The Particle Swarm Differential Evolution Algorithm for Ecological Sensor Network Coverage Optimization

2.1 Points Coverage

This paper assumes that the monitoring area is a two-dimensional plane, the isomorphic sensor nodes will be put in the area, and the sensing radius and the communication radius of all sensors will be the same. The number of the nodes is N, and the coordinates of each node are known; then, the sensing radius is r and the communication radius is R. When R = 2r, the network connectivity can be guaranteed and the wireless interference can be avoided. The sensor nodes set S = {s₁, s₂, …, s_N}, s_i = (x_i, y_i, r) and x_i, y_i are, respectively, the horizontal and vertical coordinates of the sensor node s_i. When the detecting target falls in the circle and (x_i, y_i) is the center, r is the radius of the circle, and it will indicate that the target will be covered by the sensor node s_i.

2.2 Area Coverage

The sensor nodes set S^* is the subset of set S. Moreover, Area(S^*) represents the area covered by the nodes set S^*. The area coverage rate is defined as

However, it is very complex and difficult to calculate Area(S^*) in actual practice [9]. To simplify the calculation, the monitoring area is rasterized into M₁*M₂ points. Supposing that the coordinates of the rasterized points are (x, y), then the probability of the point (x, y) covered by the node s_i will be defined as

Subsequently, the probability of the point (x, y) covered by the any node can be defined as

where “|” is the OR operation, and Area(S^*) is calculated as

and then the area coverage rate is correspondingly redefined as

3.1 Particle Swarm Optimization

In the standard PSO algorithm, each optimization problem solution is regarded as a “particle” of the search space. At first, the algorithm is initialized to a group of random particles (random solutions), and each particle has the properties of position and velocity. The particle is updated by tracking two extremes, i.e., the best previous position of the particle itself and the best particle among all the particles. The process of the algorithm is shown in Figure 1; the velocity and position of the particles are updated according to the equations as follows:

Figure 1:

The Flowchart of Particle Swarm Optimization.

where t is the number of iterations, R₁ and R₂ are uniformly distributed numbers between 0 and 1, the parameter ω is called the inertia weight, and c₁ and c₂ are acceleration constants. From the perspective of sociology, the first part of the former items is the memory item, which represents the influence of the past on the present; the second part, which is associated with a local search, is called as a self-cognitive item, and it represents that the behavior of the particle derives from its own experience; the third part, which is associated with a global search, is known as a social-cognitive item, and it reflects the collaboration and the sharing knowledge between the particles. All the above means that the movement of the particle is decided by their own experience and the experience of the best companions.

3.2 Differential Evolution

The basic idea of the DE algorithm is that the new temporary population will be generated from the current population by the mutation and crossover operation; then, the two populations will adopt the one-on-one selection operation to generate the next new population based on a greedy idea. An offspring will be generated by the mutation operation as follows:

where t is the number of the iteration, η is the scale factor, and p₁ ≠ p₂ ≠ p₃. The mutation scheme shown in Eq. (8) is also called as DE/rand/1. The extended modes of the mutation rule have been subsequently proposed in the literature [16]. Other versions of the mutation rule are also listed as follows:

DE/rand/2:

x′ij(t + 1) = xp0j(t) + η(xp1j(t) − xp2j(t) + xp3j(t) − xp4j(t)).

DE/best/1:

x′ij(t + 1) = xbest,j(t) + η(xp1j(t) − xp2j(t)).

DE/best/2:

x′ij(t + 1) = xbest,j(t) + η(xp1j(t) − xp2j(t) + xp3j(t) − xp4j(t)).

DE/current-to-rand/1:

x′ij(t + 1) = xij(t) + η(xp1j(t) − xij(t)) + η(xp2j(t) − xp3j(t)).

DE/current-to-best/1:

x′ij(t + 1) = xij(t) + η(xbest,j(t) − xij(t)) + η(xp2j(t) − xp3j(t)),

where x_best denotes the individual with the best performance among solutions of the population and p₀ ≠ p₁ ≠ p₂ ≠ p₃ ≠ p₄.

3.3 Hybrid Algorithm (PSI-DE)

The hybrid algorithms of DE and PSO could be divided into two classes [27] of which the first method is to adopt DE to improve PSO. The differential mutation is applied to make the particles escape from local optima [6], to make perturbation on the best personal positions of the particles [30], to modify the velocity replacement scheme [3], and to update the position of the particle partly [4]. The other method is to adopt PSO to improve DE. The second one is relatively fewer than the first. The DE is applied to mutate the attractor for each particle, and the attractor, which is inspired by PSO’s convergence analysis [2], is defined as a weighted average of its best personal and neighborhood positions [17].

