Proving the Existence of Solutions to the Problems of Minimizing the Energy Consumption of Sensor Networks

Abstract

Sensors and sensor networks have become widespread in today’s technological world. As with any technical system, sensors and sensor networks face certain challenges, one of which is the problem of energy efficiency, the solution of which allows for increasing the autonomy of sensors and making the network more cost-effective. This problem is relevant, and there are many studies devoted to its solution. In some of the proposed solutions, this problem is formulated as an optimization problem. However, in most cases, there is no proof of the existence of a solution to the presented optimization problem, which in turn calls into question the correctness of the proposed approaches. The article presents proof of the existence of a solution to the problem of minimizing the energy consumption of a sensor network by adjusting the sensor coverage radii. The work also proves the existence of a solution to a multi-criteria optimization problem, the criteria of which are the minimization of the network cost and the maximization of the area of the covered territory. An analysis of the energy efficiency of the network was carried out in cases of different values of the intersection of coverage zones and different values of the level of the intersection of coverage zones. Using the described approaches, the energy consumption of a specific sensor network was reduced by 7%, which confirms the effectiveness of the presented methods.

1. Introduction

Currently, sensors and sensor networks are widespread in all spheres of human activity. The use of sensors is ensured by their advantages, such as small size, various measured values, ease of software development, low power consumption, mobility and service life. Additionally, many tasks, such as collecting, analyzing, identifying and measuring data, can be solved using sensors [1,2]. In addition, sensors are an integral part of the Internet of Things and the Internet of Everything [3,4]. Additionally, sensors can be static and moveable. One of the main problems in the use of sensors and sensor networks is to ensure energy efficiency, as the size of the sensors is reduced, resulting in a decrease in the size of the battery. Most of the tasks of optimizing the energy consumption of sensor networks are formulated in the form of optimization problems. Among the factors affecting the energy consumption of a sensor network can be the energy consumption of each sensor, the number of sensors, the radius of sensor coverage, etc. Additionally, optimization criteria can be the cost of the network, the area of the territory covered by the network and the time of autonomous operation. Some problems are solved using iterative algorithmic procedures. It is also possible to use approximate methods of finding solutions to optimization problems. However, in most cases, the existence of a solution to the problem has not been proven, which calls into question the correctness of such problems. Thus, proving the existence of solutions is an integral part of the process of finding a solution that satisfies the optimality condition.

Article [5] introduces a charging planning problem for multiple chargers; namely, the Charging Energy Efficiency Maximization problem for Multi-Chargers in WRSNs is introduced. The problem aims to maximize the charging energy efficiency of the charging process by assigning the charging amount and planning the charging path. To balance the charging consumption among multiple chargers, the authors proposed Ring-Wandering Algorithm and Eight-Wandering Algorithm. In the presented article, NP-hardness of the problem is proved, but the intersection of sensor coverage areas is not taken into account, and the existence of the solution of the problem with the intersection of sensor coverage areas is not proved.

Paper [6], approaches to the allocation of resources in the Internet of Things sensor network systems applied to water-quality monitoring for the optimal and more sustainable utilization of resources are presented. An energy efficiency optimization problem is formulated for the successive wireless power sensor network system and solved by exploiting the problem structure and through a meta-heuristic algorithm. However, the existence of the solution of the described optimization problem is not proven.

In work [7], a survey on applications of the Markov Decision Process (MDP) for reducing sensor networks’ energy consumption is presented by designing the MDP framework which is a powerful tool for decision making and further providing solutions to obtain energy efficient sensor network. Various methods for solutions are discussed, and the methods are compared but do not describe the formulation of the optimization problem and prove the existence of its solution.

In article [8], power consumption of the various wireless sensor network components is analyzed. The paper includes a discussion of various factors and the way they impact the level of energy efficiency, but any optimization problem formulation is not presented.

In study [9] is a numerical study of the energy consumption and time efficiency of sensor networks with five different structural topologies and four different routing methods, regarding their performances and costs, which might provide some references and guidelines for designing sensor networks under various conditions for possible applications. However, the existence of the solution of the described optimization problem is not proven.

