WSN Nodes Placement Optimization Based on a Weighted ... - MDPI

algorithms Article

WSN Nodes Placement Optimization Based on a Weighted Centroid Artificial Fish Swarm Algorithm Hong Yin *, Ying Zhang and Xu He School of Mechatronic Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China; [email protected] (Y.Z.); [email protected] (X.H.) * Correspondence: [email protected]; Tel.: +86-931-495-5789 Received: 22 August 2018; Accepted: 26 September 2018; Published: 30 September 2018







Abstract: Aiming at optimal placement of wireless sensor network (WSN) nodes of wind turbine blade for health inspection, a weighted centroid artificial fish swarm algorithm (WC-AFSA) is proposed. A weighted centroid algorithm is applied to construct an initial fish population so to enhance the fish diversity and improve the search precision. Adaptive step based on dynamic parameter is used to jump out local optimal solution and improve the convergence speed. Optimal sensor placement is realized by minimizing the maximum off-diagonal elements of the modal assurance criterion as the objective function. Five typical test functions are applied to verify the effectiveness of the algorithm, and optimal placement of WSNs nodes on wind turbine blade is carried out. The results show that WC-AFSA has better optimization effect than AFSA, which can solve the problem of optimal arrangement of blade WSNs nodes. Keywords: wind turbine blade; optimal sensor placement; wireless sensor network (WSN); weighted centroid algorithm; artificial fish swarm algorithm (AFSA)

1. Introduction Blade is one of the key components of the wind turbine. It is responsible for converting wind energy into rotational mechanical energy of wind turbine rotors. The wind turbine blade is extremely easy to suffer varying degrees of damage under the influence of environment of wind, sand, ice, thunder and lightning, as well as a variety of stress concentrations, leading to the shutdown of the wind turbine, and serious economic losses. The volume and mass of wind turbine blade are huge, so it is difficult to maintain manually, with high maintenance costs. Therefore, in order to decrease the risk of operation, while reducing maintenance costs, it is of great significance to carry out health monitoring in the wind turbine blade structure, especially identify the damage state quickly and accurately [1,2]. In this work, modal test is made for the health monitoring of blades. Attach a certain number of sensors on the blades to extract modal parameters. Due to the relative motion between the blades and cabin, it is inconvenient to transmit data through wired mode. At the same time, owing to the large number of sensors and complex wire arrangement, it is very convenient to use wireless sensors to transmit the real-time information about blade’s health state to receiver. Wireless sensor networks (WSNs) [3] are usually deployed in complex monitoring areas where the environment is harsh and is inaccessible to people. Deployment of WSNs can enlarge the transmission distance and make the system not easily disturbed by the environment. The wireless sensor network adopts an ad hoc network and multi-hop technology, which greatly reduces the cost of the monitoring system, and makes the installation and maintenance of the sensor network very convenient and fast. By changing the topology of the network in WSNs dynamically, energy consumption is reduced, network capacity and life are increased and collisions between nodes are avoided [4,5]. The topology control of WSNs is divided into the network topology establishment stage and the network topology Algorithms 2018, 11, 147; doi:10.3390/a11100147

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maintenance stage. To ensure that the complex topology can operate, as well as the simple topology built initially, a reasonable topology control algorithm is needed [6]. WSNs technology has attracted more and more attention in recent years [7,8]. By means of WSNs, we can collect, process, and analyze the data reflecting the blade’s health state. However, there are few studies on the optimal placement of WSNs for wind turbine blades monitoring. The artificial fish swarm algorithm (AFSA) is a new kind of swarm intelligence optimization algorithm based on the behavior of fish [9,10]. The basic idea of AFSA is to imitate the fish behaviors such as prey, swarm, and follow. AFSA is simple in principle and has some good features such as robustness, and tolerance of parameter setting. This algorithm has some disadvantages yet. Non-uniform distribution deduced by randomly generating initial fish population decreases the global search ability of the algorithm. The search step of each fish is fixed. At the early iteration, the algorithm is easy to fall into local minimum. To solve the above problems, a weighted centroid algorithm (WCA) [11] is introduced to the artificial fish swarm algorithm (AFSA) to initialize the fish swarm in this work. By measuring the distance between nodes, localization compensation error and adaptive search step, the convergence speed and optimization precision are improved. 2. Mathematical Model of Blades WSNs Nodes Optimal Placement To carry out finite element model (FEM), the essential is to obtain the three mechanical properties: displacement, strain and stress. For the complex structure like wind turbine blades, after analyzing the above three mechanical properties, we can evaluate the structures’ strength and stiffness to obtain the structure’s vibration characteristics parameters such as frequency and mode shapes which provide theory support for structure health monitoring, damage identification and analysis of dynamic characteristics. Establish the FEM of wind turbine blade, and extract mode shape matrix φn×m . n denotes the number of degrees of freedom (DOFs), namely, the number of candidate sensor positions. m denotes the order number of mode shapes. Select s DOFs from n DOFs as the final sensor locations. In order to ensure the measured modal vectors being easily distinguished, modal assurance criterion (MAC) [12] is applied to evaluate the measured modal vectors. The maximum off-diagonal element of MAC is taken as the objective function of minimum problem. The mathematical model can be expressed as:  


