Power and Bandwidth Allocation Optimization in Off-Grid Renewable Mobile Base Station Using Lagrange Multiplier

The burgeoning demand for robust and high-capacity mobile networks has become a prevalent concern in the modern era of communications, precipitated by the exponential proliferation of interconnected devices and data traffic [1, 2]. To meet the data rates demand, more base stations (BSs) are geographically deployed, which guides to increased power consumption [3, 4]. Nonetheless, connecting the users in rural areas is further challenging, first because the in-grid power is not always available, second, providing the required quality service is not always easy. There are several reasons for this claim: 1- providing higher data rates is challenging due to sparse population density, geographical hurdles, topographical issues, and infrastructure constraints. These challenges encompass terrain complexities, distance, and signal propagation. 2- regulatory frameworks can be non-supportive of new technology deployment, exacerbating technological barriers such as outdated infrastructure and elevated maintenance costs. 3- higher operational expenses result from extensive coverage necessities and logistical intricacies, and 4- rural users often possess obsolete or less sophisticated devices, incompatible with advanced technologies [5, 6].

To tackle these issues, a synthesis of technological innovation, policy backing, and focused investment is demanded. However, the potential solutions may involve utilizing alternative technologies, governmental subsidies, and community-driven efforts to enhance rural connectivity. These strategies aim to ameliorate the digital divide and foster equitable access to high-speed connections. One of the solutions that relaxes the power demand is the deployment of autonomous off-grid mobile BSs that derive power from renewable energy sources, presents a viable solution to cater to the network coverage and service quality enhancement requirements, particularly in remote and under-served regions [7]. Not to mention, the anticipation is that future mobile networks must be powered by green energy to reduce their environmental impact [8].

Subsequently, the significance of off-grid BSs is amplified in locales, characterized by weak or non-existent electrical grid infrastructure in rural areas, islands, and regions affected by natural disasters. Their utilization offers a sustainable and scalable alternative to traditional grid-dependent BSs, thereby addressing the challenge of extending network coverage in a cost-effective and eco-friendly manner. Note that the International Telecommunication Union (ITU) underscores the essential nature of these solutions to bridge the digital divide and realize the goal of information and communication technologies (ICTs) [9]. A practical example of off-grid BSs has yielded substantial benefits, exemplified by the successful deployment of solar-based BSs in India’s rural expanses, which has significantly bolstered mobile connectivity while diminishing reliance on costly diesel generators. Similarly, in Africa, the integration of solar and wind-powered base stations has played a pivotal role in extending network coverage to distant communities, thereby enabling access to essential services such as mobile banking and telemedicine [10].

However, the integration of green sources with an operational framework of off-grid BSs comprises several critical components: solar panels, energy storage systems, and sophisticated power management units. Solar photovoltaic systems have emerged as the preferred choice due to their adaptability to various geographical contexts, declining costs, and scalability. Energy storage systems, primarily batteries, serve the crucial function of storing excess energy generated during peak periods [11]; for subsequent utilization during low or absent energy production. Power management systems are tasked with optimizing the exploitation of renewable energy to uphold network reliability and performance [12, 13].

Furthermore, the allocation of users’ resources within off-grid base stations, encompassing power, bandwidth, and other network elements, poses a complex challenge. This complexity arises from the nature of green supplies, the variability of user demands, and the imperative to maintain stringent QoS standards. Note that effective strategies for resource allocation must consider crucial metrics, such as

a) Energy efficiency, ensuring the judicious use of both generated and stored energy to minimize waste and guarantee uninterrupted operation.

b) Bandwidth management, provisioning adequate bandwidth to users, contingent upon their requirements, while preventing congestion and upholding high data rates. Moreover.

c) Load balancing includes distributing user connections equitably among multiple sectors to preclude overloading and guarantee consistent service quality.

d) QoS assurance by prioritizing critical services to ensure the maintenance of predefined performance thresholds.

Beyond enhancing connectivity, these implementations demonstrate a commitment to environmental sustainability by lowering greenhouse gas emissions and reducing dependence on fossil fuels.

