Optimal wireless sensor network configuration for structural monitoring using automatic-learning firefly algorithm

Abstract

Wireless sensor networks are becoming attractive data communication patterns in structural health monitoring systems. Designing and applying effective wireless sensor network–based structural health monitoring systems for large-scale civil infrastructure require a great number of wireless sensors and the optimal wireless sensor networks configuration becomes critical for such spatially separated large structures. In this article, optimal wireless sensor network configuration for structural health monitoring is treated as a discrete optimization problem, where parameter identification and network performance are simultaneously addressed. To solve this rather complicated optimization problem, a novel swarm intelligence algorithm called the automatic-learning firefly algorithm is proposed by integrating the original firefly algorithm with the Lévy flight and the automatic-learning mechanism. In the proposed algorithm, the Lévy flight is adopted to maximize the searching capability in unknown solution space and avoid premature convergence and the automatic-learning mechanism is designed to drive fireflies to move toward better locations at high speed. Numerical experiments are performed on a long-span bridge to demonstrate the effectiveness of the proposed automatic-learning firefly algorithm. Results indicate that automatic-learning firefly algorithm can find satisfactory wireless sensor network configurations, which facilitate easy discrimination of identified mode vectors and long wireless sensor network lifetime, and the innovations in automatic-learning firefly algorithm make it superior to the simple discrete firefly algorithm as to solution quality and convergence speed.

Keywords

automatic learning firefly algorithm optimal sensor placement structural health monitoring wireless sensor network

Introduction

Health monitoring technologies have been widely accepted as effective approaches to understand structural static responses and dynamic behaviors, trace structural performance deterioration and reliability reduction, and assess structural collapse risk and residual life. Structural health monitoring (SHM) systems have been implemented on a great number of existing and newly built structures throughout the world (Chen et al., 2015; Rezaiee-Pajand et al., 2018; Seo et al., 2015; Sun et al., 2014, 2017). In an SHM system, the reasonable evaluation of structural condition can only be performed if adequate data are collected by sensors (Dinh-Cong et al., 2017; Marano et al., 2011; Pan and Chen, 2017; Sun et al., 2016, 2017; Xiong and Chen, 2018). Wireless sensor networks (WSNs) have the potential to facilitate dense sensor arrays and prolific data collection in complex civil structures at costs lower than those historically associated with tethered counterparts (Chen et al., 2016; Kurata et al., 2013; Zhou and Yi, 2013a). Even so, the available wireless sensors are still far less than the number of accessible degrees of freedom (DOFs) because the behaviors of large-scale continuous structures are generally described by tens of thousands of discrete DOFs. As a consequence, optimally deploying limited number of wireless sensors in spatially separated large structures, which could result in the best characterization of structural behaviors, becomes a significant and challenging task.

In fact, determining optimal WSN configurations is an indispensable component in the design and implementation of a sophisticated WSN-based SHM system. The most important locations, where the most valuable information is contained, are expected to be explored during optimization. Besides, the highest network performance popularly represented by network connectivity and network lifetime is emphasized for the reason that wireless sensors are featured by short data transmission distance and are powered by batteries with limited capacity. In recent years, intensive research activities regarding optimal WSN configuration have been carried out using various optimization strategies. Hada et al. (2012) developed a Lagrangian heuristic algorithm to find the most suitable number and locations of relays with the lowest cost. In this research, the locations of wireless sensors used for collecting information are predetermined and only the network cost is taken into account. Soman et al. (2012) adopted the genetic algorithm to optimize the locations of wireless sensors in the WSN so that the determinant of the Fisher information matrix is minimized, which is an interesting approach to place wireless sensors for SHM but is difficult to apply in complex structures. Bhuiyan et al. (2014) proposed a sequential optimization method to place wireless sensors for a special tree network and address civil engineering requirements, communication efficiency, network complexity, and network failures. However, this method is only suitable for simple network topologies. Fu et al. (2013) addressed the problem of wireless sensor deployment as finding node locations to reliably diagnose structural health while consuming minimum energy during data collection. But, in this WSN, wireless sensors are uniformly distributed on structures and the information quality is ignored. Elsersy et al. (2015, 2016) presented a modified effective independence method and a genetic algorithm to place wireless sensors for structural monitoring. Unfortunately, the energy consumption of different wireless sensors and the network lifetime are not taken into consideration. Bhuiyan et al. (2015) employed several backup sensors to increase the degree of fault tolerance of the WSN. But the cost of the WSN is also increased. Zhou et al. (2015b) investigated the possibility of finding an optimal WSN configuration in SHM with a trade-off between parameter identification and network lifetime. In this study, the method used for judging network connectivity is difficult to be applied in self-organized network topologies. Liu et al. (2016) provided a particle swarm optimization algorithm to seek optimal heterogeneous WSNs with minimal energy consumption. The optimization algorithm shows low efficiency and easily falls into local optimum when the searching space is large. Those research works offer some meaningful references for deploying wireless sensors in SHM systems