2  φiT φj   f (φs×m ) = max MACij = .
 φiT φi φTj φj 

(1)

where 0 ≤ f (φs×m ) ≤ 1. φi and φj represent the ith and jth column vectors in matrix φ, respectively. The superscript T denotes the transpose of the vector. If the off-diagonal elements of MACij (i 6= j) tend to zero, which indicates that there is little correlation between the modal vector i and j, that is to say, the modal vector can be distinguished easily. The objective function of blade WSNs nodes optimal placement can be expressed as:  min f ( x ) = max MACi j . i6= j

(2)

3. Weighted Centroid Fish Swarm Algorithm In this paper, the weighted centroid fish swarm algorithm is used for global search. The weighted centroid algorithm and adaptive search step are introduced to improve the search capability.

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3.1. Procedure of the Weighted Centroid Fish Swarm Algorithm 3.1. Procedure of the Weighted Centroid Fish Swarm Algorithm 3.1.1. Principle of the AFSA 3.1.1. Principle of the AFSA In water, fish are most likely distributed around the field with the most abundant food. A fish In water, fish are most foraging likely distributed around the social field with the mostAFSA, abundant food.by A fish swarm completes its food process by several behaviors. inspired the swarm completes foraging process several social AFSA, inspired the behaviors behaviors of fish,itsisfood a novel method forbysearching the behaviors. global optimum. It is a by random search of fish, is abased novelon method for searching the behaviors, global optimum. is a random search algorithm based on algorithm simulating fish swarm which It include prey behavior, swarm behavior, simulating fish swarm behaviors, which include prey behavior, swarm behavior, follow behavior and follow behavior and random behavior. random behavior. Suppose the current state of an artificial fish is X = ( x1 , x2 ,..., xn ) . Visual is the perception scope current state of an artificial fish is X = ( x1 , xof . . , xn )artificial . Visual is theStep perception 2 , . each of anSuppose artificialthe fish, which determines the moving direction fish. is the scope of an artificial fish, which determines the moving direction of each artificial fish. Step is too the maximum moving step length. To avoid missing the optimal solutions, Step should not be set maximum moving step length. To avoid missing the optimal solutions, Step should not be set too large. Xv  ( x1 , x2 ,..., xn ) is the location at certain moment. Visual and Step are shown in Figure 1. large.Xv = ( x1 , x2 , . . . , xn ) is the location at certain moment. Visual and Step are shown in Figure 1. X -X Xnext = X + rand( )  Step  vX − X (3) v (3) Xnext = X + rand() × Step ×Xv - X | Xv − X | where rand ( ) produces random number between 0 and 1. where rand ( ) produces random number between 0 and 1.

X n2

X n1 Step

Visual X

X next Xv

Figure Visual and Step artificial fish. X,n1X, n2 Xn2 Locationsofoftwo twofish fish within Visual Figure 1. 1. Visual and Step of of anan artificial fish. Xn1 : :Locations within Visual

3.1.2. Binary Binary Coding Coding 3.1.2. Binary coding coding is is intuitive intuitive and and convenient convenient for for the the optimal optimal sensor sensor placement placement problem problem [13]. [13]. Let Let nn Binary denote the initial number of candidate sensors and s denote the final number of sensors determined. denote the initial number of candidate sensors and s denote the final number of sensors determined. The binary expressed as as X =X(=x0(,xx1, ,x. .,..., . , xxj , ，， . . . , xxn ))TT ,, jj ∈ 1, 2, .n. . ,. nIf}. The binary coding coding for foraasolution solutioncan canbebe expressed  {1,2,...,  0 1 j ... n If x j = 1, there is a sensor on that DOF. If x j = 0, there is no sensor on that DOF. The condition xnj = 1 , there is a sensor on that DOF. If x j = 0 , there is no sensor on that DOF. The condition ∑ x j = s is satisfied. n j= 1

is satisfied. between binary coding X and modal matrix φ = s conversion  x The s×m is the process of removing j

j 1

the rows of φn×m . In this process, remove those rows of φn×m corresponding to those  The conversion between binary coding X and modal matrix s m is the process of removing x j x j = 0, j ∈ [1, 2, . . . , n] in X. For convenience, this process is represented by Equation (4): the rows of n m . In this process, remove those rows of n m corresponding to those φs×m = g(φthis Xn×1 ) is represented by Equation (4): n×m ,process { x j x j = 0, j  [1, 2,..., n]} in X. For convenience,

