# An Algorithm for Optimal Allocation of Water Resources in Receiving Areas Based on Adaptive Decreasing Inertia Weights

Authors: Fei Li; Guang Zhang; Shaohua Hu
Journal: Journal of Advanced Transportation
 (2022)

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## Abstract

As the biggest rigid constraint for high-quality economic and social development, how to improve the carrying capacity of water resources, realize the stable and coordinated development of water resources-ecological environment-economic and social integrated system, and provide water resources guarantee for regional transformation and upgrading development is a major issue in the current social development. This paper firstly selects the minimum loss of water resources allocation as the objective function for mathematical modelling, chooses the particle swarm algorithm as the objective algorithm, and proposes a particle swarm algorithm based on the standard particle swarm algorithm with improved adaptive decreasing inertia weights. It is a time-varying process for the inertia weights and acceleration factors of the standard particle swarm algorithm so that they change nonlinearly with the continuous advancement of the iterative optimization seeking process, thus improving the convergence accuracy and speed of the algorithm and reducing the risk of falling into local optimum solutions at a later stage. Finally, based on the actual installation of the current water distribution reactive power compensation device, the shunt distributor set is selected as the reactive power compensation device, sensitivity analysis is applied to the load nodes for sensitivity calculation, and the nodes requiring compensation are connected to the shunt distributor set for flexible and optimal configuration. The stronger local search capability of the inertia weight adaptive decreasing algorithm is utilized in the generation process of new particles to perform a local search operation for particles, which avoids premature convergence and improves the search performance of the algorithm. To realize the rational allocation of water resources, a multiobjective receiving area water resources optimization allocation model with maximum water supply benefit, minimum regional water shortage, and minimum pollutant emission is established.


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## Body

## 1. Introduction

Water is a fundamental natural resource, often playing an important role in supporting the sustainable development of regional economies and societies, and is an important strategic resource. Carrying capacity belongs to the concept of ecology, which means the maximum number of certain species that can be accommodated in a certain area [1]. Since then, the application of the concept has been expanded. For example, some scholars have proposed the concept of water resources carrying capacity, which has a certain application value and can be used as an evaluation indicator to describe the regional water resources situation and also to provide support for regional production and domestic water use. Research into the optimal allocation of water resources has become increasingly sophisticated, and in general terms, research in this area has gradually matured from scratch. Due to the complexity of water resource systems, the overall trend in academic research has also shifted from the initial single-objective optimization problems to multiobjective optimization [2]. The models constructed in water allocation are generally diversified, becoming more and more precise and systematic, i.e., from a “strategy-oriented” individual decision-making model to a multiobjective group decision-making model, and this research trend is undoubtedly more instructive. The systematic analysis of the impact of water rights allocation on the carrying capacity of water resources, including the theory of water resources carrying capacity, water rights, and water rights allocation theory, focuses on the relationship between water rights allocation, optimal allocation of water resources and water resources carrying capacity, and analyses the impact mechanism of water rights allocation and agricultural water use structure system on the carrying capacity of water resources, showing that water rights allocation is a way to completely change the previous “demand-driven” approach to water resources. “It also discusses the relationship between water rights allocation, optimal water resources allocation, and water resources carrying capacity.”This time, for the optimal allocation model of water resources in the receiving area, it is necessary to effectively avoid the search falling into local minimal solutions and obtain a model with better global search capability. A particle swarm algorithm based on natural selection and nonlinearly decreasing inertia weights are proposed based on various improvement strategies of particle swarm algorithms. This paper more comprehensively analyzes and solves the water resources scheduling optimisation problem in the receiving area and proposes a particle swarm algorithm based on the standard particle swarm algorithm with adaptive decreasing inertia weights. The improvement method includes the introduction of the essence particle library to store the individual extreme value library of particles as well as the population extreme value library so that the individual extreme value and the population extreme value of particles in the update over can use the better particles in the essence particle library. This prevents the limitation of single-particle guidance in the previous algorithm. In the iterative process of the algorithm, the maximum velocity and inertia weights of the particles are nonlinearly decreasing to ensure that the global merit-seeking ability of the particles is balanced with the local ability, and the weight control factor can be adjusted to ensure the proportion of the maximum and minimum inertia weights in the process of population evolution. Further, the natural selection mechanism in the genetic algorithm is incorporated to effectively improve the algorithm’s merit-seeking accuracy. Based on the above research, we propose to integrate the particle swarm algorithm based on adaptive decreasing inertia weights into the resource optimization algorithm and build a simulated annealing particle swarm algorithm to establish a water resource optimization model with relatively optimal comprehensive benefits, to achieve sustainable development and utilization of water resources in the region.


