A Review of Optimal Design for Large-Scale Micro-Irrigation Pipe Network Systems

Abstract

Micro-irrigation pipe network systems are commonly utilized for water transmission and distribution in agricultural irrigation. They effectively transport and distribute water to crops, aiming to achieve water and energy conservation, increased yield, and improved quality. This paper presents a model for the scaled micro-irrigation pipeline network system and provides a comprehensive review of the fundamental concepts and practical applications of optimization techniques in the field of pipeline network design. This paper is divided into four main sections: Firstly, it covers the background and theoretical foundations of optimal design for scaled micro-irrigation pipeline network systems. Secondly, the paper presents an optimal design model specifically tailored for scaled micro-irrigation pipeline networks. And then, it discusses various optimization solution techniques employed for addressing the design challenges of scaled micro-irrigation pipeline networks, along with real-world case studies. Finally, this paper concludes with an outlook on the ongoing research and development efforts in the field of scaled micro-irrigation pipeline network systems. In addition, this paper establishes a fundamental model for optimizing pipeline networks, to achieve minimum safe operation and total cost reduction. It considers constraints such as pipeline pressure-bearing capacity, maximum flow rate, and diameter. The decision-making variables include pipeline diameter, length, internal roughness, node pressure, future demand, and valve placement. Additionally, this paper provides an extensive overview of deterministic methods and heuristic algorithms utilized in the optimal design of micro-irrigation pipeline networks. Finally, this paper presents future research directions for pipeline network optimization and explores the potential for algorithmic improvements, integration of machine learning techniques, and wider adoption of EPANET 2.0 software. These endeavors aim to lay a strong foundation for effectively solving complex and challenging optimization problems in micro-irrigation pipeline network systems in the future.

1. Introduction

Micro-irrigation is a precision irrigation technology that offers several advantages, including reduced field management workload and a high degree of irrigation automation. It plays a crucial role in preventing farmland pollution, managing desertification, promoting soil and water conservation, and facilitating ecological improvement [1,2,3]. In large-scale areas, particularly in the Northwest Oasis [4,5], where the micro-irrigation pipe network covers a vast expanse and encounters complex terrains, the implementation of micro-irrigation pipe network optimization technology has emerged as a pivotal factor in promoting sustainable agricultural production [6].

The micro-irrigation pipe network system, as a conduit for agricultural irrigation water, possesses inherent characteristics that make it susceptible to various factors such as water source availability, terrain topography, plot size, and the types of crops being cultivated. As the scale of the system expands, it becomes subject to even more constraints [7,8]. In general, the micro-irrigation system is mainly composed of four parts: water source project, head hub, water distribution network, and emitter [9] (Figure 1). Among them, the investment and construction of the water distribution network account for more than half of the whole irrigation project. Therefore, whether the optimal design of the pipe network in this part is reasonable directly affects whether the project investment, energy consumption, and operating costs are economical and effective [10]. Especially considering the spatial and temporal differences of each factor in the irrigation area, the extensive scale of micro-irrigation pipe networks, characterized by multiple pipeline grades, long pipelines, large control irrigation areas, high water supply flow rates, and significant pressure differences, poses a risk of pipe bursts, which can adversely impact water transmission efficiency. Consequently, researching the optimal design of micro-irrigation pipe networks becomes highly significant. Such research aims to enhance the irrigation utilization efficiency of farmland water and achieve synergistic and efficient water and fertilizer utilization within large-scale pipe networks [11,12].

The primary objective of optimizing irrigation pipe networks is to develop an economically viable, logical, and practical design scheme that ensures optimal performance. This involves optimizing the design flow and irrigation uniformity as the main objectives while considering the constraints of node flow and node pressure. The result is the determination of an optimal layout and combination of pipe diameters for the network, followed by the selection of suitable pipe specifications and sizes [13,14,15]. In the realm of complex scaled micro-irrigation pipe networks, the utilization of innovative mathematical programming methods and heuristic algorithm optimization has been consistently adopted to investigate and achieve the optimal design. This approach is necessary due to the intricacy associated with ring networks and the presence of multiple supply sources [16,17,18]. As early as the 1960s, the concept of optimizing the design of irrigation pipe networks was introduced internationally. Over the years, various traditional design methods such as linear programming, nonlinear programming, and dynamic programming have been employed. However, more recently, there has been a shift towards using optimization models and innovative intelligent algorithms to address the challenges associated with pipe network layout, pipe diameter optimization, and field water distribution [19,20,21,22,23]. Furthermore, an analysis conducted using CiteSpace 6.2.R4 identified the most frequent keywords in the text, which include optimization, design, genetic algorithm, model, and water distribution system (Figure 2). The co-occurrence of these keywords in optimization techniques indicates that genetic algorithms, as a relatively recent intelligent algorithm, have garnered significant attention and have been extensively studied and applied.

As an essential branch of optimization technology, the new intelligent optimization algorithms, particularly the swarm intelligence optimization algorithm, represent an optimization approach that draws inspiration from collective behaviors observed in nature. The widely used algorithms include ant colony optimization (ACO) [24,25], particle swarm optimization (PSO), and the less widely used algorithms include the sticky mushroom algorithm (SMA) [26], artificial bee colony (ABC) [27,28], grasshopper optimization algorithm (GOA) [29,30,31], butterfly optimization algorithm (BOA) [32], moth–flame optimization (MFO) [33], seagull optimization algorithm (SOA) [34], whale optimization algorithm (WOA) [35,36,37,38], and golden sine algorithm (GSA) [39,40]. These algorithms simulate the collaborative behavior observed in groups of organisms, where an individual’s movement is characterized by its speed and position. Each individual independently searches for an optimal solution within the search space and keeps track of its personal best. Through sharing and comparing with other individuals, the optimal individual extreme value is determined, leading to the identification of the global optimal solution [41]. While these algorithms can generate cost-effective and robust optimal design solutions for various optimization problems, it is important to note that their application has predominantly been limited to small-scale power grids and urban water supply networks. Unfortunately, there is a noticeable dearth of research that supports the utilization of intelligent optimization algorithms in the oasis region of northwestern China, where both scale and water resources are significantly scarce.