(4)

where the modal matrix φs×m is a solution placing s sensors in optimal sensor placement; Xn×(4) sfor 1 is m  g (n m , X n1 ) the binary coding corresponding to a solution. ssubstituting whereFor thea modal is a solution placing sensorT placement; X n×1 simplematrix example, n =for 4, m = 2, s s=sensors 3 and Xin4×optimal 1 = (1, 0, 1, 1) into Equation (4), m 5 is the binary coding 1corresponding to a solution.   2 6   and take φ4×2 =   as the input modal matrix of function g, the output modal matrix of g is  3 7  4 8

For a simple example, substituting n = 4 , m = 2 , s = 3 and X 41 = (1,0,1,1)T into Equation TT Fora asimple simpleexample, example,substituting substitutingn =n 4= ,4 ,mm andX X = (1,0,1,1) intoEquation Equation For = 2= ,2 ,s =s 3= 3and into 1 (1,0,1,1) 4×41×= 1 5  1 1 5 5    2 6  input  modal matrix of function g, the output modal matrix of g is  (4), and take 42 = as the =2 2 6 6 asas  3take 7 Φ4Φ (4), and take the input modal matrix function the output modal matrix g is = (4), and the input modal matrix ofof function g, g, the output modal matrix ofof g is  × 24× 2  3 3 7 7  Algorithms 2018, 11, 147  4 of 14 4 8        4 4 8 8    1 5   1   1 5       1   5  1 1 15 5 111  5111 5 55  2 6 0 2 6  1 1 5 5 1 5                                   obtained by deleting the second row of 32 = g  2 , 6  =03 7 and can 6 2 2 06   be   easily  = ,0=  22 6 6=   3 Φ7Φ  =1=g  g, =3= 3 7 and easily obtained deleting thesecond secondrow rowofof = bebe easily bybydeleting the and can be obtained easily obtained by deleting φ3×2 = g 7 can  3 ,7  7 3and can 3× 23× 2                     4 8 7 1 1  3 3 7 7    4 38   17 3 3417      4 8  8    4   8  4 4 18 8 11   4 4 8 8   4 4 8 8              42second . the row of φ4×2 . . . Φ4Φ × 24× 2

3.1.3. 3.1.3. Fish Fish Swarm Swarm Initialization Initialization through through WCA WCA 3.1.3. Fish Swarm Initialization through WCA 3.1.3. Fish Swarm Initialization through WCA The The core core of of the the centroid centroid algorithm algorithm (CA) (CA) isisthat that unknown unknown nodes nodes estimate estimate its its location location according according The core of the centroid algorithm (CA) is that unknown nodes estimateitsitslocation location according The core of the centroid algorithm (CA) is that unknown nodes estimate according to to the the geometric geometric center center combined combined with with all all its itscommunications communications anchor anchor nodes nodes [11]. [11]. The The principle principle of of to the geometric center combined with all its communications anchor nodes [11]. The principle to the geometric center combined with all its communications anchor nodes [11]. The principle ofof centroid centroid algorithm algorithmisisshown shownin inFigure Figure2.2. centroid algorithm shown Figure centroid algorithm is is shown inin Figure 2. 2.

Figure Figure 2. 2. Principle Principle of of the the centroid centroidalgorithm. algorithm. Figure . Principle the centroid algorithm. Figure 2. 2Principle ofof the centroid algorithm.

