Spot Beam Power Optimization for an Energy Efficient Multi-LEO Satellite Network

Ⅰ. Introduction

The rapid evolution of wireless communications technologies has led to an increased focus on a nonterrestrial network (NTN) as a means to achieve global connectivity^[1]. It is considered to be integral to the vision of ubiquitous coverage, especially in remote and underserved regions, beyond existing traditional terrestrial infrastructures for the upcoming sixth-generation (6G) standard^[2]. Consequently, a vast number of researchers have begun to pay attention to addressing the unique challenges of NTNs^[3].

Fundamental challenges in the management tasks of NTNs include propagation delays between satellites and ground access points (GAPs) and Doppler shifts stemming from the movement of satellites^[4]. This entails the development of advanced optimization techniques tailored for NTNs. The integration of NTNs with terrestrial networks was explored in [5], emphasizing the role of beamforming and artificial intelligence technologies. The authors in [6] highlighted challenges in satellite beamforming and inter-satellite transmissions for radio resource management, mobility management, and dynamic network slicing. Also, [7] evaluated various multi-layered NTNs, offering guidelines on achieving a balance between system flexibility and network performance with coverage and latency constraints into consideration.

Meanwhile, energy efficiency has emerged as a critical performance measure in the design and operation of NTNs^[8]. By optimizing power consumption, we can enhance the lifespan of satellite components and reduce operational costs. Various NTN management techniques, such as adaptive power control, energy- efficient routing algorithms, and dynamic resource allocation, have been proposed to address these challenges. In particular, [9] tackled energy efficiency maximization in multibeam satellite systems under total power and quality-of-service constraints. Precoding optimization algorithms were proposed based on zero forcing (ZF) and successive convex approximation (SCA), verified using real measured channel data. Furthermore, [10] investigated a two-step quadratic transformation method to address energy efficiency maximization in multibeam satellite communications. By converting the non-convex problem into an equivalent convex one, an alternating optimization algorithm can be derived that iteratively identifies effective networking solutions. In [11], an energy-efficient resource allocation in space-air-sea NTNs for maritime coverage was presented. A mixed-integer programming task was formulated which maximizes the system energy efficiency by jointly optimizing user equipment association, power control, and unmanned aerial vehicle (UAV) deployment. In addition, [12] maximized the global energy efficiency (GEE) of the whole NTN supporting multiple ground access points (GAPs) while accounting for various interference effects.

With the aid of machine learning, [13] assigned spreading factors (SFs) to end devices to minimize co- SF interference and enhance the energy efficiency. In addition, [14] explored the integration of open radio access network (O-RAN) within NTNs, focusing on optimizing the dynamic functional split between centralized units (CU) and distributed units (DU) to enhance energy efficiency. A deep Q-network method was employed, which dynamically determines the optimal network function split and selects the best NTN platform based on real-time conditions and traffic demands.

Most of the previous works considered a single satellite system. For this reason, these cannot scale effectively with the increasing number of satellites. To this end, our previous work^[15] extended the single satellite system^[12] to general multiple low Earth orbit (LEO) satellite NTNs and proposed the optimal GAP- beam assignment while assuming equal power allocation for all GAPs and beams. Such an approach has been shown to degrade the GEE performance. In particular, the GEE improvement with an increasing number of LEO satellites was marginal even though the system could exploit greater link diversity. This suggests that relying solely on GAP-beam assignment with fixed transmit power may be insufficient to effectively mitigate both inter-cell and intra-cell interference. Therefore, it is necessary to further investigate the impact of optimal downlink power control to fully leverage the benefits of multi-satellite configurations.

This paper presents an efficient algorithm for maximizing the GEE performance of multi-LEO satellite NTNs where several satellites support multiple GAPs in the downlink. We aim to optimize the spot beam power of LEO satellites considering practical challenges of NTNs such as inter-beam, inter-carrier, and terrestrial interference signals. We first construct the GEE maximization problem, which is proven to be non-convex. To handle this difficulty, the SCA technique is applied that addresses approximated convex tasks of the original non-convex GEE maximization problem. Numerical results validate the effectiveness of the proposed algorithm over baseline schemes.

Ⅱ. System Model

We consider a multi-LEO network illustrated in Fig. 1 where S satellites support K GAPs in the downlink. Each satellite projects at most M spot beams, and each GAP is assumed to be assigned to at most one spot beam. In terrestrial networks, B ground base stations (GBSs) share the same frequency spectrum with LEO satellites, e.g., Ku- or Ka-band of satellite communications and frequency range 2 of the fifth-generation (5G)^[16], which interfere GAPs. The channel state information and locations of the GAPs and GBSs can be available at the satellites by the enhanced channel estimation and localization algorithms^{[ 12]}.

