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Impact of Land Use Change on Ecosystem Services Values in Danjiangkou Reservoir Area,China in the Context of National Water Network Project Construction
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作者 LIU Linghua ZHENG Liang +3 位作者 WANG Ying LIU Chongchong ZHANG Bowen BI Yuzhe 《Chinese Geographical Science》 2025年第1期111-130,共20页
Investigating the ecological impact of land use change in the context of the construction of national water network project is crucial,as it is imperative for achieving the sustainable development goals of the nationa... Investigating the ecological impact of land use change in the context of the construction of national water network project is crucial,as it is imperative for achieving the sustainable development goals of the national water network and guaranteeing regional ecological stability.Using the Danjiangkou Reservoir Area(DRA),China as the study area,this paper first examined the spatiotemporal dynamics of natural landscape patterns and ecosystem service values(ESV)in the DRA from 2000 to 2018 and then investigated the spatial clustering characteristics of the ESV using spatial statistical analysis tools.Finally,the patch-generating land use simulation(PLUS)model was used to simulate the natural landscape and future changes in the ESV of the DRA from 2018 to 2028 under four different development scenarios:business as usual(BAU),economic development(ED),ecological protection(EP),and shoreline protection(SP).The results show that:during 2000-2018,the construction of water facilities had a significant impact on regional land use/land cover(LULC)change,with a 24830 ha increase in watershed area.ESV exhibited an increasing trend,with a significant and growing spatial clustering effect.The transformation of farmland to water bodies led to accelerated ESV growth,while the transformation of forest land to farmland led to a decrease in the ESV.Normalized difference vegetation index(NDVI)had the strongest effect on the ESV.ESV exhibited a continuous increase from 2018 to 2028 under all the simulation scenarios.The EP scenario had the greatest increase in ESV,while the ED scenario had the smallest increase.The findings suggest that projected land use patterns under different scenarios have varied impacts on ecosystem services(ESs)and that the management and planning of the DRA should balance social,economic,ecological,and security benefits.nomic,ecological,and security benefits. 展开更多
关键词 ecosystem services value(ESV) national water network project of China South-to-North water Diversion Project(SNWDP)of China multi-scenario simulation Danjiangkou Reservoir Area China
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Assessment of the hydraulic performance of water supply distribution system in Gidole Town,Ethiopia
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作者 Lemita Berisha Urto Tewaso Harneso Shoke Otoma Orkaido Garo 《River》 2025年第3期416-430,共15页
The majority of water utilities,particularly public service providers such as Gidole town,are struggling to deliver a sufficient and consistent supply of water in Ethiopia's developing towns.The primary objective ... The majority of water utilities,particularly public service providers such as Gidole town,are struggling to deliver a sufficient and consistent supply of water in Ethiopia's developing towns.The primary objective of this study was to assess the hydraulic performance of water supply distribution system in Gidole Town,Ethiopia,a representative case of the challenges facing public water utilities in developing towns.The WaterGEMS v8i hydraulic model was utilized to simulate and evaluate the distribution network's performance.The system was configured as a looped network and analyzed against standard permitted pressure and velocity values in the distribution system.The model was effectively calibrated(coefficient of determination(R^(2))=0.969)using measured and observed pressure data.The model simulation run was conducted at peak and low hourly demand with 1.9 and 0.25 hourly factors,respectively.The estimated water demand of the town is 1284.3m^(3)/day(48.4 liters per capita per day),and it would be increased to 3099.77m^(3)/day(66.03 l/c/d)by the 2037 design period.The system experiences significant non-revenue water losses(75,434.11m^(3)/year),accounting for 29.9%of total water production;as a result,the present water supply coverage of the town is only 33.6%.Hydraulic simulations under peak and low demand scenarios revealed nodes with pressures outside the normal range,indicating system-wide inefficiencies.These findings highlight a combined issue of large physical losses and insufficient capacity of the water supply in the town,which is typical of many municipal systems in developing regions.The study concludes that strategic infrastructure rehabilitation,with an emphasis on pressure management and leak reduction,is not only a town necessity but a fundamental requirement for improving water security and financial sustainability for utilities in Ethiopia and similar contexts.The findings and methodology have been forwarded to town's water supply project and institutional development departments for immediate future implementation and provide a replicable framework for evidence-based investment and planning in other struggling municipalities in similar situations. 展开更多