In ( x1x, y, 1y),) B, ( B( x2 ,xy2, )y and C( x3 , y3 ) are the anchor nodes, and hollow dot In Figure Figure 2,2,solid soliddots dotsAA( 1 1 2 2 ) and C( x3 , y 3 ) are the anchor nodes, and hollow Figure solid dotsA(A( B( ,2y) 2 )distances andC(C( arethe anchor nodes, andhollow hollow InInFigure 2, 2,solid x1x,a, )b, B(are x2x, 2ythe and x x,between ) 3 )are anchor D( X, Y )is the unknown nodes to bedots located. Dthe and A, B,nodes, and and 1y,1y 1,) ,c 3y,3y dot D( X , Y ) is the unknown nodes to be located. a, b, c are the distances3 between D and A, B, and C C respectively. dotD(D( the unknown nodes located. c are the distances between and and dot XX , Y,)Yis) is the unknown nodes toto bebe located. a, a, b,b, c are the distances between DD and A,A, B,B, and CC respectively. Select three nodes close to D to form trigon. Determine the trigon centroid. This is the location to respectively. respectively. Select three nodes close to D to form trigon. Determine the trigon centroid. This is the location be determined, which is expressed as:close Select three nodes close form trigon. Determine the trigon centroid. This the location Select three nodes toto DD toto form trigon. Determine the trigon centroid. This is isthe location to be determined, which is expressed as: determined, which expressed as: toto bebe determined, which is is expressed as: 11 nn 11 nn ( X,( XY,)Y= ( x , (5) i ∑ ) = (n  xi , n ∑ yy1ii ))1n n 1 1n n (5)  = ni=i =11 ( X( X n,i)Y ,Y =)1 (= (  xi x, i ,  yi y) i ) i =1 (5)(5) n ni =1i =1 n ni =1i =1 where wherennisisthe thenumber numberofofanchor anchornodes. nodes. where n is the number anchor nodes. where n is the number ofof anchor nodes. The the nodes to be is estimated based on the distance centroid centroid method. ( XY, )Yof ) of The location location( X, the nodes tolocated be located is estimated based on the distance ( X , Y ) The location of the nodes belocated located estimated based thedistance distancecentroid centroid ( X , Y ) The location of the nodes toto be is isestimated based ononthe Let T the uniform distributed random number between 0 and 1. Generate the initial fish swarm: method. Let T the uniform distributed random number between 0 and 1. Generate the initial fish method.Let LetT Tthe theuniform distributedrandom randomnumber numberbetween between0 0and and1. 1.Generate Generatethe theinitial initialfish fish method. (uniformdistributed swarm: [ T ×10] x = X + (− 1 ) × T × δ x swarm: swarm: (6) 10] y = xY=+X(− 1)[T[T×10] + (-1) × TT  δ×x δy[T ×[10] T ×10]  x=x X =X + (-1) × T× × T δ×xδx + (-1) (6)  y = Y + (-1)[T 10] T  δy [T ×[10] (6)(6)  T × 10]  where δx = X /2, δy = Y /2. =Y + (-1) × T× × T δ×yδy + (-1)  y =y Y Theδweighed centroid algorithm (WCA) rationally distributes the weights to make the location where x = X / 2 , δy = Y / 2 . Y/ 2/ unknown where X X/ 2/between ,2 ,δyδ=y =Y the .2 . more precise. where The lessδdistance node and anchor node, the greater the impact of x = xδ= the anchor node on the unknown node [14]. WCA can be expressed as:

Pi ( x, y) =  ( wij  Bj ( x, y)) /  wij

here wij is the weight of anchor node j to unknown node i, and wij = 1 / (dij )n ; Pi ( x, y) is

Algorithmsof 2018,unknown 11, 147 5 of node 14 timated location node i; Bj ( x, y) is the location of the anchor j; dij is

stance between the anchor node j and theN unknown node i; n is weight coefficient ranging f N ( x, y) = ∑ (wij × Bj ( x, y))/∑ wij (7) to 4; N is the number of the anchorPinodes. The average error location can be expressed as: where w is the weight of anchor node j to unknown node i, and w = 1/(d )n ; P ( x, y) is the estimated

here

ij

ij

ij

i

location of unknown node i; Bj ( x, y) is the location of the anchor node j; dij is the distance between the N 2 anchor node j and the unknown node i; n 1 is weight coefficient ranging 1 to 24; N is the number of ERROR = xi  xi  yfrom  y i i the anchor nodes. N  R i 1 The average error location can be expressed as:

 

 



N q unknown node; 2( xi , yi ) is the estimate locatio ( xi , yi ) is the actual location of the 1 2

ERROR =

N × R i∑ =1

( xi − x i ) + ( yi − yi )

(8)

e unknown node; R is the radio range of WSNs. where ( xi ,50 yi ) is the actualnodes location uniformly of the unknown distributed node; ( xi , yi ) is the estimate100 location of them unknown  100 Figure 3 shows WSNs with area. Suppose R node; R is the radio range of WSNs. . From Figure 4,Figure the 3error of CA is larger than that of WCA. The location determined by WC shows 50 WSNs nodes uniformly distributed with 100 × 100 m area. Suppose R is 50 m. Figure 4,and the error of CA is larger than that of WCA. The location determined by WCA is closer oser to actualFrom location more accurate. to actual location and more accurate.

Figure 3. Node distribution of WSNs.

Figure 3. Node distribution of WSNs.

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6 o

Figure 4. Location error comparison of WCA and CA.

Figure 4. Location error comparison of WCA and CA.

