2. System Model and Problem Formulation
In this paper, we consider a three-tier space–air–ground network as shown in
Figure 1. We assume there are
I IoT devices on the ground, denoted by
= {1, 2, …,
I}, and a satellite with an orbital altitude of
provides the full coverage to all these devices. In our setup,
U UAVs, denoted by
= {1, 2, …,
U}, are deployed to collect data from IoT devices and transmit them to the satellite. The Cartesian coordinate system is introduced, so the coordinate of IoT device
i is denoted by
. The coordinates of UAV
u are denoted by
, where
is the fixed flight altitude. In order to support communication with a large number of IoT devices at the same time, OFDMA is adopted, which has the advantages of flexible bandwidth and power allocation over users [
21]. We assume that the total bandwidth available to IoT devices is equally divided into
I resource blocks, and each IoT device is allocated a resource block. Meanwhile, we assume that each UAV can only communicate with
K IoT devices simultaneously. This means each channel between UAVs and IoT devices is divided into
K sub-channels and there is no interference between different sub-channels. Assume the sub-channel bandwidth between each UAV and IoT devices is
B and there is no interference between different channels. For convenience of discussion, let
. Similarly, it is assumed that the channel bandwidth between each UAV and the satellite is
W, and there is no interference between different channels.
2.1. IoT-UAV Data Gathering
Considering there may be occlusion between UAVs and devices, according to [
22,
23], the average path-loss between IoT device
i and UAV
u can be expressed as
where
(Hz) is the carrier frequency,
v (m/s) is the velocity of light in a vacuum,
and
(dB) are the average additive losses caused by the free space path-loss for LOS and NLOS links, respectively,
is the distance between IoT device
i and UAV
u, and
is the LOS probability, which can be expressed as
where
and
are constants greater than zero determined by the environment.
is the elevation angle between IoT device
i and UAV
u. It can be seen that the probability of the line-of-sight link is closely related to the elevation angle. The higher the elevation angle, the greater the probability of line-of-sight link, and the greater the channel gain. The expression for elevation angle
is
Therefore, the channel gain
between IoT device
i and UAV
u can be expressed as
Suppose that an IoT device can only communicate with one UAV at the same time. The rate at which IoT device
i transmits data to UAV
u can be expressed as
where
is the transmit power of IoT device
i.
is the additive white Gaussian noise (AWGN) power spectral density at the UAV receiver.
is 0, 1 binary variable.
indicates that there is communication between device
i and UAV
u, and
is the opposite.
2.2. UAV–Satellite Data Transmission
Because there are few obstacles between UAVs and the satellite, the propagation loss follows the free space loss model. Taking antenna pattern into account, the radio propagation loss between UAVs and the satellite can be expressed as
where
and
are the antenna gain of UAV and satellite, respectively.
is the distance between UAV
u and the satellite, which is approximately equal to satellite orbital height
. So the gain between UAVs and the satellite can be expressed as
Similarly, the data transmission rate between UAV
u and the satellite can be expressed as
where
is the noise power spectral density and
is the transmit power of UAV
u.
2.3. Problem Formulation
Firstly, we provide expressions for energy efficiency and spectral efficiency, and construct the objective function for the optimization problem. Energy efficiency
is the ratio of effective information transmission rate to power. Since the data rate of the system depends on the communication rate between UAVs and the satellite, the EE can be expressesd as
where
is the power required for each UAV to maintain levitation.
The spectrum efficiency
is defined as the information rate divided by the bandwidth of the communication channel, which can be expressed as
For notational convenience, the horizontal and vertical coordinates of IoT device
i and UAV
u are denoted as
and
. Letting
,
,
and
, the problem can be described as
where
represents the minimum rate requirement for IoT devices to transmit data.
and
represent the maximum transmitting power of IoT devices and UAVs respectively.
represents the minimum distance between two UAVs.
In problem , constraint stipulates that is a binary variable. means there is communication between the UAV and the IoT device, otherwise . Constraints and limit the number of communication devices. They ensure that each UAV can communicate with K IoT devices and each IoT device can communicate with at most one UAV respectively. Constraints and indicate that the transmission power of IoT devices and UAVs cannot exceed the specified maximum transmission power. Constraint guarantees the rate at which each IoT device transmits data is greater than the minimum rate required to meet the requirements of service quality. Constraint means the rate at which the IoT device transmits data to the UAV is greater than the rate at which the UAV transmits data to the satellite. Furthermore, constraint guarantees that there is adequate safety distance between two UAVs. This ensures that any UAVs will not collide with each other.