Therefore, this review establishes a mathematical model based on a scaled micro-irrigation pipe network and focuses on three typical metaheuristic algorithms selected for generalization. This review is organized into three main sections: (1) The optimization model for micro-irrigation pipe networks is succinctly described in terms of the objective function, constraint conditions, and decision variables. (2) On this basis, the optimization solution techniques used for WDSs are classified into two categories: deterministic methods and metaheuristic algorithms, and the successful cases concerning the optimization of micro-irrigation pipe networks are analyzed and compared. The metaheuristic algorithms are also improved to address the shortcomings of the application of metaheuristic algorithms in the field of pipe network optimization. (3) By analyzing the enhancement of algorithms, the integration of machine learning and algorithms, and the widespread adoption of software such as EPANET 2.0, the future development direction of micro-irrigation pipe network optimization the future development direction of micro-irrigation pipeline network optimization is envisioned.

2. Description of the Optimization Model for Micro-Irrigation Pipe Networks

2.1. Objective Function

Based on the structural complexity and long pipeline length of self-pressurized pipeline irrigation systems, two primary optimization objectives are proposed:

(1)

Ensuring the safe operation of the pipe network system: The goal is to maintain the actual pressure head at the entrance of each branch pipe within a safe range. It is desirable to minimize the deviation between the actual pressure head and the required pressure head of each branch pipe, ensuring that the actual pressure head exceeds the required pressure head by the smallest possible margin.

(2)

The objective of the pipe network system is to minimize the total cost, which includes the cost of the main water supply pipes, the cost of the main pipe for water distribution, the cost of the surge tank, and the cost of the pressure-reducing valve.

The objective function is defined as follows:

$$\min F={\displaystyle \sum _{i=1}^{N_s}{L}_{si}}{C}_{si}+{\displaystyle \sum _{j=1}^{N_d}{L}_{dj}}{C}_{dj}+{\displaystyle \sum _{\mathrm{k}=1}^{N_p}{C}_{pk}}+{\displaystyle \sum _{t=1}^{N_v}{C}_{\mathrm{vt}}}$$

(1)

$$\min F_2=H_{sub\_\max }-H_{sub\_req}$$

(2)

In the formula, F is the total cost of the main pipe system (yuan); N_s and N_d are the numbers of water supply main pipe sections and the number of water supply main pipe sections, respectively. L_si and C_si are the length (m) and unit price (yuan/m) of the first section of the water supply main pipe, respectively; L_dj and C_di are the length (m) and unit price (yuan/m) of the j-section of the main pipe, respectively. N_p and N_v are the numbers of pressure-regulating pools and pressure-reducing valves, respectively. C_pk is the total cost of the kth surge tank (yuan); C_ot is the unit price (yuan) of the tth pressure-reducing valve. The total cost (C_p) of the surge tank involves many factors. To facilitate the calculation of C_p, it is assumed that C_p is a function of the effective water storage volume (V_p) of the surge tank, C_p = f (V_p). By fitting the relationship between the total cost of the existing surge tank project and its effective water storage volume, the corresponding cost calculation formula can be obtained. F₂ is the value of the actual maximum pressure head at the entrance of all branch pipes higher than the required pressure head at the entrance of branch pipes, and the unit is m. H_{sub_max} is the actual maximum pressure head at the inlet of all branch pipes, in m; H_{sub_req} is the required pressure head at the inlet of the branch pipe, in m.

2.2. Restrictive Condition

In mathematical models, constraints on the decision scheme typically manifest as either inequalities or equations. When it comes to the optimal design of a large-scale micro-irrigation pipe network, the objective function often aims to maximize or minimize a specific value while adhering to certain constraints. The formula incorporates variables that represent the decision-making scheme, thus imposing a set of limitations on it.

The maximum pressure of each pipe section should not exceed the pressure-bearing capacity of the adopted pipeline.

$$P_{i,\max }<P_i{}_{,c}\dots \dots i=1,2,\dots ,\left(N_s+N_d\right)$$

(3)

In the formula, P_i,max is the maximum pressure (MPa) of the i-section pipeline and P_i,c is the pressure bearing capacity (MPa) of the i-section pipeline.

The pressure head at the inlet of the branch pipe must meet the following requirements:

$$H_i>H_{req}\dots \dots i=1,2,\dots ,\left(N_{sub}\right)$$

(4)

In the formula, H_i is the pressure head (m) at the entrance of the I branch pipe; H_req is the pressure head (m) required for the inlet of the branch pipe (i); N_sub is the number of branch pipes. H_i is equal to the height difference between the water demand node i (the inlet of the branch pipe i) and the surge tank supplying water to node i minus the total head loss (H_loss) in this section. The pressure at the inlet of the branch pipe is calculated step by step from the bottom to the top of the outlet at the end of the pipe (dripper, nozzle, etc.), that is, from the downstream pipe to the upstream pipe (according to the order of capillary pipe, branch pipe, and branch pipe). When the inlet pressure requirements of the branch pipe are known, solving the corresponding head pressure constraints ensures that the actual pressure head at the inlet of the branch pipe meets the required pressure head during the optimization process. Moreover, if the actual pressure head at the inlet of the branch pipe exceeds the safe allowable value, it is recommended to install a pressure-reducing valve.

Formula (5) is used to calculate the total head loss of the pipeline section.

$$H_{loss}=\lambda \times fl\frac{Q^m}{D^b}$$

(5)

where H_loss is the total head loss (m); λ is the expansion coefficient considering the local head loss and the general value is 1.1, that is, the local head loss is considered to be 10% of the head loss along the way; f is the head loss coefficient along the way; l, Q, D is the pipe length (m), flow rate (m³/h), pipe diameter (mm); m and b are coefficients related to the type of pipeline.

To prevent the deposition of impurities in the pipeline water and improve the utilization efficiency of the pipeline, the flow velocity in the pipeline must not be lower than the minimum value. In addition, to prevent excessive erosion and wear of the pipeline caused by high-speed water flow, the actual flow velocity of each pipe section should not exceed the maximum allowable value. The minimum and maximum allowable flow velocities are determined with engineering experience and hydraulic tests. The flow velocity limits of different water quality and pipeline types are also different. The velocity constraint is expressed as:

$$V_{\min }\le V_i\le V_{\max } i=1,2,\dots ,\left(N_s+N_d\right)$$

(6)

In the formula, V_i is the actual flow velocity (m/s) in the i section of the pipeline; V_min and V_max are the minimum required flow rate (m/s) and the maximum allowable flow rate (m/s), respectively.