## 2. Related Work

The PSO (Particle Swarm Optimization) algorithm is a population evolutionary algorithm that simulates the predatory behaviour of birds in nature and guides particles to continuously adjust their speed and position by memorizing and continuously sharing spatial information about the speed and position of individual particles and the optimal position experienced by individual microparticles and the optimal position of the particle population during iterative movement. This leads the particles to adjust their velocity and position information to topologies all possible optimal positions in the neighborhood space to achieve global search [3].A new particle swarm optimization mechanism is proposed by Al-Shaikhi [4], where particles can adjust their flight speed and direction according to their own and companion flight experience and thus keep moving toward the optimal position. Liu [5] fixed a particle swarm algorithm that is proposed to integrate with simulation analysis, and a simulation model is developed for the dynamic sequencing of products in a randomized assembly line. In the literature [6], the concentration suppression mechanism of the immune algorithm is used to prevent the overpopulation of particles with a high degree of adaptation and to compensate for the premature convergence of the particle swarm algorithm. Literature [7] added two update formulas that can accelerate and slow down the movement of particles, respectively, to the standard particle swarm algorithm and proposed a multiple swarm particle swarm algorithm based on parallel structure, which improved the defects of poor population diversity of the particle swarm algorithm and ensured the computational accuracy of the algorithm. Samanta [8] fixed the particle swarm algorithm period and subpopulated the particles, outputting high-quality subregions around each swarm of particles as the modelling region of the agent model and obtaining high-quality optimal solutions or even global optimal solutions by comparing the optimal values of each region. To improve the global search capability of the Dynamic Multi-Swarm Particle Swarm (DMS-PSO) algorithm, Wu [9] divides the swarm population into several small populations dynamically, and the optimal particles in each small population are used as particles of the higher populations for deep optimization by the cuckoo search algorithm, thus improving the local search capability of the swarm algorithm. In the literature [10], a particle swarm optimization algorithm based on Lévy flight is proposed. After the particles update their velocity and position, instead of directly calculating the value of the objective function, the Lévy flight is used to change the flight direction of the particles to find the optimal solution under that flight, avoiding the particles from falling into the local optimal solution prematurely. The literature [11] divided the particle swarm population into multiple subpopulations for simultaneous optimization search and improved the particle velocity update formula, used inverse trigonometric mapping to initialize the particle swarm to make the initial particle swarm distribution more homogeneous and used a variational method to mutate the external profile to avoid premature convergence and improve the search efficiency of the algorithm. In [12], based on the migration behavior of populations in nature, the population was randomly divided into several subpopulations and particle migration was carried out using race selection. Salgotra [13] uses an adaptive strategy to update inertia weights on the subpopulations and adaptively update the current record by sharing information. Rauf [14] combines the particle swarm algorithm and the differential evolution algorithm, periodically analyzing the performance of both algorithms and setting the probability of use based on the decision results to fully combine the two algorithms and improve the quality of understanding. Zhang [15] retains the worst particles in the initial population and records the historical worst positions of the particles, using this information to tract some particles to quickly escape from the local optimum; at the same time, the difference results in the better particles are used to guide the local learning, effectively balancing the particle convergence ability and population diversity. The literature [16] proposes a competitive-collaborative PSO based on an information-sharing mechanism. Once the population is trapped in a local optimum, a competitive mechanism is used to select the particle with the better evaluation result to guide its learning, while all particles in the population share the individual optimal positions to enhance the communication between understandings. Subsequently, a PSO based on multiscale selective learning and the detection-shrinkage mechanism was proposed in the literature [17], where particles choose different scales for learning according to their current evolutionary stage and use historical information to guide the optimal particles to adopt detection-shrinkage strategies in different situations, ensuring individual learning efficiency and algorithm mining capability. Afshar [18] proposes a strategy for adaptive control of population size based on a logistic model that can adaptively balance population effectiveness and diversity, and applying this strategy to PSO significantly improves particle convergence speed (Table 1).Table 1
Literature review.
NumberLiteratureDescription1This articleParticle swarm algorithm; inertia weights;2A hybrid particle swarm optimization technique for adaptive equalizationAdaptive equalization3Heterogeneous comprehensive learning and dynamic multiswarm particle swarm optimizer with two mutation operatorsDynamic multiswarm4Adaptive inertia weight Bat algorithm with Sugeno-function fuzzy searchSugeno-function fuzzy search


## 3. Particle Swarm Algorithm Based on Adaptive Decreasing Inertia Weights

### 3.1. Improved Particle Swarm Optimization Algorithm Flow

Particle swarm optimization algorithms were first applied to the optimization of nonlinear continuous functions and the training of neuronal networks, but later they were also used to solve constrained optimization problems, multiobjective optimization problems, dynamic optimization problems, and so on. In contrast to traditional optimization theory algorithms (e.g., Newton’s method, conjugate gradient method), particle swarm optimization algorithms do not require the computation of gradient vectors and Hessian matrices, nor do they require the feasible domain and positive definiteness of the objective function, while ensuring efficient and stable computational speed and accuracy. The encoding process of particles is a process of transferring the feasible solutions involved in a problem to the search space of a particle swarm optimization algorithm. This paper focuses on the effect of a change in transporter speed on the scheduling results, finding the optimum transporter speed at which the scheduling of water resources operations can be developed to maximize the utilization of each piece of equipment so that the load on each piece of equipment is balanced and capacity is maximized [19]. Therefore, the transporter speed is defined as a dimension of the particle and the range of speeds is used as the solution space. The fitness function of the improved particle swarm algorithm is defined as follows:(1)Gx=∫kx2+pxdx,where k is the learning factor and px is the individual extremum function. In the velocity update, the algorithm’s development capability is improved by disturbing the individual and population extremes through the perturbation factor δ, avoiding the algorithm from converging early due to excessive influence by the particle extremes (individual and population extremes), and introducing an inertia weight w that varies with the number of iterations to balance the search capability of the algorithm. w takes a value related to the number of evolutionary generations as shown in equation (2). w makes the algorithm achieves global search with larger inertia inheriting the current velocity of the particles in the early stages, and smaller inertia in the later stages ensures that the algorithm achieves local search near the optimal solution. The operational performance of water resources systems is generally evaluated using the economic benefit-cost ratio as an indicator, with the water supply subsystem as the core of the evaluation. The introduction of the concept of essence particle library makes the particles in the updating process no longer guided by a single individual extreme value and a single group extreme value but also by the influence of other better particles in the individual extreme value library as well as the group extreme value library, expanding the search range of the algorithm, making the algorithm search faster to the vicinity of the optimal solution, saving the search time of the algorithm and improving the search efficiency of the algorithm.(2)w=kx+δp2x+C.In the process of population evolution, there may be a situation where the initial motion of a particle is in the direction of the optimal solution, but due to the complexity of the process of finding the optimal solution, it moves in the direction of deviation from the optimal solution in one of the following generations, and if the velocity and displacement continue to be updated in the original way, guided by the wrong information generated by the random flight of some particles in the previous generation, the learning time of the particles is inevitably wasted, resulting in convergence The speed of convergence becomes slower. By combining the two strategies using different control periods, the corrective strategy can automatically monitor the evolutionary trend of the particles after eachT1 evaluation generation, increasing the convergence rate and then responding to the evaluation results at each T1 + 1 correction generation. After using the correction strategy, to further improve the convergence accuracy of the particles, a dimension-by-dimension learning strategy is used at every T2 generation cycle to reduce the complexity of the algorithm by reducing the number of runs. By using two different control cycles, the two strategies are fully combined to give a maximum advantage. The operational flow of the improved particle swarm algorithm is shown in Figure 1.Figure 1
The operational flow of the improved particle swarm algorithm.