However, in most of previous techniques, either very simple fixed network topologies are taken into account or some important requirements of structural monitoring practices are not incorporated into the evaluation criteria. Hence, there is still a great gap between theoretical methods and practical engineering applications. It is necessary to exert more efforts on the topic of wireless sensor placement for SHM. For this reason, the present study proposes a new method to optimally configure WSNs with self-organizing multi-hop network topologies, which are the most favorite network structures in WSN-based SHM systems. In the method, the mathematical model, which addresses the information requirement for modal identification and the energy consumption for network life, is developed and a novel automatic-learning firefly algorithm (ALFA) is provided to solve this complicated optimization problem. The proposed method can be applied to find optimal WSN configurations for most structures. The remainder of this article is organized as follows. First, the mathematical model for optimal WSN configuration is presented in section “Mathematical model.” And then, the details of the proposed ALFA are described in section “ALFA.” Following that, a long-span bridge is employed as a test bed to demonstrate the proposed method in section “Numerical experiment.” Finally, conclusions are drawn in section “Conclusion.”

Mathematical model

In order to reliably assess structural condition using a WSN-based SHM system, effective information collected by wireless sensors is essential. From this point of view, wireless sensors are desired to be deployed on those locations containing the most valuable information. As a consequence, information effectiveness becomes the primary requirement that should be met when configuring WSNs (Kammer and Tinker, 2004; Yi et al., 2017; Yi and Li, 2012).

Being different from traditional tethered sensor networks, wireless sensors are powered by batteries and the data transmission range is generally short. Therefore, how to minimize energy consumption and maximize network lifetime is regarded as a challenging task when applying WSNs in SHM. Meanwhile, ensuring that information collected from different parts of the structure is successfully transmitted to the sink is of great significance.

Information effectiveness

Identified structural parameters, that is, mode shapes, are generally adopted as the basic data for evaluating structural conditions (Doebling et al., 1998; Kim and Lynch, 2012). Thus, sensor configurations with the best modal identification are expected. The singular value decomposition ratio (SVDR) not only provides a good measure of the mode orthogonality but also offers a desirable metric of the condition for mode expansion and the observability of the modes (Friswell and Mottershead, 1995). The SVDR is close to 1.0 if all identified mode vectors are orthogonal and have little or no correlation, while the greater value of SVDR would be obtained if identified mode vectors have a high degree of similarity. It is therefore an ideal indicator of evaluating the effectiveness of information collected by different WSN configurations. The SVDR is computed by

SVDR = \frac{σ_{max}}{σ_{min}}

(1)

where σ_max and σ_min denote the maximal and the minimal singular values of the mode shape matrix, respectively.

Network performance

Network connectivity

The self-organizing multi-hop network, which employs a number of wireless sensors as intermediate nodes to relay data packets, is frequently used to transmit information to a destination which is beyond the radio range. In this type of network, a wireless sensor could utilize any wireless sensor within its radio range to deliver information. In such case, a wireless sensor may have numerous optional data transmission routings and the connectivity judgment for every wireless sensor in a large-scale wireless network becomes rather difficult. To cope with this problem, the judgment matrix deduced from the adjacency matrix in graph theory is adopted.