关键词 Gidole Town hydraulic performance pressure Junction water distribution network system water losses waterGEMSv8i
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Digital twin-based resilience evaluation and intelligent strategies of smart urban water distribution networks for emergency management
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作者 Hongyan Dui Taiyu Cao Fan Wang 《Resilient Cities and Structures》 2025年第1期41-52,共12页
Resilient smart urban water distribution networks are essential to ensure smooth urban operation and maintain daily water services.However,the dynamics and complexity of smart water distribution networks make its re-s... Resilient smart urban water distribution networks are essential to ensure smooth urban operation and maintain daily water services.However,the dynamics and complexity of smart water distribution networks make its re-silience study face many challenges.The introduction of digital twin technology provides an innovative solution for the resilience study of smart water distribution networks,which can more effectively support the network’s real-time monitoring and intelligent control.This paper proposes a digital twin architecture of smart water dis-tribution networks,laying the foundation for the resilience assessment of water distribution networks.Based on this,a performance evaluation model based on user satisfaction is proposed,which can more intuitively and effectively reflect the performance of urban water supply services.Meanwhile,we propose a method to quantify the importance of water distribution pipes’residual resilience,considering the time value to optimize the re-covery sequence of failed pipes and develop targeted preventive maintenance strategies.Finally,to validate the effectiveness of the proposed method,this paper applies it to a water distribution network.The results show that the proposed method can significantly improve the resilience and enhance the overall resilience of smart urban water distribution networks. 展开更多
关键词 Digital twin Smart water distribution network Resilience evaluation Importance measure
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Designing an Intelligent Control Philosophy in Reservoirs of Water Transfer Networks in Supervisory Control and Data Acquisition System Stations
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作者 Ali Dolatshahi Zand Kaveh Khalili-Damghani Sadigh Raissi 《International Journal of Automation and computing》 EI CSCD 2021年第5期694-717,共24页
In this paper, a hybrid neural-genetic fuzzy system is proposed to control the flow and height of water in the reservoirs of water transfer networks. These controls will avoid probable water wastes in the reservoirs a... In this paper, a hybrid neural-genetic fuzzy system is proposed to control the flow and height of water in the reservoirs of water transfer networks. These controls will avoid probable water wastes in the reservoirs and pressure drops in water distribution networks. The proposed approach combines the artificial neural network, genetic algorithm, and fuzzy inference system to improve the performance of the supervisory control and data acquisition stations through a new control philosophy for instruments and control valves in the reservoirs of the water transfer networks. First, a multi-core artificial neural network model, including a multi-layer perceptron and radial based function, is proposed to forecast the daily consumption of the water in a reservoir. A genetic algorithm is proposed to optimize the parameters of the artificial neural networks. Then, the online height of water in the reservoir and the output of artificial neural networks are used as inputs of a fuzzy inference system to estimate the flow rate of the reservoir inlet. Finally, the estimated inlet flow is translated into the input valve position using a transform control unit supported by a nonlinear autoregressive exogenous model. The proposed approach is applied in the Tehran water transfer network. The results of this study show that the usage of the proposed approach significantly reduces the deviation of the reservoir height from the desired levels. 展开更多
关键词 water demand forecasting water transfer network supervisory control and data acquisition water management multicore artificial neural network fuzzy inference system
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Water Supply Networks as Cyber-physical Systems and Controllability Analysis 被引量:5
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作者 Yongsong Wei Shaoyuan Li 《IEEE/CAA Journal of Automatica Sinica》 SCIE EI 2015年第3期313-319,共7页