The signal-to-interference-plus-noise ratio (SINR) of GAP k on spot beam m of satellite s, denoted by [TeX:] $$\gamma_{k, m, s,}$$ can be expressed by

where [TeX:] $$G_{\text {main }, s} \text { and } G_{\mathrm{AP}, k}$$ denote the main lobe antenna gain of satellite s and antenna gain of GAP k, respectively, Pk,m,s indicates the transmit power allocated to spot beam m of satellite s for GAP k, [TeX:] $$L_{k, s}$$ denotes channel attenuation between GAP k and satellite s, and [TeX:] $$\sigma_{z, k}^2$$ stands for the noise variance at GAP k. In addition, [TeX:] $$\mathrm{IBI}_{k, m, s} \text { and } \mathrm{ICI}_{k, m, s}$$ account for the inter-beam interference (IBI) and inter-carrier interference (ICI) by the Doppler effect for GAP k on beam m of satellite s.

These interference terms are written by

where [TeX:] $$G_{\text {side }, \mathrm{X}, m, s}$$ accounts for the sidelobe antenna gain towards ground node X by beam m of satellite s, Tsym represents the symbol duration, and fk,s denotes the Doppler shift between GAP k and satellite s. Here, the Doppler shift is calculated as

where [TeX:] $$v_s$$ equals the velocity of satellite s, [TeX:] $$C=3 \times 10^8 \mathrm{~m} / \mathrm{s}$$ is the speed of light, fc stands for the carrier frequency, C = 3 × 108 m/s is the speed of light, and [TeX:] $$\phi k, s$$ denotes the angle of the receiving direction of GAP k and moving direction of satellite s, respectively. Also, [TeX:] $$x_{k, m, s} \in\{0,1\}$$ in (2) stands for a binary indicator that equals 1 if spot beam m of satellite s is assigned to GAP k and 0 otherwise. The beam assignment is assumed to be fixed such that each beam is dedicated to a single GAP and vice versa. Thus, [TeX:] $$x_{k, m, s} \in\{0,1\}$$ is randomly chosen to satisfy

Notice that [TeX:] $$x_{k, m, s}$$ can be zero when the numbers of GAPs and beams are unequal.

Ⅲ. Problem Formulation

This section formalizes the GEE maximization task of the multi-LEO network. From the SINR (1), the achievable rate of GAP k on spot beam m of satellite s based on (1)-(3) is calculated as

By taking the binary beam assignment indicator [TeX:] $$x_{k, m, s}$$ into account, the sum-rate can be written by

We adopt the power consumption model from [17]- [20] in which both static and dynamic components are considered. Concretely, static power consumption Pcircuit,s of satellite s is drawn even when the satellite is not actively transmitting data, often due to leakage Fig. 1. System model for multi-LEO networks currents and bias voltages within the circuitry. It is influenced by factors such as device technology, chip design, and operating conditions^[21]. On top of that, dynamic power consumption occurs during active operations such as signal processing, modulation, and transmission. It is typically modelled to be proportional to the number of activated beams and a significant contributor to overall energy consumption^[22]. This leads to the total power consumption of satellite s as

where [TeX:] $$\rho_s \in(0,1]$$ indicates the efficiency of power conversion on satellite s.

Based on (8)-(10), the GEE η is defined as the ratio of the sum-rate to the total power consumption of all satellites, which can be expressed by

Our objective is to maximize the GEE by optimizing the spot beam power allocation variables [TeX:] $$\left\{P_{k, m, s}\right\}$$ for given GAP-beam-satellite assignment [TeX:] $$\left\{x_{k, m, s}\right\}$$. The corresponding optimization task can be formulated as

where [TeX:] $$\mathrm{TNI}_b$$ stands for the terrestrial network interference towards GBS b, defined as

Constraint (12b) ensures that the total spot beam power of satellite s does not exceed its power budget [TeX:] $$\bar{P}_{\mathrm{sat}, s},$$ and (12c) limits the power allocated to spot beam m of satellite s by the peak beam power constraint [TeX:] $$\bar{P}_{m, s}.$$ Also, constraint (12d) guarantees the terrestrial network interference for every GBS is under the maximum allowable level [TeX:] $$\mathrm{TNI}_{\max } .$$

It can be shown that the objective function in (P1) is non-convex with respect to the spot beam power variables, mainly due to mutual coupling in a complicated way. In the subsequent section, we first reformulate the problem into a more tractable form and then propose efficient optimization strategies.

