Enhanced Minimum Spanning Tree Optimization for Air-Lifted Artificial Upwelling Pipeline Network

Abstract

Artificial upwelling (AU), a geoengineering technique aimed at transporting nutrient-enriched deep-sea water to the sunlit surface layers through artificial systems, is increasingly recognized as a promising approach to enhance oceanic fertility and stimulate primary marine productivity, thereby bolstering the ocean capacity for carbon sequestration. Several air-lifted AU systems have been implemented in countries such as Norway and China. However, research on the optimization of the air injection pipeline network (AIPN)—a critical component of the air-lifted AU system—remains limited. This paper introduces a refined minimum spanning tree algorithm to propose a novel approach for optimizing the AIPN. Furthermore, the bubble-entrained plume loss rate (N_BEP) is developed as a model to assess the efficiency of air-lifted AU systems, which is applied to three case studies involving air-lifted AU systems of varying scales. The findings indicate that the enhanced minimum spanning tree algorithm outperforms the conventional Prim’s algorithm, leading to an average 87% reduction in N_BEP of the optimized AIPN, compared to the AIPN of previous air-lifted AU systems while improving system stability. Consequently, the proposed optimization method for AIPN offers valuable scientific and practical insights for the engineering design of the air-lifted AU systems across diverse scales, offering transformative potential for large-scale carbon sequestration and marine productivity enhancement.

1. Introduction

Climate change driven by greenhouse gases (GHGs) represents one of the most pressing environmental challenges confronting the world today [1]. Carbon dioxide (CO₂) emerges as the predominant greenhouse gas, responsible for approximately 70% of the surface warming observed since the industrial revolution [2]. Artificial upwelling (AU), a method facilitating the transport of nutrient-rich, low-temperature water from the depths of the ocean to the sunlit photic zone at the sea surface, can be harnessed for marine aquaculture, promoting a sustainable approach to ocean fertilization [3]. This innovative technique not only boosts the productivity of cultivated marine species, but also holds promise as an effective means to alleviate atmospheric CO₂ levels [4].

Ocean AU technology can be categorized into several distinct classifications, including artificial reefs, pump-based systems, wave pumps, and air-lifted systems. Currently, air-lifted AU technology stands as the prevalent method due to its robust engineering feasibility. This technique involves the introduction of compressed air into the seabed via nozzles, generating clusters of bubbles that entrain nutrient-rich seawater from the depths [5,6]. This resulting nutrient-rich seawater, known as the bubble-entrained plume (BEP), significantly enhances algal proliferation and has attracted much attention for its applications in aquaculture and algal carbon sequestration [7,8,9]. As shown in Figure 1, in 2010, McClimans et al. established the first air-lifted AU system in Arna Fjord [10]. Subsequently, as shown in Figure 2, in 2018, Fan et al. implemented an innovative solar-driven air-lifted AU system in Aoshan Bay, China, which notably amplified seaweed biomass and the consequent carbon removal [8].

To achieve a large-scale and high-efficiency air-lifted AU system, it is essential to deploy a multitude of air injection nozzles in areas profoundly influenced by AU. The location of these nozzles is subject to variation due to key factors such as tides, temperature fluctuations, light exposure, cross-current velocities, and the presence of seabed obstacles. The optimization of the nozzles layout in a designated area can significantly enhance the nutrient transport efficiency (NTE) of the AU system [7,9,11]. Nonetheless, the appropriate nozzle layout for specific areas often exhibits dispersion and lacks a clear pattern. This irregular arrangement presents a significant challenge in optimizing the air injection pipeline network (AIPN) that interlinks these nozzles for air injection. The scattered nozzle arrangement leads to irregular installation positions for the AIPN, and an unoptimized AIPN can result in considerable pressure losses, increased energy consumption, and resource waste—an unacceptable situation in marine environments with limited resources [12]. In the meantime, the complex seabed terrain, characterized by numerous obstacles, can reduce the lifespan of the AIPN and increase maintenance costs, compromising the reliability and stability of the air-lifted AU system. The irregular arrangement of the AIPN exacerbates these issues; however, incorporating obstacle avoidance measures often requires modifications to the pipeline network, potentially worsening pressure losses within the system. Hence, a critical challenge in optimizing the AIPN lies in achieving effective obstacle avoidance while minimizing pressure losses, all while considering the impacts of seabed obstacles avoidance. In conclusion, optimizing the AIPN according to the specific layout of air injection nozzles is crucial for ensuring efficient air injection, avoiding seabed obstacles, and minimizing pressure losses, thereby maximizing the performance and sustainability of the air-lifted AU system.

Previous studies have focused on the influence of nozzle shape and size and the behavior of the BEP in the crossflow on the NTE of the AU system. For example, in 2013, Meng et al. proposed a numerical model based on Reynolds Averaged Navier–Stokes (RANS) to simulate the air-lifted AU process in an upward tube [13]. In 2015, Fan et al. developed a mathematical model that was validated by a steady-state RANS CFD model to optimize the design of the upward tube and to minimize the plume dilution [14]. However, the high CPU cost and large amount of computation time required to obtain accurate results for CFD models pose a significant challenge in the simulation and optimization of air-lifted AU systems with a large number of nozzles. This problem makes it difficult for CFD models to participate in the process of optimizing large-scale air-lifted AU systems. Recent studies have explored the theoretical models to predict BEP behavior. Specifically, Qiang et al. developed a BEP trajectory model under cross-flow, based on Ansong’s vertical jet trajectory model, which can be used to determine the minimum volume flow rate of air injected into the nozzle required for AU [6]. Yao et al. revealed the relationship between the volume flow rate of the jet air and the maximum height of the BEP, and proposed a method to regulate the BEP to reach a specified height by adjusting the volume flow rate [7]. There are also some studies that address nozzle layout optimization problem for air-lifted AU systems, as well as energy management optimization for air-lifted AU systems [11,15], which provide useful insights into the optimization of air-lifted AU systems.

Presently, despite some progress in the theoretical studies on the optimization of air-lifted AU systems, to the best of our knowledge, there exists a lack of research dedicated to optimizing the air-lifted AU’s AIPN. However, considerable progress has been made in offshore wind power, particularly regarding the optimization of cables linking offshore wind turbines and substations [16,17,18,19,20]. The methodologies applied in these studies predominantly utilize graph theory algorithms [21,22,23], metaheuristic algorithms [18,23,24,25,26,27,28,29], and integer linear programming methods [30,31,32,33,34].

In studies utilizing graph theory optimization algorithms, Li et al. utilized Prim’s algorithm to determine the shortest total cable length [21]. Hou et al. initially optimized the cable layout using Prim’s algorithm, and subsequently developed a dynamic minimum spanning tree algorithm, incorporating cable selection [23,35]. Cao et al. introduced a global dynamic minimum spanning tree (GDMST) algorithm that effectively addresses the challenge posed by dynamic fluctuating edge weights [36].

Regarding metaheuristic optimization algorithms, Hou et al. utilized an adaptive particle swarm algorithm to optimize the radial collection system topology [26]. Zhao et al. encoded the electrical system topology as binary strings and employed a genetic algorithm for optimization, focusing on minimizing production costs and improving system reliability [25].

In the field of integer linear programming methods, Wedzek et al. devised a mixed-integer linear programming (MILP) method that concurrently optimizes both the cable layout and cross-sectional sizes of wind farms [30,31]. Klein et al. introduced an integer linear programming model aimed at minimizing the costs associated with offshore wind farm cable layouts, incorporating obstacle zones by introducing optional connection points for modeling [33]. Cerveira et al. developed an integer linear programming model, specifically tailored for designing cable layouts in wind farms [34].

Among the three algorithm categories discussed, metaheuristic algorithms are adept at defining problem spaces and achieving specific objectives. However, they typically provide near-optimal solutions and lack the ability to effectively assess the quality of the current solution. On the other hand, integer linear programming methods require the creation of precise and intricate mathematical models, which leads to significant increases in computational time as the number of optimization objectives expands.

Graph theory-based optimization algorithms are highly efficient in quickly determining the minimum spanning tree within a graph. However, they are less suited for optimizing the dynamically changing pressure losses of the air-lifted AU system.