### 3.2. Particle Swarm Optimization Algorithm Based on Adaptive Decreasing Inertia Weights

Although particle swarm optimization algorithms are efficient in solving complex problems with high global search capability, they are prone to early convergence. To improve the optimisation performance of the particle swarm algorithm, this paper uses the local optimization capability of adaptive decreasing inertia weights to locally optimize the temporary particles generated in each generation. The temporary particle is used as the initial solution, and the search for the optimal solution is performed by the Metropolis criterion. If the fitness value of the final optimal solution is better than the fitness value of the particle, the particle is replaced with the optimal solution. Otherwise, the particle remains unchanged. In the process of finding the optimal value, the individual particles need to memorize the experience of other particles in addition to their own.The inertia weight is a very important parameter in the PSO algorithm, which describes the effect of the particle’s velocity in the previous generation on the velocity in the current generation. When the inertia weight is large, the global search ability is stronger and the local search ability is weaker; when the inertia weight is small, the local search ability is stronger and the global search ability will be weaker [20]. Considering that the Chebyshev filter amplitude-frequency response curve model exhibits excellent transitions between linear and nonlinear behavior, a strategy based on the nonlinear variation of the Chebyshev filter inertia weights is proposed, and a weight control factor is introduced to adjust the proportion of the maximum inertia weight in the population evolution process by adjusting the size of this control factor, which can ensure that the particle population in the initial state with larger inertia. This ensures that the global exploitation search is carried out at the initial state with large inertia weights and at a later stage of the iteration with smaller fixed weights for a more refined local search. This can be expressed as follows:(3)wt=1.221+Kt2/T2+0.43,φj=tj⋅∑i=1nxij∑j=1mTj,where K is the weight control factor, t is the current evolutionary generation of the population, T is the total evolutionary generation of the population, and Tφ is the learning factor. In the initial iterations of the algorithm, the weights of the particles can achieve a maximum value of 1.65, which is conducive to the initial global search, while with the increase in the number of iterations, the weights of the particles gradually converge to a minimum 0.43 value, which allows for better local refinement of the search. The velocity direction and size of the particles in the particle swarm algorithm are not fixed but will be adjusted according to the velocity size and direction of the optimal particles and will gradually approach the optimal through iteration. There are two restrictions on the update conditions of the particles: individual extremes and global extremes. To improve the applicability of the PSO algorithm, a natural selection mechanism is added to the genetic algorithm based on the above-mentioned improvement strategies, with the basic idea that the entire particle population is sorted according to the size of the fitness value during each iteration, and the velocity and position of the particle with the best half of the fitness value in the population are replaced by the velocity and position of the particle with the worst fitness value [21]. The position and velocity of the worst half of the swarm are replaced by the velocity and position of the best half of the swarm, preserving the individual memory of the optimal value and increasing the probability of the particle being close to the optimal value. The velocities and positions of all particles of the particle swarm algorithm generate updates in each iteration with the following equation:(4)Kρall=Kρx,yMx=φ∑y∈γ∑x∈χpx,y⋅lnpx,y+Ax+Cy+λφ∑x∈χp2x⋅lnpx+Ax,Kρx,y=φ∑y∈γ∑x∈χpx,y⋅lnpx,y+Ax+Cy+λ,where Kρx,y denotes the location of the particle at the kth iteration and px denotes the inertia weight. The optimal solution is selected by comparing the calculated values with the historical optimal points, and then the optimal values of the velocity and position of the obtained optimal particles are retained and replaced. It is ensured that the values of the particles obtained after each iteration are optimal; after the individual particles are searched for, all the particles in the particle population are compared, and the optimal values are selected and retained after the comparison to ensure that the current global optimal solution can be obtained, and then the population optimal solution obtained from this calculation is compared with the historical optimal solution. The global solution after each iteration is guaranteed to be the optimal solution from the historical values; the value after the iteration is determined to be the optimal solution, and if it is the optimal solution, it is output directly.The setting of the maximum particle velocity is very important. A larger velocity is beneficial to the global exploitation of the population, but too large a velocity can cause the particles to miss the global optimum in the process of searching; conversely, a smaller velocity can help the population to perform a local search, but too small a velocity can cause the particles not to be able to fully explore the solution space, thus increasing the possibility of falling into a local optimum. To further improve the performance of the PSO algorithm and to prevent the population diversity from decreasing due to particles flying out of the search space, a nonlinear decreasing strategy for the maximum velocity of particles is then proposed. The maximum velocity of a particle decreases nonlinearly as the population iterates, allowing the particle to effectively avoid falling into the boundary region for an invalid search due to excessive velocity. It satisfies the following equation:(5)Vmaxt=vmax1+kt/T6+vmin.The process of optimization preservation is the process of ensuring that the optimal solution of a particle is not lost in the evolutionary process. The particle swarm optimization algorithm updates the velocity and position of the current particles by using the particles in the individual and group polar libraries. In this paper, we use the method of the essence particle library to update the individual polar library with the 10 best solutions experienced by each particle and store them in the essence particle library, and update the group polar library with the 4 best solutions experienced by all particles and put them in the essence particle library. In this paper, we break the limitation of the traditional single individual and group extreme value guiding particle update, use the particles in the individual and group extreme value libraries to update the current particles, and find the particles corresponding to the best fitness value as the next generation of the current particles for iterative optimization, and update the particle library in the same way as the individual and group extreme value libraries.