It is supposed that a WSN comprising a set of wireless sensors is a finite and simple graph. The connectivity of any two nodes in the WSN is characterized by the adjacency matrix A of this graph, which is (Deo, 2016)

A_{uv} = {\begin{matrix} 0 u = v \\ 1 u \neq v, d_{uv} \leq d_{R} \\ 0 u \neq v, d_{uv} > d_{R} \end{matrix}

(2)

where $A_{uv}$ represents the element in adjacency matrix A; u and v ( $u, v = 1, 2, \dots, W$ ), in which W represents the number of wireless sensors in the WSN, denote the serial numbers of wireless sensors in the WSN; $d_{uv}$ is the Euclidean distance between wireless sensor u and wireless sensor v; $d_{R}$ represents the maximal radio range.

Then, the judgment matrix D is derived by

D = \sum_{s = 1}^{W - 1} A^{s}

(3)

If no element in the judgment matrix D is equal to zero, it can be judged that the WSN is a connected network, in which a wireless sensor has at least one route to transmit data to the sink, while the case that any element is equal to zero implies that the network is unconnected. By this way, the network connectivity can be easily judged.

Balanced energy consumption

For a typical wireless sensor, the energy consumed by data transmission is generally orders of magnitude greater than data sampling and data processing (Fu et al., 2013). In the present study, the energy consumption for data sampling and data processing is neglected and only the energy consumed by data transmission is considered. The WSN lifetime is defined as the time span from the instant when the network starts work to the instant when the first wireless sensor exhausts its energy. The wireless sensors bearing heavier data transmission burden exhaust their energy quickly, which may induce the premature failure of the whole network. Thus, keeping the balance of energy consumption of all wireless sensors is an effective strategy to prolong network lifetime. The balance of energy consumption can be measured by the standard deviation of normalized energy consumption (Zhou et al., 2015).

The normalized energy consumption is equal to

{\bar{E}}_{C, u} = \frac{E_{C, u}}{E_{C, max}}

(4)

where ${\bar{E}}_{C, u}$ ( $u = 1, 2, \dots, W$ ) represents the normalized energy consumption of wireless sensor u in a sampling period; $E_{C, u}$ ( $u = 1, 2, \dots, W$ ) denotes the energy consumption of wireless sensor u in a sampling period; and $E_{C, max}$ is the maximal energy consumption of a single node in the WSN.

The index of the balance of energy consumption $E_{E}$ is formulated as

E_{E} = var ({\bar{E}}_{C, 1}, {\bar{E}}_{C, 2}, \dots, {\bar{E}}_{C, W})

(5)

where $var (\cdot)$ represents the standard deviation operator. With this definition, the smaller value of $E_{E}$ indicates the more balance of energy consumption in the network.

Optimization criteria

As mentioned before, the optimal WSN configuration is expected to collect the most useful information and to have the longest lifetime. Therefore, the optimization objective in the present study is twofold. The modal discrimination and the network performance are considered.

The first objective function involving the information effectiveness can be expressed as

f_{IE} = SVDR

(6)

The second objective function relating to the network performance is

f_{NP} = N_{C} + E_{E}

(7)

where $N_{C}$ represents the indicator of the network connectivity and is calculated by

N_{C} = {\begin{matrix} 1 If min (| D_{uv} |) > 0 \\ 100 If min (| D_{uv} |) = 0 \end{matrix}

(8)

where $D_{uv}$ represents the element in judgment matrix D. Those unconnected WSNs are assigned a great value of $N_{C}$ and would be naturally removed during optimization.

Finally, optimal WSN configuration can be modeled as a minimize optimization problem, whose objective function is

Minimize f (x) = ω α f_{IE} (x) + (1 - α) f_{NP} (x)

(9)

where x represents a specified WSN configuration, $ω$ is a adjusted coefficient, and $α$ denotes the weight of information effectiveness.

