group for fault-tolerance purposes. erefore, G � (V, E) is the fault-tolerance placement model, as shown in Figure 1.

Data Aggregation.
In the FaTPM, data are attached in a hop-by-hop manner in each group. All data are finally transmitted to CC by CM at the last relay node in the same group. For instance, there are four relay nodes in a group as demonstrated in Figure 2. Data from RN 1 to RN 4 are transmitted to CC in the hop-by-hop manner. e last relay node, RN 4 , is a sink that collects all data in this group and triggers the cellular network by CM.
For every relay node, there are three necessary and successive actions: sensing, compressing, and transmitting data. A relay node collects data from around sensors by sensing action and then compresses data to reduce redundancy due to duplicated data. After these two actions, the relay node transmits compressed data to the next relay node. erefore, for the most of relay nodes, received data come from themselves and from the previous relay node. Moreover, there is an optional action before transmitting, backing up data, if the relay node has a backup node.
Several types of sensors are considered (e.g., tension sensors and insulator degradation sensors). RN i+1 senses and collects sensor data SD i+1,j from sensor type j around pole i + 1. Consequently, the data from itself are j SD (i+1),j . After sensing and collecting data, the RN i+1 then compresses these data. If there is the backup relay node for RN i+1 , RN i+1 backs up the compressed data before these data are transmitted [9]. If r is a ratio of compression, the received data in RN i+1 can be calculated as j SD (i+1),j + r · SD i .
Suppose that sensing data (SD) are the same size in each RN, and the received data in last relay node SD q are ((q − 1) · r + 1) · SD; if q nodes are in group Gp k , k ∈ [1..m], where |Gp k | � q, therefore, the total data to be transmitted by the CM are r · q · SD.
Energy consumption is composed of at least three parts for RN i according to actions: energy for sensing En sen , compressing En comp , and transmitting En tx .
e En backup stands for optional backup energy due to fault tolerance. e energy of sensing, compressing, and backup is only relevant to data from itself, while the energy of transmitting is related to the previous relay node. erefore, for RN i , we have En sen � p s · SD i , En comp � p c · SD i , and En backup � p b · r · X i · SD i . p s , p c and p b are energy parameters of sensing, compressing, and backing up one bit of data, respectively. X i functions as an indicator if RN i has the backup node. For simplicity, we replace these parameters with a compound p 0 and use En scb in place of En sen , En comp and En backup . In general, under the assumption of a (1/d n ) path loss, we obtain where p 1 is energy/bit of the transmitter and receiver electronics, and p 2 is that for the transmit op-amp. Parameter d is transmission distance between poles. us, energy consumption of RN i is En i � En scb + En tx . From the FaTPM, we know that energy consumption is unequal or uneven for each relay node because there are hop-by-hop relay and differences for transmitting data.

Time Delay.
In a surveillance application, time delay is extremely important compared to other factors [4]. e relay can be carried out in a parallel manner in groups. We concentrate on time delay in a group. e time delay can be divided into two sections, relay time delay within a group td rt and directed link delay td dl : the first represents delay between relay nodes in the same group; the other represents transmission delay by the directed link CM. e td rt is related to the specific protocol. In [17], Zigbee is used since it has low cost and energy requirements. For wireless technology, we must consider collision avoidance owing to the hidden station problem in WSNs. e most commonly available technology is CSMA/CA. For simplicity, we refer to td at as an average channel access time with CSMA/CA. Notably, there is a unique characteristic of the CM: in contrast to a base station, CM has wider bandwidth for uplink than downlink. Consequently, for the specific group Gp j , time delay td rt within the group is the sum of td at · (|Gp j | − 1) and (r/tr zigbee ) |Gp j | k�1 SD k , k ∈ Gp j . us, we have two components of delay: the time delay within a group and the delay of direct transmission: where tr zigbee and tr cm signify a transmission rate of Zigbee and CM, respectively. D t is the threshold of time delay.