3.1.4. Swarm Behavior

An artificial fish tends to move toward the central location of fish swarm. This is the swarm

4. Swarm Behavior behavior. Swarm behavior is the main process for search local optimization, and the search time

is longer. step and the maximum iterations Nmax affect the accuracy and convergence speed of the

An artificial fish tends tostep move the central location of but fish swarm. algorithm. The larger and thetoward smaller Nmax , the faster convergence speed lower precision.This So, is the swa proper step should determined. Here, we adopt kind of adaptive step. In the earlyand iteration, havior. Swarm behavior isbethe main process for one search local optimization, the search tim large step is used to accelerate convergence speed. With the increase of the number of iterations, the ger. step andoptimum the maximum iterations N maxthenaffect the accuracy and convergence speed of solution is gradually approximated, adopt a small step to refine the search and improve

precision. orithm. The search larger step and the smaller N max , the faster convergence speed but lower precisi Assuming that the current iteration number is N < Nmax , the current state of an artificial fish

proper stepisshould befunction determined. adopt kind of adaptive In the ea xi . Its fitness is f (xi ). A Here, new statewe within visual isone x j , and the fitness function is f (step. x j ). f (x j ) < f (xi ), then we choose the moving Step = rand( ) × k x j − xi k. Adding the function rand ( ) ation, large Ifstep is used to accelerate convergence speed. With the increase of the number is to get into the local optimum in order to prevent its convergence too quickly. ations, the optimum solution is gradually approximated, then adopt a small step to refine 3.1.5. Follow Behavior rch and improve search precision. When one or several fish find a field where there is more food and it is not crowded, the fish Assuming nearby that the current iteration number is N  N behavior. , theFor current state of an artificial fis will follow and move toward the food. This is the follow max the i-fish, the procedure

of follow behavior is as follows: Its fitness function is f ( xi ) . A new state within visual is x j , and0 the fitness function is f ( x j ) Step 1: Randomly generate new location of artificial fish within xi on the interval [ xij − visual, xij + visual ], j = 1, 2, . . . , n;

x j ) < f ( x i ) , then we choose the moving Step  rand ( )  x j  xi . Adding the function rand ( ) i f ( xi 0 ) Step 2: Compute food concentration f (xi 0 ). If

nf

> δ f (xi ), the new fish moves toward a

new location; into the local optimum in order to prevent its convergence too quickly. Step 3: Repeat (1), (2) until Nmax is reached.

5. Follow Behavior

When one or several fish find a field where there is more food and it is not crowded, the arby will follow and move toward the food. This is the follow behavior. For the i-fish, ocedure of follow behavior is as follows:

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3.1.6. Prey Behavior An artificial fish follows the direction with high food concentration in prey behavior. Suppose the current state of an artificial fish is xi . Randomly select a new state x j within visual. If the food concentration is f (x j ) < f (xi ), the artificial fish moves one step toward x j ; otherwise, randomly reselect x j and judge whether it satisfies the condition of moving forward. If the condition cannot be satisfied until try_number is reached, the artificial fish randomly moves one step within visual. 3.2. Advantage Analysis of the Adaptive Step The AFSA has a strong global search ability. However, in later iterations, the fish swarm gathers near the peak value, which makes the bot search precision and convergence speed slow. If the fixed step is large, it is easy to jump out of the optimal solution, but the accuracy of calculation is low. If the fixed step is small, the precision is improved, but the convergence speed is slow, it is easy to fall into the local optimum and difficult to jump out. To solve the above problems, an adaptive step is introduced. In the initial optimization, the artificial fish, which is far away from the optimal value, adopts larger step to approach the optimum value quickly. When approaching the optimal value, make use of smaller step to improve the accuracy. The introduction of the adaptive step not only accelerates the convergence speed but also improves the accuracy of the solution. 3.3. Time Complexity Analysis The effectiveness of the algorithm can be measured by time complexity. The basic operation count is regarded as the time complexity of the algorithm. We analyze the time complexity according to the implementation procedure of WCA-AFSA.

• • • • •

Initialization of N artificial fish needs N times. Time complexity is O( N ). Initialization of other parameters assigns once and compares N − 1 times. Time complexity is O( N ). Swarm behavior computes crowding factor N times, judges once, moves once, and swarms N times. Time complexity is O( N 2 + 2 × N ). Follow behavior searches the optimal value N times, computes crowding factor N times, judges once, moves once, and follows N times. Time complexity is O(2 × N 2 + 2 × N ). The frequency of prey behavior is, at most, try_number times, at least once. N artificial fishes prey N times. Time complexity is O( N × try_number). After l iterations, time complexity is O(l × (3 × N 2 + N × try_number + 4 × N )).

