Energy Optimization with Adaptive Transmit Power Control for UAV-Assisted Data Transmission in VANETs

Abstract

Time for unmanned aerial vehicle (UAV) assisted vehicular ad hoc networks (VANETs) to promote the efficient data transmission is limited. To this end, improving the endurance of UAV has become a crucial issue. In this paper, we first propose an energy optimization model to improve the endurance of UAV, which consider not only the flying energy, but also communication energy. By considering the relative position between UAV and vehicle, adaptive transmission power is applied to communication energy consumption. Second, in order to verify the existence of the solution, we use Rolle’s theorem and the monotonicity of the function to prove the objective function, and obtain the approximate solution of the objective function by using the principle of inequality. Finally, compare with optimized algorithm and algorithm without optimized communication energy, and our proposed algorithm which performance is better than the existing energy optimization algorithms.

Supported by NSF of China under Grant No. 61772130, No. 71171045, No. 61772130 and No. 61901104; the Innovation Program of Shanghai Municipal Education Commission under Grant No. 14YZ130; and the International S&T Cooperation Program of Shanghai Science and Technology Commission under Grant No. 15220710600.

Appendix A

When $\nu >\nu _0$, the problem $E(\nu )$ is differentiable on the interval $(\nu _0, \infty )$, there is only one stable point $\nu ^\sharp$, and $\nu ^\sharp$ is the extreme value of $E(\nu )$ at $(\nu _0,\infty )$, when $E(\nu ^\sharp )$ is extreme value, the minimum value of $E(\nu )$ is $E(\nu ^\sharp )$. So the energy-minimization is $E(\nu ^\sharp )$.

Due to $\nu ^\sharp$ is the only stable point of $E(\nu )$ in the interval $(\nu _0,\infty )$, thus for any point of $\nu \in (\nu _0,\nu ^\sharp )\bigcup (\nu ^\sharp ,\infty )$, there will be $E^{'}(\nu )\ne 0$. Furthermore, we confirm that $E^{'}(\nu )$ is different symbol at both ends of $\nu ^\sharp$. If we assume $E^{'}(\nu )$ is same symbol at both ends of $\nu ^\sharp$, there will be have two point $\nu _1,\nu _2\in (\nu _0,\nu ^\sharp )\bigcup (\nu ^\sharp ,\infty )$, which can be given as

$$\begin{aligned} E^{'}(\nu _1)E^{'}(\nu _2)<0 \end{aligned}$$

(23)

According to the Rolle theorem, we can get the existence a point $\nu _e\in (\nu _1,\nu _2)$, which make

$$\begin{aligned} E^{'}(\nu _e)=0 \end{aligned}$$

(24)

From the above formula, which is contradicts the existence of only one stable point. Obviously, $E(\nu )$ is strictly monotone in these two intervals. Therefore, $E(\nu ^\sharp )$ is minimum value of UAV’s energy, and $\nu ^\sharp$ is the optimal solution to energy-minimization.

By simplifying the problem, we can get the following formula

$$\begin{aligned} E(\nu )=MR(\nu -\nu _0)+\frac{B+MR\nu _0^2}{\nu -\nu _0}+2MR\nu _0 \end{aligned}$$

(25)

Looking closely at the above formula, we found that the shape is like a check function. In order to further visualize the mathematical characteristics of the expression, we use $k=\nu -\nu _0$ to change the element. Through method of passing the mean inequality to address the problem.

$$\begin{aligned} MRk+\frac{B+MR\nu _0^2}{k}+2MR\nu _0\ge 2\sqrt{MRk\cdot \frac{B+MR\nu _0^2}{k}}+2MR\nu _0 \end{aligned}$$

(26)

When $MRk=\frac{B+MR\nu _0^2}{k}$, take the equal sign. So the energy of UAV is the minimum.

$$\begin{aligned} k=\sqrt{\frac{B+MR\nu _0^2}{MR}} \end{aligned}$$

(27)

Therefore, the optimal solution to energy-minimization is

$$\begin{aligned} \nu ^{*}=k+\nu _0=\sqrt{\frac{B+MR\nu _0^2}{MR}}+\nu _0 \end{aligned}$$

(28)