In [10], a method for optimal coverage of the territory by sensors with different coverage radii was proposed. To solve this problem, the authors modified the least squares method. The criterion for optimality is the maximum coverage of the territory with a minimum intersection of coverage areas. The existence of the solution of the described optimization problem is not proven.

In the manuscript [11], models and information technologies of coverage of the territory by sensors with energy consumption optimization are described. A modification of the method for constructing energy-efficient sensor networks using static and dynamic sensors is presented in article [12]. A Method for Maximum Coverage of the Territory by Sensors with Minimization of Cost is described in paper [13]. In the described works [11,12,13], authors find optimal sensor network parameters by a solution of multicriterial optimization problems, but the existence of the solution of these optimization problems is not proven.

Attention should also be paid to the use of distributed methods to solve current problems that arise in the design and use of wireless sensor networks. Thus, in work [14], an energy-aware method for selecting cluster heads in wireless sensor networks is presented. To select an appropriate cluster head, authors first model this problem by using multi-factor decision-making according to the four factors, including energy, mobility, distance to center and the length of data queues. Then, authors use the Cluster Splitting Process algorithm and the Analytical Hierarchy Process method in order to provide a new method to solve this problem. However, the authors do not take into account the radius of coverage of each sensor as a factor affecting energy consumption and the intersection of sensor coverage areas for data exchange between sensors. In the article [15], a wireless power transfer networks’ meanwhile maximizing the transmitted power requirement as an optimization problem is presented. In order to provide an approximated solution to this problem, the authors introduce a dual ascent-like distributed charging algorithm with its in-depth theoretical analysis, but the existence of the solution of the formulated optimization problems is not proved. In reference [16], a multi-agent distributed solution for linear programming problems with time-invariant box constraints on the decision variables and possibly time-varying inequality constraints is proposed. Such a class of linear programming problems is relevant in different multi-agent smart systems. In the proposed approach, each agent computes only a single or a few of decision variables, while convergence to the optimal solution for the overall problem is guaranteed.

Table 1. Literature review comparison.

Authors	Main Focus/Outcome	Factors Not Taken into Account
Yi Hong et al. [5]	The charging energy efficiency of the charging process maximization by assigning the charging amount and planning the charging path	The intersection of sensor coverage areas is not taken into account, and the existence of the solution of the problem with the intersection of sensor coverage areas is not proved
Segun O.Olatinwo et al. [6]	The allocation of resources in Internet of Things sensor network systems applied to water-quality monitoring for optimal and more sustainable utilization of resources	The existence of the solution of the described optimization problem is not proved
Gauri Kalnoor et al. [7]	Markov Decision Process for reducing sensor networks’ energy consumption	The formulation of the optimization problem and prove the existence of its solution
Doko Bandur et al. [8]	Discussion about various factors and the way they impact the level of energy efficiency of sensor networks	Optimization problem formulation is not presented
Fan Yan at al. [9]	Numerical study of energy consumption and time efficiency of sensor networks with five different structural topologies and four different routing methods	The existence of the solution of the described optimization problem is not proved
U. Elangovan et al. [10]	The method of optimal coverage of theterritory by sensors of differentcoverage radius.	The existence of the solution of the described optimization problem is not proved
V. Petrivskyi et al. [11,12,13]	Models and information technologies of energy-efficiency sensor networks constructing	The existence of the solution of the described optimization problems is not proved
A. Rahiminasab et al. [14]	Energy-aware method for selecting cluster heads in wireless sensor networks	The radius of coverage of each sensor as a factor affecting energy consumption and the intersection of sensor coverage areas for data exchange between sensors
K.S. Yıldırım at al. [15,16]	A wireless power transfer networks’ meanwhile maximizing the transmitted power requirement as an optimization problem	The existence of the solution of the described optimization problems is not proved

presents a comparison of the studies found in the literature review, the main focus/outcome of the study, and the factors that were not taken into account.

Based on the reviewed studies, it can be concluded that the problem of energy efficiency of sensor networks is very relevant. Different methods and approaches are used to solve this problem; in particular, they formulate this problem in the form of an optimization problem. In the theory of optimization, a necessary step in solving problems is proving the existence of a solution to the formulated problem. Unfortunately, in most modern studies, the authors do not prove the existence of a solution to the described problems. That is why, based on the studies reviewed above, it can be concluded that proving the existence of solutions of optimization problems for finding optimal parameters of sensor networks is an urgent problem.