Energy Harvesting.
Energy is one of fundamental factors in WSNs [25]. Since overhead power lines in distribution power networks are always in open areas, relatively ample sunlight is available. However, solar energy is prone to weather changes, for example, clouds and rain. Typically, it results in instability. Accordingly, we add a battery Bt to overcome instability. Previous research has proposed a wireless charging approach to solve this problem [26], but linear topology limits the effectiveness of a charging algorithm. Alternatively, magnetic-field energy harvesting can be a beneficial supplement in the absence of sunshine [27], because power lines are always surrounded by electromagnetic fields. In practice, solar panels and magnetoelectric transducers can be firmly fixed to the pole, as shown in Figure 1.
Assuming that the charging rate of solar energy is symmetric during day time, we can achieve where w is determined by the size of the solar panel, and t r and t s (hours) are the sunrise time and sunset time, respectively. Because of the unbalanced energy consumption, solar panels are naturally not all the same size, as shown in Figure 1. Furthermore, we simplify magnetoelectric energy harvesting into a time-related function; hence, it can be defined as me where ϕ is proportional to the number of coil turns and the flux gradient [19]. t a , t b are parameters with the dimension of hours. Similar to solar energy harvesting, magnetoelectric energy harvesting varies adaptively due to the uneven energy consumption of relay nodes. erefore, for the specific relay node RN i , as the maximal time of data reporting is D t , we have to show that the energy of consumption should not exceed the harvesting energy.

Coverage.
e CM is relevant to group information. Researchers have previously resolved CM numbers in detail [17]. Fault tolerance, however, is neglected from previous studies. To achieve fault tolerance, we relax boundaries of groups and double the number of CM because the strict boundaries leave no space for fault tolerance. us, different from [17], we use the last relay node as the representative node of a group to assemble data and transmit data directly, as shown in Figure 2.
As there is different spacing between poles, for CM j , if R j is a coverage range, the number of poles covered by CM j is at least e part of denominator indicates the maximum distance between adjacent poles in group Gp j .

Fault.
In open areas, the frequency at which faults occur is generally not equal in different places owing to the geographical locations. Moreover, some types of faults occur with high-frequency (e.g., partial discharge faults at the peaks of hills or mountains), while other types of faults occur with low frequency. For the purpose of fault tolerance, we cannot treat different position equally. We use weight vector wei to identify the different placement. We denote fre max and fre min as the maximum frequency and minimum frequency, respectively. fre avg represents the average frequency. e frequency is determined based on historical records. We attempt to assign a greater weight to high-frequency faults, whereas low-frequency faults are weighted based on the harmonic mean. us, the effects are less extreme for fault events that occur with low frequency. erefore, for RN i , the weight wei i is When a fault occurs, it should be tolerated by the system. Specifically, a backup relay node should take control and ensure continued operation. Given the total number of backup relay nodes (TN BRN ) as well as the total number of backup cellular modules (TN BCM ), we should carefully determine placement of backup nodes to maximize tolerance capability. Because the CM is also likely to fail, we should also back up the cellular module to keep the monitoring system functional.

Mathematical Problems in Engineering
e backup numbers of relay nodes, TN BRN and TN BCM , have been studied in previous literature [16,17].

4.7.
Cost. Cost is one of crucial factors in realizing the smart grid [28]. e cost of fault-tolerance system consists of four parts: cost for relay nodes and backup; Ex RN + X i · Ex RN ; cost for the cellular module and backup; cost for solar panels and cost for the magnetoelectric transducer. e last two parts are l · Ex SP + Ex ME , where l is a position coefficient in a group. As CM and backups for CM need a license and subscription fee sub from other operation companies, the cost is Functioning as data sensing, compressing, communication, and control, relay nodes have high cost as the equipment of energy harvesting and cellular module is closely related to them. erefore, backups for all relay nodes are impractical. But we still need to consider the fault tolerance. erefore, we determine the appropriate placement of backups to maximize capability of fault tolerance.
Since backup nodes should work exactly the same as original relay nodes in the event of a fault, the backups are also constrained by factors. As the number of backup nodes is resolved in [16], we target backup cost in this placement problem. us, the IBDP can be stated as follows.
Given the definitive backup numbers, TN BRN and TN BCM , we determine the optimal deployment placement of backup relay nodes to maximize fault tolerance. e IBDP can be formulated mathematically as follows: given system model G � (V, E), RN, CM, TN BRN , TN BCM and group information Gp, we obtain the exact solution for the following minimal optimization problem: Equation (9) constrains time delay. Energy is limited in equation (10). Equations (11) and (12) account for the coverage and fault weight, respectively.