The weight coefficient $α$ is selected according to the purpose of the WSN-based SHM system. If a system with long lifetime is desired, $α$ is set to a small value (i.e. $α < 0.5$ ). In contrast, if the information effectiveness is emphasized, a large value (i.e. $α > 0.5$ ) should be used. The adjusted coefficient $ω$ is introduced to make the right two parts in equation (9) at the same order of magnitude, so that the two parts can be synchronously optimized. Unfortunately, $f_{IE}$ and $f_{NP}$ change with the number of available sensors and structural features. As a result, providing a theoretical method to choose the value of the adjusted coefficient $ω$ is challenging work. A compromise is the trial method. $ω$ is originally set to 1 and the optimization is carried out. According to the optimization result, $ω$ is reset to make the two parts almost equal. And then, the optimization is performed again using the new $ω$ . After repeating five times, the suitable $ω$ would be extracted.

ALFA

Choosing W locations to place wireless sensors from M ( $M >> W$ ) candidate DOFs on a given structure is a combinatorial optimization problem with discrete variables and is therefore computationally intractable. The firefly algorithm (FA), which was proposed by Yang (2008), finds optimal solutions in a manner analogous to the aggregation behavior of fireflies. Brighter fireflies attract darker ones and all fireflies finally gather together. FA reveals outstanding capability in solving complicated optimization problems (Fister et al., 2013; Sahu et al., 2015) and therefore provides a possible approach to deal with the issue of optimal WSN configuration. Nevertheless, the following shortcomings still need to be overcome:

The original FA is devised to solve optimization problems with continuous variables, while the optimization process of optimal WSN configuration is to determine whether a DOF is chosen to place a wireless sensor or not. As a result, the real coding system in the Cartesian coordinate system adopted by the original FA cannot code the firefly population used for optimizing WSN configurations.

The disturbance of firefly movement in the original FA is generated by a uniform distribution or normal distribution, which shows low capability in searching unknown space. In contrast, finding optimal WSN configurations for large-scale civil structures is a typical optimization problem with extremely large candidate solution space, which easily causes the original FA to fall into a local optimum.

In the original FA, the movement of a firefly is constrained into a small region and the evolution relies on the information of itself. The excellent information about optimization carried by other fireflies nearby is not fully used. It is not difficult to image that the computational efficiency of the original FA is not high enough. Utilizing the information of neighboring fireflies to improve the convergence speed is therefore expected.

In this article, ALFA, which introduces the dual-structure coding system, the Lévy flight, and the automatic-learning mechanism to train simple fireflies in the original FA, is proposed. The dual-structure coding system provides a visualized description of the wireless sensor deployment. The Lévy flight can maximize the capability of searching unknown space and avoid premature convergence (Gandomi et al., 2013). And the automatic-learning mechanism relocates fireflies by learning from adjacent excellent fireflies and improves the movement efficiency. The schematic drawing of the proposed ALFA is shown in Figure 1. In each iteration process, the movement with Lévy flights is followed by the movement driven by the automatic-learning mechanism. And the automatic-learning mechanism is composed of five sub-steps, which are dividing into groups, assigning group leaders, packaging knowledge, learning from group leaders, and moving toward group leaders. The implementation of ALFA includes coding and initialing fireflies, movement pattern, and automatic learning.

Figure 1.

Schematic drawing of ALFA.

Coding and initialing fireflies

The issue of optimal WSN configuration is similar to the knapsack problem from the view of mathematics. A dual-structure coding system instead of the real coding system is presented to code fireflies. Each firefly represents a possible solution, that is, a WSN configuration, and is coded by a location vector and a state vector, as displayed in equation (10). The brightness of fireflies is proportional to the values of the objective function. The location vector $x$ with length of M is composed of the serial number of accessible DOFs. And the state vector $g$ is a binary string. If a sensor is distributed on the $x_{i} th$ DOF, the $i th$ element in the state vector is equal to 1. In contrast, if no sensor is placed on the $x_{i} th$ DOF, the $i th$ element is 0

C_{i} = (\begin{matrix} x_{i} \\ g_{i} \end{matrix}) = (\begin{matrix} x_{i, 1}, x_{i, 2}, \dots, x_{i, M} \\ g_{i, 1}, g_{i, 2}, \dots, g_{i, M} \end{matrix})

(10)

where $C_{i}$ ( $i = 1, 2, \dots, N$ ) represents the code of the $i th$ firefly $P_{i}$ , in which N denotes the total number of fireflies in the population.