By comparing and analyzing the individual updated formulas of the DE algorithm and the PSO algorithm, it can be seen that there are some similarities between the two algorithms, and one of which is the differential vector. In the DE algorithm, the differential vector is the difference of two random individuals or the random individual with the best individual. In the PSO algorithm, the differential vector is the difference of the current individual with the best personal individual in history, and the current individual with the global best individual. Inspired by the DE/current-to-rand/1 and the kernel concept of PSO, a novel mutation strategy is proposed as follows:

where F₁ and F₂ are the scale factors; F₁ and F₂ are, respectively, equal to R₁c₁ and R₂c₂. The formula is named as DE/current-to-the-own-best; current-to-the-global-best/1. The main idea of the Eq. (9) is to generate the offspring according to the solution’s own experience and the best experience of neighbors. The process of the hybrid DE algorithm (PSI-DE) with particle swarm intelligence is as follows:

Initialize the individuals of the population;

t←0;

N: the population size;

MAXGENS: the number of max generations;

while t < MAXGENS

 for i = 1:N

  Compute f(x_i);

 end-for

 for i = 1:N

  Update the extreme P_i;

 end-for

 Update the global extreme p_g;

 for i = 1:N

  /*mutation*/

  Generate the offspring x′ij according to Eq. (9)

  /*crossover*/

 for j = 1: the dimension of the solution

  if rand(0, 1)<crossover probability

   xoff,j = x′ij

  else

   xoff,j = xij

  end-if

 end-for

  /*selection*/

  iff(x_off) < f(x_i)

   Save the individual x_off and replace the x_i;

  end-if

 end-for

 t++

end while

4.1 The Influence of Population Size on the Coverage Performance

In this subsection, the influence of population size on the coverage performance is discussed. The number of iterations is 200; the number of the wireless sensor nodes is 20; the sensing radius is 3; and the raster size is 2*2. The population size is set to 10, 20, 30, and 40, respectively. In Table 1, the first column is the population size, the second is the best coverage rate of the 10 runs, the third is the average coverage rate of the 10 runs, the fourth is the worst coverage rate of the 10 runs, and the last is the variance of the 10 runs. Figure 2 is the coverage rate curves when the population size was set four kinds of different values. Figure 3 is the line chart of the coverage rate when the population size is changing. Figure 4 is the layout when the population size is 30. From the experimental results shown in Table 1 and Figures 2–4, it can be seen that the coverage rate is optimal when the population size is 30; in other words, the population size is not of “the bigger the better” style. The increase of the population size can improve the coverage rate to a certain extent, but the computational speed greatly reduces; therefore, in the PSI-DE algorithm, the population size should be selected appropriately.

Figure 2:

The Coverage Rate Curves When the Population Size Was Set Different Values.

Figure 3:

Three Kinds of the Line Chart of the Coverage Rate with the Population Size Changing.

Figure 4:

The Node Layout When the Population Size Was 30.

4.2 The Influence of Iteration Number on the Coverage Performance

In this subsection, the influence of iteration number on the coverage performance is discussed. The population size is 20; the number of the wireless sensor node is 20; the sensing radius is 3; and the raster size is 2*2. The iteration number is set at 50, 100, 150, 200, and 250, respectively. Table 2 shows the statistical results about the influence of the iteration number on coverage performance, and the meanings of the five columns are the same with the data in Table 1. Figure 5 is the coverage rate curves when the iteration number was set with five kinds of different values. Figure 6 is the line chart of the coverage rate when the iteration number is increasing. Figure 7 is the wireless sensor nodes layout when the iteration number is 200. When the iteration number is 200, both the best coverage rate and the average rate are the biggest; when the iteration number is 250, the worst coverage rate is the biggest and the variance is the smallest.

Figure 5:

The Coverage Rate Curves When the Iteration Number Was Set with Different Values.

Figure 6:

Three Kinds of the Line Charts of the Coverage Rate with the Iteration Number Changing.

Figure 7:

The Node Layout When the Iteration Number Was 200.

