The relentless evolution of wireless technology and the emergence of novel applications such as the Internet of Things (IoT) and autonomous driving have precipitated an insatiable demand for high-quality, ubiquitous network services. Traditional terrestrial infrastructure, often characterized by high deployment costs and limited geographical coverage, struggles to maintain service flexibility and robustness in challenging scenarios like natural disasters, large-scale temporary events, or remote area coverage. In this context, Unmanned Aerial Vehicle (UAV) assisted communication has emerged as a transformative solution.
Leveraging their high mobility, flexible deployment, and propensity for establishing Line-of-Sight (LoS) links, UAV drones can function as aerial base stations or mobile relays, providing rapid and reliable communication services to ground users. This capability not only effectively fills coverage gaps in terrestrial networks but also significantly enhances the overall flexibility and resilience of the communication ecosystem. While single-UAV systems have been extensively studied, they often fall short in scenarios with dense user populations or diverse service requirements due to limited coverage and computational resources. Consequently, the deployment of multi-UAV drone networks has become a promising trend to elevate network service quality. However, this paradigm introduces significant new challenges, primarily intense co-channel interference among UAV drones from different clusters and the complex interplay between spectrum resource allocation and three-dimensional trajectory design, both critically impacting overall network performance and energy consumption.
This article focuses on the uplink communication scenario within a multi-UAV assisted wireless network. We formulate a joint optimization problem encompassing subchannel allocation and UAV drone flight trajectory design, with the primary objective of maximizing the network’s energy efficiency (EE). The problem is subject to constraints on subchannel assignment, minimum data rate requirements for ground users (ensuring quality of service fairness), and physical limitations on UAV drone mobility. To tackle this NP-hard non-convex problem, we propose a novel alternating optimization algorithm. The core contributions are: 1) Employing the k-means++ clustering algorithm integrated with the silhouette coefficient method to adaptively partition ground users into optimal clusters, effectively matching users to UAV drones; 2) Designing a group-based fair matching algorithm for subchannel allocation that explicitly improves the lower bound of the minimum user data rate, enhancing service fairness compared to sum-rate maximizing schemes; 3) Developing a low-complexity trajectory optimization algorithm using successive convex approximation (SCA) and slack variables to transform the non-convex subproblem into a tractable convex form, solvable via standard tools like CVX.
System Model and Problem Formulation
We consider an uplink wireless network assisted by multiple UAV drones. The system comprises M UAVs serving as aerial base stations and N ground users (GUs). The set of UAVs is denoted by $\mathcal{U}=\{1,2,…,M\}$ and the set of GUs by $\mathcal{G}=\{1,2,…,N\}$. Each UAV drone provides relay services to a distinct cluster of GUs. All UAVs are assumed to fly at a fixed altitude $H$. The total flight period $T$ is discretized into I time slots of equal length $\tau = T/I$, indexed by $i \in \mathcal{I}=\{1,2,…,I\}$. The horizontal coordinate of UAV $u$ in slot $i$ is denoted by $\mathbf{q}_u[i] = (x_u[i], y_u[i])^T$, and the fixed location of GU $g$ is $\mathbf{w}_g = (x_g, y_g)^T$. The system bandwidth is divided into K orthogonal subchannels, indexed by $k \in \mathcal{K}=\{1,2,…,K\}$, each with bandwidth $B$. To avoid intra-cluster interference, GUs within the same cluster served by one UAV drone must use different subchannels. However, GUs from different clusters can reuse the same subchannel to improve spectral efficiency, which introduces inter-cluster interference.
The distance between UAV $u$ and GU $g$ in time slot $i$ is:
$$d_{u,g}[i] = \sqrt{||\mathbf{q}_u[i] – \mathbf{w}_g||^2 + H^2}.$$
Assuming dominant LoS links, the channel power gain follows the free-space path loss model:
$$h_{u,g}[i] = \frac{\beta_0}{d_{u,g}[i]^2} = \frac{\beta_0}{||\mathbf{q}_u[i] – \mathbf{w}_g||^2 + H^2},$$
where $\beta_0$ represents the channel power gain at a reference distance of 1 meter.