Cyber-physical systems (CPS) is a system of systems which consists of many subsystems that can stand alone in an individual manner and can be taken as a typical complex network. CPS can be applied in the critical infr... Cyber-physical systems (CPS) is a system of systems which consists of many subsystems that can stand alone in an individual manner and can be taken as a typical complex network. CPS can be applied in the critical infrastructures such as water supply networks, energy supply systems, and so on. In this paper, we analyze the structure of modern city water supply networks from the view of CPS theory. we use complex network theory to build an undirected and unweighted complex network model for the water supply networks to investigate the structural properties, and present the structure of the water supply networks and detect communities by a spectral analysis of the Laplacian matrix. Then, we analyze the structure and controllability of water supply networks by the structural controllability method. The results show the feasibility and effectiveness of the proposed complex network model. © 2014 Chinese Association of Automation. 展开更多
关键词 CONTROLLABILITY Embedded systems Matrix algebra Spectrum analysis Structural properties water supply
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Predictive Zone Control of Pressure Management for Water Supply Network Systems
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作者 Dong-Ming Liu Shao-Yuan Li 《International Journal of Automation and computing》 EI CSCD 2016年第6期607-614,共8页
In this paper we address the problem of pressure management in water supply system (WSS) network. The model-based predictive control (MPC) strategies have some important features to deal with WSS. By hydraulic ana... In this paper we address the problem of pressure management in water supply system (WSS) network. The model-based predictive control (MPC) strategies have some important features to deal with WSS. By hydraulic analysis of WSS, the predictive model is derived from the dynamic model and static model of WSS. Through WSS, the consumers' demands are required to be met at all times according to some operational constraints that must be satisfied. The constraints of flow through actuators, the water level of reservoirs and the consumer areas' pressure demand are determined by a specific system. In this work, we develop a constrained MPC controller that considers the zone control of the pressure outputs and incorporates steady state economic targets in the control cost function. The designed management strategies are applied to a case study and simulation results, covering different aspects, are provided. The output nodal pressure can be controlled in the desired zone by optimal scheduling the actuators of the WSS. If the variation range of reservoir's water level is broader, the rate of flow through the actuators is gentle, and vice versa. 展开更多
关键词 water supply system (WSS) network mathematical modelling zone control pressure control model predictive control(MPC).
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Application of an expert system using neural network to control the coagulant dosing in water treatment plant 被引量:3
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作者 HangZHANG 《控制理论与应用(英文版)》 EI 2004年第1期89-92,共4页
The coagulation process is one of the most important stages in water treatment plant, which involves many complex physical and chemical phenomena. Moreover, coagulant dosing rate is non-linearly correlated to raw wate... The coagulation process is one of the most important stages in water treatment plant, which involves many complex physical and chemical phenomena. Moreover, coagulant dosing rate is non-linearly correlated to raw water characteristics such as turbidity, conductivity, PH, temperature, etc. As such, coagulation reaction is hard or even impossible to control satisfactorily by conventional methods. Based on neural network and rule models, an expert system for determining the optimum chemical dosage rate is developed and used in a water treatment work, and the results of actual runs show that in the condition of satisfying the demand of drinking water quality, the usage of coagulant is lowered. 展开更多
关键词 water treatment Process control Expert system Neural network Rule models
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Nerual Network Expert System and Their Application to Forecasting Water Invasion of Colliery
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作者 Zhang Jing & Li Renhou (Computer & Application Group, Xi’an University of Technology, Xi’an 710048, China)(System Engineering Institute of Xi’an JiaoTong University, Xi’an 710049, China) 《Journal of Systems Engineering and Electronics》 SCIE EI CSCD 1995年第2期52-57,共6页
In this paper, we propose a formal definition, general structure and work principle of the Neural Network Expert System (NNES) based on joint-type knowledge representation, and show a practical application example usi... In this paper, we propose a formal definition, general structure and work principle of the Neural Network Expert System (NNES) based on joint-type knowledge representation, and show a practical application example using NNES for forecasting the water invasion of coal mine. 展开更多
关键词 Neural network Expert system water calamity Forecasting.