## 3.1. Improved Particle Swarm Optimization Algorithm Flow

Particle swarm optimization algorithms were first applied to the optimization of nonlinear continuous functions and the training of neuronal networks, but later they were also used to solve constrained optimization problems, multiobjective optimization problems, dynamic optimization problems, and so on. In contrast to traditional optimization theory algorithms (e.g., Newton’s method, conjugate gradient method), particle swarm optimization algorithms do not require the computation of gradient vectors and Hessian matrices, nor do they require the feasible domain and positive definiteness of the objective function, while ensuring efficient and stable computational speed and accuracy. The encoding process of particles is a process of transferring the feasible solutions involved in a problem to the search space of a particle swarm optimization algorithm. This paper focuses on the effect of a change in transporter speed on the scheduling results, finding the optimum transporter speed at which the scheduling of water resources operations can be developed to maximize the utilization of each piece of equipment so that the load on each piece of equipment is balanced and capacity is maximized [19]. Therefore, the transporter speed is defined as a dimension of the particle and the range of speeds is used as the solution space. The fitness function of the improved particle swarm algorithm is defined as follows:(1)Gx=∫kx2+pxdx,where k is the learning factor and px is the individual extremum function. In the velocity update, the algorithm’s development capability is improved by disturbing the individual and population extremes through the perturbation factor δ, avoiding the algorithm from converging early due to excessive influence by the particle extremes (individual and population extremes), and introducing an inertia weight w that varies with the number of iterations to balance the search capability of the algorithm. w takes a value related to the number of evolutionary generations as shown in equation (2). w makes the algorithm achieves global search with larger inertia inheriting the current velocity of the particles in the early stages, and smaller inertia in the later stages ensures that the algorithm achieves local search near the optimal solution. The operational performance of water resources systems is generally evaluated using the economic benefit-cost ratio as an indicator, with the water supply subsystem as the core of the evaluation. The introduction of the concept of essence particle library makes the particles in the updating process no longer guided by a single individual extreme value and a single group extreme value but also by the influence of other better particles in the individual extreme value library as well as the group extreme value library, expanding the search range of the algorithm, making the algorithm search faster to the vicinity of the optimal solution, saving the search time of the algorithm and improving the search efficiency of the algorithm.(2)w=kx+δp2x+C.In the process of population evolution, there may be a situation where the initial motion of a particle is in the direction of the optimal solution, but due to the complexity of the process of finding the optimal solution, it moves in the direction of deviation from the optimal solution in one of the following generations, and if the velocity and displacement continue to be updated in the original way, guided by the wrong information generated by the random flight of some particles in the previous generation, the learning time of the particles is inevitably wasted, resulting in convergence The speed of convergence becomes slower. By combining the two strategies using different control periods, the corrective strategy can automatically monitor the evolutionary trend of the particles after eachT1 evaluation generation, increasing the convergence rate and then responding to the evaluation results at each T1 + 1 correction generation. After using the correction strategy, to further improve the convergence accuracy of the particles, a dimension-by-dimension learning strategy is used at every T2 generation cycle to reduce the complexity of the algorithm by reducing the number of runs. By using two different control cycles, the two strategies are fully combined to give a maximum advantage. The operational flow of the improved particle swarm algorithm is shown in Figure 1.Figure 1
The operational flow of the improved particle swarm algorithm.


## 3.2. Particle Swarm Optimization Algorithm Based on Adaptive Decreasing Inertia Weights

Although particle swarm optimization algorithms are efficient in solving complex problems with high global search capability, they are prone to early convergence. To improve the optimisation performance of the particle swarm algorithm, this paper uses the local optimization capability of adaptive decreasing inertia weights to locally optimize the temporary particles generated in each generation. The temporary particle is used as the initial solution, and the search for the optimal solution is performed by the Metropolis criterion. If the fitness value of the final optimal solution is better than the fitness value of the particle, the particle is replaced with the optimal solution. Otherwise, the particle remains unchanged. In the process of finding the optimal value, the individual particles need to memorize the experience of other particles in addition to their own.The inertia weight is a very important parameter in the PSO algorithm, which describes the effect of the particle’s velocity in the previous generation on the velocity in the current generation. When the inertia weight is large, the global search ability is stronger and the local search ability is weaker; when the inertia weight is small, the local search ability is stronger and the global search ability will be weaker [20]. Considering that the Chebyshev filter amplitude-frequency response curve model exhibits excellent transitions between linear and nonlinear behavior, a strategy based on the nonlinear variation of the Chebyshev filter inertia weights is proposed, and a weight control factor is introduced to adjust the proportion of the maximum inertia weight in the population evolution process by adjusting the size of this control factor, which can ensure that the particle population in the initial state with larger inertia. This ensures that the global exploitation search is carried out at the initial state with large inertia weights and at a later stage of the iteration with smaller fixed weights for a more refined local search. This can be expressed as follows:(3)wt=1.221+Kt2/T2+0.43,φj=tj⋅∑i=1nxij∑j=1mTj,where K is the weight control factor, t is the current evolutionary generation of the population, T is the total evolutionary generation of the population, and Tφ is the learning factor. In the initial iterations of the algorithm, the weights of the particles can achieve a maximum value of 1.65, which is conducive to the initial global search, while with the increase in the number of iterations, the weights of the particles gradually converge to a minimum 0.43 value, which allows for better local refinement of the search. The velocity direction and size of the particles in the particle swarm algorithm are not fixed but will be adjusted according to the velocity size and direction of the optimal particles and will gradually approach the optimal through iteration. There are two restrictions on the update conditions of the particles: individual extremes and global extremes. To improve the applicability of the PSO algorithm, a natural selection mechanism is added to the genetic algorithm based on the above-mentioned improvement strategies, with the basic idea that the entire particle population is sorted according to the size of the fitness value during each iteration, and the velocity and position of the particle with the best half of the fitness value in the population are replaced by the velocity and position of the particle with the worst fitness value [21]. The position and velocity of the worst half of the swarm are replaced by the velocity and position of the best half of the swarm, preserving the individual memory of the optimal value and increasing the probability of the particle being close to the optimal value. The velocities and positions of all particles of the particle swarm algorithm generate updates in each iteration with the following equation:(4)Kρall=Kρx,yMx=φ∑y∈γ∑x∈χpx,y⋅lnpx,y+Ax+Cy+λφ∑x∈χp2x⋅lnpx+Ax,Kρx,y=φ∑y∈γ∑x∈χpx,y⋅lnpx,y+Ax+Cy+λ,where Kρx,y denotes the location of the particle at the kth iteration and px denotes the inertia weight. The optimal solution is selected by comparing the calculated values with the historical optimal points, and then the optimal values of the velocity and position of the obtained optimal particles are retained and replaced. It is ensured that the values of the particles obtained after each iteration are optimal; after the individual particles are searched for, all the particles in the particle population are compared, and the optimal values are selected and retained after the comparison to ensure that the current global optimal solution can be obtained, and then the population optimal solution obtained from this calculation is compared with the historical optimal solution. The global solution after each iteration is guaranteed to be the optimal solution from the historical values; the value after the iteration is determined to be the optimal solution, and if it is the optimal solution, it is output directly.The setting of the maximum particle velocity is very important. A larger velocity is beneficial to the global exploitation of the population, but too large a velocity can cause the particles to miss the global optimum in the process of searching; conversely, a smaller velocity can help the population to perform a local search, but too small a velocity can cause the particles not to be able to fully explore the solution space, thus increasing the possibility of falling into a local optimum. To further improve the performance of the PSO algorithm and to prevent the population diversity from decreasing due to particles flying out of the search space, a nonlinear decreasing strategy for the maximum velocity of particles is then proposed. The maximum velocity of a particle decreases nonlinearly as the population iterates, allowing the particle to effectively avoid falling into the boundary region for an invalid search due to excessive velocity. It satisfies the following equation:(5)Vmaxt=vmax1+kt/T6+vmin.The process of optimization preservation is the process of ensuring that the optimal solution of a particle is not lost in the evolutionary process. The particle swarm optimization algorithm updates the velocity and position of the current particles by using the particles in the individual and group polar libraries. In this paper, we use the method of the essence particle library to update the individual polar library with the 10 best solutions experienced by each particle and store them in the essence particle library, and update the group polar library with the 4 best solutions experienced by all particles and put them in the essence particle library. In this paper, we break the limitation of the traditional single individual and group extreme value guiding particle update, use the particles in the individual and group extreme value libraries to update the current particles, and find the particles corresponding to the best fitness value as the next generation of the current particles for iterative optimization, and update the particle library in the same way as the individual and group extreme value libraries.