To ensure good hydraulic performance and safe operation of the irrigation pipeline, the diameter of the current section of the water supply pipe or the water transmission pipe must not be less than the diameter of the next section along the flow direction. The pipe diameter constraint is expressed as:

$$D_i\ge D_{i+1} i=1,2,\dots ,N$$

(7)

In the formula, D_i is the inner diameter (mm) of the i section of the main pipe, D_i+1 is the inner diameter (mm) of the i + 1 section, and N is the number of pipe sections.

The optimization of micro-irrigation network design involves considering various conditions and constraints. These include factors such as pipeline pressure capacity, the maximum allowable flow rate, and pipe diameter limitations, among others. Additionally, specific constraints relevant to the project’s unique circumstances should be incorporated into the design optimization system.

2.3. Decision Variables

Decision variables are variables used to represent different choices or decisions in decision-making problems. These variables have a direct impact on the final decision. The selection of decision variables is typically based on the characteristics and objectives of the decision problem, for example, pipe diameter [22], pipe length [23,42], pipe roughness [21], node pressure [20], future node demand [43], valve setting [44], etc. The decision variables are categorized according to the different elements of the pipe network from six aspects: water quality, nodes, pumps, tanks, valves, and pipelines, which are shown in Figure 3.

In mathematical modeling, symbols can be employed to denote these decision variables. By defining the constraint conditions and objective function based on the specific problem, the optimal value of the decision variables is determined through an optimization algorithm, leading to the identification of the optimal decision scheme. The selection and value of decision variables play a crucial role in problem solving, as they have a direct impact on the final decision. In practical applications, it is necessary to determine the appropriate decision variables based on the specific situation and problem objectives. By combining mathematical modeling and algorithms, we can effectively address the problem and find a solution.

The optimization of a micro-irrigation pipe network at scale is contingent on the mathematical model selected based on the specific problem at hand. Furthermore, the choice of optimization algorithms can vary depending on the expertise of the professionals involved and their familiarity with the relevant software and tools [45]. The next section of this paper delves into these optimization algorithms in detail.

3. Solution Techniques and Examples for WDS Optimization

The optimization of micro-irrigation pipe networks is extensively applied in the field of water distribution and irrigation. Researchers have employed various analysis methods to design and arrange water distribution systems (WDSs). Figure 4 illustrates the arrangement of a basic self-pressurized drip irrigation network system. The optimization design of water distribution systems (WDSs) now incorporates the concept of traditional economic flow rate for the first time [46,47]. Due to the intricate calculation process and high complexity, this technology is gradually being replaced by alternative optimization techniques. In recent years, deterministic and heuristic methods have increasingly been employed in the optimization of micro-irrigation pipe networks. Researchers have studied the evolution of various algorithms based on new intelligent algorithms, aiming to approach the optimal solution of the scheme. Successful cases of micro-irrigation pipe network optimization are summarized in the following section, considering both deterministic techniques and heuristic methods.

3.1. Deterministic Approach

The deterministic method refers to an approach that solves the objective function by following specific steps and accurately processing the data, thus obtaining the required solution for the problem. The optimization problem of a pipe network is typically unconstrained. Researchers often aim to obtain the optimal solution for the optimization by using the maximum or minimum objective function under unconstrained conditions. When dealing with constraint problems related to decision variables, mathematical programming methods are commonly employed to achieve local or global optimization. In the optimization of micro-irrigation pipe networks, deterministic algorithms such as linear programming (LP) and nonlinear programming (NLP) are commonly employed.

3.1.1. Linear Programming

The linear programming method is characterized by an objective function that is linear and constraint conditions that can be either equal or inequal. The objective function is optimized to maximize or minimize its value. Linear programming models frequently involve multiple decision variables, such as pipeline diameter and pump capacity [49], reservoir elevation, and operating parameters [50]. The objective function is formulated based on the functional relationship and the decision variables. Additionally, the decision variables need to satisfy various constraints. Thus, it is crucial to determine the constraints that the decision variables must adhere to.

The use of linear programming became widespread in various fields such as economics, management, engineering, and logistics at the end of the 20th century, especially in water distribution systems [51,52,53] and distribution network systems [54]. It is used to solve the optimization of resource allocation, production planning, supply chain management, and other issues. The methods used to solve linear programming problems include the simplex method [55], interior point method [56], support vector machine [57], and so on. These methods allow for finding the optimal solution or a set of optimal solutions, thereby obtaining the best solution under the given constraints.

The linear programming gradient method as a type of linear programming is often used for WDS optimization design. The early linear programming gradient method opened the door to the optimal design of pipe networks. This method was first proposed by Alperovits and Shamir [50] and has been promoted and developed in the past decade. During that period, the primary challenges in pipe network optimization encompassed the following aspects: (a) given multiple operational modes and a fixed pipe network layout, the objective was to determine the pipe network diameter that minimized costs; (b) based on the foundation of network component design optimization, it was assumed that branch network processing would handle the network layout.

However, when dealing with multivariable decision-making involving mutual relations, the linear programming method requires a significant amount of computational resources. Additionally, linear programming must satisfy multiple constraints. Sometimes, reducing the size of the constraint set can be effective, but it does not guarantee global optimality of the pipe network. It can only achieve local optimality [58]. The pipe network problem in actual projects is typically more complex and often cannot be represented by simple linear relationships. The linear programming method can only provide approximate solutions to the problem. In this process, certain errors may occur, and the accuracy of the data cannot be guaranteed.