This environmental consciousness resonates with global directions towards combating the climate change. In summary, there are two main metrics to be considered to guarantee effective resource allocation within the energy efficiency problem while considering renewable energy source, data rate optimisation and power consumption mitigation. To tackle the intricacies of resource allocation, a multitude of optimization techniques have been found, spanning from classical optimization Methods to cutting-edge machine learning and artificial intelligence algorithms. These methodologies aim to enhance the efficiency and sustainability of resource utilization in these systems. For instance, the resource allocation is optimised at the edge users [14, 15]. The resource block allocations in enhanced to improve the QoS among users in Kukade et al. [16], and the LTE service is enhanced in Chai et al. [17]. In the context of energy utilization, modelling, planning and configuration, review papers such as Hussain et al. [18], and Siddaiah and Saini [19] are beneficial. In addition, the voltage profiles during off-peak to peak periods are enhanced by Ahmadi et al. [20]. A hybrid energy grid was used in the LTE system in Yaacoub’s study [21].

Based on that, the optimisation of resource allocation in off-grid green mobile base stations is a critical area of research that combines renewable energy technologies with advanced optimization algorithms.

Based on the previous discussion, there are several questions to be answered:

a). How can renewable energy sources be effectively integrated into off-grid base stations?

b). What are the optimal resource allocation strategies for power and bandwidth in off-grid stations?

c). How do different optimisation algorithms, such as SQP, active set, and interior-point methods, compare in terms of performance and efficiency?

d). What impact do geographical and topographical challenges have on the deployment of off-grid mobile BSs for rural areas?

e). How can dynamic user behaviour and variability in renewable energy supply be managed over time? 6. What are the environmental benefits of deploying off-grid green mobile base stations?

f). How can advanced power management systems optimize the exploitation of renewable energy sources?

g). What are the key factors influencing the carbon footprint?

To answer these questions, this research endeavor seeks to establish and substantiate a holistic optimization framework for the allocation of resources of off-grid green mobile BSs. The proposed work presents a multi-objective nonlinear problem, containing energy efficiency, CO₂ emission, maintenance and QoS metrics.

The essential objective of the proposed work relies on enhancing energy efficiency by optimizing the use of green energy under operational network parameters, such as bandwidth and user power. Integrating renewable energy systems, that is a combination of solar panels and batteries facilitates energy storage, managed by a sophisticated power management system; and plays a pivotal role in optimizing power systems that are adaptively allocated considering user demand and environmental dynamics. The objective is to sustain network efficiency through judicious distribution of both renewable and stored energy, thereby achieving energy optimization, carbon reduction, and preserving QoS. Furthermore, it aims to diminish carbon emissions by integrating green energy sources and energy management practices that are both effective and efficient. To ensure QoS, the framework will optimize power and bandwidth to users in a manner that is commensurate with their specific requirements.

Additionally, the framework is designed to dynamically adjust these allocations in response to fluctuations in user demand, bandwidth, and channel conditions, thereby facilitating balanced load distribution and mitigating the risk of network congestion. The overarching goals of this framework are to minimize carbon footprint, guarantee QoS, manage the variability of renewable energy supply over time, and sustain high data rates along with reliable connectivity and guaranteed maintenance. The study is designed to ascertain the most proficient algorithms capable of addressing the inherently non-linear optimization challenges that arise in the context of such BSs.

The Lagrange multiplier method has been implemented to find the optimal solution mathematically. In addition, three algorithms are used and compared to tackle this problem numerically. The following points highlight the contributions of this research:

a). Formulating an optimization framework designed for off-grid green mobile base stations integrates solar energy to serve rural telecommunication networks with unreliable grid connections. The framework’s uniqueness arises from its emphasis on adaptive resource allocation and dynamic responsiveness to varying user demands, with the ultimate objective of optimizing power and bandwidth distribution while ensuring energy efficiency and service quality.

b). Contrary to prior research that has either examined renewable energy generally or considered mobile base station deployment without explicitly tackling off-grid challenges, this work distinguishes itself by comparing the efficacy of three numerical algorithms: SQP, active set, and interior point in addressing the specific non-linear optimization problems inherent to such a scenario. Through empirical analysis, it highlights the performance of these algorithms in these context studies, the superiority of SQP and interior point algorithms in managing such complexities is established, thereby contributing to enhanced network performance and resource utilization in remote settings.

c). Moreover, the research underscores practical significance by presenting case studies of real-world solar and wind-powered base station implementations in rural environments. These examples exemplify not only technical optimization but also environmental sustainability, particularly in the reduction of operational costs and carbon emissions.