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Application of Dynamic Programming Method in Optimization of Water Distribution Pipe Network 被引量:2
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作者 樊鹏 王莹 +1 位作者 杨建波 杨青伟 《Agricultural Science & Technology》 CAS 2014年第4期703-705,共3页
With a water-supply network by dynamic programming. The minimal as an example, the network was optimized annual discounted costs were taken as an objective function and node pressure etc. as constraint conditions. The... With a water-supply network by dynamic programming. The minimal as an example, the network was optimized annual discounted costs were taken as an objective function and node pressure etc. as constraint conditions. The alternative pipe diameters were optimized as per enumeration method and the group allowing objective function with the least values would be the optimized one. It is proved the optimized pipe network reduced by 11.49% in terms of cost and the optimized ben- efits proved much significant. 展开更多
关键词 DYNAMICPROGRAMMING water distribution pipe network OPTIMIZATION
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STUDY ON OPTIMAL CONTROL OF MUNICIPAL WATER DISTRIBUTION NETWORK 被引量:1
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作者 张宏伟 杨芳 庄健 《Transactions of Tianjin University》 EI CAS 2001年第3期167-171,共5页
A systematic investigation is made on the problems which are related to the optimal control of the municipal water distribution network.A mathematical model of forecasting the water short term demand is proposed using... A systematic investigation is made on the problems which are related to the optimal control of the municipal water distribution network.A mathematical model of forecasting the water short term demand is proposed using the time series trigonometric function analysis method;the service discharge based macroscopic model of network performance is established using the network structuring method;a relatively satisfactory mathematical model for the optimal control of water distribution network is put forward in view of security and economy,and solved by the constrained mixed discrete variable complex arithmetic.The model is applied in many examples and the results are satisfactory. 展开更多
关键词 water distribution network water demand forecast macroscopic model optimal control
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Global Optimization for the Synthesis of Integrated Water Systems with Particle Swarm Optimization Algorithm 被引量:9
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作者 罗袆青 袁希钢 《Chinese Journal of Chemical Engineering》 SCIE EI CAS CSCD 2008年第1期11-15,共5页
The problem of optimal synthesis of an integrated water system is addressed in this study, where water using processes and water treatment operations are combined into a single network such that the total cost of fres... The problem of optimal synthesis of an integrated water system is addressed in this study, where water using processes and water treatment operations are combined into a single network such that the total cost of fresh water and wastewater treatment is globally minimized. A superstructure that incorporates all feasible design alterna- tives for wastewater treatment, reuse and recycle, is synthesized with a non-linear programming model. An evolutionary approach--an improved particle swarm optimization is proposed for optimizing such systems. Two simple examples are .Presented.to illustrate the global op.timization of inte.grated water networks using the proposed algorithm. 展开更多
关键词 integrated water network water minimization particle swarm optimization
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Improving Energy Performance of Water Allocation Networks Through Appropriate Stream Merging 被引量:9
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作者 冯霄 李育才 余新江 《Chinese Journal of Chemical Engineering》 SCIE EI CAS CSCD 2008年第3期480-484,共5页
Water-using operations in the process industry have demands for water usually both on water quality and temperature, and the existing mathematical models of heat exchange networks cannot guarantee the energy performan... Water-using operations in the process industry have demands for water usually both on water quality and temperature, and the existing mathematical models of heat exchange networks cannot guarantee the energy performance of a water network optimal. In this paper, the effects of non-isothermal merging on energy performance of water allocation networks are analyzed, which include utility consumption, total heat exchange load, and number of heat exchange matches. Three principles are proposed to express the effects of non-isothermal merging on energy performance of water allocation networks. A rule of non-isothermal merging without increasing utility consumption is deduced. And an approach to improve energy performance of water allocation network is presented. A case study is given to demonstrate the method. 展开更多
关键词 water allocation network energy performance non-isothermal merging
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Functionality analysis of an urban water supply network after strong earthquakes 被引量:4
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作者 Li Jichao Shang Qingxue +2 位作者 Hou Guanjie Li Quanwang Wang Tao 《Earthquake Engineering and Engineering Vibration》 SCIE EI CSCD 2021年第2期291-302,共12页