## 4. Application of Adaptive Decreasing Particle Swarm Optimization Algorithm Based on Inertia Weights for Optimal Water Resource Allocation

Water allocation is the rationalization of the social water cycle to meet the needs of multiple users, objectives, and levels of water supply. The definition of water allocation, in a broad sense, is the study of how to allocate water resources for maximum benefit, which includes the development, use, and conservation of water resources. Due to the uneven distribution of water resources in space and time, the amount of water resources varies greatly from region to region, and for areas where water resources are relatively scarce, there is a strong urgency to combine water resources allocation with the construction of water conservancy projects. The implementation of rational allocation of water resources will not only solve the problem of mismatch between the natural spatial and temporal distribution of water resources and socioeconomic development but will also allow for the classification of water quality, alleviating the demand for water quality and quantity between different users, and will allow for a more scientific and sustainable approach to the development and utilization of water resources, alleviating the ecological problems currently faced by the region due to overexploitation [22].The sustainable connotation of water resources carrying capacity is reflected in the two aspects of water quantity and water quality: in terms of water quantity, it is expressed in the sustainable exploitation of water resources, in the process of water consumption, human beings should respect and protect nature, and the speed and extent of exploitation of water resources should be limited to the capacity of water cycle recovery and renewal, to achieve the sustainable use of water resources and support the sustainable development of the economy; in terms of water quality, the rate and extent of pollutant discharge should be lower than the self-purification capacity of the water body.Firstly, the carrying capacity of water resources is based on the current or expected future level of economic technology development, and differences in socioeconomic levels inevitably affect the carrying capacity of water resources. Secondly, the water resources carrying capacity is the limit of water resources utilization based on the optimal allocation of water resources. The role of the supply chain is not only dependent on the corresponding performance management mechanism but also requires an efficient supply chain structure. The supply chain structure is dynamic but stable in relative time. Although the supply chain structure has dynamic characteristics, it is stable in relative time, and the enterprises in the supply chain will not change easily. The enterprises in the supply chain will not change easily. With the improvement of the water rights system, water resources appear in the form of commodities in the water market, and the market realizes the optimal allocation of water resources. Therefore, the water resources carrying capacity has a significant socioeconomic connotation.From a systems theory perspective, any area that can be treated as a water resource carrying capacity study can be regarded as a coupled socioeconomic, water resources, and ecological environment system. The three interact with each other, and their coupling relationship is shown in Figure2.Figure 2
Water resources-ecology-economy-socioeconomic systems.According to the principles and ideas of water rights allocation, a multiobjective water rights allocation model based on adaptive decreasing inertia weights is constructed, using objective functions including maximum water supply benefits, maximum coordination between supply and demand, and minimum pollutant emissions, as described below. (1) Maximum water supply benefits: The economic benefits are characterized by the water supply benefits of all water sources in the region, and the product of the water supply volume and water supply benefits is used to characterize them, in which the water supply order coefficient and the water use equity coefficient are used to guide the model to find the best according to the principles of water rights allocation, from which the objective function of economic benefits in water rights allocation can be obtained as follows:(6)maxf1x=∑j=1J∑i=1Ibij−cijxijαjβj.Thisbij is expressed as a benefit factor, cij a cost factor, xij a water supply quantity, αj a water equity factor, and βj a water supply sequence factor. The social benefits are expressed in terms of the degree of coordination between supply and demand in the region during the year, which indicates that the various sectors of water use are coordinated with each other and that the sum of the difference between water demand and water supply for all regions and all users is used, with higher values indicating greater social benefits. The formula for calculating this is as follows:(7)maxf2x=∑k=1K∑i=1Ixjkt−∑j=1Jxijktcij.Unlike ordinary commodities, which have two-way flow characteristics, water resources have one-way flow characteristics. Water resources are unique in the supply chain of the receiving area, mainly through engineering channels for the transport of water resources, while in the transport of water resources, regional differences, pumps, and other factors help to achieve, usually from west to east transport; because the western terrain is high, the eastern terrain is low, the use of terrain differences can reduce transport costs, improve transport efficiency, which also makes the transport of water resources has a single characteristic. In simple terms, when water resources are not fully utilized downstream, the corresponding water source cannot be recovered, in which case either the excess water is stored, the sales supply is increased, or it flows directly downstream, resulting in a waste of water resources and reduced profitability if too much is supplied [23].Water allocation models are one of the most important tools for studying the social water cycle. They can describe the supply, consumption, and discharge of water resources in society, taking into account the water cycle between different sectors and different water users. From another way, water allocation models can objectively describe the social water cycle, and on the other hand, they can simulate the objectives and constraints and allocate water to individual water-using units to achieve the optimal solution. Figure3 shows the technical route for the optimal allocation of water resources in the receiving area based on adaptive decreasing inertia weights.Figure 3
A technical route for optimal allocation of water resources in receiving areas based on adaptive decreasing inertia weights.The generalization of system network diagrams in water allocation models is similar to that of landscape ecology, which also uses correspondence and logical relationships between points and lines to form a network graph. From the definition of the water transfer system (lines) in the network diagram of the optimal water resources allocation system, the concept of nodes and corridors based on graph theory-landscape ecology is expanded based on the water transfer relationships included in the artificial lateral water cycle system, combined with the evaluation indexes of the level of production and domestic water use, i.e., nodes include river origin intersections, water demand nodes, water supply nodes, and water transfer nodes, and corridors include river systems [24]. The corridors include river systems, surface water supply systems, external water supply systems, groundwater supply systems, wastewater treatment and reuse systems, and drainage systems; thus, the five indicators of urban per capita comprehensive domestic water consumption/L/(person-d), water consumption per unit of GDP/m³/million yuan, leakage rate of pipeline networks, urban wastewater treatment rate/%, urban wastewater recycling rate/%, etc. are selected to describe the artificial lateral branch water system in terms of water use, water extraction, and drainage [25]. The KMO (Kaiser-Meyer-Olkin) test statistic is used in factor analyses dealing with multivariate statistics and can evaluate the simple and partial correlation coefficients between variables; the KMO statistic takes values between 0 and 1, with 1 indicating a strong correlation between variables and the resulting sum of squares of simple correlation coefficients being much larger than the sum of squares of partial correlation coefficients, indicating that these variables are suitable for factor analysis. In contrast, when the value is close to 0, the correlation between the variables is very weak and not suitable for factor analysis [3].Principal Component Analysis (PCA) is a statistical method used to determine the correlation and contribution of variables to the overall picture. By using SPSS, PCA can be used to eliminate interactions between evaluation indicators, thus reducing the workload of selecting indicators [26]. The use of principal component analysis can reflect the contribution of variables and the weight of information and can overcome the error of subjective determination of weights, making the results more objective and reasonable.