Fault-Tolerance System
In this section, we first introduce requirements of the faulttolerance system. en, we provide an analysis of tolerance requirements. Next, the IBDP is reformulated in compliance with N-1. Finally, we propose the dynamic optimal deployment placement (DODP) algorithm to solve the faulttolerance placement problem.

Tolerance Requirements.
For a reliable power system, N-1 plays a significant role in fault tolerance [29]. N-1 implies that if any independent component fails or is removed, it cannot cause power system failure among N components of a power system [12]. In this subsection, we discuss N-1 requirements for communication system.

N-1 Implications for Communication Networks.
For communication networks, we focus on node faults since many link faults are caused by relay nodes in wireless technologies. Further, a fault involves time of occurrence and duration time, besides the frequency and type, which are already discussed. More accurately, fault tolerance means that the backup really works for the time in which a fault has occurred and continued. erefore, to fully understand faults, we introduce two variables, TTR and TBF, which are related to the fault time t [30]. TTR i stands for the time required to repair faults on i pole, while TBF i signifies the time between adjacent failures on i pole. e state of each RN i is either TTR i or TBF i . us, the N-1 can be transcribed as follows: ∀j in n poles, Equation (9) suggests that the probability of TTR j be as small as possible to be neglected.

Proof.
Suppose that there is a linear network  RN 1 , RN 2 , . . . , RN n composed of n poles. f j indicates that a fault has occurred on the j th pole, j ∈ [1..n]. According to Definition 1, ∀t, there is always a sufficiently small positive number ε that makes the expression lim t⟶+∞ P TTR i (t) < ε � 1 hold. is expression indicates that the event TTR i (t) < ε is an inevitable event as time passes. e inevitable event implies that when the fault f j has occurred on the j th pole, fortunately, there is always a backup relay node if we neglect the switching time from the fault node to backup node. As j is arbitrary, we can conclude that each pole requires a backup relay node to comply with the requirements of N-1.

Relaxation.
A relay node is more expensive than an ordinary sensor node. It is impractical to back up all relay nodes as the above Proposition 1 claims. erefore, N-1 requirements are too strong to apply for an actual fault-tolerance system. Naturally, we attempt to relax N-1 for practical use. Here, we perform the relaxation based on two assumptions: (1) a threshold Thr is allowed for a monitoring application in a power system, and (2) TTR is limited and not long time. us, we can translate the N-1 as follows: in observation time T, Equation (11) makes compliance of the communication network with the N-1 requirements feasible. For the tolerance system, N-1 requires the probability P(TTR i (t) ≥ Thr) to be as close to 0 as possible, even if a fault occurs in practice.
If we look closely at it, TTR i can be an accumulation of multiple faults in i pole for several types of sensors. at is, TTR i � j TTR i,j . Generally, the different fault types on a pole are not independent, whereas the faults in different poles are independent. An occurring fault is always relevant to the TTR i,j . Suppose that k is a fault number (FN); the event TTR i ≥ Thr can be reformulated according to the total probability theorem, j TTR i,j is the cumulative time of the fault, and k is the specific fault number. en, (12) can be rearranged as follows: We assume that FN follows a Poisson distribution with λ and TTR i,j follows a normal distribution [30]. us, (13) is the probability of sum of nonindependent random variables, essentially, that is, convolution of computing. In fact, (13) is not helpful in making decisions since it involves complex integration. We resort to a generating function of the random variables to achieve approximate results. Here, we directly provide a proposition. More details can be found in the Appendix. B(n, (λ i /n)) with the constraints that λ i is very small and μ < 0. at is,

Proposition 2. (binomial approximation). For a fault that has occurred, N-1 can be approximately computed by the binomial distribution
The proposition signifies that if the occurring fault is a rare event and TTR i is very small, we can approximately Mathematical Problems in Engineering replace the fault time with the fault number, which is subject to a binomial distribution with constraints.

