Efficient energy resource utilization in a wireless sensor system for monitoring water quality

Abstract

In this paper, a new approach to energy harvesting and data transmission optimization in a heterogeneous-based multi-class and multiple resource wireless transmission wireless sensor network system that focus on monitoring water and its quality is presented. Currently, energy is a scarce resource in wireless sensor networks due to the limited energy budget of batteries, which are typically employed for powering sensors. Once the available energy of a particular sensor node battery is depleted, such sensor node becomes inactive in a network. As a consequence, such node may not be able to participate in the transmission of the application signal in the uplink stage of the network, resulting in a lack of ability to communicate vital signals in a timely manner. Energy scarcity has been a long standing problem in wireless sensor network applications. To address this problem, energy harvesting from intended radiofrequency power source is considered in this work. However, wirelessly powered wireless sensor network systems are confronted by unfairness in resource allocation problem, as well as interference problem in multiple energy resource transmissions. These problems adversely impact the performance of the system in the context of the harvested energy by the sensor nodes, sensors information transmission rates, and the overall system throughput rate. These problems are tackled in this paper by formulating a sum-throughput maximization problem to reduce system energy consumption and enhance the system overall throughput rate. The throughput optimization problem is formulated as a non-convex function. Through the exploitation of the problem structure, it is converted to a convex function. The mathematical models of the optimization problem are validated through numerical simulations. The simulation results reveal that the proposed wireless powered sensor network system outperforms an existing wireless powered sensor network system, by comparison of the numerical simulations of this work to the numerical simulations of the existing WPSN system, regardless of the distances of the sensor nodes to the IPS and the base station. Also, the newly proposed method performs efficiently using parameters that include path-loss exponent impact, performance comparison of systems, convergence based on iteration, comparison based on unequal network distances to the BS, transmission power impact on the attainable throughput and on fraction of energy consumed on information transmissions, and influence of different number of nodes in the network classes.

Appendices

Appendix 1

Problem P1 transformation convexity proof

The parameters of (21) such as log function, rendered the optimization problem as a non-convex function, and it was transformed through problem structure exploration technique by changing variable t_cξ₀ to ξ_{0, c}, and obtaining the natural log of the log function in (9) and (18) respectively, to obtain a new problem (P2). From (9) and (18), (27) and (28) were derived as follows:

$$ {R}_a\left(j,\upzeta \right)=\frac{\upzeta_a}{\mathrm{In}2}\mathrm{I}n\left(1+{\alpha}_a\frac{\sum_{c=1}^C{j}_c}{\upzeta_a}\right),\forall a=1,2,\dots, \mathrm{A} $$

(27)

$$ {R}_b\left(\upxi \right)=\frac{\upxi_b}{\mathrm{In}2}\mathrm{I}n\left(1+{\gamma}_b\frac{\sum_{c=1}^C{\upxi}_{0,c}}{\upxi_b}\right),\forall a=1,2,\dots, \mathrm{B} $$

(28)

where:

$$ j={j}_1,{j}_2,\dots, {j}_C,\upzeta ={\upzeta}_1,{\upzeta}_2,\dots, {\upzeta}_A $$

and

$$ \upxi ={\upxi}_{0,1},{\upxi}_{0,2},\dots, {\upxi}_{0,M},{\upxi}_1,{\upxi}_2,\dots, {\upxi}_B $$

The sum-throughput of the overall sensor nodes in class A and Class B, which is defined by R_{a, b} is:

$$ {R}_{a,b}\left(j,\upzeta, \upxi \right)={\sum}_{a=1}^A{R}_a\left(j,\upzeta \right)+{\sum}_{b=1}^B{R}_b\left(\upxi \right) $$

(29)

Consequently, the optimization problem (P1) is transformed to a new problem defined as (P2).