3. Problem Solution
Problem
has two objective functions, including energy efficiency and spectral efficiency. According to [
24], because the two objective functions conflict with each other, the MOO problem is hard to solve. The authors in [
25] introduced a multi-objective optimization framework for optimizing URLLC design in the presence of decoding complexity constraints. They solved the proposed multi-objective optimization problem through two scalarization methods. The purpose of scalarization is to transform the objective into a single objective function and simplify the problem into a constrained single objective optimization problem. We apply this method to the SAG-IoT network, using the
-constraint method to convert multi-objective optimization functions into single objective optimization functions. We retained energy efficiency as the objective function, transformed spectral efficiency into constraint conditions. Therefore, we obtain a single objective optimization problem, which can be described as
where
represents the coefficient of trade off between SE and EE.
Although converted into a SOO problem, the problem is still difficult to solve because of the non-convex function in fractional form containing integer variables. To solve problem , we can divide the problem into three sub-problems.
3.1. Sub-Problem 1: Sub-Channel Selection
First, we consider the establishment of communication between IoT devices and UAVs. Appropriate sub-channels need to be selected, so the first sub-problem is called sub-channel selection, which is related to the integer variable
. We fix the variable
,
, and
, the problem can be described as
Observing
, we can see that the value of the objective function is fixed for any
that meets the constraint conditions. Therefore, when meeting the constraints, we consider optimizing the throughput of the IoT devices, which can be described as
Due to the presence of integer variables, the problem is complex to solve. A matching algorithm is used to solve
, as shown in Algorithm 1. IoT devices and UAVs construct preference lists based on descending communication rate between IoT devices and UAVs, respectively. Taking the selection of the UAV sub-channel by IoT device
as an example. Assume the preference list of
is
, where
means the communication rate between IoT device
and UAV
.
first selects
and calculates the number of devices
that have communicated with
. If
is less than
K, the match is successful. Otherwise, look at the preference list of
, and assume the list is
. Based on the preference list of
, compare the lowest priority of the
K devices
that already match
to
. If
has a higher priority, match
with
. Meanwhile, delete
from the preference list of
and rematch
. If
has a lower priority, delete
from the preference list of
and rematch
. If there is communication between UAV
u and IoT device
i,
, otherwise
. According to the matching algorithm, we can obtain
.
Algorithm 1 A matching algorithm for solving subchannel selection. |
- 1:
Input: Power allocation and , UAVs positon . - 2:
Let the set of IoT devices to be matched be and the set of devices that have been matched UAV u be . - 3:
Initial all to be zero. - 4:
Calculate . Construct preference lists of IoT devices and UAVs . - 5:
whiledo - 6:
for IoT device i⊆ do - 7:
if then - 8:
Select the most preferred UAV . - 9:
Count the number of devices that have been matched to UAV u, denoted as . - 10:
if then - 11:
Select the worst matched IoT in . - 12:
if i has a higher priority than in then - 13:
Replace with i in and , then delete u from . - 14:
Swap and . - 15:
else - 16:
Delete u from . - 17:
end if - 18:
else - 19:
Add i into , and delete i from . - 20:
, . - 21:
end if - 22:
end if - 23:
end for - 24:
end while - 25:
Output: Subchannel selection .
|
3.2. Sub-Problem 2: Power Allocation
The second sub-problem is power allocation, including transmit power of IoT devices and UAVs. For the given
and
, power allocation sub-problem can be described as
This is a non-convex problem because objective function and constraint
are non-convex. For
,
is a concave function of
, so we can obtain its upper bounder at
through first-order Taylor expansion
Denote the upper bounder as
, which is linear respect to
. Constraint
can be described as
which is convex. For the objective function, which is in fractional form, Dinkelbach algorithm is adopted. Introduce a non-negative parameter
, which is
The objective function can be transformed into a parametric subtractive form as
which is continuous and strictly monotonic decreasing in
and has an unique root. Denote
is the the root of
, which can be derived using the Dinkelbach method. Substituting (19) for the objective function of
, the optimal solution set of the problem is the same as that of
with
.
According to the above derivation, the transformed sub-problem can be obtained as
Containing two variables, problem
can be solved by Lagrange dual algorithm because it is a convex problem. The Lagrange function of (20) can be described as
where
are Lagrangian multipliers related to the constraints.