3.4. Procedure of WCA-AFSA Step 1: Initialize fish swarm. N is the size of artificial fish swarm. Set the number of maximum iterations, try the number of prey behavior, perception scope, initial step, and factor of crowding degree as Nmax , try_number, visual, a, and δ respectively. Step 2: Obtain the maximum off-diagonal element of MAC. On the basis of modal matrix φn×m , compute the objective function of each artificial fish according to Equation (2). Step 3: Execute swarm and prey behavior. Move toward the direction of small objective function. If two behaviors both cannot satisfy the condition of moving forward, perform prey behavior. If try_number is reached and still cannot satisfy the condition of moving forward, carry out a random behavior. Step 4: Re-compute the objective function value of each fish. If Nmax is reached then execute (5); otherwise execute (3). Step 5: Determine the optimal solution according to Equation (2). The flow chart of WCA-AFSA is depicted in Figure 5.

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f ( xi ')   f ( xi ) nf

f ( xi  xi )   f ( xi ) nf

f ( xi ')

f ( xi  xi )

xi  x j

f ( xi )  f ( x j )

f ( xi  xi )  f ( xi ')

xi  xi  xi

xi  xi '

N max

Figure 5. Flow chart of WCA-AFSA.

Figure 5. Flow chart of WCA-AFSA.

4. WCA-AFSA Performance Test

benchmark functions, 4. WCA-AFSAFive Performance Test which include the Sphere function, Ackley function, Rastrigin function, Griewank function, and Rosenbrock function, are applied to verify the performance of the WCA-AFSA.

Five These benchmark which ofinclude functionsfunctions, all have the features multiple the peaksSphere and highfunction, dimension. Ackley function, Rastrigin function, Griewank (1) Sphere function, function: and Rosenbrock function, are applied to verify the performance of the n WCA-AFSA. These functions all have the features of multiple peaks and high dimension. 2 f1 (x) =

(1) Sphere function: (2)

∑ xi ,

xi ∈ [−100, 100], f min = 0

i =1

Ackley’s function:

n

f1 (s x) =  xi2 , xi  [-100,100] , fmin = 0 f 2 ( x ) = −20 exp(−0.2

(2) Ackley’s function: (3)

1 n i=1 2 1 n xi ) − exp( ∑ cos(2πxi )) + 20 + exp(1), xi ∈ [−32.768, 32.768], f min = 0 ∑ n i=1 n i=1

Generalized Rastrigin’s function: n n

f2 ( x) = -20exp(-0.2

1 1 xi 2 ) - exp(   cos(2πxi )) + 20 + exp(1), xi  [-32.768, 32.768], fmin = 0 n n i =1 n 2i =1 f3 (x) =

∑ ( xi

− 10cos(2πxi ) + 10), xi ∈ [−5.12, 5.12], f min = 0

i =1

(3) Generalized Rastrigin’s function: n

f3 (x) =  ( xi2 - 10cos(2πxi ) + 10) , xi  [-5.12, 5.12] , fmin = 0 i=1

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(4)

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Generalized Griewank’s function:

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n x x f 4 ( x ) = ∑ i − ∏ cos( √i ) + 1, xi ∈ [−600, 600], f min = 0 4000 i=1 i i =1

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n

(5)

f5 ( x) =  [100( xi+1 - xi2 )2 + (1- xi )2 ] , xi  [-30, 30] , fmin = 0 i=1 function: Generalized Rosenbrock’s

speed and precision are single modal functions with f1 ( x) and f2 ( x) used for testing convergence n ( xifor xi2 )2 + the (1 −global xi )2 ] search ability are multiple single extreme values. f3 ( x) fand used f4 (∑ x) [100 5 (x) = +1 −testing i =1

modal functions with nonlinear multiple extreme values. f5 ( x) is an abnormal function, which is applied algorithm performance. andtesting f 2 ( x ) the used for testing convergence speed and precision are single modal functions f 1 ( x ) for Set single the size of thevalues. artificial swarm = 20,for initial stepthe = global 0.5, and the number of multiple maximum with extreme f 3 (fish x ) and f 4 ( x )Nused testing search ability are N max = 120. iterations averagemultiple value isextreme listed invalues. Table 1.f 5 ( x ) is an abnormal function, which is modal functions withThe nonlinear applied for testing the algorithm performance. Table 1. Test results.