2. Materials and Methods

Let’s consider the problem of finding the radii of coverage of the sensors of the sensor network, which achieves the minimization of energy consumption under the condition of the intersection of coverage areas. According to [11], this problem has the form (1)–(5):

$$E\left(r_1,r_2,\dots ,r_n\right)\to \underset{r\in G}{\min } ,$$

(1)

$$G:\{\begin{array}{c}{r}_1\in \left[r_1^{min};r_1^{max}\right]\\ r_2\in \left[r_2^{min};r_2^{max}\right]\\ \dots \\ r_n\in \left[r_n^{min};r_n^{max}\right]\end{array},$$

(2)

$$a_{i,j}\left(r_i+r_j-\sqrt{{\left(x_i-x_j\right)}^2+{\left(y_i-y_j\right)}^2}-c\right)\le 0,$$

(3)

$$-a_{i,j}<0,$$

(4)

$$i=\overline{1,n}, j=\overline{i+1,n},$$

(5)

where (1)—the objective function, (2)—the area of definition of the function, (3)—the conditions for ensuring the minimum allowable cross section of a pair of sensors whose coverage areas intersect, (4)—intersection matrix elements $a_{i,j}=\{\begin{array}{c}1, l_{i,j}<r_i+r_j\\ 0,l_{i,j}\ge r_i+r_j\end{array}$, $l_{i,j}$—Euclidean distance between sensors $i$ and $j$, and $n$—count of sensors.

The energy consumption of the network (1) calculates as the sum of the energy consumption of network elements:

$$E\left(r_1,r_2,\dots ,r_n\right)={\displaystyle \sum }_{i=1}^n\left(Er\left(r_i\right)+E{s}_i+E{a}_i\right).$$

(6)

According to [12], the energy consumption of each sensor is calculated as:

$$E=Er+Es+Ea,$$

(7)

where $Er$—energy consumption of data reception, $Es$—energy consumption of data transmission, and $Ea$—energy consumption monitoring of the covered area.

In turn, according to [17], energy consumption monitoring will be calculated as:

$$Ea=r^2.$$

(8)

For the existence of the solution of the described problem, the goal function must be convex, and the domain of the function must be compact. Let us prove the existence of a solution to the formulated problem.

Lemma 1.

The function$E\left(r_1,r_2,\dots ,r_n\right)$reaches its minimum value in the domain$G$.

Proof of Lemma 1.

The function $E\left(r_1,r_2,\dots ,r_n\right)$ is continuous as the sum of continuous functions (6). The domain G is bounded because all variables have minimum and maximum values of $r_i\in \left[r_i^{min};r_i^{max}\right]$, $i=\overline{1,n}$. Additionally, the domain G is closed, since for all variables the segment $\left[r_i^{min};r_i^{max}\right]$ is closed. Thus, the domain G is compact, and, according to Weierstrass’s theorem [18], the function $E\left(r_1,r_2,\dots ,r_n\right)$ reaches its minimum in the domain G. Lemma is proved. □

Theorem 1.

The problems (1)–(5) has a solution.

Proof of Theorem 1.

The function $E\left(r_1,r_2,\dots ,r_n\right)$ is convex, continuous and differentiated. According to Lemma 1, the domain G is compact; then, according to the Kuhn–Tucker theorem [19], there is a saddle point of the Lagrange function, and it is an extremum. Thus, problems (1)–(5) has a solution. The theorem is proved. □

Since the function $E\left(r_1,r_2,\dots ,r_n\right)$ is convex and differentiated, we use the gradient descent method to find the minimum of the objective function. The solution of problems (1)–(5) is the vector $r=\left(r_1,r_2,\dots ,r_n\right)$, the elements of which are the radii of coverage of the corresponding sensors, which minimizes energy consumption at a given level of the intersection of coverage areas.