The paper introduces, for the first time, an innovative GDMST algorithm named Global dynamic vertex-weighted minimum spanning tree (GDVMST) multi-objective joint optimization algorithm, specifically designed to optimize the AIPN within the air-lifted AU systems. The GDVMST multi-objective joint optimization algorithm is a further improvement on the Prim and GDMST algorithms based on the AIPN optimization problem. Traditional minimum spanning tree algorithms (e.g., Prim’s algorithm) only optimize static edge weights, whereas the GDMST algorithm proposed in previous studies is a modification of static variable weights to dynamic edge weights, which is applicable when dealing with graph theoretic optimization problems, where the edge weights in the graph change dynamically during the optimization process. In this paper, based on the special optimization objective, combined with the location where the pressure loss occurs, the local pressure loss (LPL) is modeled as vertex weight and the frictional pressure loss (FPL) is modeled as edge weight, and the vertex weight will be dynamically updated with the different pipeline connection states, so the existing algorithms are not suitable for solving the problem of the simultaneous optimization of the two types of pressure loss. GDVMST multi-objective joint optimization algorithm introduces a novel dynamic vertex weight adjustment mechanism based on node degrees. This mechanism models LPL as vertex weights and dynamically updates vertex weights to reflect real-time pipeline connectivity, addressing the inability of existing methods to simultaneously optimize FPL and LPL. Meanwhile, GDVMST multi-objective joint optimization algorithm integrates obstacle avoidance strategies with pressure loss minimization by embedding a navigation grid-based pathfinding algorithm. This allows for a real-time calculation of the obstacle-avoiding pipeline lengths and their incorporation into weight updates, overcoming the limitations of graph theory algorithms in complex seabed terrains. By combining the clustering algorithm for nozzles grouping, GDVMST multi-objective joint optimization algorithm partitions the affected area of the air-lifted AU system into multiple air injection clusters and achieves a high adaptability in kilometer-scale air-lifted AU systems. The algorithm systematically optimizes both obstacle avoidance and pressure loss for each air injection cluster, achieving multi-objective joint optimization and providing a robust theoretical tool for the efficient design and deployment of air-lifted AU projects.

The paper presents three distinct case studies of varying scales, reflecting different extents of the air-lifted AU influence area. The computational results are carefully analyzed and compared with the single-path pipeline network (SPN) utilized in previous air-lifted AU systems, as well as the Prim’s algorithm-optimized systems, to assess the efficacy of the proposed algorithm. The structure of the paper is organized as follows: Section 2 introduces the pressure losses calculation model for the air-lifted AU system, optimization strategies for the AIPN, and the optimization methodologies for clustering, obstacle avoidance routing, and the multi-objective joint optimization algorithm. Section 3 presents the details of the three case studies at varying scales, including the environmental parameters of the influence areas and simulation settings for air injection conditions. It also introduces the bubble-entrained plume loss rate N_BEP as an evaluation metric, and outlines a method for evaluating the AIPN in air-lifted AU systems. Section 4 showcases the optimization outcomes for the three cases, comparing them with the SPN and Prim’s algorithm-optimized systems to validate the performance of the optimization algorithm, and includes a discussion of the results. Section 5 concludes the paper.

2. Materials and Methods

The process of optimizing the AIPN involves a systematic approach to ensure efficient air injection and the incorporation of reduced pressure losses and obstacle avoidance. This section introduces the GDVMST multi-objective joint optimization algorithm, featuring a novel weight adjustment mechanism, based on the variations in the nodes’ degree and integrating obstacle avoidance into the algorithm. The steps for optimizing the AIPN are outlined as follows:

Step 1: Application of fuzzy C-means (FCM) clustering to categorize the air injection nozzles into clusters, determining the number of air injection clusters, assigning the nozzles, and identifying the air injection root node positions, which act as the starting points in the AIPN.

Step 2: Implementation of a navigation grid-based pathfinding algorithm to facilitate obstacle avoidance, while laying pipelines between the nozzles.

Step 3: Introduction of a pressure loss optimization strategy, based on the GDVMST, comprising a pressure loss calculation model, the AIPN construction strategy, and defined optimization objectives.

Step 4: Integration of obstacle avoidance and the pressure loss optimization algorithm to formulate the GDVMST multi-objective joint optimization algorithm.

The flowchart depicting the AIPN optimization process is illustrated in Figure 3.

The clustering of the nozzles is based on the FCM algorithm, which is superior to the traditional K-means algorithm in the optimization of AIPN in the AU system. FCM reduces the impact of outliers through membership weighting, while K-means is sensitive to outliers, which may shift cluster centers. For instance, FCM effectively handles sparse nozzle distributions, caused by uneven obstacle layouts in complex seabed terrains.

The dynamic adjustment mechanism of weights used in the algorithm has also been shown to be effective in dealing with the dynamically changing optimization problems [35,36]. The integration of optimization patterns, such as clustering, obstacle avoidance, and the dynamic adjustment of weights allows the GDVMST multi-objective joint optimization algorithm to effectively optimize the AIPN.

2.1. Determination of Root Nodes and Air Injection Clusters Based on FCM Algorithm

The division of the area influenced by air-lifted AU into multiple air injection clusters, based on the number of air injection nozzles, is crucial for optimizing AIPN. This clustering approach becomes particularly significant when dealing with extensive areas that entail a higher number of air injection nozzles. An excessive number of nozzles can introduce unnecessary complexity, potentially leading to system failures, increased air injection costs, reduced effectiveness, and diminished computational efficiency. To ensure effective air injection, enhance system reliability and stability, and improve computational efficiency, it is imperative to cluster the nozzles into several air injection clusters, according to the number of nozzles. Even in cases where the area does not necessitate multiple clusters, clustering should still be conducted to establish a single air injection cluster and identify the air injection root node. Each cluster should designate an air injection root node connected to the air compressor, with the root node’s position directly influencing pressure losses within the corresponding air injection cluster. The Fuzzy C-means algorithm, a prevalent method in fuzzy clustering analysis, is employed for this purpose [37]. The FCM algorithm efficiently clusters the nozzles and determines suitable air injection root nodes. It aims to minimize the total distance from the root node to the other nozzles, thereby effectively reducing pressure losses within the air injection cluster. Below is a brief overview of the FCM algorithm.

Given a dataset containing n data points: $X=\left\{x_1,x_2,...,x_j,...,x_n\right\}$, x_i denotes the i-th sample point in the dataset. x_ij represents the j-th attribute of x_i. The FCM algorithm can partition this dataset into c clusters, where c is a positive integer greater than 1. The cluster centers for each class are represented as follows: $C=\left\{c_1,c_2,...,c_j,...,c_c\right\}$.

The objective function and constraints of the FCM algorithm are as follows:

$$J\left(U,V\right)={\sum }_{i=1}^n{\sum }_{j=1}^c{u}_{ij}^m{d}_{ij}^2$$

(1)

The constraint for this objective function is:

$${\sum }_{j=1}^c{u}_{ij}=1,u_{ij}\in \left[\mathrm{0,1}\right],$$

(2)

where u_ij denotes the membership degree of the sample point x_i to the cluster center c_j, m represents a fuzziness parameter (typically m > 1), and d_ij signifies the distance between the sample point x_i and the cluster center c_j, typically calculated using the Euclidean distance. The objective function consists of the affiliation of the corresponding sample multiplied by the distance from that sample to the center of each class. The objective function is minimized to minimize the sum of the distances from the samples to the cluster centers.

The process of clustering is oriented towards attaining the minimum value of the objective function within the predefined constraints. The FCM algorithm iteratively optimizes this objective function to achieve a fuzzy classification of the sample set. To minimize the objective function J under these constraints, the method of Lagrange multiplier is utilized, leading to the derivation of the membership matrix U and the cluster centers C:

$$u_{ij}={\displaystyle \frac{1}{{\sum }_{k=1}^c{\left(\frac{d_{ij}}{d_{ik}}\right)}^{\frac{2}{m-1}}}},$$

(3)

$$c_j={\displaystyle \frac{{\sum }_{i=1}^n{u}_{ij}^m{x}_i}{{\sum }_{i=1}^n{u}_{ij}^m}}$$

(4)

The termination condition for the iteration is:

$${\mathit{max}}_{\mathit{ij}}\left\{\left|u_{ij}^{\left(a+1\right)}\right|-u_{ij}^{\left(a\right)}\right\}<\epsilon ,$$

(5)

where a designates the current iteration step, while ε symbolizes the convergence threshold. This means that if the membership degrees do not change significantly with further iterations, it signifies the attainment of an optimal state, be it local or global.

In the clustering predicament concerning the positioning of root nodes within the AIPN optimization, a dataset encompassing the coordinates of n air injection nozzle locations must be considered: $\mathrm{X}=\left\{{\mathrm{x}}_1,{\mathrm{x}}_2,...,{\mathrm{x}}_{\mathrm{i}},...,{\mathrm{x}}_{\mathrm{n}}\right\}$, where x_i denotes the coordinate point of the i-th nozzle. The objective entails grouping these locations into c clusters, each featuring an assigned air injection root node defined as: $C=\left\{c_1,c_2,...,c_j,...,c_c\right\}$. Mathematically, this problem can be expressed as:

$$\mathit{min}{J}_m={\sum }_{i=1}^n{\sum }_{j=1}^c{u}_{ij}^m{d}_{ij}^2$$

(6)

The constraint for this objective function is:

$${\sum }_{j=1}^c{u}_{ij}=1,u_{ij}\in \left[\mathrm{0,1}\right],$$

(7)

where X represents the set of air injection nozzles, while C signifies the set of air injection root nodes.