Set the size of the artificial fish swarm N = 20, initial step = 0.5, and the number of maximum Optimal Average Value Average Time/s Optimal iterations Nmax = 120. The average value is listedActual in Table 1. Function TWCA- AFSA TAFSA fWCA- AFSA fAFSA Theoretical Value

f1 ( x ) f 2 ( x) Function f 3 ( x) f 4 ( x) f1 (x) f ( fx5)( x)

0

Table 0.00351. Test results.0.0099

49.203655

50.245561

0 0.0012 0.0040 48.945043 50.062352 Optimal Actual Average Value Average Time/s Optimal 0 Value 0.0427 0.1498 49.671079 50.784103 Theoretical fWCA−AFSA fAFSA TWCA−AFSA TAFSA 5 5 0 49.660634 49.887099 4.0878  10 5.9343  10 0 0.0035 0.0099 49.203655 50.245561 0 0.0052 0.0123 49.686631 50.670954 0 0.0012 0.0040 48.945043 50.062352 2 f3 (x) 0 0.0427 0.1498 49.671079 50.784103 f 4From ( x ) Table 1, the0 optimal actual 49.887099 4.0878 × 10−5value5.9343 × 110−5 average by WCA-AFSA is49.660634 smaller than that by AFSA. It f ( x ) 0 0.0052 0.0123 49.686631 50.670954 5 demonstrates WCA-AFSA has higher convergence accuracy. The average time of WCA-AFSA is

shorter than that by AFSA. It indicates WCA-AFSA is more effective. Figure 6 shows the From Table optimal actual average value by WCA-AFSA is smaller than that by AFSA. comparison of 1, thethe results. It demonstrates WCA-AFSA higher convergence accuracy. WCA-AFSA is WCA-AFSA has betterhas convergence precision and speedThe thanaverage AFSA. time This of means WCA-AFSA shorter than search that byability. AFSA. It indicates WCA-AFSA is more effective. Figure 6 shows the comparison has better of the results. (a) f1

(b) f2

Figure 6. Cont.

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(c) f3

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(d) f4

(e) f5

Figure 6. Results comparison of WCA-AFSA with AFSA. Figure 6. Results comparison of WCA-AFSA with AFSA.

better convergence precision and speed than AFSA. This means WCA-AFSA has 5. WCA-AFSA WSNs Nodeshas Optimal Placement Based on WCA-AFSA better search ability. 5.1. Example of a Wind Turbine Blade 5. WSNs Nodes Optimal Placement Based on WCA-AFSA For a one megawatt level horizontal axis wind turbine blade, the length is 29 m . The blade 5.1.material Exampleisofglass a Wind Turbine Blade plastic. The density is 1950 kg m 3 . The modulus of elasticity is fiber reinforced

16.5 . The Poisson ratio 0.305. Theaxis firstwind five mode shapes arethe shown in Figure five ForGPa a one megawatt levelishorizontal turbine blade, length is 29 m.7. The first blade material is glass fiber reinforced plastic. density 1950inkg/m mode frequencies and their mode shapeThe features are is listed Table3 .2.The modulus of elasticity is 16.5 GPa. The Poisson ratio is 0.305. The first five mode shapes are shown in Figure 7. The first five mode frequencies and their mode shape features are listed in Table 2.

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(a) First mode shape

(b) Second mode shape

(c) Third mode shape

(d) Fourth mode shape

(e) Fifth mode shape Figure 7. 7. First First five five mode mode shapes. shapes. Figure Table Table 2. 2. First First five five mode mode frequencies frequencies and andtheir theirmode modeshape shapefeatures. features. Order Frequency/Hz Frequency/Hz Order 1 5.4011 1 5.4011 2 7.2602 2 7.2602 3 34.108 3 4 34.108 46.286 4 5 46.286 87.176 5 87.176

Mode Feature ModeShape Shape Feature Waving vibration Waving vibration Oscillating vibration Oscillating vibration Waving + Oscillating Waving + Oscillating Waving + Oscillating Waving + Oscillating Waving + Oscillating

Waving + Oscillating

The low order modes have major mode shapes contribution and can describe the structure The low order modes have major mode shapes contribution and can describe the structure dynamic characteristics. Thus, the first five modes are selected as the target modes. The low order dynamic characteristics. Thus, the first five modes are selected as the target modes. The low order modes are mainly in the form of waving vibration. The sensor placement focuses on the waving modes are mainly in the form of waving vibration. The sensor placement focuses on the waving