(P2):

$$ {}_{j,\upzeta, \upxi\ }{}^{\max }{R}_{a,b}\left(j,\upzeta, \upxi \right) $$

(30)

s.t:

$$ {\sum}_{c=1}^C{j}_c+{\sum}_{c=1}^C{\upxi}_{0,c}+{\sum}_{a=1}^A{\upzeta}_a+{\sum}_{b=1}^B{\upxi}_b\le 1 $$

(31)

$$ {j}_c\ge 0,\forall c=1,2,..,\mathrm{C} $$

(32)

$$ {\upzeta}_a\ge 0,\forall a=1,2,..,\mathrm{A} $$

(33)

$$ {\upxi}_b\ge 0,\forall b=1,2,..,\mathrm{B} $$

(34)

The concavity of problem (P2) is shown as follows through the application of partial derivative

$$ {R}_{a,b}\left(j,\upzeta, \upxi \right)=\frac{\upzeta_a}{\mathrm{In}2}\mathrm{I}n\left(1+{\alpha}_a\frac{\sum_{c=1}^C{j}_c}{\upzeta_a}\right)+\frac{\upxi_b}{\mathrm{In}2}\mathrm{I}n\left(1+{\gamma}_b\frac{\sum_{c=1}^C{\upxi}_{0,c}}{\upxi_b}\right) $$

(35)

The second derivative of R_{a, b}(j, ζ, ξ) and j_c is:

$$ \frac{\partial^2{R}_{a,b}\left(j,\upzeta, \upxi \right)}{\partial {j_c}^2}=\frac{-{\alpha}_a^2{\upzeta}_a}{\mathrm{I}n2{\left({\upzeta}_a+{\alpha}_a{\sum}_{c=1}^C{j}_c\right)}^2} $$

(36)

Since ${\alpha}_a^2$, In2, and $\left({\upzeta}_a+{\alpha}_a\sum \limits_{c=1}^C{j}_c\right)$ are positive, then,

$$ \frac{\partial^2{R}_{a,b}\left(j,\upzeta, \upxi \right)}{\partial {j_c}^2}=\frac{-{\alpha}_a^2{\upzeta}_a}{\mathrm{I}n2{\left({\upzeta}_a+{\alpha}_a{\sum}_{c=1}^C{j}_c\right)}^2}\le 0 $$

(37)

if ζ_a ≥ 0. Hence, R_{a, b}(j, ζ, ξ) is concave with respect to j_c.

The second derivative of R_{a, b}(j, ζ, ξ) vis-à-vis ξ_{0, c} gives:

$$ \frac{\partial^2{R}_{a,b}\left(j,\upzeta, \upxi \right)}{\partial {\upxi_{0,c}}^2}=\frac{-{\gamma}_b^2{\upxi}_b}{\mathrm{I}n2{\left({\upxi}_b+{\gamma}_b{\sum}_{c=1}^C{\upxi}_{0,c}\right)}^2} $$

(38)

Since ${\gamma}_b^2$, In2, and $\left({\upxi}_b+{\gamma}_b\sum \limits_{c=1}^C{\upxi}_{0,c}\right)$ are positive, then,

$$ \frac{\partial^2{R}_{a,b}\left(j,\upzeta, \upxi \right)}{\partial {\upxi_{0,c}}^2}=\frac{-{\gamma}_b^2{\upxi}_b}{\mathrm{I}n2{\left({\upxi}_b+{\gamma}_b{\sum}_{c=1}^C{\upxi}_{0,c}\right)}^2}\le 0 $$

(39)

if ξ_b ≥0. Hence, R_{a, b}(j, ζ, ξ) is concave with respect to ξ_{0, c}.

The second derivative of R_{a, b}(j, ζ, ξ) and ζ_a gives:

$$ \frac{\partial^2{R}_{a,b}\left(j,\upzeta, \upxi \right)}{\partial {\upzeta_a}^2}=\frac{-{\alpha}_a}{\ {\upzeta}_a^2\mathrm{I}n2}{\left(1+{\alpha}_a\frac{\sum_{c=1}^C{j}_c}{\upzeta_a}\right)}^{-2}.{\left({\sum}_{c=1}^C{j}_c\right)}^2 $$

(40)