Ⅳ. Proposed Solutions

4.1 Problem Reformulation

This section presents solution approaches for addressing the non-convex GEE maximization problem in (P1). To this end, we first simplify it by applying the change of variables. Without loss of generality, we rearrange the indices of all spot beams as an integer set [TeX:] $$\mathscr{M}=[1, S M],$$ where [TeX:] $$[a, b] \triangleq\{a, a+1, \ldots, b-1, b\}$$ indicates a set of integers between a and b. Then, the spot beams of satellite s can be represented by a set [TeX:] $$\mathscr{M}_s \triangleq[(s-1) M+1, s M] \subset \mathscr{M} .$$ As a result, the power allocation variables [TeX:] $$\left\{P_{k, m, s}\right\}$$ and the constants [TeX:] $$\left\{\boldsymbol{x}_{k, m, s}\right\}$$ can be equivalently converted into [TeX:] $$\left\{P_{k, \tilde{m}}\right\} \text { and }\left\{x_{k, \tilde{m}}\right\},$$ which are respectively defined as

It is worth noting that the new index [TeX:] $$\tilde{m}$$ ranges from 1 to SM, and it virtually combines the multiple satellites into a single satellite projecting at most S groups of M beams. Similarly, we can newly define the following quantities:

We next re-express the (P1) in terms of (14)-(23) as

with modified η and [TeX:] $$\mathrm{TNI}_b$$ as

where

Furthermore, since GAP k is paired with a unique and exclusive spot beam ??????, we can define the effective spot beam power as

Also, let [TeX:] $$\mathscr{A} \triangleq\left\{(k, \tilde{m}) \mid x_{k, \tilde{m}}=1, \forall k, \tilde{m}\right\}$$ and [TeX:] $$\mathscr{A}_{\text {beam }} \triangleq \left\{\tilde{m} \mid x_{k, \tilde{m}}=1, \forall k, \tilde{m}\right\}$$ be the sets of active GAP-spot beam pairs and active spot beams, respectively. Consequently, (25)-(29) can be further simplified into

Based on these results, (P2) can be equivalently transformed as

Compared to (P1) which consists of KMS optimization variables [TeX:] $$\left\{P_{k, m, s}\right\}$$, (P3) now contains [TeX:] $$\left|\mathscr{A}_{\text {beam }}\right|=\min (K S, M S)\lt K M S$$ variables, thereby reducing the problem complexity. Still, however, (P3) invokes the non-convex fractional form in the objective function (37a). Therefore, it is highly difficult to identify the globally optimal solution to (P3) using off-the-shelf convex optimization software.

4.2 Proposed Algorithm

We propose an efficient optimization algorithm that tackles the non-convexity of (P3). We first reformulate it by introducing an auxiliary variable λ as

It is readily seen that any [TeX:] $$\lambda \in[0, \hat{\lambda}]$$ is feasible for any feasible [TeX:] $$\hat{\lambda}$$ for (P4). This enables the use a two-layer optimization approach, in which an optimal [TeX:] $$\lambda^{\star}$$ for (P4) is found by the bisection method in the outer layer, while a feasible set of [TeX:] $$\left\{P_{\tilde{m}} \mid \tilde{m} \in \mathscr{A}_{\text {beam }}\right\}$$ is identified with each fixed λ in the inner layer.

It is not difficult to see that the feasible set [TeX:] $$\left\{P_{\tilde{m}} \mid \tilde{m} \in \mathscr{A}_{\text {beam }}\right\}$$ can be attained by determining the maximum of the left-hand side of constraint (38b). Therefore, in the inner layer, we solve the following optimization problem for a given λ :

The existence of feasible [TeX:] $$\left\{P_{\tilde{m}} \mid \tilde{m} \in \mathscr{A}_{\text {beam }}\right\}$$ for a certain λ is easily determined if the optimal value of (P4-inner) is greater than or equal to 0. Otherwise, it indicates that λ should be decreased in the outer bisection layer.