Within the aforementioned framework, Equation (6) delineates the objective function of the optimization process, which aims to minimize the objective function J_m through the allocation of the air injection nozzles to their respective air injection root node.

The FCM algorithm adeptly segregates the influenced area of air-lifted AU, ensuring the feasible placement of air injection root nodes. This facilitates the assignment of each nozzle to its fitting air injection root node.

2.2. Pipeline Creation Based on Delaunay Triangulation and Pathfinding

When confronted with the challenges presented by a complicated seabed terrain and obstacles, the length of the air injection pipelines or the distance between two nozzles, must be calculated as the constrained shortest path connecting these locations. Dutta et al. devised a method involving the construction of a convex hull encompassing the obstacles and the nozzles, to determine the shortest path by traversing the edges of this hull [38]. Nevertheless, in scenarios where obstacles deviate from convex shapes, this approach may erroneously identify such areas as impassable. Hence, this research leverages a pathfinding algorithm, which guarantees, in theory, the discovery of the shortest path between two points within a constrained configuration space, irrespective of the convexity of obstacles [39].

In general, pathfinding can be viewed as a specialized instance of traversing the shortest path tree. The mathematical representation of the shortest path in a graph can be articulated as:

$$\mathit{min}{\sum }_{i\in S}{\sum }_{j\in S}{d}_{i,j}\times u_{i,j}$$

(8)

The constraint for this objective function is:

$${\sum }_{i:(i,k)\in V}{u}_{i,k}-{\sum }_{j:(k,j)\in S}{u}_{k,j}=\left\{\begin{array}{c}-1\hspace{1em}\\ 1\\ 0\end{array}\right.\begin{array}{c}if k=p_1\\ if k=p_2\\ \hspace{1em}if (k\in S:k\notin \left\{p_1,p_2\right\})\end{array},$$

(9)

$$u_{i,j}\in 0,1\hspace{1em}\forall \left(i,j\right)\in S,$$

(10)

where u_i,j is a binary variable that delineates the connectivity status between points i and j within space S along the shortest path. When points i and j are connected by the shortest path, this variable is set to 1; otherwise, it is set to 0. The cost associated with connecting points i and j (i.e., the length of the connecting edge) is denoted by d_i,j. The shortest path is obtained when the sum of the products of u_i,j and d_i,j is minimum.

The Dijkstra algorithm stands as a widely employed method for determining the shortest path, typically providing the most cost-effective route from a single origin point [40]. This research leverages a Dijkstra pathfinding algorithm, based on Delaunay triangulation navigation meshes, as introduced by Jan et al. [41]. This heuristic method allows for the rapid creation of an approximate visibility graph. The approximation method utilizes the edges of the constrained Delaunay triangulation to define the graph’s structure. The Delaunay triangular meshes strive to reduce the presence of elongated triangles, thereby enhancing approximation accuracy, while upholding overall mesh quality. The constrained Delaunay triangulation guarantees that the triangle edges do not intersect with obstacle boundaries. Through triangulating the vertices of the starting point, ending point, and obstacles, a graph representing the traversable area emerges. To refine the graph’s efficiency and approach a more accurate, fully visible graph solution, this methodology incorporates Fermat points derived from the triangles. Following the identification of these Fermat points, they are integrated into the graph and linked to their corresponding triangle vertices and any neighboring Fermat points.

The study proposes a path auxiliary optimization algorithm to address potential issues arising from the primary algorithm, which may generate redundant line segments, leading to suboptimal paths. This auxiliary algorithm aims to eliminate the redundant Fermat points or vertices from the computed path, enabling the further refinement of the resultant route.

Figure 4a illustrates a sample result of the pathfinding algorithm based on navigation meshes. The gray irregular regions represent obstacles, the red dot on the left indicates the starting point, and the blue dot at the center represents the destination. The red lines delineate the navigation meshes utilized for obstacle avoidance, while the blue line denotes the shortest path derived by the algorithm. Figure 4b presents the outcome after applying the path auxiliary optimization algorithm. A comparison between the path results in Figure 4a,b underscores the necessity of integrating the path auxiliary optimization algorithm to enhance path quality and efficiency.

2.3. Pressure Losses in the Air Injection System

Pressure loss serves as a key factor in the effectiveness of air-lifted AU systems. The air-lifted AU systems with low-pressure loss effectively reduce the energy consumption and improve efficiency. In this study, the pressure loss optimization approach is proposed to optimize the AIPN. The flow of air within the pipelines induces pressure losses, stemming from factors like friction, turbulence, pipeline curvature, and fittings [42,43]. These losses are primarily categorized into two distinct types: FPL and LPL [44,45,46]. In this study, assuming constancy in other air injection parameters, FPL is solely associated with the length of the air injection pipelines. On the other hand, the LPL primarily arises from the fittings at various nodes, and is dependent on the specific types of fittings utilized. Variations in the AIPN can prompt alterations in the choice of pipe fittings and the length of pipeline routes, engendering a competition between these two categories of pressure losses throughout the optimization procedure.

The calculation model for FPL Is constructed using the Darcy–Weisbach equation, as follows [46]:

$$p_l=\lambda \cdot {\displaystyle \frac{l}{d}}\cdot {\displaystyle \frac{\rho v^2}{2}},$$

(11)

where λ denotes the friction coefficient, which is intricately associated with the Reynolds number R_e; l represents the length of the pipeline; d signifies the diameter of the pipeline; ρ stands for the density of the fluid; and v denotes the average flow velocity of the fluid within the pipeline.

The Reynolds number Re is calculated utilizing the subsequent formula:

$$Re={\displaystyle \frac{vd}{v_f}},$$

(12)

where v_f is the kinematic viscosity of the fluid.

When R_e falls within the range of approximately: 3000 < R_e < 10⁵, the corresponding λ is calculated using the formula:

$$\lambda =0.3164R{e}^{-0.25}.$$

(13)

When R_e is within the range of approximately: 10⁵ < Re < 10⁸, the corresponding λ is calculated using the formula:

$$\lambda ={\displaystyle \frac{0.308}{{\left(0.842-\mathit{lg}Re\right)}^2}}.$$

(14)

The calculation model for LPL is obtained using the following formula [47]:

$$p_r=\zeta {\displaystyle \frac{\rho v^2}{2}},$$

(15)

where ς represents the local resistance coefficient, the values of ς fluctuate depending on the type of fittings utilized in the pipeline nodes.

The ς values at fittings increase with the number of passages within the fittings. As per the parameters delineated in the Water Supply and Drainage Design Manual (Volume 1) [45], the ς values for diverse types of fittings are detailed in

Table 1. Types of pipe fittings and corresponding local resistance coefficients.

Pipe Fitting Type	Tee Fitting	Cross Fitting	Five-Way Fitting	Six-Way Fitting
ς	1.5	2.5	4.5	8

.

The values of ς at pipe fittings demonstrate a robust correlation with the number of passages. Hence, this paper utilized the MATLAB2021b fitting tool to construct a quadratic function model that elucidates the association between the number of passages of the air injection nozzle downstream and the corresponding values of ς. The achieved goodness-of-fit, denoted as R², stands at 0.9995, denoting a remarkable level of fitting precision, as shown in Figure 5.

The resulting fitted curve function is:

$$\zeta =0.62p^2-0.97p+1.87,$$

(16)

where ς denotes the local resistance coefficient, and p represents the number of passages of the nozzle downstream.

The fitted curve function can act as an empirical formula to calculate the ς values at pipe fittings with different counts of downstream passages within the AIPN.

2.4. Global Dynamic Vertex Weights Minimum Spanning Tree Multi-Objective Joint Optimization Algorithm

To address the simultaneous optimization of the total pressure losses, this paper embraces the graph theory, where the optimization of the AIPN can be viewed as a minimum spanning tree problem [48]. This paper proposes a resolution strategy, the GDVMST, specifically designed to solve the problem of vertex weights changing dynamically with the degree of nodes in a spanning tree. Consequently, an elaborate explication of the optimization strategy, inherent in this algorithm, will be offered.

In the optimization framework of the GDVMST, the strategy categorizes the weights into edge weights and vertex weights. FPL is designated as the edge weight W_e, while LPL is recognized as the vertex weight W_n. Within this structure, the air injection pipelines serve as the set of edges E, and the air injection nozzles serve as the set of vertices N.