Before the iteration process is performed, it is necessary to randomly distribute fireflies in the possible solution space, which is termed as initialization. Previous study indicates that a well-distributed initialized population is helpful to keep diversity and accelerate convergence. Thus, the location vector and the state vector of each firefly are initialized by two different methods. The location vector is generated by the shuttle method, while the state vector is figured out by the random method (Zhou et al., 2014). To ensure the number of ones in any initialized state vector is equal to the number of predetermined wireless sensors, those state vectors are normalized. After that, elements in the state vector are reordered according to their corresponding elements in the location vector. As a result, elements in the state vector are sequenced by DOFs. Thus, the location vector can be ignored in the following iteration process and the algorithm becomes simpler.

Movement pattern

The firefly movement is defined by the step length and step direction. In the original FA, the distance between two fireflies is simply defined by the Euclidean distance because fireflies are located in the Cartesian coordinate system. In the dual-structure coding system, the Euclidean distance is no longer applicable and the Hamming distance is introduced (Zhou et al., 2015). Lévy flights are employed to determine the movement direction. Lévy flights are typical foraging routes of animals and insects in nature and the following movement is selected according to the current location and the transition probability. The movement of firefly $P_{i}$ toward firefly $P_{j}$ is calculated by

d_{H, ij}^{M} = [0.5 \times {d_{H}}_{, ij} \times L é vy (λ)]

(11)

where $d_{H, ij}^{M}$ denotes the movement size, d_{H, ij} is the Hamming distance, and Lévy(λ) represents the Lévy distribution.

In order to keep the number of ones in the binary string a constant, the movement is judiciously devised and performed by relocating $d_{H, i j}^{M}$ wireless sensors. The wireless sensors in $P_{i}$ whose locations are different with wireless sensors in $P_{j}$ , or the wireless sensors in $P_{j}$ , whose locations are different with wireless sensors in $P_{i}$ , are named as incongruous wireless sensors. For the problem at hand, it is impossible to predict the contribution of each DOF to the objective function, so the movement operation is implemented by replacing randomly selected $d_{H, ij}^{M}$ incongruous wireless sensors in firefly $P_{i}$ with randomly selected $d_{H, ij}^{M}$ incongruous firefly sensors in firefly $P_{j}$ .

Automatic learning

After the movement driven by Lévy flights is finished, a bright firefly is generally surrounded by several darker ones. It is obviously that some wireless sensors in the bright firefly locate good positions and make great contributions to the objective function. The automatic-learning mechanism is then carried out. The information about those good wireless sensors is packaged as knowledge and learned by nearby darker fireflies. And those darker fireflies would become brighter. The convergence speed is therefore improved. To prevent excessively learning and premature convergence, the population is separated into several groups. The implementation of automatic learning is constrained in groups and listed as follows:

Step 1: Divide the firefly population into several groups ( $G_{1}, G_{2}, \dots, G_{E}$ ) where E is the number of groups and assign the firefly with the best objective function as the group leader ( $G_{L, 1}, G_{L, 2}, \dots, G_{L, E}$ ).

Step 2: Delete a wireless sensor $S_{L, i, u}$ from the WSN configuration $x_{L, i}$ ( $i = 1, 2, \dots, E$ ), where $x_{L, i}$ represents the WSN configuration of the group leader $G_{L, i}$ and $S_{L, i, u}$ is the $u th$ wireless sensor in the WSN configuration $x_{L, i}$ . The WSN configuration with $W - 1$ wireless sensors is denoted as $x'_{L, i}$ .

Step 3: Calculate the increment of the objective function because of the deletion of a wireless sensor, which is

I_{L, i} = | f ({x'}_{L, i}) - f (x_{L, i}) |

(12)

where $I_{L, i}$ represents the increment of the objective function of the group leader.