3.1.2. Nonlinear Programming Method

Nonlinear programming (NLP) refers to a mathematical programming problem where the objective function or constraint includes at least one nonlinear function. In contrast to linear programming, which involves linear models, nonlinear programming deals with curve models, making the problem’s solution more complex and challenging. Nonlinear programming plays a crucial role in addressing a wide range of practical engineering problems. It is mainly used in energy supply systems [59,60] (such as natural gas [61,62,63], petroleum, electricity, etc.) and pipeline water distribution systems [64,65,66] (such as urban and rural water distribution networks and water supply networks). The computational complexity of solving the nonlinear programming problem is increased due to the necessity of employing complex mathematical methods and algorithms [66]; the derivatives of objective functions and constraint functions need to be calculated in each iteration [67] and solved by constructing nonlinear models. In addition, nonlinear programming problems often exhibit multiple locally optimal solutions, in contrast to linear programming problems where a globally optimal solution is typically present. The presence of multiple local extrema in the search space can be attributed to the nonconvex nature of the nonlinear function [20]. Finally, the solution to a nonlinear programming problem can be either continuous [22] or discrete [68,69]. This implies that the values assigned to decision variables are not restricted to integers or sets of real numbers, as they can be arbitrary.

Theocharis et al. [70] proposed a new simplified nonlinear programming formulation. The new simplified nonlinear programming method is employed as a substitute for the conventional pipe network optimization method, considering the intricate network branches and substantial computational load. This alternative method is evaluated in conjunction with a general nonlinear planning approach applied to a specific irrigation network. The comparison demonstrates that both methods yield identical outcomes, leading to the conclusion that the proposed method can be equally effective when employed in the classical approach.

The discrete nonlinear model proposed by Cheng et al. [69] focused on a pressurized tree distribution network utilized in rural water supply projects. This optimization model incorporates head loss and flow rate constraints, utilizing pump head and pipe diameter as dual decision variables. To solve the model, the authors integrated an enhanced decomposition–dynamic programming aggregation (DDPA) algorithm. Practical applications of the model demonstrate the relationship between operating costs and engineering investments, while also offering an effective approach to obtaining optimal solutions for the proposed scheme.

Cassiolato et al. [68] focused on optimizing circular distribution networks in industrial and urban water distribution systems to minimize costs. To achieve this, they imposed constraints on mass balance in the nodes, energy balance in the circulation, and hydraulic equations, considering known pipe lengths and a discrete set of commercially available diameters. By incorporating these constraints, the researchers aimed to eliminate the requirement for additional software to determine suitable pressure drops and water velocities.

In general, the advantages of using deterministic methods are mainly reflected in the following aspects:

(1)

Deterministic methods provide a viable solution for optimizing complex pipeline networks. By employing these methods, it becomes possible to identify the optimal decision-making scheme within the given constraints. This facilitates the optimization of resource utilization, enhances efficiency, and reduces costs.

(2)

Deterministic methods offer the capability to provide both global and local optimal solutions to problems, even in complex multidimensional spaces. These methods utilize mathematical models and algorithms to identify the best decision-making solutions. Consequently, decisionmakers are empowered to select the optimal solution for pipe network optimization from a range of decision variable values.

(3)

The deterministic method possesses a well-defined mathematical structure, characterized by its simplicity and intuitive nature. Its straightforward form and utilization of concepts and techniques from linear algebra and geometry contribute to a more accessible and comprehensible problem-solving process. This approach enhances the accuracy and clarity of the solutions obtained.

(4)

The deterministic method offers significant economic benefits by optimizing the utilization and allocation of resources. Through the reduction in costs and improvement in efficiency, this method can substantially enhance the profitability and competitiveness of enterprises. In practical applications, the deterministic method has demonstrated notable economic benefits.

The deterministic method holds a crucial position in the domain of operational research and pipe network optimization. It exerts significant and widespread influence in addressing optimization problems by offering global or local optimal solutions. Moreover, its simplicity and intuitive nature contribute to its effectiveness, while also yielding economic benefits.

3.2. Metaheuristic Algorithm

To address the limitations associated with deterministic methods in practical problem solving, the late 20th century witnessed the rise in the popularity of metaheuristic algorithms [71]. These algorithms, derived from optimal algorithms, offer a mathematical approach to obtaining optimal solutions to mathematical problems. As an artificial intelligence algorithm, it is capable of generating a feasible solution for single-objective or multi-objective combinatorial optimization problems within a given time and space constraint. However, it is important to note that this feasible solution may not be comparable to the optimal solution. Metaheuristic algorithms, such as the genetic algorithm, particle swarm optimization algorithm, and whale optimization algorithm, have emerged as prominent research approaches to solving complex problems. These algorithms mimic natural processes and can accurately identify the global optimal solution based on algorithmic and heuristic evaluation criteria, eliminating the need for extensive training computations. Furthermore, this algorithm possesses the capability to dynamically update parameters in real time to adapt to changes in instances. It proves particularly valuable in solving the problem of scaled pipe networks, where it may be the only viable solution available, despite the considerable amount of computational effort required.

3.2.1. Genetic Algorithm

The genetic algorithm is widely recognized as a metaheuristic algorithm commonly employed in water resources planning and management [72]. It serves as an optimization algorithm that emulates the principles of natural selection [73] and genetic mechanisms observed in biological evolution. By simulating operations such as crossover, mutation, and gene selection, this algorithm iteratively searches for the optimal solution to a given problem. The iterative process occurs generation by generation, using a fitness function to evaluate the fitness of each individual and guide the search for the optimal solution. At the core of genetic algorithms are the concepts of individuals and populations. An individual represents a candidate solution within the search space, and a population refers to a collection of multiple individuals. By utilizing these core concepts, genetic algorithms effectively explore the solution space to find near-optimal or optimal solutions for complex optimization problems in water resources planning and management.

The basic steps of the genetic algorithm include:

(1)

The process of initializing the genetic algorithm involves randomly generating a group of individuals to form the initial population.

(2)

A specific number of individuals are chosen as parents through the evaluation of the fitness function.

(3)

Cross-operation is performed on the selected parent individuals to generate a new set of individuals.

(4)

New individuals undergo variation operations, which introduce a certain level of randomness (Figure 5).

(5)

New individuals are evaluated using a fitness function that assesses the fitness of each individual.

(6)

Based on the fitness scores, a specific number of individuals are chosen to form the subsequent generation of the population.

(7)

The termination condition is considered satisfied when specific criteria are met, such as reaching the maximum number of iterations or achieving a fitness threshold.