Problem Reformulation.
e fault-tolerance system integrates optimal placement and fault tolerance. We can reformulate the problem based on the IBDP. Given system model G � (V, E) and related parameters, we obtain a faulttolerance system that complies with the N-1 requirements. erefore, when a fault has occurred, the fault-tolerance system can be reformulated as follows:

Dynamic Algorithm.
In operation of networks, we compute the optimal backup placement to provide the best locations to obtain fault tolerance within observation time T.
In the next observation time, we can repeat this process and achieve the new optimal placement. We need to record the TTR of faults. us, from sequential time series, we can compute the optimal placement dynamically at multiple observation times. In addition, the TTR time of faults can further accumulate data of prior probability and facilitate the IBDP, as demonstrated in Algorithm 1.
e algorithm can be run in a server of the control center (CC). CC sends control commands to relay nodes by control flow, as shown in Figure 2.

Complexity and Confidence
Interval. We provide complexity of above algorithm with respect to network scale n. e complexity of Step 1 is O(n). For steps 3 to 7, complexity heavily depends on the 0-1 integer linear program (ILP) owing to Step 5. Since each binary variable is either 0 or 1, the complexity is O(k * 2 m ) theoretically in the worst case, where m is the number of binary variables, and k is the number of constraints. According to (8), we have m ≤ n since the number of variables m is definitely not more than the network scale n. erefore, complexity of the algorithm is O(k * 2 n ) in the worst case.

Experimental Configuration.
We use Matlab to perform experiments to validate the model and algorithm. We ran codes on a CPU with a 2.5 GHz Intel i5 processor, 4 GB memory, and Linux Mint 19.1 operating system. e configuration data are mainly from the testbed and are slightly modified in terms of fault tolerance [9]. We consider a maximum of 100 poles/towers spanning approximately 40km, with an average span length of 1300ft. Additionally, t at is 41ms, and D t is 2s. e parameters p 1 and p 2 are (50nj/bit) and (100pj/(bit/m 2 )), respectively. t r and t s are defined as 9 : 00 and 16 : 00 of each day. e cost ratio of Ex SP , Ex EM and Ex CM is 5: 10: 20. e distance between poles follows a normal distribution. We run experiments 50 times and calculate the average value. We experimentally verify the proposed DODP algorithm and compare it with random deployment and the BPBP algorithm from [16].