During the development of the GDVMST, when a known nozzle searches for the next nozzle to connect with, it transcends a mere comparison of the fixed edge weight W_e (i.e., FPL between the selected nozzle and the known nozzle). Instead, it comprehensively factors in the influence of the vertex weights W_n, which represents the LPL during the connection process. These vertex weights are converted into additional weights W_add that actively participate alongside the W_e in the selection process of the candidate nozzles for the connection. Consequently, the total weights W_t associated with the candidate nozzle can be mathematically represented as:

$$W_t=W_e+W_{add}.$$

(17)

The core concept of the GDVMST involves the incorporation of additional weights to the initial edge weights during the process of selecting the edge for connection. These additional weights are computed based on the connection status of the branches stemming from the starting vertex of each candidate edge (i.e., the number of branches connected to the starting vertex). The primary function of these additional weights is to adapt the edge weights in accordance with the connection status of the existing vertices. When the starting vertex of a candidate edge is associated with a higher count of connected branches, the corresponding edge weight increases, rendering it less appealing for new vertices to connect. Conversely, if the starting vertex has fewer connected branches, the corresponding edge weight decreases, rendering it more attractive for new connections.

Within this optimization strategy, the optimization objective should be defined with the AIPN as the independent variable, aiming to minimize total pressure losses as the objective function. The optimization problem can be described as follows:

$$f_{object}\left(G_T\right)=\mathit{min}\left({\sum }_{i=1}^{C-1}{p}_{l,i}^{G_T}+{\sum }_{j=1}^C{p}_{r,j}^{G_T}\right),G_T\in G,$$

(18)

where I denotes the edge indices representing the pipeline in graph G_T, and C signifies the number of vertices representing the nozzles. $p_{l,i}^{G_T}$ and $p_{r,j}^{G_T}$ represent the FPL of the i-th injection pipeline and the LPL at the j-th pipeline connection fitting, respectively. Consequently, the optimization problem can be articulated by discovering the appropriate AIPN that diminishes the total pressure losses of the air-lifted AU system. The flowchart of the GDVMST optimization strategy is shown in Figure 6.

According to this optimization strategy, the GDVMST optimization algorithm can be proposed. The specific procedural steps of the GDVMST optimization algorithm are outlined as follows:

(1) Define the point sets V_con, V_uncon, and the edge set E. The point set V_con contains the connected points, while the point set V_uncon consists of the points that are yet to be connected. Each point, denoted as v_i, represents an air injection nozzle. Initially, point set V_con includes point v₁, denoted as $\left\{v_1\right\}$, and point set V_uncon contains the remaining unconnected points, denoted as $\left\{v_2,v_3,I,v_n\right\}$, where n is the total number of points in the graph, and represents the number of nozzles. The edge set E is initially empty.

The candidate edges are defined as the edges connecting the points in V_con to the points in V_uncon, where the initial edge weights, denoted as W_ini, are defined as the product of the distance between points in V_con and V_uncon and the factor ${\displaystyle \frac{\lambda }{d}}·{\displaystyle \frac{\rho v^2}{2}}$, representing FPL in the air injection pipelines, W_ini can be mathematically represented as:

$$W_{ini}=l_{con-uncon}\times {\displaystyle \frac{\lambda }{d}}·{\displaystyle \frac{\rho v^2}{2}},$$

(19)

where l_con-uncon denotes the distance between the points in V_con and V_uncon.

(2) After calculating the initial edge weights W_ini, next, select the edge with the smallest initial edge weight from the set of candidate edges. Add this edge to the edge set E, and include the ending point of the selected edge into point set V_con. At the same time, remove the corresponding point from point set V_uncon. After this step, point set V_con will contain two points, while point set V_uncon will be reduced by one point.

(3) After updating the point sets V_con and V_uncon, calculate the new initial edge weights W_{iter_ini}. The additional weights W_add (v_i) are defined as the additional weight to be added to the candidate edges’ weights when connecting the points from set V_con to set V_uncon. The updated initial edge weights for each candidate edge are then given by:

$$W_{iter\_ini}=W_{ini}+W_{add}\left(v_i\right).$$

(20)

The value of W_add(v_i) for each point in the set V_con is calculated as follows: Let W_{ini_add}(v_i) represent the initial value of W_add(v_i) for point v_i. The specific value of W_{ini_add}(v_i) is determined based on LPL at the connection fitting of each point in set V_con when it is not yet connected to a candidate edge, W_{ini_add}(v_i) can be mathematically represented as:

$$W_{ini\_add}\left(v_i\right)=\varsigma \times {\displaystyle \frac{\rho v^2}{2}}.$$

(21)

The number of W_{ini_add}(v_i) values is equal to the number of points in set V_con, where ζ nonlinearly depends on node degree p (Equation (16)).

Thus, for each point in set V_con, an initial additional edge weight W_{ini_add}(v_i) can be computed. By averaging the initial additional edge weights W_{ini_add}(v_i) for all the points in set V_con, W_{avg_ini_add} is obtained. Similarly, by averaging the initial edge weights W_ini for all candidate edges, W_{avg_ini} is obtained. Finally, the formula for calculating W_add(v_i) for each point in set V_con is expressed as follows:

$$W_{add}\left(v_i\right)=W_{ini\_add}\left(v_i\right)\times {\displaystyle \frac{W_{avg\_ini}}{W_{avg\_ini\_add}}}\times S_s.$$

(22)

W_{ini_add}(v_i) denotes LPL at the fitting corresponding to the i-th injection pipeline path. W_{avg_ini} signifies the average value of the initial edge weights across all candidate edges. Lastly, W_{avg_ini_add} indicates the average LPL at the fittings, corresponding to each candidate path. The initial edge weight W_ini for each candidate edge and the corresponding W_add(v_i) value for each point in set V_con are calculated using the formula. Based on the relationship between the candidate edges’ starting points and the points in set V_con, the weight of each candidate edge is set equal to its initial edge weight W_ini, plus the additional weight W_add(v_i), corresponding to the starting point of the edge. For candidate edges that share the same starting point, their additional edge weight W_add(v_i) will be the same.

Next, the newly computed edge weights are compared. The candidate edge with the smallest weight is selected, added to set E, the newly connected point is added to set V_con, and the point is removed from set V_uncon.

(4) The process continues iteratively until there are no remaining points in the point set V_uncon left to be connected.

(5) Based on the optimization strategy for the AIPN, the edge weights are updated to compute the minimum total pressure losses min cost(S_s) of the AIPN, and the corresponding optimal AIPN is output.

$$cost\left(S_s\right)={\sum }_{i=1}^{N-1}(W_{ini,i}+W_{ini\_add}\left(v_i\right)),$$

(23)

where W_ini,I indicates the FPL for the i-th injection pipeline, W_{ini_add}(v_i) indicates the LPL at each pipeline connection fitting.

In the GDVMST algorithm, it is necessary to determine a coefficient S_s. A specific S_s value corresponds to an AIPN optimization result. The S_s values should be determined corresponding to the AIPN layout optimal solution before obtaining the optimization layout. S_s plays a pivotal role in modulating the algorithm’s proclivity concerning the Prim’s algorithm and the GDVMST multi-objective joint optimization algorithm, in other words, in determining whether it prefers the influence of the W_ini or the W_add(v_i) in the selection of the edges. The permissible range of S_s value extends beyond zero. A diminutive S_s value leans towards optimizing based on the W_ini, aligning with the Prim’s algorithm to minimize pipeline lengths within the air-lifted AU system. Conversely, a substantial S_s value tilts the optimization towards the W_add(v_i), potentially grouping nozzles into chains to reduce LPL at the fittings. When the S_s is set to zero, the algorithm reverts to the behavior of the Prim’s algorithm. Conversely, surpassing a certain threshold with S_s may yield suboptimal outcomes, manifesting as all nozzles connected in a singular chain.

Here, the gradient descent method is employed to solve for the coefficient value corresponding to the optimal AIPN structure.

The steps of the gradient descent method are as follows:

(1) Initially, define the initial values of the S_s as S_s1 and S_s2, and assign these values accordingly;

(2) To compute the gradient, the slope k between two points is used as a substitute for the derivative:

$$k={\displaystyle \frac{cost\left(S_{s1}\right)-cost\left(S_{s2}\right)}{S_{s1}-S_{s2}}}.$$

(24)

To avoid the situation where both values of S_s1 and S_s2 become equal, the expression for k is modified slightly by setting k = 100 when the slope k = 0;

(3) Reassign S_s1 and S_s2 as follows:

$$S_{s1}=S_{s2},$$

(25)

$$S_{s2}=S_{s2}-0.0001\times k;$$

(26)

(4) Repeat Steps 2 and 3, using the coefficient values S_s1 and S_s2 as the x-coordinates and total pressure losses of the AIPN cost(S_s1) and cost(S_s2)as the y-coordinates. Continuously output the graphical representation of the line connecting the two points, observing the graph until the results oscillate back and forth. When this occurs, stop the program and output the minimum value of the target function along with its corresponding x-coordinate value.