According to [13], the problem of minimizing the cost of the sensor network while maximizing the coverage is formulated in the form (9)–(11):

$$\{\begin{array}{c}P\left(r,c\right)\to min\\ S_{uncovered}\left(r,c\right)\to min\end{array},$$

(9)

$$r_{min}\le r\le r_{max},$$

(10)

$$0\le c\le c_{max},$$

(11)

where $r$—coverage radius, $c$—intersection level, $S_{uncovered}={(2r-c)}^2-\pi r^2$, $r_{min}$—minimum coverage radius and $r_{max}$—maximum coverage radius.

In problems (9)–(11), the possibility of regulating the radius of coverage (10) and the level of the intersection of coverage areas (11) are taken into account. The result of solving this problem will be a set of Pareto-optimal pairs of radii of coverage and levels of the intersection of coverage areas with which the maximum coverage of the territory is achieved at the minimum cost of the network.

Using the method of convolution of criteria, construct the objective function:

$$F\left(r,c\right)=\alpha P\left(r,c\right)+\left(1-\alpha \right)S_{uncovered}\left(r,c\right),$$

(12)

where $\alpha$—expert assessment of the importance of parameters.

Using function (12) present problem (9)–(11) in the form:

$$F\left(r,c\right)\to \underset{\left(r,c\right)\in G}{\min } ,$$

(13)

$$G:\{\begin{array}{c}r\in \left[r_{min},r_{max}\right]\\ c\in \left[c_{min},r_{max}\right]\end{array}.$$

(14)

Let us prove the existence of a solution to the described problem.

Lemma 2.

Objective function$F\left(r,c\right)$ (13) reaches its minimum value in the domain G.

Proof of Lemma 2.

The function $\mathit{F}\left(\mathit{r},\mathit{c}\right)$ is continuous as the sum of continuous functions (12). The domain G is bounded because all variables have minimum and maximum values $\mathit{r}\in \left[{\mathit{r}}_{\mathit{m}\mathit{i}\mathit{n}},{\mathit{r}}_{\mathit{m}\mathit{a}\mathit{x}}\right]$ and $c\in \left[c_{\mathit{m}\mathit{i}\mathit{n}},{\mathit{r}}_{\mathit{m}\mathit{a}\mathit{x}}\right]$. The domain G is also closed, since for the variables $\mathit{r}$ and $c$ the segments $\left[{\mathit{r}}_{\mathit{m}\mathit{i}\mathit{n}},{\mathit{r}}_{\mathit{m}\mathit{a}\mathit{x}}\right]$ and $\left[c_{\mathit{m}\mathit{i}\mathit{n}},{\mathit{r}}_{\mathit{m}\mathit{a}\mathit{x}}\right]$ are closed. Thus, the domain G is compact, and, according to Weierstrass’s theorem, the function $\mathit{F}\left(\mathit{r},\mathit{c}\right)$ reaches its minimum value in the domain G. Lemma is proved. □

Theorem 2.

Problems (13) and (14) has a solution.

Proof of Theorem 2.

The function $\mathit{F}\left(\mathit{r},\mathit{c}\right)$ is a convex and linear combination of convex functions, continuous and differentiated. According to Lemma 2, the domain G is compact; then, according to the Kuhn–Tucker theorem, there is a saddle point of the Lagrange function and it is an extremum. Thus, problem (13) and (14) has a solution. The theorem is proved. □

The parameter $\alpha$ can be included in the function $F\left(r,c\right)$ as an unknown variable. Thus, we obtain the problem in the form (15) and (16):

$$F_1\left(r,c,\alpha \right)\to \underset{\left(r,c,\alpha \right)\in G}{\min } ,$$

(15)

$$G:\{\begin{array}{c}r\in \left[r_{min},r_{max}\right]\\ c\in \left[c_{min},r_{max}\right]\\ \alpha \in \left[0,1\right]\end{array}.$$

(16)

Let us prove the existence of a solution to the described problem.

Lemma 3.

The function$F_1\left(r,c,\alpha \right)$ (15) reaches its minimum value in the domain G (16).

Proof of Lemma 3.