Let $p_g[i]$ be the transmit power of GU $g$ in slot $i$, which is assumed fixed in this work. Let $\mathcal{G}_k[i]$ denote the set of GUs transmitting on subchannel $k$ in slot $i$. The Signal-to-Interference-plus-Noise Ratio (SINR) for the link from GU $g$ to its associated UAV $u$ on subchannel $k$ in slot $i$ is:
$$\gamma_{u,g}^k[i] = \frac{p_g[i] h_{u,g}[i]}{\sigma^2 + \sum_{g’ \in \mathcal{G}_k[i], g’ \neq g} p_{g’}[i] h_{u,g’}[i]},$$
where $\sigma^2$ is the additive white Gaussian noise power. The corresponding achievable data rate is:
$$R_{u,g}^k[i] = B \log_2\left(1 + \gamma_{u,g}^k[i]\right).$$
We define the subchannel allocation indicator $a_{u,g}^k[i] \in \{0,1\}$, where $a_{u,g}^k[i]=1$ indicates that GU $g$ is associated with UAV $u$ and transmits on subchannel $k$ in slot $i$, and $a_{u,g}^k[i]=0$ otherwise. The subchannel allocation matrix is denoted by $\mathbf{A} = \{a_{u,g}^k[i]\}$. The trajectory of all UAVs is denoted by $\mathbf{Q} = \{\mathbf{q}_u[i]\}$.
The total achievable sum rate of the network over the flight period is:
$$R_{total}(\mathbf{A}, \mathbf{Q}) = \sum_{i=1}^{I} \sum_{u=1}^{M} \sum_{g=1}^{N} \sum_{k=1}^{K} a_{u,g}^k[i] R_{u,g}^k[i].$$
The total power consumption includes the transmit power of all GUs and the propulsion power of all UAVs, denoted as $P_{fly}$ (assumed constant for simplicity):
$$P_{total} = \sum_{i=1}^{I} \sum_{g=1}^{N} p_g[i] + M \cdot P_{fly}.$$
Consequently, the network energy efficiency (EE) in bits/Joule is defined as:
$$EE(\mathbf{A}, \mathbf{Q}) = \frac{R_{total}(\mathbf{A}, \mathbf{Q})}{P_{total}}.$$
Our goal is to maximize the network EE by jointly optimizing subchannel allocation $\mathbf{A}$ and UAV trajectories $\mathbf{Q}$, subject to the following constraints:
1. Subchannel Allocation Constraints:
C1: $a_{u,g}^k[i] \in \{0,1\}, \forall i,u,g,k.$
C2: $\sum_{u=1}^{M} \sum_{k=1}^{K} a_{u,g}^k[i] = 1, \forall i,g.$ (Each GU uses one subchannel per slot)
C3: $\sum_{g=1}^{N} a_{u,g}^k[i] \leq 1, \forall i,u,k.$ (No intra-cluster interference)
C4: $a_{u,g}^k[i] = 0, \forall g \notin \mathcal{C}_u, \forall i,u,k.$ (GU only connects to its cluster’s UAV)
2. Quality of Service (QoS) Constraint:
C5: $\sum_{u=1}^{M} \sum_{k=1}^{K} a_{u,g}^k[i] R_{u,g}^k[i] \geq R_{min}, \forall i,g.$ (Minimum rate requirement)
3. UAV Trajectory Constraints:
C6: $\mathbf{q}_u[1] = \mathbf{q}_u[I], \forall u.$ (Cyclic trajectory, return to start)
C7: $||\mathbf{q}_u[i+1] – \mathbf{q}_u[i]|| \leq V_{max} \tau, \forall i, u.$ (Maximum speed constraint)
The joint optimization problem (P1) is formulated as:
$$
\begin{aligned}
& (\text{P1}): \max_{\mathbf{A}, \mathbf{Q}} \quad EE(\mathbf{A}, \mathbf{Q}) = \frac{R_{total}(\mathbf{A}, \mathbf{Q})}{P_{total}} \\
& \text{s.t.} \quad \text{C1, C2, C3, C4, C5, C6, C7.}
\end{aligned}
$$
Problem (P1) is a mixed-integer non-convex optimization problem due to the binary variables in $\mathbf{A}$ and the non-convexity of the rate function $R_{u,g}^k[i]$ with respect to $\mathbf{q}_u[i]$ in constraints C5 and the objective. It is generally NP-hard. To obtain a high-quality suboptimal solution, we propose an alternating optimization framework that decouples (P1) into two subproblems solved iteratively.
Proposed Alternating Optimization Methodology
1. User Clustering and Initialization
Before communication resource allocation, GUs must be partitioned into $M$ clusters, each served by one UAV drone. Considering the spatial distribution of GUs, we employ the k-means++ clustering algorithm due to its efficiency and tendency to produce compact clusters. A key challenge is determining the optimal number of clusters $M$. We utilize the silhouette coefficient method, which evaluates clustering quality based on intra-cluster cohesion and inter-cluster separation. The average silhouette coefficient is computed for a range of candidate $M$ values, and the value yielding the highest coefficient is selected, allowing adaptive and optimal cluster formation without manual presetting.