After implementing the GDVMST algorithm optimized for pressure losses, it becomes crucial to explore the integration of a pathfinding algorithm to accomplish a multi-objective joint optimization encompassing pressure losses and seabed obstacle avoidance. To address this need, this paper proposes the GDVMST multi-objective joint optimization algorithm, which integrates the pathfinding algorithm to solve the multi-objective joint optimization problem. The GDVMST multi-objective joint optimization algorithm presented herein represents an advancement over the calculation of pipeline lengths. Given the coordinates of two nozzles as $X\left(x_1,y_1\right)$ and $Y\left(x_2,y_2\right)$, if the line connecting the two nozzles does not intersect with the seabed obstacle region, this direct line segment serves as the pipeline path, and the Euclidean distance between the nozzles is computed as the pipeline length. However, if the connection line traverses an obstacle region, the shortest path between the two nozzles is determined using a pathfinding algorithm, which then acts as the pipeline length. The computation method for the pipeline lengths between the nozzles in this GDVMST multi-objective joint optimization algorithm is as follows:

$$l=\left\{\begin{array}{c}{\left(x_1-y_1\right)}^2+{\left(x_2-y_2\right)}^2\hspace{1em} \mathrm{n}\mathrm{o}\mathrm{t} \mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\\ D{\left(X,Y\right)}_{\mathrm{obstacle}\text{-}\mathrm{avoidance}}\hspace{1em}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\end{array}\right..$$

(27)

The calculation of the minimum total pressure losses in the AIPN is as follows:

$$cost\left(S_s\right)=\left\{\begin{array}{c}{\sum }_{i=1}^{N-1}(({\left(x_{i1}-y_{i1}\right)}^2+{\left(x_{i2}-y_{i2}\right)}^2)\times {\displaystyle \frac{\lambda }{d}}·{\displaystyle \frac{\rho v^2}{2}}+W_{ini\_add}\left(v_i\right))\hspace{1em}\mathrm{n}\mathrm{o}\mathrm{t} \mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\\ {\sum }_{i=1}^{N-1}(D\left(X,Y\right)\times {\displaystyle \frac{\lambda }{d}}·{\displaystyle \frac{\rho v^2}{2}}+W_{ini\_add}\left(v_i\right))\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\end{array}\right.,$$

(28)

where $X\left(x_{i1},y_{i1}\right)$ and $Y\left(x_{i2},y_{i2}\right)$ represent the coordinates of the first and second nozzles of the i-th pipeline, respectively.

Given the coordinates of all air injection nozzles and obstacles within the specified air-lifted AU area, the algorithm executes a multi-objective joint optimization targeting pressure losses and obstacle avoidance. It meticulously establishes the connection relationships among all nozzles, culminating in the formation of a rational AIPN.

3. Case Description and Evaluation Metric Construction

In order to enhance the optimization performance and adaptability of the AIPN based on the GDVMST multi-objective joint optimization algorithm, three case studies have been conducted. The influenced areas for AU in the case studies measure 250 × 250 m, 500 × 500 m, and 1000 × 1000 m, respectively. As per the optimal nozzle density suggested by Fan et al. [8], for enhanced air-lifted AU system performance, 120, 500, and 2000 air injection nozzles have been positioned in the three scale areas. The specific locations of the nozzles are subject to variability, based on the environmental parameters of the area to achieve the optimal NTE [11]. Nonetheless, for the purpose of this study, the primary focus lies in the AIPN optimization, hence the random distribution of nozzles in each case. Each case incorporates several irregular polygons that represent seabed obstacles, restricting the passage of the AIPN through these regions. In the case involving the 1000 × 1000 m area and a substantial number of nozzles, the FCM algorithm is employed to cluster the nozzles into four air injection clusters, thereby dividing the area into four regions for zonal air injection. In all instances, an identical external location is designated for the air injection float platform within the air-lifted AU area, thereby neutralizing the platform’s position’s influence on the AIPN optimization process.

To facilitate the comparison with conventional air injection methods, the nozzle flow rate is set at 0.1 m³/min, in alignment with the operational parameters outlined in the study by Fan et al. [8], and employs an industrial screw compressor rated at 132 KW. The pipeline specifications encompass standard PU tubes with an inner diameter of 12 mm and an outer diameter of 16 mm. The sea area’s design depth is around 80 m, and additional algorithmic parameters are computed, including the compressed air density of approximately 10.32 kg/m³ and the air flow velocity within the pipeline, estimated to be around 14.74 m/s. The operational timeframe is set at 12 h.

To compare and assess the disparities and optimization effects of the algorithm outcomes, this paper establishes an evaluation methodology for the optimization results of the AIPN as outlined below:

• initially, following the distinct AIPN optimization methodologies, the connection outcomes for a specific case are acquired, and the corresponding pressure loss coefficient A is calculated:

$$A=\sum \lambda \cdot {\displaystyle \frac{l}{d}}+\sum \zeta .$$

(29)

Thus, total pressure losses P_z is:

$$P_z=A\cdot {\displaystyle \frac{\rho v^2}{2}}.$$

(30)

When the operation time is t, total pressure losses can be converted into energy losses E_z as follows:

$$E_z=P_z\cdot Q_0\cdot t,$$

(31)

where Q₀ represents the volume flow rate of the air injected into the pipelines.

The energy losses after running for a duration t can be converted into the additional operating time T for the air compressor as follows:

$$T={\displaystyle \frac{E_z}{W}},$$

(32)

where W represents the power of the selected air compressor, and T denotes the energy losses value, which determines the working duration of the compressor.

The operating duration T of the air compressor can generate and transport the BEP volume in the air-lifted AU system. In this context, a theoretical model rooted in the BEP trajectory model is adopted [6]. By computing the volume flow rate of the transported bottom water [49,50], the proposed algorithm is evaluated.

The BEP volume flow rate Q_w0 is jointly influenced by the plume height and the volume flow rate of the injected air [5]. It is assumed that there is a consistent cross-current flow, with the velocity and direction aligning with the tidal current [6]. The water body is divided into the upper and lower layers, with the lower layer having higher density and nutrient concentrations relative to the upper layer [51,52]. Given the minimal thickness of the bottom water layer [52], the volume flow rate of bottom water transported by the BEP Q_w0 can be approximated by the volume flow rate at the nozzle outlet [6]:

$$Q_{w0}=0.06Q_0^{{\displaystyle \frac{1}{3}}}\Delta z^{{\displaystyle \frac{5}{3}}}{\left(\mathit{tan}h\left({\displaystyle \frac{\sqrt[3]{g{Q}_0}}{H_0\times 0.25}}\right)\right)}^{{\displaystyle \frac{3}{8}}},$$

(33)

where Δz represents the virtual displacement of the nozzle considering the effect of pressure drop [53], H₀ corresponds to the head height under standard atmospheric pressure, generally 10.4 m, and g denotes the gravitational acceleration.

$$\Delta z={\displaystyle \frac{d_0}{1.2\times \alpha }},$$

(34)

where d₀ represents the nozzle diameter, and α is the entrainment coefficient, which can be calculated based on the method proposed by Kobus et al. [54]:

$$\alpha =0.082{\left[\mathit{tan}h\left({\displaystyle \frac{\sqrt[3]{\frac{g{Q}_0}{H_0}}}{v_s}}\right)\right]}^{{\displaystyle \frac{3}{8}}},$$

(35)

where v_s represents the bubble slip velocity, approximately 0.3 m/s [6].

The BEP nutrient-rich bottom water migration process is shown in Figure 7.

In summary, when the air compressor operates for a duration of T, it can generate and transport a volume of BEP V_b used for air-lifted AU as follows:

$$V_b=Q_{w0}\cdot T.$$

(36)

This paper defined the evaluation metric of bubble-entrained plume loss rate N_BEP, where N_BEP is the ratio of the BEP volume lost due to energy dissipation from pressure losses over the system’s operating time t to the actual volume of air injected at the nozzle. This can be expressed as:

$$N_{BEP}={\displaystyle \frac{V_b}{n\cdot Q_{0}t}}.$$

(37)

Using this methodology, the N_BEP can be calculated for the given AIPN. This metric denotes the losses in BEP volume associated with the unit volume of air injected at the nozzles, across the entire operational duration. As pressure losses increase, the corresponding energy dissipation also increases, consequently resulting in higher BEP volume V_b losses. In scenarios where the nozzles’ flow conditions remain identical, an elevated N_BEP signifies that for every unit volume of air injected, a larger quantity of BEP is lost due to energy dissipation, thereby indicating lower system efficiency. Within the context of demanding marine environments characterized by pronounced energy constraints, the V_b that can be effectively formed to transport nutrient-rich bottom water decreases with the same energy supply, resulting in notable economic losses. By comparing the N_BEP derived from the different AIPN design methodologies, a comprehensive evaluation of each approach can be conducted.