It can be concluded that the greater the number of iteration is, the more stable the performance of the PSI-DE algorithm will be. However, the increase of the iteration number did not contribute to the improvement of the coverage rate; on the contrary, an additional iteration number increases the algorithm time, which causes the need for a suitable value for the iteration number.

4.3 The Influence of Node Sensing Radius on the Coverage Performance

In this subsection, the influence of node sensing radius on the coverage performance is discussed. The number of iteration is 200; the population size is 20; the number of the wireless sensor node is 20; the raster size is 2*2; and then the sensing radius is 2.5, 3, 3.5, 4, 4.5, 5, and 5.5, respectively.

The experimental results in Table 3 are the coverage rate and variance in the case of the different node sensing radius. From Table 3 and Figure 8, it can be seen that the coverage rate is gradually increasing with the increasing of sensor node sensing radius, and when the sensing radius is 5.5, the area has been completely covered. According to the simulation experiment data in Table 3, the line chart about the coverage rate with the node sensing radius changing was drawn by using Matlab, as shown in Figure 9. In addition, Figure 10 is the node layout when the node sensing radius was 5. From Table 3 and Figures 8 and 9, the coverage rate increased with the node sensing radius is increasing. When the node sensing radius reaches 4.5, the rising slope of the curve will become smaller. It describes that the speed of the coverage rate improvement slows down when the sensing radius increases, which can be confirmed by Figure 9. The repeated coverage increased when the sensing radius increased, which can be proved by Figure 10.

Figure 8:

The Coverage Rate Curves When the Node Sensing Radius Was Set Different Values.

Figure 9:

The Line Chart of the Coverage Rate with the Node Sensing Radius Changing.

Figure 10:

The Node Layout When the Node Sensing Radius Was 5.

4.4 The Influence of Raster Size on the Coverage Performance

In this subsection, the influence of node sensing radius on the coverage performance is discussed. The number of iteration is 200; the population size is 20; the number of the wireless sensor node is 20; the sensing radius is 3; and the length of the raster side is 1, 2, 2.5, 4, and 5, respectively. The experimental results in Table 4 are the coverage rate and variance in the circumstances of different raster sizes. Figure 11 is the coverage rate curves when the raster size was set with five kinds of different values. Figure 12 is the line chart of the coverage rate when the raster size is changing. Figures 13 and 14 are the wireless sensor nodes layouts when the raster size is 1*1 and 4*4, respectively. The difference of the uncovered area size between Figures 13 and 14 is not obvious. When the raster size is 1*1 and 4*4, the coverage rate is correspondingly 63.25% and 76%, and the difference between the two coverage rates is relatively obvious. When the length of the raster side increases, the coverage rate gradually increases as well. However, it does not mean that the actual coverage area increases because of the calculation method of coverage rate.

Figure 11:

The Coverage Rate Curves When the Raster Was Set in Different Size.

Figure 12:

The Line Chart of the Coverage Rate with the Raster Size Changing.

Figure 13:

The Node Layout When the Raster Size Was 1*1.

Figure 14:

The Node Layout When the Raster Size Was 4*4.

4.5 The Influence of the Number of Nodes on the Coverage Performance

In this subsection, the influence of the number of nodes on the coverage performance is discussed. The number of iteration is 200; the population size is 20; the sensing radius is 3; the raster size is 2*2; and the number of the wireless sensor node is 10, 20, 30, 40, 50, and 60, respectively. The experimental results in Table 5 are the coverage rate and variance in the situation of the different number of nodes. Figure 15 is the coverage rate curves when the number of nodes was set with six kinds of different values. Figure 16 is the line chart of the coverage rate when the number of nodes is changing. Figure 17 is the wireless sensor nodes layout when the number of nodes is 40. From Table 5 and Figures 15 and 16, the coverage rate increased with the number of nodes increasing. When the number of nodes reaches 40, the rising slope of the curve will become smaller. It describes that the speed of the coverage rate improvement slows down when the number of nodes increases, which can be confirmed by Figure 15. When the number of nodes increases, the repeat coverage will increase, and it can be proved by Figure 16. It indicates that there is a large number of redundant nodes. Therefore, it is necessary to reduce or to close some nodes. When the number of nodes increases, the number of dimensions will increase, and the calculation amount of the algorithm will increase as well.

Figure 15:

The Coverage Rate Curves When the Number of Nodes Was Set in Different Size.

Figure 16:

The Line Chart of the Coverage Rate with the Number of Nodes Changing.

Figure 17:

The Node Layout When the Number of Nodes Was 40.