Step 4: Find the best wireless sensor $S_{L, B}$ corresponding with the maximal $I_{L, i}$ .

Step 5: Package the information about the best wireless sensors as knowledge.

Step 6: Delete a wireless sensor $S_{M, i, u}$ from the WSN configuration $x_{M, i}$ ( $i = 1, 2, \dots, R$ ), where $x_{M, i}$ denotes the WSN configuration of the group member $G_{M, i}$ , $S_{M, i, u}$ is the $u th$ wireless sensor in the WSN configuration $x_{M, i}$ , and R represents the number of group members. The WSN configuration with $W - 1$ wireless sensors is denoted as $x'_{M, i}$ .

Step 7: Compute the increment of the objective function because of the deletion of a wireless sensor, which is

I_{M, i} = | f ({x'}_{M, i}) - f (x_{M, i}) |

(13)

where $I_{M, i}$ represents the increment of the objective function of the group member.

Step 8: Find the worst wireless sensor $S_{M, W}$ corresponding to the minimal $I_{L, i}$ .

Step 9: Repeat Steps 6–8 until all of the worst wireless sensors in all group members are extracted.

Step 10: Replace the worst wireless sensors $S_{M, W}$ with the best wireless sensor $S_{L, B}$ by learning if the DOF is not occupied.

Step 11: Repeat Steps 2–10 until the learning in all groups is finished.

The pattern of elite inheritance is employed to assure steadily convergence. The current population is compared with the last population. Only those worse fireflies in the last population are replaced by fireflies in the current population and those better fireflies in the last population are preserved and passed to the next iteration. In such case, fireflies with distinguished features are inherited.

The predetermined iteration number is used as the stop criteria because of the insurmountable difficulty during finding reasonable iterative error. The flow chart of ALFA is displayed in Figure 2.

Figure 2.

Flow chart of ALFA.

Numerical experiment

To demonstrate the effectiveness of the proposed method, the numerical experiment is performed on a long-span bridge. The simple discrete firefly algorithm (SDFA) (Zhou et al., 2015) is also carried out in the experiment so that the results optimized by the two algorithms can facilitate comparison and the effectiveness of the improvements based on SDFA is validated. The following two cases are tested:

Case I: 25 wireless sensors are used and $ω$ and $α$ are set to 0.04 and 0.5, respectively.

Case II: 35 wireless sensors are used and $ω$ and $α$ are set to 0.03 and 0.5, respectively.

Experiment environment

The long-span bridge is composed of a main span and two equal side spans, which are 1490 and 470 m, respectively. The two side spans are simply supported by piers and separated from the main span. The steel box-girder and two reinforced concrete towers are employed to support the main span. Two main cables and 182 suspenders are adopted to transfer loads from the main girder to two towers (Zhou and Yi, 2013b). An updated finite element model is established to investigate structural dynamic characteristics. The main girder is separated by 92 beam elements and 93 nodes. The left end of the main girder is numbered by 1 and the right end is numbered by 93. The length of the beam element is about 16.1 m. Except two end nodes, the other 91 nodes are candidate locations. The first 20 vertical mode shapes are arbitrarily selected as the target modes considering that the numerical experiment is just an example. In real SHM systems, the number of selected modes is determined by many factors, such as the complexity of the structure, the methodology used for condition evaluation, the technology of damage identification, and the methodology for updating finite element models (Pickrel, 1999). At present, there is no universal criterion to determine how many target modes should be selected. One suggestion is that the number of target modes should provide us sufficient information about the structural stiffness or other parameters of interest to us. The topic of determining model number deserves further investigation.

All of the available wireless sensors are distributed on the main girder and formed a multi-hop linear WSN. One sink is used and installed on the right girder end. Wireless sensors used in the test are uniform and can adjust their transmission power within the radio range. A wireless sensor only transfers data forward to the nearest wireless sensor. It is assumed that the WSN collapses if any wireless sensor depletes its power. The path loss communication model is introduced to describe the energy consumption in the WSN (Kalpakis et al., 2003) and parameters are listed in Table 1 (Li and Mohapatra, 2007). In the table, $λ$ represents the size of the data packet, and $ε_{T}$ , $ε_{R}$ , and $η_{T}$ are energy cost coefficients.