Unfortunately, the log-minus-log form of [TeX:] $$R_{k, \tilde{m}}$$ with respect to [TeX:] $$\left\{P_{\tilde{m}} \mid \tilde{m} \in \mathscr{A}_{\text {beam }}\right\}$$ in (39a) exhibits nonconvexity, making (P4-inner) difficult to handle in the current form. Hence, we resort to the SCA technique which iteratively addresses convex-approximated tasks of the original formulation (P4-inner)^[23]. Let us define the solution vector p and a function [TeX:] $$g_{l, \tilde{m}}(\mathbf{p})$$ as

with [TeX:] $$\tilde{m}_{\min } \triangleq \min _{\tilde{m} \in \mathscr{A}_{\text {beam }}} \tilde{m} \text { and } \tilde{m}_{\max } \triangleq \max _{\tilde{m} \in \mathscr{A}_{\text {beam }}} \tilde{m} .$$ According to the first-order Taylor expansion, an approximated convex expression of Rk,mﾘ at a given solution p0 can be written by

As a consequence, the convex approximated inner layerproblem (P4-inner) is formulated as

Problem (P4-inner-SCA) now becomes convex, which can be easily tackled using existing convex solvers, e.g., CVXPY^[24].

We summarize the proposed spot beam power allocation algorithm in Algorithm 1. As discussed, the overall procedure consists of the outer layer for obtaining λ using the bisection method and the inner layer addressing (P4-inner) using the SCA algorithm. We first initialize the power allocation solution [TeX:] $$\mathbf{p}_0$$ and the bound of [TeX:] $$\lambda \in\left[\lambda_{\mathrm{LB}}, \lambda_{\mathrm{UB}}\right] .$$ Also, we set the tolerance δ and the number of maximum iterations of the inner layer [TeX:] $$I_{\max } .$$ At each iteration of the outer layer, a new λ is set to the median of the predetermined bound as [TeX:] $$\lambda \leftarrow \frac{\lambda_{\mathrm{LB}}+\lambda_{\mathrm{UB}}}{2} .$$ Then, the SCA algorithm is adopted which solves (P4-inner-SCA) repeatedly. Denoting [TeX:] $$\tilde{\eta}$$ as the optimal value of (P4-inner-SCA) at the convergence, the bisection method is employed to update the lower and upper bounds of λ . Such a procedure is repeated until λ converges within the predefined tolerance δ.

Ⅴ. Numerical Results

We evaluate the proposed spot beam power optimization algorithm through numerical simulations. We incorporate the rain fading model for satellite-toground links as specified in [25], [26]. Thus, the small- scale fading component between ground node X and satellite s is modeled as

where ξ denotes the small-scale channel gain whose dB scale value [TeX:] $$\xi_{\mathrm{dB}}=20 \log _{10}(\xi)$$ follows the log-normal distribution with mean μ and variance [TeX:] $$\sigma^2,$$ i.e., [TeX:] $$\ln \left(\xi_{\mathrm{dB}}\right) \sim \mathscr{N}(\mu, \sigma), \text { and } \phi \sim \mathscr{U}(0,2 \pi)$$ stands for the random phase changes. Notice that the parameters μ and σ depend on environmental factors such as receiver location, operation frequency, polarization, and the elevation angle towards the satellite. The overall channel attenuation, denoted by [TeX:] $$L_{X, s,}$$ is then calculated as [12]

where [TeX:] $$d_{Xs}$$ indicates the distance between ground node X and satellite s.

We adopt the satellite antenna gain model described in ITU-R S.672-4 as

in dBi where [TeX:] $$\theta_{\mathrm{X}, m, s}$$ indicates the angle between ground node X and the direction of beam m from satellite s, [TeX:] $$\bar{G}_{\mathrm{sat}, m, s}$$ denotes the maximum main lobe gain of beam m at satellite s,[TeX:] $$\bar{\theta}_{m, s}$$ signifies the angle at which the gain falls 3 dB below the maximum, [TeX:] $$\check{\theta}_{m, s}$$ marks the angle at which the third equation in (46) becomes 0 dBi, and Ψ determines the required near-in-side-lobe level relative to peak gain^[27]. In this work, we set [TeX:] $$\alpha=2 a=2.58, b=6.32, \Psi=-20, G_{\text {main }, s}=\bar{G}_{\text {sat }, m, s} \forall m,$$ s^[25], [TeX:] $$G_{\text {side }, \mathrm{X}, m, s}=G_{\text {sat }}\left(\theta_{\mathrm{X}, m, s}\right), \quad \forall \mathrm{X}, m, s \text {, and } \bar{\theta}_{\mathrm{X}, m} =0.5^{\circ}, \forall \mathrm{X},$$ m^[28].

It is assumed that [TeX:] $$K_s$$ ground terminals are randomly located within a horizontal range of [TeX:] $$d_{\mathrm{X}, s}$$ from satellite s. The satellites are positioned in a circular formation with radius [TeX:] $$r_s$$ from the origin, each separated by angle [TeX:] $$\theta_s=2 \pi / S .$$ System parameters are set as outlined in Table 1, unless otherwise stated. Fig. 2 illustrates an example of node placement when [TeX:] $$K_s=10,$$ B = 10, and S = 3.