An urban water supply network(WSN)is a crucial lifeline system that helps to maintain the normal functioning of modern society.However,the hydraulic analysis of a significantly damaged WSN that suffers from pipe break... An urban water supply network(WSN)is a crucial lifeline system that helps to maintain the normal functioning of modern society.However,the hydraulic analysis of a significantly damaged WSN that suffers from pipe breaks or leaks remains challenging.In this paper,a probability-based framework is proposed to assess the functionality of WSNs in the aftermath of powerful earthquakes.The serviceability of the WSN is quantified by using a comprehensive index that considers nodal water flow and nodal pressure.This index includes a coefficient that reflects the relative importance of these two parameters.The demand reduction(DR)method,which reduces the water flow of nodes while preventing the negative pressure of nodes,is proposed.The difference between the negative pressure elimination(NPE)method and the DR method is discussed by using the example of a WSN in a medium-sized city in China.The functionality values of the WSN are 0.76 and 0.99 when nodal pressure and nodal demands are used respectively as the index of system serviceability at an intensity level that would pertain to an earthquake considered to occur at a maximum level.When the intensity of ground motion is as high as 0.4 g,the DR method requires fewer samples than the NPE method to obtain accurate results.The NPE method eliminates most of the pipes,which may be unrealistic. 展开更多
关键词 water supply network system serviceability hydraulic analysis negative pressure elimination benchmark city
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Serviceability evaluation of water supply networks under seismic loads utilizing their operational physical mechanism 被引量:4
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作者 Miao Huiquan Li Jie 《Earthquake Engineering and Engineering Vibration》 SCIE EI CSCD 2022年第1期283-296,共14页
The serviceability of water supply networks(WSNs)under seismic loads has significant importance for estimating the probable losses and the impact of diminished functionality on affected communities.The innovation pres... The serviceability of water supply networks(WSNs)under seismic loads has significant importance for estimating the probable losses and the impact of diminished functionality on affected communities.The innovation presented in this paper is suggesting a new strategy to evaluate the seismic serviceability of WSNs,utilizing their operational physical mechanism.On one hand,this method can obtain the seismic serviceability of each node as well as entire WSNs.On the other hand,this method can dynamically reflect the propagation of randomness from ground motions to WSNs.First,a finite element model is established to capture the seismic response of buried pipe networks,and a leakage model is suggested to obtain the leakage area of WSNs.Second,the transient flow analysis of WSNs with or without leakage is derived to obtain dynamic water flow and pressure.Third,the seismic serviceability of WSNs is analyzed based on the probability density evolution method(PDEM).Finally,the seismic serviceability of a real WSN in Mianzhu city is assessed to illustrate the method.The case study shows that randomness from the ground motions can obviously affect the leakage state and the probability density of the nodal head during earthquakes. 展开更多
关键词 water supply networks seismic serviceability nodal water pressure stochastic ground motions probability density evolution method
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Design of Water Network with Multiple Contaminants and Zero Discharge 被引量:2
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作者 李英 都健 姚平经 《Chinese Journal of Chemical Engineering》 SCIE EI CAS CSCD 2003年第5期559-564,共6页
The paper presents a procedure to design water network. First of all, water reuse system, water regeneration reuse system (including regeneration recycle) and wastewater treatment system are designed separately. But t... The paper presents a procedure to design water network. First of all, water reuse system, water regeneration reuse system (including regeneration recycle) and wastewater treatment system are designed separately. But the interaction between different parts demands that each part is designed iteratively to optimize the whole water network. Therefore, on the basis of the separated design a water netvrork superstructure including reuse, regeneration and wastewater treatment is established from the system engineering point of view. And a multi-objective adaptive simulated annealing genetic algorithm is adopted to simultaneously integrate the overall water netvrork to balance the economic and environmental effects. The algorithm overcomes the defect of local optimum of simulated annealing (SA), avoids the pre-maturation of genetic algorithm (GA) and finds a set of solutions (pareto front) in acceptable computer time. Prom the pareto front, a point with minimum fresh water consumption will be extended to zero discharge as our ultimate goal. 展开更多
关键词 water network wastewater treatment REUSE regeneration reuse multi-objective adaptive simulated annealing genetic algorithm zero discharge
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Synthesis of Water Utilization System Using Concentration Interval Analysis Method ( Ⅰ ) Non-Mass-Transfer-Based Operation 被引量:3