## 5. Experimental Verification and Conclusions

### 5.1. Model Adaptability and Water Carrying Capacity Analysis

To further demonstrate the superiority of the adaptive decreasing particle swarm algorithm based on inertia weights, under the recommended water-saving scheme, the water supply guarantee rateP = 75% is used as an example, and the inertia weight-based adaptive decreasing particle swarm algorithm and the basic particle swarm algorithm are used to solve the optimal water allocation model respectively.Analysis of Figure4 shows that the inertia-weight-based adaptive decreasing particle swarm algorithm has better individual adaptation values and is faster than the basic particle swarm algorithm in solving the above configuration model. It can be seen that the inertia weight-based adaptive decreasing particle swarm algorithm is scientifically sound and has a certain degree of reliability and operability in solving the optimal water resource allocation scheme under different conditions. Bidirectionality is another distinctive feature of the supply chain structure, which can be analyzed in two ways. This feature can be analyzed in two aspects. First, at the horizontal level, there are cooperative and competitive relationships between subjects of the same products and raw materials. If the basic profile of the study area changes, the model parameters and algorithm parameters can be adjusted to further rationalize the use of water resources in different study areas. In summary, the above water resource allocation model and the adaptive decreasing particle swarm algorithm based on inertia weights used in the solution process are highly adaptable and generic and can provide a basis for decision-making for those involved in water resource planning.Figure 4
The evolutionary process of optimal individual adaptation values.The Water Resources Carrying Capacity Analysis module is used to analyse the extent to which water resources can be exploited under the current conditions in the study area and to explore the water resources potential of the study area, which is an important guarantee for the water resources optimization allocation module and an important guide for decision-makers to measure the degree of sustainable development of the region. The implementation of the water resources carrying capacity analysis module is based on the water resources carrying capacity evaluation system established in the paper, data processing of the loaded data, then based on MATLAB to write the principal component analysis (PCA) program, extraction of feature values, calculation of the cumulative contribution of the winning components, and finally the results of the principal component analysis and the water resources carrying capacity analysis results, as shown in Figure5.Figure 5
Results of the water resources carrying capacity analysis.As can be seen from Figure5, when the regional water resources carrying level is in a serious overload state, water resources will seriously restrict the orderly development of social and economic development, and the water ecological environment is damaged and difficult to restore; when the regional water resources carrying level is in a moderate overload state, the water administration department should pay attention to it and strengthen the management of water resources and the comprehensive management of water environment; when the regional water resources carrying level is in a light overload state. When the regional water resources carrying capacity is in a reasonable state, it means that the national economic development will be constrained by both water resources and the water environment, and the scale of social and economic development is in a saturated state; when the regional water resources carrying capacity is in a good state, it means that the region is strong in regulation and control, can effectively resist disturbance, and the environment can change automatically. When the regional water resources carrying capacity is in a good carrying capacity, it means that the region has a strong regulatory capacity, can effectively resist disturbances, the environment can recover quickly, the degree of water resources development and utilization is not high, and the social economy has more room for development.