Also, the second derivative of R_{a, b}(j, ζ, ξ) and ξ_b gives:

$$ \frac{\partial^2{R}_{a,b}\left(j,\upzeta, \upxi \right)}{\partial {\upxi_b}^2}=\frac{-{\gamma}_b}{\ {\upxi}_b^2\mathrm{I}n2}{\left(1+{\gamma}_b\frac{\sum_{c=1}^C{\upxi}_{0,c}}{\upxi_b}\right)}^{-2}.{\left({\sum}_{c=1}^C{\upxi}_{0,c}\right)}^2 $$

(41)

$$ \frac{\partial^2{R}_{a,b}\left(j,\upzeta, \upxi \right)}{\partial {j}_c\partial {\upzeta}_a}=\frac{\alpha_a^2}{\ {\upzeta}_a^2\mathrm{I}n2}{\left(1+{\alpha}_a\frac{\sum_{c=1}^C{j}_c}{\upzeta_a}\right)}^{-2}.{\sum}_{c=1}^C{j}_c $$

(42)

$$ \frac{\partial^2{R}_{a,b}\left(j,\upzeta, \upxi \right)}{\partial {\upxi}_{0,c}\partial {\upxi}_b}=\frac{\gamma_b^2}{\ {\upxi}_b^2\mathrm{I}n2}{\left(1+{\gamma}_b\frac{\sum_{c=1}^C{\upxi}_{0,c}}{\upxi_b}\right)}^{-2}.{\sum}_{c=1}^C{\upxi}_{0,c} $$

(43)

$$ \frac{\partial^2{R}_{a,b}\left(j,\upzeta, \upxi \right)}{\partial {j}_c\partial {\upxi}_b}=0;\frac{\partial^2{R}_{a,b}\left(j,\upzeta, \upxi \right)}{\partial {\upxi}_{0,c}\partial {\upzeta}_a}=0;\frac{\partial^2{R}_{a,b}\left(j,\upzeta, \upxi \right)}{\partial {\upzeta}_a\partial {\upxi}_b}=0; $$

(44)

From the second derivative test, R_{a, b}(j, ζ, ξ) is a concave function and it can be solved through any known standard convex method.

Appendix 2

Minimization of energy harvesting unfairness

Using (22a), that is:

$$ {\sum}_{c=1}^C{j}_c+{\sum}_{c=1}^C{t}_c=1 $$

(45)

If

$$ {\sum}_{c=1}^C{P}_c{n}_a{j}_c+{\sum}_{c=1}^C{P}_c{u}_b{t}_c{\upxi}_0={\sum}_{c=1}^C{P}_c{n}_a{j}_c+{\upxi}_0{\sum}_{c=1}^C{P}_c{u}_b{t}_c $$

(46)

Then, (45) is true.

By substituting (46) in (4) and (15), (47) is derived as:

$$ {E}_{a,b}={\varepsilon}_a{\sum}_{c=1}^C{P}_c{n}_a{j}_c+{\varepsilon}_b{\upxi}_0{\sum}_{c=1}^C{P}_c{u}_b{t}_c,a=1,2,\dots, A,b=1,2,\dots, B $$

(47)

Let ${E}_{a,b}^{\ast }$be the optimal value for E_a and E_b due to ${j}_c^{\ast }$, ${t}_c^{\ast }$, and ${\upxi}_0^{\ast }$.

Since ${j}_c^{\ast }$ represents the optimal value for j_c , while ${t}_c^{\ast }$and ${\upxi}_0^{\ast }$represents the optimal values for t_c and ξ₀. Therefore, from (47), we have:

$$ {E}_{a,b}^{\ast }={\varepsilon}_a{\sum}_{c=1}^C{P}_c{n}_a{j}_c^{\ast }+{\varepsilon}_b{\upxi}_0^{\ast }{\sum}_{c=1}^C{P}_c{u}_b{t}_c^{\ast },a=1,2,\dots, A,b=1,2,\dots, B $$

(48)

From (48), ${j}_c^{\ast }$remains constant. Similarly, ${t}_c^{\ast }$ remains constant regardless of ${\upxi}_0^{\ast }$.