The current investigation aims to fill the identified lacunae within the existing literature through a comprehensive analysis of the following areas:

1. This research provides a comparative evaluation of three sophisticated algorithms, specifically designed to address the multi-objective, non-linear optimization challenges inherent to off-grid base stations (BSs). This approach diverges from earlier studies by focusing on algorithms tailored to the nuanced requirements of such systems.

2. Unlike the prevalent static resource allocation paradigms presented in existing research, this study incorporates dynamic user behavior patterns, real world fluctuations in renewable energy sources, and multi-tiered BS architectures to enhance the practicality and relevance of the proposed solutions.

3. Incorporating environmental metrics, such as carbon emissions, into the optimization framework represents a significant advancement over prior research that has frequently neglected such environmental impacts. 

4. To substantiate the theoretical foundations of the study, real-world case studies are presented, detailing the implementation of solar- and wind-powered BSs. These empirical analyses serve to demonstrate the feasibility and sustainability of the proposed optimization framework, thereby bridging the gap between abstract models and tangible implementations.

1.1 Related works

The reviewed literature provides an extensive overview of the currently available works, highlighting both theoretical advancements and practical implementations. Continued innovations and collaborations are essential to unleash the full potential of off-grid green mobile base stations in achieving sustainable and inclusive mobile network coverage. Several studies have explored different aspects of this field, offering insights into effective strategies and methodologies. In a thorough academic investigation, the researchers scrutinized the incorporation of solar and wind power into mobile BSs, emphasizing the substantial prospects for diminishing carbon emissions and operational expenses. They presented a conceptual framework for a hybrid solar-wind configuration supplemented with battery storage, aiming to guarantee an uninterrupted power supply even amidst periods characterized by limited renewable energy production, but there is no consideration for the CO₂ emission [22].

Subsequently, a stochastic queue model is proposed to evaluate the solar-powered BSs, focusing on metrics such as energy utilization, outage probability, and discharge depth. A novel CAPEX minimization algorithm was used to achieve about 12.1% more performance in comparison with traditional systems [23]. In addition, a wireless communication network driven by renewable energy sources was proposed, comprising a capacity-bound green energy supplier with anticipated mobile user demand. The BS operates on a mix of renewable and conventional power. A queuing model is introduced to analyse decentralized decision-making and detect inefficiencies. Furthermore, an incentive-compatible mechanism is presented to ensure accurate energy demand reporting [24].

The energy consumption in mobile networks is escalating, prompting operators to enhance energy efficiency through cell deactivation during low-traffic periods. This presents a challenging optimisation scenario, which can be framed as binary integer programming with constant interference. A switch-off scheme based on a genetic algorithm was used, demonstrating linear complexity and superiority over the NP-hard bin-packing problem [25]. Moreover, Hammadi et al. [26] examined reinforcement learning for adaptive resource management in renewable energy-driven base stations. Their findings indicated that these algorithms effectively optimise real-time allocation, responding to varying network demands and energy supply.

Subsequently, green energy models for network architecture have been analysed and optimised, outlining fundamental principles and challenges in optimizing these eco-friendly systems proposed by Han and Ansari [8]. Based on solar energy in South Africa, Aderemi et al. [27] studied that solar irradiance levels of 4.5-6.5 kWh/m² can lower operational costs by 49% in contrast to diesel generators. However, in Asia, particularly in northern Pakistan. The study constructs models for solar panels and wind turbines, considering temporal fluctuations and traffic dynamics. An energy cooperation scheme is proposed to optimize cost savings across various BSs [28].

Whereas in Bangladesh, the viability of biomass hybrid and solar photovoltaic systems was analysed. Implementing an energy-sharing mechanism and a low resistance path enhances efficiency for Mobile BSs [29]. Subsequently, green energy is used to diminish the carbon footprint and operational expenses of the telecommunications sector. An algorithm for time-based allocation of collected green energy is proposed, incorporating both delay and green energy considerations. The methodology exhibits enhanced performance relative to established benchmarks across BS deployment scenarios [30].