The function ${\mathit{F}}_{\mathbf{1}}\left(\mathit{r},\mathit{c},\mathbf{\alpha }\right)$ is continuous as the sum of continuous functions (12). The domain G is bounded because all variables have minimum and maximum values $\mathit{r}\in \left[{\mathit{r}}_{\mathit{m}\mathit{i}\mathit{n}},{\mathit{r}}_{\mathit{m}\mathit{a}\mathit{x}}\right]$, $c\in \left[c_{\mathit{m}\mathit{i}\mathit{n}},{\mathit{r}}_{\mathit{m}\mathit{a}\mathit{x}}\right]$ and $\mathbf{\alpha }\in \left[\mathbf{0},\mathbf{1}\right]$. The domain G is also closed, since for the variables $\mathit{r}$, $c$ and $\mathbf{\alpha }$ the segments $\left[{\mathit{r}}_{\mathit{m}\mathit{i}\mathit{n}},{\mathit{r}}_{\mathit{m}\mathit{a}\mathit{x}}\right]$, $\left[c_{\mathit{m}\mathit{i}\mathit{n}},{\mathit{r}}_{\mathit{m}\mathit{a}\mathit{x}}\right]$ and $\left[\mathbf{0},\mathbf{1}\right]$ are closed. Thus, the domain G is compact, and, according to Weierstrass’s theorem [18], the function ${\mathit{F}}_{\mathbf{1}}\left(\mathit{r},\mathit{c},\mathbf{\alpha }\right)$ reaches its minimum value in the domain G. Lemma is proved. □

Theorem 3.

Problems (15)–(16) has a solution.

Proof of Theorem 3.

The function ${\mathit{F}}_{\mathbf{1}}\left(\mathit{r},\mathit{c},\mathbf{\alpha }\right)$ is a convex and linear combination of convex functions, continuous and differentiated. According to Lemma 3, the domain G is compact; then, according to the Kuhn–Tucker theorem [19], there is a saddle point of the Lagrange function and it is an extremum. Thus, problems (15) and (16) has a solution. The theorem is proved. □

To find the solutions of problems (13)–(16), use the gradient descent method. The solutions of the problems will be a pair of values $\left(r_{opt},c_{opt}\right)$, which will be the value of the optimal radius of coverage and the value of the intersection of coverage areas where maximum coverage is achieved while minimizing energy consumption.

3. Results

Consider a sensor network, the elements of which have the following parameters (

Table 2. Input sensor network parameters.

Sensor	X Coordinate	Y Coordinate	Coverage Radius, m	Battery Volume, mAh
Sensor 1	15	15	13	9800
Sensor 2	27	18	12	10,000
Sensor 3	30	35	12	12,500
Sensor 4	48	43	15	15,000
Sensor 5	70	25	20	12,000
Sensor 6	49	60	15	15,000
Sensor 7	80	50	14	13,000

):

The schematically described above network can be presented as follows (Figure 1):

The coordinates of the sensors presented in Table 1 represent the positions of the sensors of the sensor network transferred to the coordinate plane (Figure 1).

Using the proposed approach with the level of the intersection of coverage areas c = 2, we obtained a network that schematically can be presented in the next form (Figure 2):

According to the obtained results, the parameters of the sensors are presented in

Table 3. Result sensors’ parameters.

Sensor	Sensor Coordinates	Optimal Coverage Radius, m	Optimal Energy Consumption, mAh	Input Lifecycles Number	Result Lifecycles Number
Sensor 1	(15; 15)	6.37	40.69	57.98	240.83
Sensor 2	(27; 18)	7.99	63.84	69.44	156.62
Sensor 3	(30; 35)	11.27	127.06	86.80	98.37
Sensor 4	(48; 43)	10.42	108.68	66.66	138.01
Sensor 5	(70; 25)	20	400	30	30
Sensor 6	(49; 60)	8.60	74.02	66.66	202.62
Sensor 7	(80; 50)	8.92	79.67	66.32	163.17

.

The number of lifecycles was calculated by dividing the battery volume by the amount of energy consumption.

It should be noted that changing the location of the sensors will lead to a change in the result, a change in the parameters of the resulting network (sensor coverage radius, energy consumption, number of life cycles).

4. Discussion

To assess the effectiveness of the proposed approach, we calculated the energy consumption of each sensor depending on the level of the intersection of the coverage areas (Figure 3). The interval of change in the level of the intersection of coverage areas was from zero to seven and was in increments of 0.5. The upper limit of the studied interval was experimental.