The origin of genetic algorithms can be traced back to the early 1960s. However, it was not until the 1980s that the limitations of deterministic methods, characterized by high computational requirements and numerous constraints, prompted the need for improvement in optimization problems or obtaining optimal solutions. As a result, optimization techniques for pipe network design gradually transitioned towards metaheuristic algorithms. Professor Holland of the University of Michigan in the USA has made significant contributions to the research of genetic algorithms. He and his students were the first to propose the concept of genetic algorithms, and Professor Holland further introduced the pattern theory, which holds immense importance in the theoretical study of genetic algorithms. Additionally, he explored the application of genetic algorithms in complex adaptive systems [73].

The genetic algorithm exhibits inherent parallelism and adaptability. It has demonstrated superior optimization capabilities in various problems, including the traveling salesman problem, vehicle path planning, and machine learning. Moreover, it excels in scenarios characterized by a large search space and high complexity [75]. Unlike deterministic methods, genetic algorithms do not rely on linear or nonlinearization assumptions [76]. They also eliminate the need for computing partial derivatives. By simulating the evolutionary mechanism of organisms and iteratively converging towards the globally optimal solution, genetic algorithms efficiently identify improved solutions within the search space [77].

However, genetic algorithms have certain limitations. They are primarily designed for single-objective optimization problems and may struggle when faced with multi-constraint problems [78]. The convergence of the algorithm can be slow, resulting in relatively low computational efficiency [79]. Additionally, genetic algorithms are susceptible to becoming trapped in local optima, which can hinder their ability to find the globally optimal solution [77]. In recent years, researchers and scholars have integrated genetic algorithms with various optimization approaches and introduced enhancements to address the challenges posed by complex scaled pipeline networks. This has resulted in the development of new genetic algorithm-based variants that aim to overcome the limitations of traditional genetic algorithms. To provide a comprehensive overview of these advancements,

Table 1. Summary of improvements and variants of genetic algorithms.

Author	Year	Improvements	Results
Murphy et al. [80]	1994	Genetic algorithm optimization model combined with steady-state hydraulic simulation model.	Demonstrate the flexibility of the new methodology with significant design cost savings.
Savic and Walters [77]	1997	GA added to the development of the computer model GANET.	Optimization of the distribution network was addressed.
Montesinos et al. [81]	1999	Genetic algorithms for improvement in selection and mutation processes.	Significantly increases the convergence of the algorithm.
Vairavamoorthy and Ali [79]	2000	Genetic algorithm modeling of strings encoded with real variables.	Avoid redundant state problems often found when using binary (and Gray) encoding schemes.
Wu and Simpson [82]	2002	An adaptive boundary search strategy applied to genetic algorithms is proposed.	Increased efficiency in reaching or approaching optimal solutions.
Prasad and Park [78]	2004	A constrained treatment technique is proposed that does not require a penalty factor and applies to water distribution systems.	The proposed method generates a set of Pareto-optimal solutions in the search space of cost and network elasticity.
Van Zyl et al. [75]	2004	GA methods combined with climber search strategies.	Whether applied to a test problem or a large existing water distribution system, hybrid methods can outperform pure GAs in finding good solutions quickly.
Keedwell and Khu et al. [83]	2005	A new approach called CANDA-GA was proposed in this paper.	Consistently outperforms traditional non-heuristic GA methods in producing more economically designed distribution networks.
Atiquzzaman et al. [84]	2006	Multi-objective optimization algorithm (NSGA-II) combined with water distribution network simulation software (EPANET).	Provides much-needed Pareto front end for cost and node pressure deficits.
Jin et al. [85]	2008	Solving altered multi-objective optimization models using nondominated sorting genetic algorithm-II (NSGA-II)	The contradiction between the single fitness value of the standard genetic algorithm (SGA) and the multi-objective restoration problem is resolved by controlling the uncertainty associated with the use of weighting coefficients or penalty functions in the traditional approach.

presents a compilation of examples highlighting the improvements and variants of genetic algorithms.

Genetic algorithms have made significant contributions to solving the optimization problems associated with water transmission and distribution networks. As one of the most widely utilized metaheuristic algorithms, genetic algorithms have proven to be effective in addressing these challenges. Genetic algorithms have made significant contributions to solving the optimization problems associated with water transmission and distribution networks. Particularly for large-scale and complex pipeline network systems, genetic algorithms offer several advantages. Firstly, their ability to explore a large search space ensures the accuracy of the optimization results. Additionally, by maintaining a balance between genetic operators, genetic algorithms are capable of obtaining the global optimal solution. Over the years, extensive research has led to numerous advancements and variations in genetic algorithms. Among these, the NASA-II nondominated sorting genetic algorithm and the SMGA structured messy genetic algorithm have emerged as prominent examples. Additionally, genetic algorithms have been integrated with a range of optimization techniques to overcome the limitations associated with traditional genetic algorithms when applied in practical engineering scenarios. Through the utilization of information technology, the genetic algorithm has been successfully integrated with EPANET water distribution simulation software and computer models. This integration has enabled the simulation of complex pipe networks, resulting in a substantial improvement in optimization efficiency. By combining the power of genetic algorithms with advanced software tools, it becomes possible to analyze and optimize intricate water transmission and distribution systems more effectively.

3.2.2. Particle Swarm Optimization

Particle swarm optimization (PSO) represents a bionic optimization algorithm that takes inspiration from the collective behavior observed in various natural systems, such as flocks of birds or schools of fish [86]. This algorithm emulates the movement and information exchange among particles within the solution space, constantly updating their positions to approach both the current optimal position and the global optimum. By iteratively refining particle positions, PSO effectively explores the solution space to identify the global optimal solution. Figure 6 presents a visual representation of the PSO process.

The basic principle of the PSO algorithm is as follows:

(1)

Randomly generate a set of particles, each with a position and a velocity.

(2)

For each particle, a fitness value is calculated according to a problem-specific evaluation function, which is used to measure the quality of its solution.

(3)

For each particle, the individual optimal solution is updated based on the current position and the historical optimal position (i.e., the individual optimal solution).

(4)

Select the global optimal solution based on the individual optimal solutions of all particles.

(5)

Update the velocity and position of the particles based on the current velocity and position, as well as the individual and population optimal solutions.

(6)

According to the preset termination conditions (such as reaching a certain number of iterations, finding a satisfactory solution, etc.), determine whether to end the algorithm or not.

(7)

Return the optimal solution obtained during the iteration process as the output of the algorithm.