Given the user clusters, we initialize the trajectory for each UAV drone as a circular path centered at its cluster’s centroid, with a radius set to cover half of the cluster’s GUs. The initial subchannel allocation is determined via a greedy heuristic based on channel gains.
2. Subchannel Allocation via Group-Based Fair Matching
With fixed UAV trajectories $\mathbf{Q}$, problem (P1) reduces to a subchannel allocation subproblem. Our proposed algorithm aims not only to maximize sum rate but also to improve fairness by boosting the minimum user rate. The algorithm operates per time slot $i$.
Step 1: User Grouping. For slot $i$, based on the current UAV positions, we calculate the channel gain between each GU and its associated UAV drone. A sorted list $L_L[i]$ is created for these gains in descending order. To protect fairness, the bottom 10% of GUs (with the poorest channel gains) are assigned dedicated subchannels, meaning they do not share with others. The remaining GUs are grouped to share subchannels. The maximum number of GUs per shared subchannel, $N_{max}$, is calculated based on the number of remaining GUs and subchannels. Let $K_{ded}$ be subchannels for dedicated users and $K_{shr}=K-K_{ded}$. Let $N_{rem}=0.9N$ be remaining GUs. Then $N_{max} = \lceil N_{rem} / K_{shr} \rceil$. The actual grouping results in some groups with $N_{max}$ users and others with $N_{max}-1$ users.
Step 2: Matching and Swapping. We iteratively form user groups for shared subchannels. The GU with the highest gain in $L_L[i]$ that is unassigned is selected as the “primary user.” To minimize co-channel interference for this primary user, we search among GUs from *other* clusters (potentially reusing the same subchannel) for the one that causes the least interference to the primary user’s associated UAV drone (i.e., has the smallest channel gain to that UAV). This GU is the “match candidate.” If the candidate is unassigned, they are grouped. If the candidate is already in a group, we calculate the sum rate of the candidate’s current group ($R_{prev}$) and the sum rate if the candidate joins the primary user’s group ($R_{new}$). If $R_{new} > R_{prev}$, we perform a swap, moving the candidate to the new group. The vacated spot in the old group is then filled by reconsidering its members. This process continues until all GUs are assigned. This matching strategy actively manages interference to improve the rates of disadvantaged users.
3. UAV Trajectory Optimization via Successive Convex Approximation
With fixed subchannel allocation $\mathbf{A}$, problem (P1) reduces to a trajectory optimization subproblem (P2):
$$
\begin{aligned}
& (\text{P2}): \max_{\mathbf{Q}} \quad \frac{R_{total}(\mathbf{Q})}{P_{total}} \\
& \text{s.t.} \quad \text{C5, C6, C7.}
\end{aligned}
$$
(P2) remains non-convex due to the rate function $R_{u,g}^k[i]$ in the objective and constraint C5. We employ SCA to iteratively approximate (P2) as a convex problem.
First, note that for a given link, the rate can be decomposed as:
$$
R_{u,g}^k[i] = B\log_2\left(\sigma^2 + \sum_{g’ \in \mathcal{G}_k[i]} p_{g’}h_{u,g’}[i]\right) – B\log_2\left(\sigma^2 + \sum_{g’ \in \mathcal{G}_k[i], g’\neq g} p_{g’}h_{u,g’}[i]\right).
$$
Let the first term be $\overline{R}_{u,g}^k[i]$ and the second term be $\underline{R}_{u,g}^k[i]$. The term $\overline{R}_{u,g}^k[i]$ is a concave function of the distances $d_{u,g’}[i]$. Using a first-order Taylor expansion at a local point $\mathbf{Q}^t$ from iteration $t$, we can obtain a concave lower bound $\overline{R}_{u,g}^k[i]^{lb}$.
The term $\underline{R}_{u,g}^k[i]$ is non-convex. To handle it, we introduce slack variables $s_{u,g’}[i]$ such that:
$$ s_{u,g’}[i] \geq ||\mathbf{q}_u[i] – \mathbf{w}_{g’}||^2 + H^2, \quad \forall i,u,g’.$$
Then, $\underline{R}_{u,g}^k[i]$ is replaced by $\widetilde{R}_{u,g}^k[i] = B\log_2\left(\sigma^2 + \sum_{g’ \neq g} \frac{p_{g’}\beta_0}{s_{u,g’}[i]}\right)$, which is convex with respect to $s_{u,g’}[i]$ as its Hessian is positive semidefinite. To ensure the equivalence, it can be shown that at the optimum, the inequality for $s_{u,g’}[i]$ holds with equality.