### 5.2. Results of the Optimal Allocation of Water Resources in the Receiving Area

A Pareto is an optimal set of solutions formed by combining multiple solutions. Pareto optimal solutions have better individual performance than all feasible solution sets, and there is no good or bad Pareto optimal solution among them; all solutions are optimal and only need to be selected depending on the focus of the decision-maker. Figure6 shows the distribution of Pareto solutions searched by adaptive decreasing of inertia weights in the objective function space, with each scatter point being the Pareto optimal solution obtained by weighing the three objective functions.Figure 6
Distribution of moral Pareto solution sets for adaptive decreasing search with inertia weights.In this study, three typical scenarios were selected from the Pareto optimal solution set for comparative analysis, and the specific results are shown in Figure7. As can be seen from Figure 7, the three typical scenarios focus on the three objectives of maximum yield, the minimum average depth of groundwater decline and maximum economic efficiency. Option 1 has the greatest crop yield and the greatest average cumulative depth of groundwater drop, and the economic benefits are between Option 2 and 3, which is not conducive to the economic development of the irrigation area; Option 2 has the greatest benefits for the irrigation area, and the crop yield and average cumulative depth of groundwater drop are between Options 1 and 3; Option 3 has the smallest average cumulative depth of groundwater drop, and the smallest crop yield and economic benefits for the sustainable development of groundwater in the irrigation area. It is the most beneficial to the sustainable development of groundwater in the irrigation area.Figure 7
Comparison of the three water allocation options.In scenario 1, the crop yield is 53,567.81 kg/mu, the economic benefit is RMB 20,234,848.15/mu, and the average cumulative depth of groundwater loss is 0.86 m. In times of drought, when the irrigation water supply is insufficient, the water requirement of food crops needs to be prioritized to maximize yield, followed by cash crops. The water deficit for summer maize is the smallest at 0.16%, while the water deficit for vegetables is the largest at 3.43%. In Option 2, the crop yield was 33368.55 kg/mu, the economic benefit was 157,565.5 yuan/mu, and the average cumulative depth of groundwater loss was 0.84 m. During dry periods when irrigation water supply was insufficient, priority was given to irrigating fruit trees to minimize the cumulative depth of groundwater loss. The water deficit rate for all crops was 10% or more. Winter wheat had a low water deficit of 13.95% and vegetables had the highest water deficit of 23.80%. In Option 3, the crop yield was 52778.41 kg/mu, the economic benefit was 208217.73 Yuan/mu, and the average cumulative depth of groundwater loss was 0.85 m. In times of drought, to maximize the economic benefit, priority should be given to summer maize and fruit trees, followed by winter wheat and vegetables. Winter wheat has a low water deficit of 0.21%, while vegetables have the highest water deficit of 1.45%. The three objectives compete with each other, and improvement in one is at the expense of the other two. Decision-makers can judge the direction of their preference, choosing Option 1 if they are considering increasing crop yields, Option 2 if they are focusing on groundwater conservation, and Option 3 if they are inclined to increase large economic benefits.The water supply is supplied to each customer with 90% satisfaction of the water supply as the actual water supply. In real life, water is in the balance between supply and demand, and therefore, the actual water supply is equal to the actual water consumption. As the result of the multiobjective optimization is a Pareto-optimal solution set and the water supply varies from solution to solution, solutions at multiple locations were selected for comparison. The configuration results and their water shortage ratios are shown in Figure8.Figure 8
Results of optimal water allocation with 90% water supply satisfaction.Each of the four options is to meet the different water demands of different users. Scenario 1 is to meet ecological water demand, with a water deficit of 0.3%. This scenario has a greater proportion of water deficit for agricultural and residential uses, 14.6% and 13.1%, respectively, and a smaller proportion of water deficit for industrial uses. Scenario 2 is to meet the water deficit for agriculture, with a deficit of 2.6%. Under this scenario, the water shortages for industry, residential, and ecology are relatively large, at 18.9%, 20%, and 8.2%, respectively. Scenario 3 is for industrial water, with a deficit of 0.1%. The water shortages for residential and ecological uses are also relatively small, at 2% and 3.7%, respectively. The proportion of water shortage for agriculture is larger, at 17.7%. Scenario 4 is for residential water use, with a shortage of 0.7%. As can be seen from the proportion of water allocated to each water user in the table, the proportion of water shortages for industry, residential and ecological water use all increase as water demand for agriculture increases. The proportion of water shortages in agriculture is greater in all three scenarios for meeting industrial, residential, and ecological water demand, in contrast to Scenario 4. Agriculture accounts for 50.9% of the total water demand, which is the highest of the four users. The reason for the same total water deficit is that agricultural water demand is much greater than that of the three users, industrial, residential, and ecological, increasing the proportion of water deficit for the other three users when agricultural water demand is met.


## 5.1. Model Adaptability and Water Carrying Capacity Analysis

To further demonstrate the superiority of the adaptive decreasing particle swarm algorithm based on inertia weights, under the recommended water-saving scheme, the water supply guarantee rateP = 75% is used as an example, and the inertia weight-based adaptive decreasing particle swarm algorithm and the basic particle swarm algorithm are used to solve the optimal water allocation model respectively.Analysis of Figure4 shows that the inertia-weight-based adaptive decreasing particle swarm algorithm has better individual adaptation values and is faster than the basic particle swarm algorithm in solving the above configuration model. It can be seen that the inertia weight-based adaptive decreasing particle swarm algorithm is scientifically sound and has a certain degree of reliability and operability in solving the optimal water resource allocation scheme under different conditions. Bidirectionality is another distinctive feature of the supply chain structure, which can be analyzed in two ways. This feature can be analyzed in two aspects. First, at the horizontal level, there are cooperative and competitive relationships between subjects of the same products and raw materials. If the basic profile of the study area changes, the model parameters and algorithm parameters can be adjusted to further rationalize the use of water resources in different study areas. In summary, the above water resource allocation model and the adaptive decreasing particle swarm algorithm based on inertia weights used in the solution process are highly adaptable and generic and can provide a basis for decision-making for those involved in water resource planning.Figure 4
The evolutionary process of optimal individual adaptation values.The Water Resources Carrying Capacity Analysis module is used to analyse the extent to which water resources can be exploited under the current conditions in the study area and to explore the water resources potential of the study area, which is an important guarantee for the water resources optimization allocation module and an important guide for decision-makers to measure the degree of sustainable development of the region. The implementation of the water resources carrying capacity analysis module is based on the water resources carrying capacity evaluation system established in the paper, data processing of the loaded data, then based on MATLAB to write the principal component analysis (PCA) program, extraction of feature values, calculation of the cumulative contribution of the winning components, and finally the results of the principal component analysis and the water resources carrying capacity analysis results, as shown in Figure5.Figure 5
Results of the water resources carrying capacity analysis.As can be seen from Figure5, when the regional water resources carrying level is in a serious overload state, water resources will seriously restrict the orderly development of social and economic development, and the water ecological environment is damaged and difficult to restore; when the regional water resources carrying level is in a moderate overload state, the water administration department should pay attention to it and strengthen the management of water resources and the comprehensive management of water environment; when the regional water resources carrying level is in a light overload state. When the regional water resources carrying capacity is in a reasonable state, it means that the national economic development will be constrained by both water resources and the water environment, and the scale of social and economic development is in a saturated state; when the regional water resources carrying capacity is in a good state, it means that the region is strong in regulation and control, can effectively resist disturbance, and the environment can change automatically. When the regional water resources carrying capacity is in a good carrying capacity, it means that the region has a strong regulatory capacity, can effectively resist disturbances, the environment can recover quickly, the degree of water resources development and utilization is not high, and the social economy has more room for development.