Metric.
We use a tolerance ratio (Tr) to evaluate the fault tolerance capability, as defined in (16). A correct placement, cp i ∈ 0, 1 { }, means that a fault has occurred on a pole i that has a backup node. A false placement, fp i ∈ 0, 1 { }, denotes that a fault has occurred on a pole i that has no backup in (16). e higher Tr indicates greater tolerance. Meanwhile, we also consider overheads and time delay.
6.3. Results. First, we show how sensor data (SD) affect time delay with respect to different numbers of poles. SD varies from 2K, 4K to 8K. As shown in Figure 3, the time delay strictly increases with the number of poles, which makes sense because a greater number of poles represent a longer distance and more time consequently. e parameter r varies from 0.125, 0.25 to 0.5, as shown in Figures 3(a)-3(c), respectively. e time delay is 6.2049s where SD is 8K and r is 0.5. A large amount of data undoubtedly consumes plenty of time in the sensing, compressing, and transmitting processes. However, time always has a deadline in monitoring applications; therefore, a large amount of data requires careful design. On the basis of data in Figure 3, we conclude that a reasonable data volume and a high compression ratio r are two goals of the system. However, a high compression ratio r will lead to greater energy consumption for the relay nodes. Figure 4 evaluates the ratio of energy consumed in the sensing, compressing, backing up, and transmitting process. According to experimental data, the energy of backing up is nearly 0, which indicates that we can ignore the backing up process when considering energy consumption. In Figure 4, we compare energy consumption of sensing and compressing (En scb ) with that of transmitting (En tr ). Furthermore, we display the parameter n in (1), n � 2 in Figure 4(a), n � 2.5 in Figure 4(b), and n � 3 in Figure 4(c). e comparison reveals that transmitting energy is the main source of energy consumption and En scb energy is a minor source in Figure 4(a). However, in Figure 4(c), these roles are reversed. e results confirm that the index n in (1) is strongly related to energy consumption. us, in design of the communication system, we should focus on the outdoor transmission path. e accurate time delay for different groups is shown in Table 2 since groups are indispensable. In contrast to the previous study [17], we double groups in the FaTPM based on fault tolerance. e same conclusion can be drawn from group 9 and group 10. erefore, even though time delay can be decreased by increasing the cellular modules, it is not reduced substantially when the group number increases.
We now consider the overheads in Figure 5 with group information Gp. Generally, many groups lead to much heavier overheads. e overheads are displayed in Figure 5(a) with SD 8K and in Figure 5 Require: G, observation T, threshold time Thr, and other parameters including location vector RN Ensure: optimal backup location vector RN with N-1 (1) Initial location vector RN; (2) t � 0 (3) while t ≤ T do (4) When a fault f i occurs, we access parameters including TTR i , TBF i . (5) Compute the optimal location with (15) and output X i , Y j (6) Update the backup location RN adaptively and configure the backup nodes (7) end while (8) Output RN, TTR i and TBF i for the step 1.
Mathematical Problems in Engineering the condition of 2 second time delay. Figure 5(a) shows much higher expense than that of Figure 5(b) since more data need to be transmitted in the restricted time. Accordingly, much more energy is consumed for transmission. e energy constraint becomes sensitive in the optimization, which consequently leads to more cost owing to the solar and magnetoelectric transducer. Figures 5(c) and 5(d) show different fault probabilities from 3% to 5%. In general, a scenario with high fault probability needs more backup nodes for the relay nodes. e fault-tolerance rate Tr is listed in Table 3. We compare the DODP algorithm with random deployment and based prior backup placement (BPBP) algorithm [16] in 14 groups. e BPBP algorithm outperforms the random algorithm because it appends prior information of faults. Moreover, the DODP algorithm achieves the highest fault-   Table 3, the average Tr is improved by 8.646%, 6.472%, and 6.423% in DODP compared to BPBP for 1-, 5-, and 9-month periods, respectively. erefore, we conclude that the fault-tolerance ratio in DODP improves at least 6% compared to that of the previous BPBP algorithm.

Discussion and Conclusion
Surveillance of overhead transmission lines by wireless sensor networks, which is one of smart grid applications, faces challenges. In common wireless sensor networks, researchers have developed cutting-edge algorithms and traded off in many factors, such as energy, time delay, and cost. Moreover, with the linear topology and fault tolerance requirements, the surveillance application has posed greater challenges. We have proposed a faulttolerance framework to deal with this tough questions. In the framework, we have explored backup method to provide fault tolerance. When we examine the backups, numbers of backup, placement of backup, and the adaptive switching algorithm should be determined.  In this paper, we follow the fault-tolerance framework and analyze deployment of relay nodes to monitor transmission power lines via wireless sensor networks. Based on a fault-tolerance placement model, we propose an initial backup deployment problem to determine the optimal backup locations of relay nodes. To build a flexible communication network, we ensure that the fault-tolerance complies with the N-1 of the power system and propose a dynamic optimal deployment placement algorithm. e numerical results present interesting highlights and confirm that the dynamic optimal deployment placement algorithm exceeds the performance of previous algorithms by at least 6%.
In the future work, we will study adaptive switching following the fault-tolerance framework. Specifically, adaptive switching from a faulty node to a backup node to ensure sustained data acquisition also represents additional challenges for researchers.

Data Availability
Some or all data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.