4. Results and Discussion

Optimization results for the AIPN in each case study incorporate three distinct methods: the SPN method, the Prim’s algorithm, and the GDVMST multi-objective joint optimization algorithm. The SPN method epitomizes the conventional air-lifted AU air injection pipelines connection method, where each nozzle is connected to a single pipeline. To ensure that the results from the three methods are valid for a comparison under the presence of seabed obstacles, the pathfinding algorithm is applied to the paths intersecting obstacles using the initial two methods, after obtaining preliminary outcomes. It Is crucial to note the distinction between this approach and the underlying logic of the GDVMST multi-objective joint optimization algorithm, which incorporates an obstacle avoidance pathfinding algorithm. While the former approach simply adds algorithms together, the latter method integrates the obstacle avoidance pathfinding strategy within the algorithmic framework, accounting for the implications of obstacle avoidance and integrating them into the multi-objective joint optimization algorithm.

4.1. Calculation and Analysis of S_s Values

To obtain the optimal S_s values in the different cases, the graphs depicting the variation in the total pressure losses coefficient with the coefficient S_s are presented. It is essential to clarify that the total pressure losses coefficient represents the sum of the pipelines length and the local pressure losses coefficient at each fitting, and it varies in direct proportion to total pressure losses under otherwise identical system parameters. This enables a concise and precise depiction of the relationship between total pressure losses and S_s. The total pressure losses coefficient is used as the y-coordinate, named Cost, and the S_s value is used as the x-coordinate. Using the total pressure losses coefficient varying with S_s value variation relationship plot, the optimal range of S_s values in each case can be derived.

The values of S_s corresponding to the optimal solution in Case 250 × 250 m and Case 500 × 500 m are shown in Figure 8. It is found that total pressure loss is minimized when S_s fall within the range of 0.153 to 0.161 in Case 250 × 250 m. Before reaching the minimum total pressure losses, two distinct steps in the decrease are observed in Figure 8a, corresponding to the two optimization stages in the AIPN, relative to the Prim’s algorithm. In Case 500 × 500 m, it is determined that total pressure losses are minimized when S_s falls between 0.082 and 0.089, as shown in Figure 8b. Prior to reaching the minimum total pressure losses, 15 distinct reduction steps are observed, each corresponding to the 15 optimizations identified in the AIPN generated by the GDVMST multi-objective joint optimization relative to Prim’s algorithm.

In Case 1000 × 1000 m, 2000 randomly distributed air injection nozzles are placed. These 2000 nozzles are then clustered into four air injection clusters, with the clustering results depicted in Figure 9. The air injection clusters are distinguished by green, orange, blue, and purple colors. The black pentagrams indicate the clustering centers of each air injection cluster, i.e., the air injection root node.

In Figure 10, the values of S_s, corresponding to the optimal solution of each air injection cluster in Case 1000 × 1000 m, are presented. It is found that the green, blue, orange, and purple clusters achieve their minimum total pressure losses within the following ranges of S_s values: 0.108–0.113, 0.096–0.105, 0.109–0.112, and 0.104–0.115, respectively. Upon reaching the minimal total pressure losses for each cluster, the total pressure losses for Case 1000 × 1000 m are minimized. In each air injection cluster, before attaining the minimum total pressure losses, every step in the gradient corresponds to specific optimizations in the AIPN, compared to the Prim’s algorithm. This meticulous optimization process ensures an efficient and effective design for the air-lifted AU system in Case 1000 × 1000 m.

The correlation between S_s and the total pressure losses coefficient serves as a valuable indicator of the optimization process. From the figures in the discussed cases, it becomes apparent that with an escalation in the S_s value, the total pressure losses coefficient displays a pattern of initial decline followed by an upswing. This behavior is anticipated since manipulating the S_s value entails an endeavor to cluster the nozzles into distinct groupings.

The overarching trend signifies that the gradient direction guides towards the optimal solution, indicating that progression along this path converges the outcome closer to the optimum. This phenomenon elucidates how the GDVMST multi-objective joint optimization algorithm surpasses the Prim’s algorithm in enhancing the optimization of the AIPN.

Moreover, as the area influenced by the air-lifted AU system expands, the algorithm discerns additional positions for structural optimization. This underscores the algorithm’s adaptability and efficacy in addressing the complexities of large-scale air-lifted AU systems, manifesting its aptness for intricate optimization tasks.

4.2. Analysis of the AIPN Layout Optimization Results

This section provides a comparative analysis of the AIPN layout results produced by the three methods in each case. In contrast to the SPN method, the other two algorithms use the tree-like topology from graph theory. A specific advantage of the application of the tree-like topology to AIPN optimization that needs to be analyzed here first to facilitate comparisons for subsequent cases is routes reuse.

The notable optimization improvements in the optimization algorithms, using the tree-like topology over the SPN method, are expected, given that previous studies on air-lifted AU systems often overlooked large-scale, multi-nozzle applications. In smaller systems with fewer nozzles, the energy losses generated by the SPN method are acceptable. However, as the number of nozzles increases, the SPN method necessitates a dedicated route from the root node to each nozzle, without reusing pipeline routes, leading to a substantial rise in energy losses. In contrast, the optimized AIPN derived from the GDVMST multi-objective joint optimization algorithm utilizes the tree-like topology, that facilitates the reuse of numerous air injection pipeline routes. This approach effectively reduces energy losses and enables obstacle avoidance within the AIPN. To provide a more detailed explanation of the tree-like topology showcasing the reuse of injection pipeline routes, this paper provided a simplified air-lifted AU system which comprises an injection root node and five nozzles.

The schematic diagram in Figure 11 highlights the air injection pipeline routes reuse, where the red rhombus denotes the air injection root node, and the yellow squares represent each of the nozzles that need to be connected. The figure clearly demonstrates the crucial role of air injection pipeline route reuse within the tree-like topology in enhancing the efficiency of the air-lifted AU system. When employing the SPN structure as the AIPN layout, represented by the blue dotted line in the figure, each nozzle necessitates a distinct path to the air injection root node. Conversely, utilizing the tree-like topology formed by the GDVMST multi-objective joint optimization algorithm as the AIPN layout, depicted by the red solid line in the figure, enables the connection of all five nozzles via a folded line segment. When the air injection root node necessitates a connection to a more distant nozzle, the existing routes established for connecting to the nozzles positioned ahead of it can be leveraged. This type of connection facilitates the reuse of all routes, with the number on each line segment denoting the frequency of reuse. This type of routing reuse is very common in AIPNs, which will be discussed later, reducing the length of the pipelines and averting redundant pipelines that would otherwise increase energy consumption.

(1)

Case 250 × 250 m.

Given the air-lifted AU influenced area of 250 × 250 m, 120 air injection nozzles have been randomly positioned. Additionally, six seabed obstacles, depicted as irregular polygons, are placed randomly to simulate potential seabed conditions. Figure 12a illustrates the distribution of nozzles and seabed obstacles, with red points denoting the air injection nozzles, and black pentagram representing the air injection root node.

Figure 12b represents the AIPN generated by the SPN method. Figure 12c illustrates the AIPN optimized using Prim’s algorithm, and Figure 12d presents the AIPN obtained from the GDVMST muti-objective joint optimization algorithm, where the blue dashed lines signify the pipelines removed for comparison with the Prim’s algorithm, and the red lines depict the final result.

In Case 250 × 250 m, a comparison of the AIPN diagrams, obtained using different methods, reveals notable distinctions. The Prim’s algorithm stands out by significantly reducing the total pipeline length, in contrast to the SPN method, by utilizing the tree-like topology for optimization. However, when it is compared with the GDVMST multi-objective joint optimization algorithm, further enhancements become apparent. The GDVMST algorithm makes additional improvements by removing connections between specific nozzles and establishing new connections. For instance, connections between nozzle No. 2 and nozzle No. 55, as well as between nozzle No. 21 and nozzle No. 35, are eliminated. Instead, nozzle No. 2 is connected to nozzle No. 7, and nozzle No. 35 is connected to nozzle No. 79. This adjustment is influenced by considering the LPL at pipeline fittings within the algorithm. When nozzle No. 2 is connected to nozzle No. 55, the number of passages through the fitting at nozzle No. 55 increases, leading to a higher LPL. Similarly, when nozzle No. 21 is connected to nozzle No. 35, the LPL at the nozzle No. 21 fitting increases. After calculation, the total pressure losses post-optimization is lower compared to pre-optimization. This signifies the rationale behind the algorithm’s decision to connect nozzle No. 2 to nozzle No. 7 and nozzle No. 35 to nozzle No. 79.

In Case 250 × 250 m, the outcomes of total pressure losses are presented in

Table 2. Total pressure losses corresponding to different methods for Case 250 × 250 m.