The original particle swarm optimization (PSO) algorithm was proposed by Kennedy, a social psychologist, and Eberhart, an electrical engineer, in 1995 [88,89]. This algorithm facilitates global optimization by promoting mutual coordination and balance among particles through their interactions in an initially disordered state. In recent decades, the algorithm has demonstrated its effectiveness in solving various optimization tasks in real-world problems. For example, it has been successfully applied to job shop scheduling problems [90,91], power system problems [92], and wireless network problems [93]. It has wide adaptability in evaluating the scheme of global optimal solutions.

The algorithm is primarily designed for steady-state problems, offering several advantages in its application. These advantages include a robust self-organizing function, a minimal number of adjustable parameters, efficient parallel computation, reduced computational requirements, and the ability to perform a global search effectively. In recent years, researchers have focused on addressing the limitations of the algorithm, particularly when applied to high-dimensional search spaces and various large-scale complex datasets. These limitations include sensitivity to parameters, slow convergence, premature convergence, poor computing quality, and a tendency to become trapped in local optimization. To overcome these shortcomings, efforts have been made to hybridize, improve, and introduce variations to traditional PSO algorithms. An analysis of the literature review on this algorithm in the past decade is presented in Table 2. This table provides valuable insights into the research advancements and application areas of PSO during this period, assisting researchers in gaining a comprehensive understanding of the progress made in the field (Figure 7).

In the field of large-scale micro-irrigation pipe networks, particle swarm algorithms have gained significant popularity for their efficient and effective optimization capabilities. The utilization of these algorithms is mainly observed in three key areas: Environmentally, Aghel et al. [94] successfully predicted and modeled water quality parameters by combining two data-driven models with the algorithm, providing new ideas for water treatment and pollution management; thus, irrigation water can be used to detect desired design metrics (e.g., water quality, pollution indices, etc.) through the algorithm. Furthermore, Güvengir et al. [95] applied PSO for the first time to a large-scale problem in the context of flood prevention, demonstrating that the algorithm is capable of early warning and control of floods. To ensure the cost-effective operation of industrial production practices, a new deep learning-driven particle swarm optimization approach to optimize energy utility was proposed by Qin et al. [96]. In addition, the algorithm also plays an equally important role in signal processing, economic dispatch, machine control, and wireless sensor networks. In data processing, a more efficient and feasible implementation of the particle swarm algorithm is obtained by applying it to function optimization, neural network training, model classification, and fuzzy systems; e.g., Song et al. [97] applied the proposed algorithm to a typical dataset based on the K-nearest neighbor classifier (K-NN). Combined with the researcher’s examples, it can be seen that based on retaining the core idea of the algorithm—randomly selecting initial nodes in the optimization space to iteratively converge to a more optimal solution—the algorithm can be widely used in the optimal design of scaled micro-irrigation pipeline network systems by improving and developing the algorithm.

Table 2. Literature review on particle swarm optimization algorithms in the last decade.

Author	Year	Focused Area
Gad [87]	2022	PSO hybrids, improvements, and variants; PSO applications in healthcare, environment, industry, commerce, smart cities, and, in general, the accuracy of PSO; assessing environmental and proposed case studies; the effectiveness of different PSO methods and applications.
Houssein et al. [98]	2021	PSO issues related to convergence, diversity, and stability; organizing and summarizing information on basic PSO algorithms as well as the recent introduction of some noteworthy developments and trends; paradigms, theories, hybrids, coverage of parallelization, diverse applications of the algorithms, current pressing issues, and open challenges that plague PSO.
Jahandideh-Tehrani et al. [99]	2020	Basic search strategies of PSO algorithms; application and performance analysis of PSO in water resources engineering optimization problems; revealing 22 different PSO algorithms (characteristics of each PSO variety and their use in different areas of water resources engineering); comparing the performance of different PSO variants with other evolutionary algorithms (EA) and mathematical optimization methods in terms of rate of convergence, diversity of computational solutions, and other properties.
Tian and Shi [100]	2018	Solve the problem of PSO local optimization and premature convergence and propose improved particle swarm optimization with chaos-based initialization and robust update mechanism.
Sengupta et al. [86]	2018	Basic development, deployment, and improvement and some recent applications; overview of concepts and directions for selection of inertia weights, constraint factors, cognitive and social weights, as well as perspectives on convergence, parallelization, elitism, coupled and discrete optimization, and neighborhood topology; attempts to hybridize PSO with other evolutionary and population paradigms.
El-Shorbagy and Hassanien [101]	2018	PSO solving various optimization problems; PSO behavior; basic concepts and developments; PSO inertia weights; contraction factors and parameter settings; selection and tuning; dynamic environments and hybridization; limitations of solutions; and future research directions
Hajihassani et al. [102]	2018	Application of PSO to slope stability analysis; piling and foundation engineering; geotechnical mechanics and design of tunnels and underground spaces; application of PSO to solve complex multidimensional problems; applicability, advantages, and limitations of PSO in different geotechnical engineering disciplines.
Jain et al. [103]	2018	PSO research progress; basic PSO progress; improvements; modifications and applications 1995-2017.
Harrison et al. [104]	2018	An analytical and empirical study of the convergence behavior of 18 adaptive particle swarm optimization (SAPSO) algorithms; empirical examination of whether the adaptive parameters reach a stable point and whether the final parameter values comply with well-known convergence criteria.
Wang et al. [105]	2017	The origin and background of PSO; theoretical analysis of PSO; analysis of its current research and application status and problems in algorithm structure, parameter selection, topology, discrete PSO algorithm, and parallel PSO algorithm; multi-objective optimization of PSO and its engineering applications; and proposed future research directions.
Benuwa et al. [106]	2016	Conducted a comprehensive study of PSO, proposed a theoretical framework for improved implementation, and suggested important conclusions and possible directions for future PSO research.
Kulkarni et al. [107]	2015	Applications of the PSO algorithm to mechanics, including optimal weight design of gear trains, simultaneous optimization of design and machining tolerances, optimization of process parameters in casting, and machine scheduling problems; describing improved versions of the PSO algorithm, i.e., hybrid PSO, multi-objective PSO, adaptive PSO, and discrete PSO.
Alam et al. [108]	2014	Provides an overview of the most commonly cited techniques for PSO-based data clustering, emphasizes the performance of the different techniques about contemporary clustering techniques, outlines a PSO-based hierarchical clustering approach (HPSO clustering), and compares it to traditional hierarchical agglomerative clustering (HAC), K-means, and PSO clustering.
Khare and Rangnekar [109]	2013	An extensive review of the literature on the concept, development, and modification of particle swarm optimization; discusses the concept and development of PSO; discusses the modification of inertia weights and contraction factors; discusses issues related to parameter tuning, dynamic environments, stagnation, and hybridization; and a brief review of selected work on particle swarm optimization and the application of PSO to solar photovoltaics.
Aote et al. [110]	2013	Solving single-peak and multimodal problems as well as two-dimensional to multidimensional problems; working on communication topology, parameter tuning, initial distribution of particles, and efficient problem-solving capabilities; elaborating on the limitations of PSO and current work.