Another algorithm was proposed in this context, targeting the mitigation of electricity grid burdens and operational costs. The proposed approach assesses the energy efficiency of both hybrid and homogeneous storage systems. Numerical examples are provided to illustrate the comprehensive analysis conducted by Kuaban et al. [31], yet the environmental impact was ignored in these systems. Subsequently, an innovative approach to optimize the deployment of large-scale solar panels and base stations within urban landscapes was proposed, reducing BS deployment costs by up to 35% and 52% in connecting cable costs by Deheyab et al. [32]. Moreover, an off-grid green cellular BS integrating solar power was proposed, in which the storage system’s discharging and charging dynamics were analysed by Kuaban et al. [33]. In addition, a flexible resource optimization approach aimed at minimizing operational costs for 5G BS was proposed with constraints and diverse decentralized resources. The research demonstrates that 5G BS involvement in demand side management decreases overall energy consumption and costs [34]. In the same context, energy cooperation work was modelled as a multi-objective linear programming problem while managing and balancing the future BS load and energy harvesting [35].

Despite significant progress, the extant literature underscores certain omissions with respect to the synthesis of environmental metrics, the consideration of multifaceted BS challenges, and the optimization of systems under real-world dynamic conditions. This research endeavor aims to mitigate these shortcomings by introducing a comprehensive optimization framework that is specifically designed to:

1. Enhance energy efficiency within the system architecture.

2. Diminish the carbon footprint associated with the operation of these structures.

3. Ensure the maintenance of Quality of Service (QoS) parameters under varying demand patterns and energy scenarios.

4. Exhibit the efficacy of the proposed algorithms in practical, operational contexts.

By systematically addressing these gaps, the current study significantly advances the state of the art and offers tangible recommendations for the implementation of sustainable and high-performance off-grid BSs.

By considering two off-grid green mobile BSs, it is possible to encapsulate the interactions that occur among various network constituents. This holistic approach enables the optimization of resource allocation within a more complicated system, thereby addressing the delicate balance between energy efficiency, carbon footprint reduction, operational expenditures, and user QoS. The nature of the optimization challenge remains multi-objective, which is essential for simultaneously considering the multifaceted aspects of the network’s performance.

The subsequent elaboration of this formulation incorporates the presence of two distinct off-grid mobile BSs, which are identified as BS1 and BS2. To account for the dual nature of this configuration, the mathematical constructs, including the relevant equations and variables, are adapted to reflect the interplay between both BSs. Accordingly, the following parameters are assumed: P_o,1(ς) is the BS1 power consumed at ς; P_o,2(ς) is BS2 power consumed at ς; E_storage,1(ς) denotes the BS1 battery’s stored energy at ς;

E_storage,2(ς) is the BS2 battery’s stored energy at time ς; and λ₁ symbolises the Lagrange multiplier for BS1. In addition, λ₂is the Lagrange multiplier for BS2; U_o,1(ς) presents the users count of BS1 at ς; while U_o,2(ς) presents the users count of BS2 at ς. Following, P_user,1,u(ς)= uth user allocated power in BS1; P_user,2,j(ς)= user jth allocated power in BS2; B_user,1,u(ς)= and width allocated to user u at BS1; B_user,2,j(ς)= user’s bandwidth j at BS2. And so on for the other formulation after one BS scenario. For instance:

$\begin{gathered} { Psolar, } 1(\varsigma)=A 1 \cdot G { max }(\varsigma) \cdot(1-G {var}(\varsigma)) \cdot \eta p v \\ { Psolar, } 2(\varsigma)=A 2 \cdot G { max }(\varsigma) \cdot(1-G {var}(\varsigma)) \cdot \eta p v \\ { Estorage, } 1(\varsigma+1) = { Estorage, } 1(\varsigma)+(P{ solar }, 1(\varsigma)-P o, 1(\varsigma) \cdot \Delta \varsigma \cdot \eta \\ { Estorage, } 2(\varsigma+1) = { Estorage, } 2(\varsigma)+(P { solar }, 2(\varsigma)-P o, 2(\varsigma)) \cdot \Delta \varsigma \cdot \eta\end{gathered}$

3.1 Optimization of two base stations

The optimization problem now involves maximizing the overall system efficiency and minimizing carbon emissions for both BSs, while ensuring the constraints are met, as follows:

$\begin{aligned} \max \left(\alpha\left(\eta_{ {eff }, 1}+\right.\right. & \left.\eta_{ {eff }, 2}\right)-\beta\left(C O_{\text {2emitted }, 1}+C O_{ {2emitted }, 2}\right) \\ & -\gamma\left(C_{ {op }, 1}(\varsigma)+C_{ {op }, 2}(\varsigma)\right) +\delta\left(\sum_{i=1}^{U_{o, 1}(\varsigma)} \frac{R_{ {user }, 1, i}(\varsigma)-R_{ {min }, u}(\varsigma)}{R_{ {min }, u}(\varsigma)}\right. \\ & \left.+\sum_{j=1}^{U_{o, 2}(\varsigma)} \frac{R_{ {user }, 2, j}(\varsigma)-R_{\text {min }, j}(\varsigma)}{R_{ {min }, j}(\varsigma)}\right)\end{aligned}$

Subject to:

$R_{ {user }, 1, u}(\varsigma)=B_{ {user }, 1, u}(\varsigma) \log 2\left(1+\frac{S_{ {user }, 1, u}(\varsigma)}{N_{ {user }, 1, u}(\varsigma)+I_{ {user }, 1, u}(\varsigma)}\right)$

$R_{ {user }, 2, j}(\varsigma)=B_{ {user }, 2, j}(\varsigma) \log 2\left(1+\frac{S_{ {user }, 2, j}(\varsigma)}{N_{ {user }, 2, j}(\varsigma)+I_{ {user }, 2, j}(\varsigma)}\right)$

$R_{o, 1}(\varsigma)=\sum_{i=1}^{U_{o, 1}(\varsigma)} R_{u s e r, 1, u}(\varsigma)$

$R_{o, 2}(\varsigma)=\sum_{j=1}^{U_{o, 2}(\varsigma)} R_{u s e r, 2, j}(\varsigma)$

$P_{o, 1}(\varsigma)=\sum_{u=1}^{U_{o, 1}(\varsigma)} \frac{R_{u s e r, 1, u}(\varsigma)}{\eta_{e f f}}$

$P_{o, 2}(\varsigma)=\sum_{j=1}^{U_{o, 2}(\varsigma)} \frac{R_{\text {user }, 2, j}(\varsigma)}{\eta_{ {eff }}}$

$C O_{2  { emitted }, 1}=P_{o, 1}(\varsigma) \cdot \Delta \varsigma \cdot E F_{{grid }}$

$C O_{2  { emitted }, 2}=P_{o, 2}(\varsigma) \cdot \Delta \varsigma \cdot E F_{t {grid }}$

$\eta_{e f f, 1}=\frac{R_{o, 1}(\varsigma)}{P_{o, 1}(\varsigma)}$

$\eta_{e f f, 2}=\frac{R_{o, 2}(\varsigma)}{P_{o, 2}(\varsigma)}$

$R_{ {user }, 1, u}(\varsigma) \geq R_{\min , u}(\varsigma)$

$R_{ {user }, 2, j}(\varsigma) \geq R_{ {min}, j}(\varsigma)$

The optimal allocations of power and bandwidth can be ascertained to re- solve the KKT conditions. The resulting expressions for P_user,1,u(ς), P_user,2,j(ς), B_user,1,u(ς), and B_user,2,j(ς) offer a methodology to distribute resources in a manner that equilibrates energy efficiency, carbon emissions, operational costs, and user QoS.

Nonetheless, the intricacy of the equations and the necessity for iterative numerical resolutions often necessitate the utilization of optimization software or tailored algorithms capable of addressing the nonlinear character of the problem in practical implementations. This mathematical framework serves as a robust foundation for such computational approaches.

For simplicity, let us assume that each base station supports two users. This assumption can be generalized to accommodate a greater number of users analogously. The Lagrangian function becomes as follows:

$\begin{aligned} & \mathcal{L}=\alpha\left(\eta_{ {eff }, 1}+\eta_{ {eff }, 2}\right)-\beta\left(C_{O_{ {emitted }, 1}}+C O_{ {2emitted }_{ {emin }}}\right) -\gamma\left(C_{ {op }, 1}(\varsigma)+C_{ {op }, 2}(\varsigma)\right) +\delta\left(\sum_{u=1}^{U_{o, 1}(\varsigma)} \frac{R_{\text {user }, 1, u}(\varsigma)-R_{\text {min }, u}(\varsigma)}{R_{\text {min }, u}(\varsigma)}\right. \left.+\sum_{j=1}^{U_{o, 2}(\varsigma)} \frac{R_{\text {user }, 1, j}(\varsigma)-R_{\text {min }, j}(\varsigma)}{R_{\text {min }, j}(\varsigma)}\right) )\end{aligned}$