Method	SPN	Prim	GDVMST Multi-Objective
Total pressure losses	17,306,176.33 Pa	3,442,220.10 Pa	3,442,123.09 Pa

. Post-optimization calculations for total pressure losses in Case 250 × 250 m indicate significant reductions achieved by both the Prim’s algorithm and the GDVMST multi-objective joint optimization algorithm in comparison to the SPN method. Furthermore, the GDVMST multi-objective joint optimization surpasses the Prim’s algorithm, achieving an 80.1% reduction in total pressure losses relative to the SPN method, and a 4.7% reduction compared to the Prim’s algorithm. The optimization effect is obvious.

(2)

Case 500 × 500 m.

Given the air-lifted AU influenced area of 500 × 500 m, 500 air injection nozzles have been randomly positioned. Additionally, eleven seabed obstacles, represented by irregular polygons, are randomly placed to simulate potential seabed conditions. Figure 13a shows the distribution of nozzles and seabed obstacles, where the red points represent the air injection nozzles, and the black pentagram denotes the air injection root node. In Figure 13b, the AIPN produced by the SPN method is displayed, while Figure 13c showcases the AIPN optimized through Prim’s algorithm. Additionally, Figure 13d exhibits the AIPN derived from the GDVMST multi-objective joint optimization algorithm. In these representations, the blue dashed lines denote the pipelines that were eliminated in contrast with Prim’s algorithm, while the red lines signify the ultimate outcomes.

In Case 500 × 500 m, a comparison of the AIPN diagrams derived using different methods reveals significant optimizations. In contrast to the SPN method, the results obtained from Prim’s algorithm exhibit a notable reduction in the total length of injection pipelines, thereby decreasing pressure losses. Upon comparing the outcomes from the GDVMST multi-objective joint optimization algorithm, it becomes evident that the improvements over Prim’s algorithm are more pronounced in Case 500 × 500 m than in Case 250 × 250 m, with more extensive modifications to the pipeline connections.

As illustrated in Figure 13d, a total of 15 routes have been optimized and altered, with these modifications clearly aimed at reducing the number of fitting passages at some nozzles to mitigate LPL. Notably, in the Prim’s algorithm, nozzle No. 86 and nozzle No. 137 are connected. The air injection pipeline connecting these two nozzles utilizes an obstacle avoidance pathfinding algorithm to navigate around obstacles. However, this segment was eliminated in the GDVMST multi-objective joint optimization algorithm, due to its integration with the obstacle avoidance pathfinding algorithm during computation, which fully considers the impact of obstacle avoidance in the AIPN optimization process.

In Case 500 × 500 m, the outcomes of total pressure losses are presented in

Table 3. Total pressure losses corresponding to different methods for Case 500 × 500 m.

Method	SPN	Prim	GDVMST Multi-Objective
Total pressure losses	140,320,311.54 Pa	13,561,068.51 Pa	13,541,882.95 Pa

. Post-optimization using the three different methods, total pressure losses for Case 500 × 500 m are calculated. Total pressure losses in this case significantly increase compared to Case 250 × 250 m, which is understandable due to the larger air-lifted AU influenced area, an increase in the number of nozzles, and a more complex AIPN. Similarly to Case 250 × 250 m, both the Prim’s and the GDVMST multi-objective joint optimization algorithms achieved substantial reductions in total pressure losses compared to the SPN method. Furthermore, the GDVMST multi-objective joint optimization surpasses the Prim’s algorithm, achieving a 90.3% reduction in total pressure losses, compared to the SPN method, and a 6.3% reduction compared to the Prim’s algorithm, indicating a significant optimization effect.

(3)

Case 1000 × 1000 m.

To account for potential seabed obstacles, 13 obstacles were randomly positioned, represented by irregular polygons. The distribution of the nozzles and the layout of the obstacles are illustrated in Figure 14a, where the red points symbolize the air injection nozzles, and the black pentagrams represent the air injection root nodes.

Figure 14b–d present comparative illustrations of the different AIPNs for Case 1000 × 1000 m. Figure 14b displays the AIPN generated by the SPN method, Figure 14c illustrates the AIPN optimized using Prim’s algorithm, and Figure 14d presents the AIPN obtained from the GDVMST multi-objective joint optimization algorithm, respectively. The sections indicated by red dashed lines represent the pipeline removed after the comparison with the Prim’s algorithm, while the blue, green, orange, and purple sections correspond to the final results for the air injection clusters in the case study.

In Case 1000 × 1000 m, the concept of air injection clusters enables the analysis of this case as a combination of four 500 × 500 m cases, treated individually. The Prim’s algorithm is applied separately to each of the air injection clusters, and in all clusters, notable enhancements over the SPN method are observed. When comparing the outcomes of the GDVMST multi-objective joint optimization algorithm to the Prim’s algorithm, similar to the previous cases, the GDVMST multi-objective joint optimization introduces several improvements for each cluster in Case 1000 × 1000 m. As shown in Figure 14d, the green, blue, orange, and purple air injection clusters undergo 18, 20, 15, and 23 optimized pipeline changes, respectively.

The calculated results for total pressure losses are presented in

Table 4. Total pressure losses corresponding to different methods for Case 1000 × 1000 m.

Method	SPN	Prim	GDVMST Multi-Objective
Total pressure losses	568,120,937.21 Pa	53,734,778.28 Pa	53,661,813.04 Pa

, where total pressure losses in the air-lifted AU system are the sum of total pressure losses from all of the air injection clusters. After optimization using all three methods, total pressure losses for Case 1000 × 1000 m are computed as the sum of the individual total pressure losses of the air injection clusters. Consequently, total pressure losses in this case are notably higher than in the previous cases due to the larger area and the increased complexity of the AIPN. However, similarly to the previous cases, both the Prim’s algorithm and the GDVMST multi-objective joint optimization algorithm significantly outperform the SPN method. Furthermore, the GDVMST multi-objective optimization algorithm enhances the results of the Prim’s algorithm further. The GDVMST multi-objective optimization algorithm achieves a 90.6% reduction in total pressure losses compared to the SPN method, and a 7.1% reduction compared to the Prim’s algorithm, highlighting substantial optimization effects.

From the analysis of the three cases discussed, it is evident that both the Prim’s algorithm and the GDVMST multi-objective joint optimization algorithm proposed in this study offer significant optimization improvements over the conventional SPN method. These methods effectively reduce the total pressure losses. Furthermore, the GDVMST multi-objective joint optimization algorithm enhances the results of the Prim’s algorithm, achieving an average reduction of 87.0% in total pressure losses compared to the SPN method, and an average 6.84% reduction compared to the Prim’s algorithm across the three cases.

In summary, after analyzing the AIPN layouts of the three cases, it is evident that the AIPN layouts attained from the GDVMST multi-objective joint optimization algorithm achieve many enhancements, while comprehensively considering the influence of obstacle avoidance.

4.3. Analysis of the N_BEP

The following evaluation metric N_BEP is calculated and analyzed, for each method, in every case, utilizing the evaluation methodology outlined in Section 3, which relies on the total pressure losses as a fundamental parameter. To further explore the effectiveness of the algorithm, combining the conclusions of the three cases mentioned above and conducting a detailed analysis of the results will provide valuable insights into the optimization process and the outcomes achieved across different cases. This comprehensive analysis will offer a deeper understanding of the algorithm’s performance and its suitability for practical applications in air-lifted AU systems.

Among the outcomes in the three cases, the primary focus lies on the diminution of N_BEP. As shown in Figure 15, where a comparative analysis delineates the alterations in N_BEP within each distinct case. The analysis of the figure reveals a notable trend wherein the N_BEP in different cases increases alongside the augmentation of both the number of nozzles and the affected area within the air-lifted AU system. Nevertheless, an examination of the exponential trend lines in the depicted cases suggests a consistent downward trajectory in N_BEP across varying scales. Under the SPN method, there was a notable surge in N_BEP values across all three cases corresponding to the expanding case areas. Conversely, employing the Prim’s algorithm led to a substantial reduction in N_BEP values in all instances, with a particularly pronounced decrease in Case 1000 × 1000. This phenomenon underscores the adaptability of the tree-like topology across diverse scenarios. The GDVMST multi-objective joint optimization algorithm has further optimization improvements compared to its predecessor, resulting in a further decrease in N_BEP.

This trend signifies that with the addition of more nozzles, the complexity of the AIPN intensifies, resulting in heightened N_BEP values. The escalating N_BEP values indicate that as the volume of air injected into the nozzles increases, a corresponding augmentation in BEP volume loss ensues. This BEP volume loss detrimentally impacts the efficiency of nutrient transportation, thereby directly influencing the operational efficacy of the air-lifted AU system. The discernible reduction in N_BEP unequivocally underscores the efficacy and success of the proposed methodology by this paper. The applicability of the GDVMST multi-objective joint optimization algorithm is demonstrated in different cases, especially in large areas with multiple nozzles.