The Lagrangian dual expression is
Therefore, the Lagrangian dual problem can be described as
Next, we derive the optimal solution with given
,
,
,
,
. Due to
, the optimal solution
should satisfy the condition
Similarly, the optimal solution
should satisfy the condition
Because the Lagrangian dual function in (20) may be non-differentiable, the subgradient method is used to update the Lagrangian multipliers. We can update the Lagrangian multipliers
,
,
,
,
as follows
where
n is the iterative index,
is the step length, and
.
We adopt the adaptive step size shown below
In summary, Algorithm 2 gives the solution process of the power allocation sub-problem.
Algorithm 2 Dinkelbach algorithm based on Lagrangian duality. |
- 1:
Input: Subchannel selection , UAVs positon , tradeoff coefficient , and the maximum number of iterations N. - 2:
Initialize Lagrangian multipliers , , , , , variable , and tolerance , . - 3:
while and do - 4:
while and do - 5:
Use (24) and (25) to obtain the optimal power and . - 6:
Use (26)∼(31) to obtain update multipliers , , , , . - 7:
Calculate the difference . - 8:
. - 9:
end while - 10:
with and . - 11:
. - 12:
end while - 13:
Onput: Power allocation and .
|
3.3. Sub-Problem 3: UAV Position Deployment
After solving the sub-channel selection and power control, we need to consider the UAV position deployment sub-problem. According to sub-problems 1 and 2, we have
,
, and
. On this basis, the objective function of
has been fixed. Like sub-problem 1, we consider maximizing the device transmission rate, so the UAV position deployment sub-problem can be described as
Obviously,
is a non-convex problem. In order to solve this problem, we introduce a new variable
.
can be redescribed as
Then, to address the non-affine constraint
[
26], we can relax the problem into
where
,
,
,
and objective function are non-convex.
Using SCA algorithm for constraint
, the lower bounder of
at
and
can be expressed as
In addition, it is also necessary to convert non-convex constraints , and objective function into convex constraints. Denote as x and as y. We can konw that and are convex for variables x and y, as shown below.
Proof. Obviously, both
x and
y are greater than 0. Under normal circumstances,
is much larger than
. Therefore,
can be expressed as
where
,
, and
are constants greater than 0.
Therefore,
can be expressed as
where
and
are constants greater than 0.
Then, calculate the Hessian matrix
of data transmission rate
For any
, the following formula can be obtained
According to (39), is a convex function of x, y. □
Therefore, according to the Taylor expansion,
at point
,
can be lower-bounded by
where
and
are the partial derivatives of
against
x and
y, respectively, which can be expressed as
For the non-convex constraints
, we can know it is a convex function of
. Therefore,
at point
can be lower-bounded by
where
According to (35), (40), and (43), the problem can be described as
which is a convex optimization problem.
So far, we have obtained the solutions of the three sub-problems. On this basis, we jointly optimize the sub-channel selection, power allocation, and UAV position deployment through the BCD process. First, for a given suitable , , and , solve sub-problem 1 to obtain . Then, according to the obtained and fixed , solve sub-problem 2 to obtain and , and then solve sub-problem 3 to obtain . According to , , and , solve sub-problem 1 to obtain the new . Alternately solve the three sub-problems like this until the error tolerance is met.
4. Results
In this section, numerical results are given to demonstrate the performance of our proposed algorithm. We assume IoT devices are randomly distributed in a square area of 1 km × 1 km. Assume that the minimum transmission rate requirement of the IoT device is 15 kbps, and the maximum transmission power is 0.1 W. UAVs collect data from IoT devices and transmit them to the low-orbit satellite, which orbital height is 500 km. The antenna array element of UAV is generally considered to be omni-directional, so , and the antenna gain of the satellite is assumed to be 15 dbi. The minimum safety distance between UAVs is assumed to be 50 m. Bandwidth between each UAV and IoT devices and between each UAV and the satellite are both 1 MHz, and carrier frequency is 5 GHz. Assume the noise power spectral density is −169 dBm/Hz. The values of variables related to the environment are .
First, we verify the convergence of the proposed algorithm. Assuming there are two UAVs,
, and the number of sub-channels is 4, so there are eight IoT devices,
. The flight altitude of UAVs is 100 m, and the maximum transmission power is 8 W.
Figure 2 illustrates how spectral efficiency (SE) varies with the number of iterations for different values of
. As seen in the
Figure 2, SE converges as the number of iterations increases. When
, the system achieves maximum energy efficiency (EE), and the SE is greater than 0.35. However, when
and
, the system’s SE is limited to 0.40 and 0.45, respectively, since the system has to sacrifice some of its energy efficiency to achieve the desired SE.