Table 2. Literature review on particle swarm optimization algorithms in the last decade.

Author	Year	Focused Area
Gad [87]	2022	PSO hybrids, improvements, and variants; PSO applications in healthcare, environment, industry, commerce, smart cities, and, in general, the accuracy of PSO; assessing environmental and proposed case studies; the effectiveness of different PSO methods and applications.
Houssein et al. [98]	2021	PSO issues related to convergence, diversity, and stability; organizing and summarizing information on basic PSO algorithms as well as the recent introduction of some noteworthy developments and trends; paradigms, theories, hybrids, coverage of parallelization, diverse applications of the algorithms, current pressing issues, and open challenges that plague PSO.
Jahandideh-Tehrani et al. [99]	2020	Basic search strategies of PSO algorithms; application and performance analysis of PSO in water resources engineering optimization problems; revealing 22 different PSO algorithms (characteristics of each PSO variety and their use in different areas of water resources engineering); comparing the performance of different PSO variants with other evolutionary algorithms (EA) and mathematical optimization methods in terms of rate of convergence, diversity of computational solutions, and other properties.
Tian and Shi [100]	2018	Solve the problem of PSO local optimization and premature convergence and propose improved particle swarm optimization with chaos-based initialization and robust update mechanism.
Sengupta et al. [86]	2018	Basic development, deployment, and improvement and some recent applications; overview of concepts and directions for selection of inertia weights, constraint factors, cognitive and social weights, as well as perspectives on convergence, parallelization, elitism, coupled and discrete optimization, and neighborhood topology; attempts to hybridize PSO with other evolutionary and population paradigms.
El-Shorbagy and Hassanien [101]	2018	PSO solving various optimization problems; PSO behavior; basic concepts and developments; PSO inertia weights; contraction factors and parameter settings; selection and tuning; dynamic environments and hybridization; limitations of solutions; and future research directions
Hajihassani et al. [102]	2018	Application of PSO to slope stability analysis; piling and foundation engineering; geotechnical mechanics and design of tunnels and underground spaces; application of PSO to solve complex multidimensional problems; applicability, advantages, and limitations of PSO in different geotechnical engineering disciplines.
Jain et al. [103]	2018	PSO research progress; basic PSO progress; improvements; modifications and applications 1995-2017.
Harrison et al. [104]	2018	An analytical and empirical study of the convergence behavior of 18 adaptive particle swarm optimization (SAPSO) algorithms; empirical examination of whether the adaptive parameters reach a stable point and whether the final parameter values comply with well-known convergence criteria.
Wang et al. [105]	2017	The origin and background of PSO; theoretical analysis of PSO; analysis of its current research and application status and problems in algorithm structure, parameter selection, topology, discrete PSO algorithm, and parallel PSO algorithm; multi-objective optimization of PSO and its engineering applications; and proposed future research directions.
Benuwa et al. [106]	2016	Conducted a comprehensive study of PSO, proposed a theoretical framework for improved implementation, and suggested important conclusions and possible directions for future PSO research.
Kulkarni et al. [107]	2015	Applications of the PSO algorithm to mechanics, including optimal weight design of gear trains, simultaneous optimization of design and machining tolerances, optimization of process parameters in casting, and machine scheduling problems; describing improved versions of the PSO algorithm, i.e., hybrid PSO, multi-objective PSO, adaptive PSO, and discrete PSO.
Alam et al. [108]	2014	Provides an overview of the most commonly cited techniques for PSO-based data clustering, emphasizes the performance of the different techniques about contemporary clustering techniques, outlines a PSO-based hierarchical clustering approach (HPSO clustering), and compares it to traditional hierarchical agglomerative clustering (HAC), K-means, and PSO clustering.
Khare and Rangnekar [109]	2013	An extensive review of the literature on the concept, development, and modification of particle swarm optimization; discusses the concept and development of PSO; discusses the modification of inertia weights and contraction factors; discusses issues related to parameter tuning, dynamic environments, stagnation, and hybridization; and a brief review of selected work on particle swarm optimization and the application of PSO to solar photovoltaics.
Aote et al. [110]	2013	Solving single-peak and multimodal problems as well as two-dimensional to multidimensional problems; working on communication topology, parameter tuning, initial distribution of particles, and efficient problem-solving capabilities; elaborating on the limitations of PSO and current work.

In conclusion, the particle swarm optimization (PSO) algorithm has been extensively improved and applied in various applications. By making appropriate enhancements and adjustments, the algorithm can be effectively utilized to address complex optimization problems in large-scale micro-irrigation pipe network design. This approach aims to achieve higher-quality solutions and optimize the overall performance of the system.

3.2.3. Whale Optimization Algorithm

The whale optimization algorithm (WOA) is a novel metaheuristic optimization algorithm that mimics the foraging behavior of humpback whales [111]. It treats each candidate solution as an individual humpback whale and optimizes it by simulating the search and tracking behavior of these whales. In this algorithm, each whale represents a candidate solution, and its position and fitness values indicate the solution’s location and quality. The algorithm incorporates three strategies to simulate humpback whale behavior: a roundup strategy, a spiral bubble net attack, and a search for prey strategy. For a visual representation of the algorithm’s flow, refer to Figure 8.