For comparison, we consider the following baseline methods that address the inner layer problem (P4- inner).

· Sequential least squares programming (SLSQP)^[29]: This method solves a sequence of quadratic subproblems to handle nonlinear optimization problems with both equality and inequality constraints.

· Trust-region algorithm for constrained optimization (Trust-constr )^[30]: We leverage a trust- region interior point approach, introducing slack variables and solving a sequence of equality- constrained barrier problems with decreasing barrier parameters.

· Equal power allocation (EPA)^[15]: From (24b)- (24d), we consider the following equally distributed power allocation as

which can be verified to be always feasible.

Fig. 3 shows the average GEE as a function of the number of satellites S for different number of GAPs per satellite Ks with random spot beam-GAP assignment. The proposed SCA-based algorithm clearly out- performs the baseline algorithms for all S and Ks . We also observe that the GEE generally improves with higher S in most schemes except for the equal power allocation (EPA) strategy. This implies that the adjustment of spot beam power becomes even more important when there are more number of satellites. The figure also shows that the GEE increases with Ks for every scheme. This performance gain stems from the increased diversity in the channel links.

Fig. 4 compares the average GEE in terms of the number of GAPs per satellite Ks with different number of satellites S. We first note that the proposed SCA method provides the highest η across all values of S and Ks , which emphasizes the effectiveness over other algorithms. While the GEE of the schemes that optimize the spot beam power increases for different S, it is the opposite for the EPA strategy. This is in line with our interpretation in Fig. 3.

Fig. 5 compares the average GEE performance by varying the maximum main lobe gain [TeX:] $$\bar{G}_{\mathrm{sat}, m, s} .$$ The GEE of all schemes gradually increases as [TeX:] $$\bar{G}_{\mathrm{sat}, m, s}$$ grows since the desired signal power gets larger. The proposed SCA method outperforms other benchmark schemes regardless of [TeX:] $$\bar{G}_{\mathrm{sat}, m, s}.$$ The gain is particularly significant in the high main lobe gain regime, where the interference management becomes more important. Thus, it can be concluded that the proposed approach is efficient in handling interference through optimized power allocation strategies.

Fig. 6 examines the average GEE with respect to the power budget [TeX:] $$\bar{P}_{\mathrm{sat}, s}=\bar{P}_{m, s} .$$ We can first see that the proposed SCA exhibits the highest GEE performance compared to benchmark schemes. The optimization- based schemes including the SCA, SLSQP, and Trust- constr converge to specific GEE values as the transmit power gets larger. This is because the increase in the IBI and ICI limits the sum-rate per unit power consumption. Therefore, in terms of the GEE performance, it is beneficial not to exhaust all the available power budget. Since the EPA always utilizes the total power constraint, its GEE performance decreases after a certain power. This demonstrates the necessity of the power control algorithms for maximizing the GEE in the multi-LEO satellite networks.

Fig. 7 presents the average GEE with respect to the static circuit power [TeX:] $$P_{\text {circuit, } s}.$$ It is clear that the GEE monotonically decreases with [TeX:] $$P_{\text {circuit, } s}.$$ The proposed SCA is superior to other baseline schemes, proving its effectiveness for arbitrary static circuit power consumption.

Fig. 8 investigates the average GEE as a function of the velocity of satellites vs . We observe that a faster movement of satellites degrades the GEE performance due to the increase in the ICI. Moreover, the proposed SCA algorithm outperforms all the other compared schemes. Thus, it is concluded that the proposed approach is effective for any given satellite mobility.

Ⅵ. Conclusion

In this paper, we studied a spot beam power control of multiple LEO satellites for downlink transmission. To be specific, we optimized the sum-rate and total power consumption to maximize the GEE of the satellites, each of which supports associated GAPs. We first formulated a GEE maximization problem taking all the inter-beam, inter-carrier, and terrestrial network interference into account. To ease the complexity of the problem, we further simplified the expressions of the GEE and interference and then proposed a SCAbased spot beam power optimization method. Numerical results validated its effectiveness over baseline algorithms in different system setups.

To further enhance the GEE performance, a joint optimization of GAP-beam-LEO satellite scheduling and spot beam power control can be an interesting extension. We can also delve into dynamic scenarios with temporal variations in GAP locations and service demands, which lead to imperfect and outdated channel state information at the satellites. Considering the load of inter-satellite link (ISL) message exchange and developing a decentralized control of multiple LEO satellites can also be a potential future work.