The constraint $ s_{u,g’}[i] \geq ||\mathbf{q}_u[i] – \mathbf{w}_{g’}||^2 + H^2$ is non-convex because the right-hand side is convex. We convexify it using its first-order Taylor expansion at $\mathbf{Q}^t$:
$$ s_{u,g’}[i] \geq ||\mathbf{q}_u^t[i] – \mathbf{w}_{g’}||^2 + 2(\mathbf{q}_u^t[i] – \mathbf{w}_{g’})^T (\mathbf{q}_u[i] – \mathbf{q}_u^t[i]) + H^2.$$
This renders the constraint linear (and thus convex) in $\mathbf{q}_u[i]$ and $s_{u,g’}[i]$.
After these transformations and introducing the slack variables $\mathbf{S}=\{s_{u,g}[i]\}$, we obtain the convex approximation of (P2) for iteration $t+1$:
$$
\begin{aligned}
& (\text{P2-Approx}): \max_{\mathbf{Q}, \mathbf{S}} \quad \frac{\sum_{i,u,g,k} a_{u,g}^k[i] (\overline{R}_{u,g}^k[i]^{lb} – \widetilde{R}_{u,g}^k[i])}{P_{total}} \\
& \text{s.t.} \quad \sum_{u,k} a_{u,g}^k[i] (\overline{R}_{u,g}^k[i]^{lb} – \widetilde{R}_{u,g}^k[i]) \geq R_{min}, \forall i,g. \\
& \quad \quad \text{C6, C7,} \\
& \quad \quad s_{u,g’}[i] \geq ||\mathbf{q}_u^t[i] – \mathbf{w}_{g’}||^2 + 2(\mathbf{q}_u^t[i] – \mathbf{w}_{g’})^T (\mathbf{q}_u[i] – \mathbf{q}_u^t[i]) + H^2, \forall i,u,g’. \\
& \quad \quad s_{u,g’}[i] \geq H^2, \forall i,u,g’.
\end{aligned}
$$
(P2-Approx) is a convex optimization problem that can be efficiently solved using interior-point methods via tools like CVX. The solutions $\mathbf{Q}^{t+1}$ and $\mathbf{S}^{t+1}$ are then used as the Taylor expansion points for the next iteration. This SCA process guarantees convergence to a locally optimal point of the original subproblem (P2).
4. Overall Algorithm and Complexity Analysis
The overall joint optimization algorithm alternates between solving the subchannel allocation subproblem (using the fair matching algorithm) and the trajectory optimization subproblem (using the SCA-based algorithm).
Algorithm Outline:
1. Initialize: Determine optimal number of clusters $M$ and user clusters via k-means++ and silhouette coefficient. Initialize UAV trajectories $\mathbf{Q}^{(0)}$ as circular paths.
2. Repeat until convergence or max iterations:
a. Subchannel Allocation: Given current trajectories $\mathbf{Q}^{(n)}$, solve for $\mathbf{A}^{(n+1)}$ using the Group-Based Fair Matching algorithm (Algorithm 1).
b. Trajectory Optimization: Given $\mathbf{A}^{(n+1)}$, solve for $\mathbf{Q}^{(n+1)}$ by iteratively solving (P2-Approx) via SCA until convergence (Algorithm 2).
c. $n \leftarrow n + 1$.
3. Output the final subchannel allocation $\mathbf{A}^*$ and UAV trajectories $\mathbf{Q}^*$.
Convergence: Let $EE^{(n)} = EE(\mathbf{A}^{(n)}, \mathbf{Q}^{(n)})$. In step (a), the matching algorithm is designed to improve (or at least not decrease) the sum rate, and since $P_{total}$ is fixed when $\mathbf{A}$ changes, $EE^{(n+1/2)} = EE(\mathbf{A}^{(n+1)}, \mathbf{Q}^{(n)}) \geq EE^{(n)}$. In step (b), the SCA procedure generates a sequence of non-decreasing objective values for (P2), so $EE^{(n+1)} = EE(\mathbf{A}^{(n+1)}, \mathbf{Q}^{(n+1)}) \geq EE^{(n+1/2)}$. Therefore, $EE^{(n+1)} \geq EE^{(n)}$, guaranteeing that the overall alternating algorithm yields a non-decreasing sequence of EE values, ensuring convergence.