$\begin{aligned} & R_{ {user }, 1, u}(\varsigma) \geq R_{\min , u}(\varsigma) \\ & R_{ {user }, 2, j}(\varsigma) \geq R_{min , j}(\varsigma)\end{aligned}$

for each base station, the KKT conditions are set as follows:

$\frac{\partial \mathcal{L}}{\partial P_{u s e r, 1, u}(\varsigma)}=0 \forall u, \frac{\partial \mathcal{L}}{\partial P_{u s e r, 2, j}(\varsigma)}=0 \forall j$,

$\begin{gathered}\frac{\partial \mathcal{L}}{\partial B_{ {user }, 1, u}(\varsigma)}=0 \forall u, \frac{\partial \mathcal{L}}{\partial B_{ {user }, 2, u}(\varsigma)}=0 \\ \forall j, \frac{\partial \mathcal{L}}{\partial \lambda_1}=0, \quad \frac{\partial \mathcal{L}}{\partial \lambda_2}=0, \quad \lambda_1 \geq 0, \quad \lambda_2 \geq 0\end{gathered}$

5.1 One base station

Figure 1 elucidates the efficiencies and limitations of SQP, Interior Point, and Active Set algorithms in a single BS power allocation scenario using three types of allocations: equal power, proportional and priority-based methods. The equal distribution method uniformly assigns power across all users, thereby ensuring equity yet potentially sacrificing optimal performance for high-demand users due to its indifference towards user-specific requirements. Conversely, the proportional distribution tailors power allocation to individual user needs, which enhances system efficiency for high-demand users, albeit at the expense of equity for lower-demand users.

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Figure 1. Power allocation comparison for three algorithms concerning user index, while using three types of allocations

The interior point method distinguishes itself in this context by allocating above W of power, indicating potential superiority in optimizing for high-power-requiring users. However, the priority distribution scheme allocates most of the power to critical users, rendering it most suitable for scenarios where, certain users’ performance is paramount. While the interior point algorithm exhibits superiority in adaptive resource management under varying user demands, both SQP and active set strike a more balanced equilibrium between equity and efficiency, particularly under the priority strategy.

The selection of an algorithm and corresponding strategy hinges on system-specific imperatives: if fairness is paramount, the equal strategy is preferred; for efficiency, the proportional strategy is more apt; and for prioritization, the priority strategy is ideal. Note that more details about the allocation types and strategies, along with different user behaviors, can be found in Section 6.

Note that for the maintenance, cost, and energy storage, the three algorithms show similar behaviour, with 100.4 CO₂ kg of CO₂ emission, 240 (Cost Unit) maintenance costs, and 2.876 Wh of the energy storage per user when using equal power allocation. These results are obtained using the values of Table 1. Note that the two BSs scenarios have been ignored as knowing the individual and multi-BSs system behaviour is sufficient to conclude the middle case scenario.

Table 1. Parameters and values of one BS

5.2 Multi base station

This scenario becomes more nonlinear than one BS because of the existence of multi-tier allocations. This analysis focuses on the performance of SQP, interior point, and active set methods. SQP is known for its robustness in handling nonlinear optimization problems, as it iteratively solves a sequence of quadratic subproblems to find the optimal solution. It typically converges quickly to a local optimum due to its iterative nature and is effective in handling complex constraints and nonlinearities. However, it may struggle with highly non-convex problems and be computationally intensive. The interior point approach is designed to efficiently tackle large-scale problems by cross passing the feasible region’s interior. They scale well with the size of the problem and can handle a wide range of constraints effectively. However, they may converge slowly near the optimal solution and require good initial feasible points to perform optimally. Active set methods iteratively explore the active constraints for finding its optimal solution. They are efficient for problems with a relatively small number of active constraints and can quickly identify and exploit the structure of the problem more efficiently. Hence, the behaviour of these algorithms reflects on the obtained results, as shown in Figures 2 and 3 after changing the total power allocated to each base station from 10 to 15 W, users’ to 14 and bandwidth to 200 Hz, while using the values of Table 2.