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direction (Y direction in Figure 7) which is the main degrees of freedom (DOFs). There are a total of 2018,Considering 11, 147 1073 nodes Algorithms (DOFs). the difficulty of placing sensors in blade tip, the 1236of 14DOFs in the blade tip are negligible. Therefore, there are a total of 1037 nodes (DOFs) which participate in direction (Y direction in Figure 7) which main degrees of freedom (DOFs). There are a total of computation. The first five mode matrix Φ isisthe a 1037 × 5 matrix. 1073 nodes (DOFs). Considering the difficulty of placing sensors in blade tip, the 36 DOFs in the blade Here, we place 10 WSNs nodes sensors. Figure 8a,b shows the MAC bar graph by AFSA and tip are negligible. Therefore, there are a total of 1037 nodes (DOFs) which participate in computation. WCA-AFSA, Their off-diagonal elements are 0.122655 and 0.008995 Therespectively. first five mode matrix φ is amaximum 1037 × 5 matrix. Here, we place 10 WSNs nodes sensors. Figure 8a,b shows theshown MAC baringraph by AFSA and respectively. The values of MAC by AFSA and WCA-AFSA are Tables 3 and 4. The above WCA-AFSA, respectively. Their maximum off-diagonal elements are 0.122655 and 0.008995 respectively. result shows that the improved algorithm has better performance. The values of MAC by AFSA and WCA-AFSA are shown in Tables 3 and 4. The above result shows that the improved algorithm has better performance.

(b) MAC bar graph by WCA-AFSA

(a) MAC bar graph by AFSA

Figure 8. MAC bar graph.

Figure 8. MAC bar graph. Table 3. The values of MAC by AFSA. 1

0.101833 0.000211 0.122655 Table0.003755 3. The values of MAC by AFSA.

0.003755

1

0.113484

0.002971

0.004244

10.101833 0.003755 0.101833 0.000211 0.055712 0.122655 0.113485 1 0.008627 0.000211 0.002971 0.008627 1 0.058377 0.003755 1 0.113484 0.002971 0.004244 0.122655 0.004245 0.055712 0.058377 1 0.101833 0.113485 1 0.008627 0.055712 4. The values of MAC by WCA-AFSA. 0.000211 Table 0.002971 0.008627 1 0.058377 0.122655 0.004245 0.055712 0.058377 0.0059401 1 0.000093 0.003196 0.000084 0.000093 0.003196 Table 0.000084 0.005940

1 0.007748 0.006421 0.000097 0.007748 1 0.008995 0.005681 4. 0.006421 The values0.008995 of MAC by 1WCA-AFSA. 0.001002 0.000097 0.005681 0.001002 1

1 0.000093 0.003196 0.000084 0.005940 5.2. Optimal Sensor Placement Scheme1 0.000093 0.007748 0.006421 0.000097 0.003196 1 0.008995 0.005681 According to engineering0.007748 practice and test experience, acceleration sensor is usually employed for the monitoring health state0.006421 of the blade [15]. The scheme of placing 100.001002 WSNs node sensors on the 0.000084 0.008995 1 blade is listed in Table 5. 0.005940 0.000097 0.005681 0.001002 1 5.2. Optimal Sensor Placement Scheme According to engineering practice and test experience, acceleration sensor is usually employed for the monitoring health state of the blade [15]. The scheme of placing 10 WSNs node sensors on the blade is listed in Table 5. Table 5. Sensor placement scheme.

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Table 5. Sensor placement scheme. Sensor No.

Node No.

Sensor No.

Node No.

1 2 3 4 5

129 300 861 746 997

6 7 8 9 10

633 877 56 741 508

The sensors in this placement scheme distribute dispersedly. The scheme avoids the sensors’ local aggregation. Here, sensors are placed in the blade root and other large deformable airfoil locations, which are most in poor conditions. Data from these locations can comprehensively reflect the blade health state. 6. Conclusions Aiming at the WSN’s node optimal sensor placement for the health monitoring of a wind turbine blade, a weighted centroid fish swarm algorithm is put forward. There are better optimization effects when optimizing the sensor placement of the blade. The conclusions are as follows: (1)

(2)

WCA is introduced to construct the initial fish swarm to enhance the ergodicity of artificial fish optimization and avoid falling into local optimal solution. The fixed step size is improved to the adaptive step size, which speeds up the optimization speed and increases the convergence accuracy. WCA-AFSA is used to optimize the WSNs node of the blade. The results show that the WSNs nodes are dispersed, which ensures that the WSNs nodes are located at the most unfavorable position of the blade, and the health information of the blade can be obtained comprehensively.

Author Contributions: The original idea of the article was provided by H.Y. and X.H. X.H. designed the algorithm and realized its MATLAB program. H.Y. and Y.Z. wrote the manuscript. H.Y. and Y.Z. jointly verified the WC-AFSA algorithm and analyzed the results. Funding: This research was funded by the Natural Science Foundation of Gansu Province (No. 1610RJZA041), Collaborative Innovation Team Project of Universities in Gansu Province (No. 2018C-12) and Talent Project of Lanzhou City (No. 2017-RC-66). Conflicts of Interest: The authors declare no conflicts of interest.

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