Complexity: The complexity of the subchannel allocation per slot is approximately $\mathcal{O}(N M K C)$, where $C$ is related to swap operations. The complexity of one SCA iteration for trajectory optimization is polynomial, dominated by the convex solver, roughly $\mathcal{O}((M I)^{3.5})$. The overall complexity per outer iteration is manageable for moderate-sized networks.
Simulation Results and Performance Evaluation
We conduct simulations in a 600m × 600m area with $N=120$ GUs randomly distributed in 4 cluster-like regions. The main simulation parameters are summarized in Table 1.
| Parameter | Value |
|---|---|
| Area Size | 600 m × 600 m |
| Number of GUs ($N$) | 120 |
| Number of UAVs ($M$) | 4 |
| UAV Altitude ($H$) | 100 m |
| Number of Subchannels ($K$) | 86 |
| Subchannel Bandwidth ($B$) | 20 MHz |
| GU Transmit Power | 1 W |
| UAV Propulsion Power ($P_{fly}$) | 2.2 W |
| Noise Power Spectral Density | -174 dBm/Hz |
| Maximum UAV Speed ($V_{max}$) | 30 m/s |
| Flight Period ($T$) | 40 s |
| Number of Time Slots ($I$) | 8 |
| Channel Gain at 1m ($\beta_0$) | 1.4×10⁻⁴ |
1. Clustering and Initial Trajectory: The silhouette coefficient analysis correctly identified $M=4$ as the optimal cluster count. The k-means++ algorithm formed well-separated clusters. Initial circular trajectories were set for each UAV drone.
2. Subchannel Allocation Performance: We compare our proposed Group-Based Fair Matching (GBFM) algorithm against a conventional Greedy Max-Rate Matching (GMRM) algorithm. Figure 1 shows the worst-user SINR (i.e., the minimum SINR among all GUs) across different time slots. The GBFM algorithm consistently achieves a higher worst-user SINR than GMRM, particularly in the initial slots (e.g., a 0.27 improvement in slot 1). This demonstrates its effectiveness in enhancing user fairness and boosting the minimum data rate, which is crucial for meeting QoS constraints (C5). Figure 2 compares the network EE per slot. While both algorithms aim for efficiency, GBFM often achieves comparable or slightly better EE than GMRM, proving that fairness and system efficiency can be improved simultaneously through intelligent interference-aware matching.
3. Trajectory Optimization Performance: The optimized trajectories for the four UAV drones are depicted in Figure 3. Compared to the initial circular paths, the optimized trajectories are more condensed and adaptive. Each UAV drone moves within a smaller region closer to its user cluster’s centroid, but not statically, as it dynamically adjusts its position to balance between improving desired channel gains for its own users and reducing interference caused to/users in other clusters. This results in reduced flight distances and lower propulsion energy consumption contribution to the denominator of EE, while maintaining or improving the sum rate in the numerator.
4. Convergence and Overall Energy Efficiency: Figure 4 illustrates the convergence of the overall alternating optimization algorithm, comparing it against a baseline using fixed circular trajectories and a Particle Swarm Optimization (PSO)-based trajectory optimization benchmark. Our proposed algorithm converges rapidly within about 5 outer iterations, achieving a significant 14.15% gain in overall network EE over the fixed trajectory baseline. In contrast, the PSO-based method requires many more iterations (over 50) to find a solution superior to the initial trajectory, highlighting the efficiency and effectiveness of our SCA-based convex optimization approach. The final optimized system delivers superior performance by jointly managing interference through fair subchannel allocation and energy-efficient trajectory design for the multi-UAV drone network.
Conclusion
This article has addressed the critical problem of joint subchannel allocation and trajectory optimization in a multi-UAV assisted uplink communication network with the goal of maximizing network energy efficiency. We proposed a comprehensive alternating optimization framework to solve this challenging non-convex problem. Key innovations include: 1) An adaptive user clustering method using k-means++ and silhouette coefficient for optimal UAV drone-user matching; 2) A novel group-based fair matching algorithm for subchannel allocation that explicitly enhances the minimum user rate, promoting service fairness; 3) An efficient trajectory optimization algorithm leveraging successive convex approximation to handle non-convexity. Simulation results validated that our proposed framework significantly improves both the network energy efficiency and the worst-user data rate compared to benchmark schemes, ensuring reliable and efficient communication service from a fleet of UAV drones. Future work may consider dynamic environments with user mobility, integration with non-orthogonal multiple access (NOMA), and robust optimization against channel uncertainty.