Table 1.

The energy parameters of wireless sensors.

Parameter	$λ$	$d_{R}$	$ε_{T}$	$η_{T}$	$ε_{R}$
Value	100	130	45 × 10^–9	10 × 10^–11	60 × 10^–9
Unit	bit	m	J/bit	J/bit/m²	J/bit

There are three problem-specific parameters, that is, the firefly number, the group number, and the iteration number, should be carefully chosen so that the proposed ALFA is able to achieve its best performance. The optional firefly number is [60, 90, 120,…, 240], the optional group size is [6, 8, 10,…, 20], and the optional iteration number is [100, 125, 150,…, 400]. Parametric studies are performed by changing these parameters one by one. The results indicate that the best values of the firefly number, the group number, and the iteration number are 150, 15, and 100, respectively. Similar to ALFA, the optimal values of the population size and the iteration number for SDFA are 150 and 300, respectively.

Results and discussions

Based on the mode shapes of the long-span bridges, ALFA and SDFA are implemented to select optimal WSN configurations for Case I and Case II. Considering the randomness of searching, each algorithm is independently run 10 times with different initial firefly populations in each case and the best result is adopted. The iterative processes in both cases are displayed in Figures 3 and 4. And the performance comparison between ALFA and SDFA in Case I and Case II is summarized in Table 2. It can be seen from the two figures that the minimum objective function in the firefly population tends to a constant and the average value and the maximum value steadily approach the minimum value along with the increasing of iteration number. Both ALFA and SDFA are capable of converging to optimal results. But the convergence features show significant difference. In Case I and Case II, ALFA spends 42 cycles and 46 cycles in finding the optimal solutions, respectively, while SDFA requires 183 cycles and 237 cycles, respectively, which is more than four times. And the computational time spent by ALFA in the two cases is 48 and 53 s, respectively. The corresponding time used by SDFA is 202 and 273 s, which is also about four times of that consumed by ALFA.

Figure 3.

The iterative process in Case I: (a) ALFA and (b) SDFA.

Figure 4.

The iterative process in Case II: (a) ALFA and (b) SDFA.

Table 2.

Performance comparison between ALFA and SDFA.

Tested case	Computational efficiency				Optimization results
	CPU time (s)		Iteration time		f_IE		f_NP		f(x)
	ALFA	SDFA	ALFA	SDFA	ALFA	SDFA	ALFA	SDFA	ALFA	SDFA
Case I	48	202	42	183	1.5254	1.6154	0.0187	0.0247	0.0416	0.0489
Case II	53	273	46	237	1.5521	1.6021	0.0252	0.0304	0.0485	0.0544

ALFA: automatic-learning firefly algorithm; SDFA: simple discrete firefly algorithm.

Two reasons cause this difference. First of all, ALFA employs the Lévy flights to expand the searching direction and improve the global searching ability in unknown solution space. When fireflies gather in a local optimal position, they can easily jump out and move to a better position. In contrast, fireflies in SDFA frequently fall into a local optimum and difficultly detach from. As a result, the minimal objective function frequently keeps a constant in the following dozens of iterations, which demonstrates many long terraces in the iterative processes, as marked by curly braces in Figures 3(b) and 4(b). And then, fireflies in ALFA automatically learn information about good wireless sensor locations within their groups and quickly move to the group leader. As a consequence, fireflies in population gather in a small region after 25 iterations, as shown in Figures 3(a) and 4(a). Being different from ALFA, SDFA moves fireflies only using the limited information of themselves and their targets. Fireflies move slowly and spend more than 60 times to gather into a small area. Therefore, it is confirmed that the automatic-learning mechanism dramatically improves the convergence speed and results in a better convergence characteristic.