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Figure 2. Power allocated concerning the user index

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Figure 3. Bandwidth allocated concerning the user index

Table 2. Parameters and values of multi-BSs

Figure 2 illustrates a comparative analysis of power allocation among three optimization algorithms across three distinct strategies: equal, proportional, and priority. The equal strategy maintains a uniform power allocation of 1W per user. In contrast, the Proportional strategy reveals a higher degree of variability, particularly within the SQP and interior point methods according to individual user demands, leading to a more dynamic system response. The interior point algorithm is notably adept at allocating power efficiently with pronounced peaks, indicating superior performance in scenarios characterized by significant demand fluctuations. The SQP algorithm, on the other hand, demonstrates a more balanced approach by optimizing power distribution while avoiding extreme allocations, thus catering to systems that necessitate a moderate degree of fairness and adaptability. Lastly, the active set algorithm, which is less reactive than its counterparts, appears to adopt a conservative strategy with fewer substantial power shifts, suggesting a preference for systems where stability and predictability are paramount. The Priority strategy further underscores the algorithmic differences, particularly favoring the interior point approach for its aggressive prioritization of certain users. While the interior point algorithm is well-suited for scenarios demanding intense optimization, SQP provides a smoother equilibrium, and the active Set algorithm is more appropriate for systems requiring modest user-specific adjustments.

Subsequently, Figure 3 illustrates the contrast in bandwidth allocation among user indices under three optimization and three distinct strategies. The equal strategy delivers a uniform bandwidth of approximately 20 Hz across all users, emphasizing equity but lacking adaptability to varying user requirements. Conversely, the proportional strategy manifests greater variability, with certain users receiving substantially larger allocations up to 40 Hz, particularly under active set and interior point, which is beneficial in systems characterized by diverse user demands.

Further information can be seen in Figure 4, where a flow chart that shows the key parameters, constraints and objectives with details.

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Figure 4. Flow chart showing the key parameters, constraints and objectives

The interior method showcases pronounced bandwidth fluctuations, prioritizing high-demand users at the expense of lower-demand users, thus being suitable for scenarios where performance maximization for key individuals is paramount. But, SQP offers a more tempered approach, achieving a balance between equitable distribution and optimization with less extreme allocation variances. When considering the priority method, both interior-point and SQP algorithms demonstrate a strong bias towards high-priority users, with interior points exhibiting more assertive allocation for these individuals. Active set, however, maintains a relatively balanced spread with less responsiveness to high-priority demands.

The key parameters, constraints and objectives can be summarized as follows:

1- solar power generation $\left(P_{  {solar }}(\varsigma)\right)$ is contingent upon solar panels' performance at a specific instant ($\varsigma$), influenced by variables including panel area, solar irradiance, and photovoltaic system efficiency. The greater the solar irradiance, the more substantial the energy yield. However, solar power output fluctuates due to weather variability.

2- total power consumption $\left(P_o(\varsigma)\right)$ encompasses user demands and station operational necessities, with the goal of upholding QoS without surpassing power limitations.

3- battery energy storage $\left(E_{  {storage }}(\varsigma)\right)$ serves as a backup energy source, with efficiency dictating the amount of energy preserved and accessible during periods of diminished solar irradiance.

4- bandwidth allocation $\left(B_{{user }, u}(\varsigma)\right)$ pertains to the specific bandwidth designated to a user, contingent on demand and allocation strategy, which may be uniform, proportional, or prioritized.

5- user data rate $\left(R_{u s e r, u}(\varsigma)\right)$ is contingent on the allocated bandwidth, signal strength, and channel conditions, directly impacting user QoS.

Subsequently, the constraints within this framework include:

1- energy balance, which precludes power consumption exceeding the amalgamated solar generation and battery capacity to avoid shortages.

2- power and bandwidth allocation must be managed judiciously, adhering to total constraints while ensuring equitable or prioritized distribution.

3- a minimum QoS $\left(R_{min , u}(\varsigma)\right)$ is guaranteed for each user, maintaining a baseline data rate for consistent service.

4- the system aims to minimize the carbon footprint by reducing energy-related emissions.

On the other hand, the objectives are fourfold: firstly, maximizing energy efficiency seeks to optimize renewable energy utilization and reduce reliance on backup power. Secondly, minimizing carbon emissions prioritizes green energy sources to curb greenhouse gases. Thirdly, ensuring user QoS focuses on reliable connectivity and adequate data rates for all users. Lastly, minimizing operational costs involves reducing energy consumption and system maintenance expenses. This approach integrates environmental considerations and resource optimization into the design and operation of the base station.