Figure 3 displays the variation of energy efficiency with the number of iterations for different
, which shows that EE reaches a stable value as the number of iterations increases, indicating convergence. Moreover,
Figure 3 demonstrates that different values of
affect the tradeoff between energy efficiency and spectral efficiency. Specifically, increasing SE may lead to a decrease in EE. The curves for
and
in the plot are very similar, suggesting that when
, the system achieves the maximum EE while maintaining an SE greater than 0.35.
Considering the large system bandwidth, although the EE gap is small, it has a greater impact on the total amount of transmitted data in the system. The spectral efficiency is the same. The total transmission rate between IoT devices and UAVs is shown in
Figure 4. According to
Figure 2 and bandwidth, we can obtain the total transmission rate between UAVs and the satellite, which is close to the transmission rate in
Figure 4. Considering the UAV capacity, the transmission quality can be guaranteed.
Then, assuming that the flight altitudes of UAVs are 100 m, 500 m, and 1000 m, we have obtained the curves of the energy efficiency varies with
under three flight altitudes, as shown in
Figure 5, and the curves of the spectral efficiency varies with
under three flight altitudes, as shown in
Figure 6.
Figure 5 and
Figure 6 show the tradeoff between the system energy efficiency and spectral efficiency. As shown in
Figure 5, the energy efficiency initially remains constant but then decreases as the tradeoff coefficient
increases. On the other hand, in
Figure 6, the spectral efficiency remains constant and then increases to the same level as the tradeoff coefficient increases. There exists a optimal value of the tradeoff coefficient where the energy efficiency and spectral efficiency of the system remain constant, and the maximum energy efficiency can be attained while fulfilling the spectral efficiency requirements. However, beyond this value, increasing the tradeoff coefficient will result in the system sacrificing some energy efficiency to achieve the desired spectral efficiency. Thus, a suitable tradeoff coefficient should be chosen based on the specific requirements of the system.
In addition,
Figure 5 shows that the flight altitude of UAVs has an impact on the EE. It can be seen that the higher the flight altitude of the aircraft, the lower the energy efficiency of the system. An increase in aircraft altitude results in a decrease in energy efficiency because an increase in altitude results in a decrease in the channel gain between the IoT devices and the UAVs. IoT devices need to increase its transmission power to maintain a reliable connection, which leads to a decrease in EE.
Figure 6 shows the impact of flight altitude on SE. In contrast to the impact on EE, an increase in flight altitude will lead to an increase in SE, which is also consistent with theoretical derivation.
Next, we observe the impact of UAV transmission power limitations on system energy efficiency and spectral efficiency, and assume the power increases from 3 W to 10 W. The curves of the energy efficiency changing with the maximum transmission power of the UAV at different flying heights of the UAV are shown in
Figure 7. In addition to the impact of flight altitude on system EE,
Figure 7 also shows that the EE first increases and then remains constant as the UAV transmission power increases. This is because there is an optimal UAV transmission power that results in the highest EE. Further increasing the transmission power beyond this optimal point will not bring about any performance improvements in terms of EE.
The curves in
Figure 8 illustrate the variation of SE concerning different UAV transmission powers and flying heights. It can be seen that with the increase of the transmission power of the UAV, SE increases until it reaches the optimal value and remains unchanged.
5. Conclusions and Future Works
In this paper, we construct a SAG-IoT model, equating the channel between IoT devices and UAVs to a line of sight channel. Due to the limited energy of drones, we focus on energy efficiency, but an increase in energy efficiency can lead to a decrease in spectral efficiency. Therefore, we consider both energy efficiency and spectral efficiency as objective functions to construct optimization problems. We improve the -constraint to transform the multi-objective optimization problem into a single-objective optimization problem with additional constraints. Additionally, we introduce a BCD-based method to jointly optimize sub-channel selection, IoT device transmission power, UAV transmission power, and UAV location deployment to solve the complex single-objective optimization problem. Simulation results demonstrate the convergence of our proposed algorithm. At the end of the paper, we provide simulations to demonstrate the convergence of the proposed algorithm, analyz the effects of different tradeoff coefficients on the energy efficiency and spectral efficiency of the system, and analyz the effects of factors such as UAV flight altitude and maximum launch power on the algorithm performance. However, our SAG-IoT model only includes one satellite, and we should consider the case of multiple satellites in the future. In addition, we only considered the location deployment of UAVs without considering the trajectory. We will carry out further research on this.