The whale optimization algorithm (WOA) was initially proposed by Mirjalili and Lewis in 2016 [113]. In addition to the theoretical investigation of this algorithm, researchers have conducted analyses on its properties, including convergence and search capability. The whale optimization algorithm (WOA) is inspired by the behavior of whale populations and offers several advantages, including a similar structure, robustness, fast convergence, a reduced number of parameters, and low population diversity [114]. These characteristics enable the algorithm to efficiently and effectively solve optimization problems by employing simple steps and powerful functions. Several subsequent studies have been conducted on the whale optimization algorithm (WOA). Like other metaheuristic algorithms, the WOA faces challenges, including premature convergence [37], an imbalance problem [115], a poor search strategy [116], the issue of local optimality, and low population diversity [117,118]. Therefore, researchers have optimized whale optimization algorithms [119], starting from improved, hybridized, binary optimization, and single/multi-objective optimization methods, and proposed Lévy flight techniques, chaotic mapping techniques, mutation strategies and crossover operators, and hybridization with evolutionary algorithms, physics-based algorithms, and population intelligence algorithms. Consequently, the algorithm has witnessed gradual advancements in both its application and development [112,120,121,122].

In addition to this, the whale optimization algorithm has been applied to optimization problems in several domains, as shown in Figure 9, including:

• Single/multi-objective optimization [38,118,123,124], to solve single and multi-objective function optimization problems, parameter optimization, and optimization problems searching for multiple nondominated solutions.
• Machine learning [125,126,127], for tuning the parameters of machine learning models such as neural networks, support vector machines, etc.
• Image processing [128,129,130], e.g., image segmentation, image recognition, and image enhancement problems.
• Power system optimization [131], used for optimization problems in power systems, such as power load scheduling, optimal allocation of power equipment, etc.
• Intelligent control [132], for optimization of intelligent control systems, such as intelligent robot path planning, automated system control, etc.
• Data mining [133], for tasks such as cluster analysis, association rule mining, anomaly detection, etc.

In just a few years, the combination of the whale optimization algorithm (WOA) with other techniques has led researchers to achieve significant advancements in optimization results. Consequently, the application and development of the WOA have experienced substantial growth. The whale optimization algorithm has demonstrated its effectiveness in various problem domains. However, it is important to note that the suitability and performance of this algorithm can vary depending on the specific characteristics and constraints of the problem at hand. Therefore, it is crucial to carefully consider both the nature of the problem and the inherent characteristics of the algorithm when selecting an appropriate optimization approach. A comprehensive evaluation of these factors is necessary to ensure the most suitable algorithm is chosen for a given problem.

To further enhance and promote the application of the whale optimization algorithm across diverse fields, it is essential to continue refining and advancing its implementation. This can be achieved through ongoing research and development efforts. By conducting thorough investigations, we can unlock the algorithm’s full potential and enable its effective utilization in various domains. This iterative process of improvement and promotion contributes to the algorithm’s wider adoption and success in tackling complex optimization problems.

4. Future Directions

The optimization of water distribution systems (WDSs) primarily revolves around developing mathematical models, mathematical planning, and optimization algorithms. However, considering the intricacy of real-world problems and the presence of numerous uncertain factors, it becomes necessary to adapt the design continually for actual pipeline network engineering challenges. This involves incorporating mathematical models into the optimization problem and employing flexible design approaches that can effectively address the complexities of WDSs. Therefore, through an exploration of the future development direction of WDS optimization, this study aims to offer insights and recommendations for future research in this field, as listed below:

(1)

Improvement and development of algorithms play a crucial role in solving the optimization problem of scaled micro-irrigation pipe networks. Researchers have the opportunity to enhance existing optimization algorithms or create new ones to achieve better results. For instance, many previous studies have presented modified and evolved versions of genetic algorithms, and simulated annealing algorithms, differential evolution algorithms, and particle swarm algorithms. By combining metaheuristic algorithms with other optimization techniques, it is possible to design novel variants of optimization algorithms or enhance existing ones. These advancements aim to improve the optimality of solutions and effectively address the complexities involved in solving complex WDS design optimization problems.

(2)

The integration of machine learning and optimization algorithms holds significant promise in addressing complex problems. Currently, support vector machines and genetic algorithms are being combined to solve optimization problems in micro-irrigation pipe networks. Researchers can explore further by combining machine learning methods with optimization algorithms to enhance the effectiveness of WDS optimization approaches. One approach is to utilize machine learning techniques to model and predict problem features, which can guide the search direction and strategy of the optimization algorithm.

(3)

Improvement and enhancement of EPANET 2.0 Software. EPANET 2.0 is a software tool utilized for optimizing and planning pipe networks. It facilitates the determination of optimal pipe diameters, valve positions, and pump station scheduling strategies based on predefined objective functions and constraints. Additionally, it enables the simulation of pipe network operation under various scenarios. However, the current version of EPANET is only available in English and is limited to usage in a few universities in China. Therefore, it is imperative to develop a localized version of this software and promote its widespread adoption.

5. Conclusions

The optimization of micro-irrigation pipe networks has emerged as a significant aspect of agricultural irrigation for the future. Through the optimization of these systems, it becomes feasible to meet the requirements of irrigation design while minimizing investment and operational costs. This, in turn, enables the attainment of optimal performance and high efficiency in addressing the problem at hand. This paper offers a comprehensive examination of pipe network optimization, encompassing mathematical models and optimization methods. It provides a detailed overview of the subject, discusses the scope of application and limitations of micro-irrigation pipe network optimization technology, and concludes by outlining future directions for optimization. This paper presents a comprehensive description of the micro-irrigation pipe network model, focusing on its objective function, constraints, and decision variables. Furthermore, this paper categorizes optimization methods into two distinct groups: deterministic methods and metaheuristic techniques. To delve deeper into the relationship between the objective function and the optimization process, two deterministic methods were specifically chosen for analysis: the linear planning method and the nonlinear planning method. This paper also explores metaheuristic algorithms as part of the optimization methods. Specifically, it focuses on the analysis of typical genetic algorithms, particle swarm algorithms, and whale optimization algorithms. Additionally, this paper proposes future directions for the optimization design of large-scale micro-irrigation pipeline network systems. These directions include enhancing and advancing the algorithms, integrating machine learning with optimization algorithms, and upgrading EPANET 2.0 software to improve work efficiency effectively.