According to Table 2, the optimal SVDRs obtained by ALFA and SDFA in Case I are 1.5452 and 1.6154, respectively. And in Case II, the optimal SVDRs obtained by the two methods are 1.5521 and 1.6021, respectively. Hence, the ALFA provides mode shapes with better orthogonality than the SDFA. In the two cases, the optimal results computed by ALFA are 0.0416 and 0.0485, respectively, and that obtained by SDFA are 0.0489 and 0.0544, respectively. More than 12% reduction in the objective function is gained. All optimal results are far less than 100, which indicate that all optimal configurations are connected. The strong capability of movement with Lévy flights in exploring global optima solutions is verified and the better performance of ALFA in converging to the global optimum is also validated.

The normalized energy consumption in optimal WSN configurations extracted by ALFA and SDFA in both cases is shown in Figures 5 and 6. From the two figures, it can be found that the optimal WSN configurations worked out by SDFA shows notably nonuniform energy consumption. The 13th wireless sensor in Case I and the 13th wireless sensor in Case II exhaust their power quickly, which in turn causes the early collapse of the WSNs and the waste of energy resources equipped in other wireless sensors. On the contrary, the WSNs extracted by ALFA present more balanced energy consumption and longer WSN lifetime becomes possible.

Figure 5.

The normalized energy consumption of optimal WSN configurations in Case I: (a) ALFA and (b) SDFA.

Figure 6.

The normalized energy consumption of optimal WSN configurations in Case II. (a) ALFA and (b) SDFA.

Table 3 lists the optimal WSN configurations found by ALFA in the two experimental cases. More sensors are distributed near the right girder end to share responsibility of transmitting data and achieve balanced energy consumption.

Table 3.

Optimal WSN configurations in the two cases.

Case	Parameter	Location
Case I	Sensor no.	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	DOF no.	5	10	14	18	22	26	30	34	37	41	45	49	53	57	60	64	68	72
	Sensor no.	19	20	21	22	23	24	25
	DOF no.	75	78	81	84	87	89	91
Case II	Sensor no.	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	DOF no.	3	7	10	13	17	21	24	27	30	33	36	39	42	45	48	51	54	57
	Sensor no.	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35
	DOF no.	59	61	63	65	67	69	71	73	75	77	79	81	83	85	87	89	91

WSN: wireless sensor network; DOF: degree of freedom.

Conclusion

It is a quite complex issue to find optimal WSN configurations for WSN-based SHM systems. Ingenious methodology with high searching efficiency is necessary to cope with this problem. In this article, the problem of optimal WSN configuration is well formulated and perfectly solved by ALFA, which is proposed by integrating FA with Lévy flights and the automatic-learning mechanism. A long-span bridge is adopted as the test bed to examine ALFA. Conclusions are summarized as follows:

The weighted summation of the SVDR, which represents the information effectiveness, and the standard deviation of energy consumption, which is adopted as a measure of network lifetime, is formulated as the optimization criteria within the constrain of network connectivity. The judgment matrix derived from the adjacent matrix in graph theory is introduced to directly identify network connectivity.

The original FA is improved in terms of the coding system and movement pattern and ALFA is proposed. A dual-structure coding system is employed to code fireflies so that the deployment of wireless sensors is visually described. The movement with Lévy flights is devised to drive fireflies to efficiently search the unknown space and avoid premature convergence. The automatic-learning mechanism dramatically improves the convergence speed.

Numerical experiment is carried out to test the optimization criteria and the proposed ALFA. The results clearly indicate that the optimization criteria can provide a reasonable measure of the WSN performance and ALFA remarkably outperforms SDFA as far as convergence speed and solution quality are concerned. The computation speed is improved at least four times. ALFA is particularly effective in solving complex optimization problem like optimal WSN configuration and shows potential of being extended to a variety of discrete optimization problems.

It should be noted that the load and the congestion of wireless sensors are left out of consideration in this study. The network lifetime would be prolonged and the time delay of data transmission would be relieved if those routings with low load and congestion are selected by optimizing, which is our future research work.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research work was jointly supported by the National Natural Science Foundation of China (Grant Nos 51625802 and 51678218), the 973 Program (Grant No. 2015CB060000), and the Science Fund for Excellent Young Scholars of Jiangsu Province (BK20170097).

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