## 5.2. Results of the Optimal Allocation of Water Resources in the Receiving Area

A Pareto is an optimal set of solutions formed by combining multiple solutions. Pareto optimal solutions have better individual performance than all feasible solution sets, and there is no good or bad Pareto optimal solution among them; all solutions are optimal and only need to be selected depending on the focus of the decision-maker. Figure6 shows the distribution of Pareto solutions searched by adaptive decreasing of inertia weights in the objective function space, with each scatter point being the Pareto optimal solution obtained by weighing the three objective functions.Figure 6
Distribution of moral Pareto solution sets for adaptive decreasing search with inertia weights.In this study, three typical scenarios were selected from the Pareto optimal solution set for comparative analysis, and the specific results are shown in Figure7. As can be seen from Figure 7, the three typical scenarios focus on the three objectives of maximum yield, the minimum average depth of groundwater decline and maximum economic efficiency. Option 1 has the greatest crop yield and the greatest average cumulative depth of groundwater drop, and the economic benefits are between Option 2 and 3, which is not conducive to the economic development of the irrigation area; Option 2 has the greatest benefits for the irrigation area, and the crop yield and average cumulative depth of groundwater drop are between Options 1 and 3; Option 3 has the smallest average cumulative depth of groundwater drop, and the smallest crop yield and economic benefits for the sustainable development of groundwater in the irrigation area. It is the most beneficial to the sustainable development of groundwater in the irrigation area.Figure 7
Comparison of the three water allocation options.In scenario 1, the crop yield is 53,567.81 kg/mu, the economic benefit is RMB 20,234,848.15/mu, and the average cumulative depth of groundwater loss is 0.86 m. In times of drought, when the irrigation water supply is insufficient, the water requirement of food crops needs to be prioritized to maximize yield, followed by cash crops. The water deficit for summer maize is the smallest at 0.16%, while the water deficit for vegetables is the largest at 3.43%. In Option 2, the crop yield was 33368.55 kg/mu, the economic benefit was 157,565.5 yuan/mu, and the average cumulative depth of groundwater loss was 0.84 m. During dry periods when irrigation water supply was insufficient, priority was given to irrigating fruit trees to minimize the cumulative depth of groundwater loss. The water deficit rate for all crops was 10% or more. Winter wheat had a low water deficit of 13.95% and vegetables had the highest water deficit of 23.80%. In Option 3, the crop yield was 52778.41 kg/mu, the economic benefit was 208217.73 Yuan/mu, and the average cumulative depth of groundwater loss was 0.85 m. In times of drought, to maximize the economic benefit, priority should be given to summer maize and fruit trees, followed by winter wheat and vegetables. Winter wheat has a low water deficit of 0.21%, while vegetables have the highest water deficit of 1.45%. The three objectives compete with each other, and improvement in one is at the expense of the other two. Decision-makers can judge the direction of their preference, choosing Option 1 if they are considering increasing crop yields, Option 2 if they are focusing on groundwater conservation, and Option 3 if they are inclined to increase large economic benefits.The water supply is supplied to each customer with 90% satisfaction of the water supply as the actual water supply. In real life, water is in the balance between supply and demand, and therefore, the actual water supply is equal to the actual water consumption. As the result of the multiobjective optimization is a Pareto-optimal solution set and the water supply varies from solution to solution, solutions at multiple locations were selected for comparison. The configuration results and their water shortage ratios are shown in Figure8.Figure 8
Results of optimal water allocation with 90% water supply satisfaction.Each of the four options is to meet the different water demands of different users. Scenario 1 is to meet ecological water demand, with a water deficit of 0.3%. This scenario has a greater proportion of water deficit for agricultural and residential uses, 14.6% and 13.1%, respectively, and a smaller proportion of water deficit for industrial uses. Scenario 2 is to meet the water deficit for agriculture, with a deficit of 2.6%. Under this scenario, the water shortages for industry, residential, and ecology are relatively large, at 18.9%, 20%, and 8.2%, respectively. Scenario 3 is for industrial water, with a deficit of 0.1%. The water shortages for residential and ecological uses are also relatively small, at 2% and 3.7%, respectively. The proportion of water shortage for agriculture is larger, at 17.7%. Scenario 4 is for residential water use, with a shortage of 0.7%. As can be seen from the proportion of water allocated to each water user in the table, the proportion of water shortages for industry, residential and ecological water use all increase as water demand for agriculture increases. The proportion of water shortages in agriculture is greater in all three scenarios for meeting industrial, residential, and ecological water demand, in contrast to Scenario 4. Agriculture accounts for 50.9% of the total water demand, which is the highest of the four users. The reason for the same total water deficit is that agricultural water demand is much greater than that of the three users, industrial, residential, and ecological, increasing the proportion of water deficit for the other three users when agricultural water demand is met.


## 6. Conclusion

This paper provides a comprehensive analysis and solution to the problem of water resources scheduling optimization in the receiving area and proposes a particle swarm algorithm based on the standard particle swarm algorithm with adaptive decreasing inertia weights, including the introduction of a particle library to store the individual and group extreme values of the particles, so that the individual and group extreme values of the particles can be updated using the better particles in the particle library. This prevents the limitations of single-particle guidance, improves the search performance of the swarm algorithm, adds a perturbation factor and inertia weights that change with the number of generations to guide the individual and group poles during the evolutionary process to avoid overreliance on individual and group poles in the process of particle updating, ensuring that the algorithm can perform a large global search in the early evolutionary stage and a small local search in the later evolutionary stage. At the same time, the strong local search capability of the inertia weight adaptive decreasing algorithm is used in the process of generating new particles to perform local search operations on the particles, avoiding premature convergence of the algorithm and improving the search performance of the algorithm. To achieve a rational allocation of water resources, a multiobjective receiving area water resources optimization allocation model with maximum water supply benefits, minimum regional water shortage, and minimum pollutant emissions is established. To enhance the effectiveness and calculation accuracy of the optimal allocation model, water allocation model constraints are proposed, specifically including five aspects such as non-negative constraints and water demand constraints. The supply chain structure of water resources in the receiving area is also graphically illustrated, and the rationality of the model is initially investigated through the establishment of a receiving area water resources optimization model and case studies. In this paper, the redistribution of benefits among members was not considered in the water allocation design, resulting in some limitations in the conclusions obtained. It is difficult to effectively measure member contribution rate. It is difficult to effectively measure the level of member benefits, and the member trust relationship is not explored, which is the focus of future research.



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*Source: 3329628-2022-04-16.xml*