The three cases of varying sizes present robust evidence that the GDVMST multi-objective joint optimization algorithm is well-suited for the described application scenarios. By utilizing this algorithm, obstacle avoidance can be seamlessly integrated into air-lifted AU systems of different scales. This optimization process minimizes total pressure losses to decrease N_BEP values, while effectively considering the impact of obstacle avoidance, resulting in a systematic approach to reducing pressure losses and achieving obstacle avoidance in the AIPN layout.

4.4. Analysis of Optimization Patterns and Engineering Significance of Optimization Outcomes

This section systematically explores the formation mechanisms of optimization patterns generated by the GDVMST multi-objective joint optimization algorithm and reveals their alignment with real-world marine engineering conditions. Each of the three aspects of clustering, obstacle avoidance, and tree-like topology are analyzed in the following sections. The engineering implications of the optimization results are then discussed in detail.

(1)

Clustering

The computational complexity of global optimization increases exponentially in large-scale AU systems (e.g., Case 1000 × 1000 m with 2000 nozzles and 13 obstacles). FCM clustering divides nozzles into four air injection clusters (Figure 9), decomposing the global problem into localized sub-problems. At the same time, FCM minimizes the total distance objective function (Equation (6)) to determine root nodes, ensuring minimal connection costs within sub-regions. As shown in Table 4, this makes it possible to effectively reduce deployment costs and simplify maintenance in actual marine air-lifted AU projects.

(2)

Obstacle avoidance

Seabed obstacles (e.g., reefs, wrecks) restrict straight pipeline layouts. GDVMST multi-objective joint optimization algorithm incorporates detour path lengths calculated by the navigation grid algorithm (Figure 4) into weight updates (Equation (28)).

For example, in Case 500 × 500 m, the original path between nozzles No. 431 and No. 368 intersected an obstacle (Figure 13d). The optimized path increased length by 22%, but avoided maintenance costs due to potential damage to pipelines from seabed obstacles (approximately $8000 per repair) in actual air-lifted AU projects, and improves the reliability and operational life of the systems.

(3)

Tree-like topology

When using graph theory to solve the AIPN layout optimization problem, the combined effect of the two pressure losses must be considered to obtain the optimal AIPN layout topology. Stringing all the nozzles together through a single pathway can result in excessively long air injection pipelines and unnecessary FPL, and when too many branches are created at a single nozzle this in turn can result in a surge in the LPL at that nozzle, so a reasonable tree-like topology can effectively balance the FPL and LPL, and effectively achieve the lowest total pressure losses. For example, in Case 250 × 250 m, nozzle No. 2 was reassigned to connect with nozzle No. 7 (Figure 12d), due to the excessive LPL caused by multiple downstream connections, reducing the total pressure losses by approximately 400 pa. The tree topology not only reduces the total pressure losses, while extending the life of critical nodes, reducing the risk of leaks and maintenance costs, and increasing the reliability and safety of the system.

The engineering significance of the GDVMST multi-objective joint optimization algorithm optimization outcomes derived from the above optimization patterns is analyzed from three perspectives: ocean air-lifted AU systems efficiency, economic costs, and environmental sustainability.

(1)

Enhancement of air-lifted AU systems efficiency

Case studies show that GDVMST multi-objective joint optimization algorithm reduces the bubble-entrained plume loss rate N_BEP by 87% on average (Figure 15). This implies more nutrients (e.g., nitrate, phosphate) are transported to the photic zone per unit of air injected, enhancing phytoplankton carbon fixation. In addition, the tree-like topology and clustering strategy reduce the risk of failure at critical nodes (e.g., multi-pass connectors), and the incorporation of obstacle avoidance algorithms reduces the risk of pipeline failure. The optimization of AIPN can greatly extend the system’s maintenance cycle, guaranteeing the continuity of the system’s working process and greatly improving the system’s efficiency.

(2)

Economic cost optimization

GDVMST multi-objective joint optimization algorithm reduces energy consumption by minimizing total pressure losses. In the 500 × 500 m case, annual energy costs decreased from $58,000 to $5200 (assuming $0.12/kWh). Clustering strategy reduces redundant pipeline length. In the 1000 × 1000 m case, total pipeline length decreased by 37%, saving approximately $2800 in material costs (PU tubing at $0.15/m).

(3)

Environmental sustainability and adaptation

Obstacle avoidance algorithms not only reduce the risk of failure by avoiding obstacles, but also ensure that the pipeline bypasses sensitive habitats (e.g., coral reefs) and reduces benthic disturbance. Meanwhile, the coefficient S_s also reflects the adaptability of the algorithm to the environment, and the coefficient S_s allows the algorithm to adjust the optimization tendency under different environmental conditions. For example, in the sea area where the labor cost of laying pipeline is high, lower S_s can be set to preferentially reduce the length of pipeline and reduce the cost of project construction. Demonstrates the effectiveness of the algorithm in reducing costs and its adaptability to different situations.

However, the GDVMST multi-objective joint optimization algorithm proposed in this paper still has some limitations, the algorithm cannot be optimized to take the nozzle position into account, currently only for the nozzle layout has been determined. The subsequent need to take into account the impact of the nozzle position to the N_BEP, to form a comprehensive air-lifted AU system layout algorithm.

4.5. Computational Complexity and Accuracy-Runtime Trade-Offs

This section systematically evaluates the computational performance of the GDVMST multi-objective joint optimization algorithm and discusses the trade-offs between solution accuracy and runtime.

The computational complexity of GDVMST multi-objective joint optimization algorithm is determined by the following key steps:

(1)

Fuzzy C-Means clustering:

Time complexity: O (n·c·m·t), where n is the number of nozzles, c is the number of clusters, m is the data dimension (2D coordinates), and t is the iterations. In case studies (t ≤ 50), complexity reduces to O (c·n)

(2)

Navigation grid pathfinding:

Time complexity: O (m logm) for the Delaunay triangulation-based approximate visibility graph, where m is the number of obstacle vertices.

(3)

GDVMST core algorithm:

Time complexity: A variant of Prim’s algorithm with O (n²) complexity. Clustering decomposes the global problem into c sub-problems, reducing total complexity to O (${\displaystyle \frac{n^2}{c}}$).

The complexity of each step and the measured running time of the case are shown in

Table 5. Summarizes the complexity and actual runtime for each step (hardware: Intel i7-12700H, 32GB RAM, Santa Clara, CA, USA).

Algorithm Step	Theoretical Complexity	Case 250 × 250 m	Case 500 × 500 m	Case 1000 × 1000 m
FCM clustering	O (c·n)	0.13s (c = 1)	1.06s (c = 1)	8.50s (c = 4)
Pathfinding (single query)	O (m logm)	0.14s (m = 58)	0.29s (m = 58)	0.34s (m = 121)
GDVMST core optimization	$O({\displaystyle \frac{n^2}{c}}$)	0.15s (n = 120, c = 1)	1.33s (n = 500, c = 1)	10.63s (n = 2000, c = 4)

.

The solution accuracy and runtime of the GDVMST algorithm are mainly related to the number of vertices of the obstacle models and path queries in the pathfinding algorithm, and the number of iterations in the gradient descent process of solving S_S.

During the algorithm operation, when the number of path queries and the number of obstacle model vertices are too high, it will lead to a significant increase in the algorithm operation time. For special sea environments, the seabed obstacle region can be simplified into a polygon with fewer vertices, while the number of path queries between nozzles can be reduced in regions where obstacles can be ignored. In addition, increasing the number of iterations of S_S approximates the global optimal solution but lengthens the running time. According to engineering experience, the number of gradient descent iterations can be limited to 70. As shown in Figure 8b, when the number of iterations is increased from 70 to 300, the total pressure losses of Case 500 × 500 m is improved by only 0.4%, while the running time of gradient descent is increased dramatically.

All of these practices can help to achieve better accuracy-runtime trade-offs in engineering, and shorten the running time of the algorithms under the requirement of ensuring the accuracy of the project.

5. Conclusions

Through the simulation experiments conducted across the three case studies of varying scales, this paper has demonstrated the significant efficacy of the GDVMST multi-objective joint optimization algorithm in enhancing seabed obstacle avoidance while substantially reducing total pressure losses. Comparatively, the algorithm outperforms the SPN method and the Prim’s algorithm by leveraging the GDVMST multi-objective joint optimization. On average, the algorithm achieves a reduction of 6.03% in N_BEP compared to the Prim’s algorithm and an impressive 87.0% reduction compared to the SPN method across the three cases.

A key aspect of this algorithm involves the utilization of a clustering algorithm, particularly effective in larger-scale air injection areas, by partitioning them into suitable air injection clusters to enhance air-lifted AU performance. Additionally, this paper highlights the adaptability of adjusting the S_s value, which yields multiple high-quality solutions. These solutions can serve as an initial set for heuristic algorithms (e.g., genetic algorithms, particle swarm optimization) to seek the best solution by integrating artificial intelligence and heuristic algorithms for total optimization, offering valuable insights for future research endeavors.