Also, we calculated the number of life cycles of each sensor depending on the level of intersection of the coverage areas (Figure 4) for the values of the level of the intersection of the coverage areas similar to the values in Figure 3.

Analyzing the obtained results (Figure 3 and Figure 4), we can conclude that the minimum value of energy consumption is achieved in the absence of the intersection of coverage areas. Additionally, in this case, the maximum number of life cycles of sensors is reached. In addition, it can be concluded that achieving the required level of the intersection of coverage areas is achieved by adjusting the coverage radii of sensors 2, 4 and 7 with coordinates (27; 18), (48; 43) and (80; 50), respectively. Adjusting the radii of coverage of other sensors does not allow to achieve the required level of the intersection of the coverage areas of all sensors simultaneously. It is also worth noting that the change of coverage radii and optimization of energy consumption for this network is achieved when the level of the intersection of coverage areas is less than 6.5 m. When comparing the change in energy consumption values for different values of the level of the intersection of coverage areas, we can conclude that using the steamed approach managed to reduce energy consumption by at least 7%. According to Figure 3, the energy consumed by sensor 5 remains constant regardless of the situation of other sensors. This is due to the fact that when changing the coverage radius of this sensor, it is impossible to achieve the required level of the intersection of coverage zones with other sensors, which leads to the lack of a solution to the problem. This also explains the lack of change in the energy consumption of this sensor (Figure 4).

We considered the case of two sensors with coordinates (33; 28) and (50; 28) and radii of coverage of 15 m. The minimum coverage radii are $r_1^{min}=3$ and $r_2^{min}=2$. The distance between the sensors is $l=17$ m. At the level of intersection of the coverage zones $c=3$ according to Formula (3), the optimal radii of coverage must satisfy condition (17).

$$r_1+r_2=l+c, \forall r_1\in \left[r_1^{min},r_1^{max}\left],\forall r_2\in \right[r_2^{min},r_2^{max}\right],$$

(17)

In the case of two sensors, the energy consumption function will be a function of two variables $E\left(r_1,r_2\right)$, the graph of which is the second order surface. Fixing the value of $r_1=5$ according to Formula (17) at the value of $l=17$ and $c=3$, we obtained $r_2=15$. Using the values of $r_1$ and $r_2$, we calculated the energy consumption $E\left(r_1,r_2\right)=250$. We constructed the plane $E=250$ and the value of energy consumption $E\left(r_1,r_2\right)$ for $r_1\in \left[3, 15\right],r_2\in \left[2, 15\right]$ (Figure 5).

However, there are some cases when a corresponding optimization problem does not have a solution:

• Sensor coverage areas do not overlap (Figure 6a);
• The values of the coverage radius and/or the intersection level of the coverage zones do not have a clear upper and lower limit;
• The coverage area of one sensor completely overlaps the coverage area of at least one other sensor (Figure 6b).

5. Conclusions

The article examines the issue of proving the existence of solutions to the problems of minimizing the energy consumption of a sensor network by adjusting the coverage radii and minimizing the cost of the network while maximizing the coverage area. These problems are formulated in the form of optimization problems, so proving the existence of a solution is a necessary condition for using the proposed approaches. The proof of the existence of solutions is based on the use of the Weierstrass and Kuhn–Tucker theorems. The case of data exchange between network elements is considered, the main condition of which is the intersection of sensor coverage zones. An analysis of the effectiveness of the proposed approach in the case of different values of the intersection level of the coverage zones was carried out. Based on the analysis of the energy consumption function, it was concluded that the energy consumption function forms a plane in the n-dimensional Euclidean space, where n is the number of sensors in the network. The values of the squares of the optimal coverage radii are located at the intersection of this plane with the plane equal to the optimal value of energy consumption. This statement is illustrated in the case of a network consisting of two sensors. With the use of the presented methods, the energy efficiency of a particular network increased by 7%, which confirms the effectiveness of the proposed methods.

The paper presents a Big Data good practice related to designing sensor network with optimal coverage and energy consumption. The work was fulfilled within the framework of the project iBigWorld [20], which is devoted to gaining skills for developing innovative solutions based on Big Data in real-world applications.