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作者 刘永健 袁希钢 罗祎青 《Chinese Journal of Chemical Engineering》 SCIE EI CAS CSCD 2007年第3期361-368,共8页
A strategy for water and wastewater minimization is developed for continuous water utilization systems involving fixed flowrate(non-mass-transfer-based)operations,based on the fictitious operations that is introduced ... A strategy for water and wastewater minimization is developed for continuous water utilization systems involving fixed flowrate(non-mass-transfer-based)operations,based on the fictitious operations that is introduced to represent the water losing and/or generating operations and a modified concentration interval analysis(MCIA) technique.This strategy is a simple,nongraphical,and noniterative procedure and is suitable for the quick yields of targets and the identification of pinch point location.Moreover,on the basis of the target method,a heuristic-based approach is also presented to generate water utilization networks,which could be demonstrated to be optimum ones. The proposed approaches are illustrated with example problems. 展开更多
关键词 water minimization water utilization network targeting method concentration interval table
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Design Energy Efficient Water Utilization Systems Allowing Operation Split 被引量:5
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作者 廖祖维 武锦涛 +2 位作者 蒋斌波 王靖岱 阳永荣 《Chinese Journal of Chemical Engineering》 SCIE EI CAS CSCD 2008年第1期16-20,共5页
This article deals with the design of energy efficient water utilization systems allowing operation split. Practical features such as operating flexibility and capital cost have made the number of sub operations an im... This article deals with the design of energy efficient water utilization systems allowing operation split. Practical features such as operating flexibility and capital cost have made the number of sub operations an important parameter of the problem. By treating the direct and indirect heat transfers separately, target freshwater and energy consumption as well as the operation split conditions are first obtained. Subsequently, a mixed integer non-linear programming (MINLP) model is established for the design of water network and the heat exchanger network (HEN). The proposed systematic approach is limited to a single contaminant. Example from literature is used to illustrate the applicability of the approach. 展开更多
关键词 water utilization network heat integration wastewater minimization operation split
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Identification of contamination source in water distribution network based on consumer complaints 被引量:3
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作者 陶涛 黄海东 +1 位作者 信昆仑 刘书明 《Journal of Central South University》 SCIE EI CAS 2012年第6期1600-1609,共10页
A new methodology was proposed for contamination source identification using information provided by consumer complaints from a probabilistic view.Due to the high uncertainties of information derived from users,the ob... A new methodology was proposed for contamination source identification using information provided by consumer complaints from a probabilistic view.Due to the high uncertainties of information derived from users,the objective of the proposed methodology doesn't aim to capture a unique solution,but to minimize the number of possible contamination sources.In the proposed methodology,all the possible pollution nodes are identified through the CSA methodology firstly.And then based on the principle of total probability formula,the probability of each possible contamination node is obtained through a series of calculation.According to magnitude of the probability,the number of possible pollution nodes is minimized.The effectiveness and feasibility of the methodology is demonstrated through an application to a real case of ZJ City.Four scenarios were designed to investigate the influence of different uncertainties on the results in this case.The results show that pollutant concentration,injection duration,the number of consumer complaints nodes used for calculation and the prior probability with which consumers would complaint have no particular effect on the identification of contamination source.Three nodes were selected as the most possible pollution sources in water pipe network of ZJ City which includes more than 3 000 nodes.The results show the potential of the proposed method to identify contamination source through consumer complaints. 展开更多
关键词 water distribution network contamination source IDENTIFICATION consumer complaints
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Optimal Dispatching of Large-scale Water Supply System 被引量:3
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作者 吕谋 SONG Shuang +1 位作者 Zhao Hongbin ZHANG Tuqiao 《High Technology Letters》 EI CAS 2003年第2期21-26,共6页
This paper deals with the use of optimal control techniques in large-scale water distribution networks. According to the network characteristics and actual state of the water supply system in China, the implicit model... This paper deals with the use of optimal control techniques in large-scale water distribution networks. According to the network characteristics and actual state of the water supply system in China, the implicit model, which may be solved by utilizing the hierarchical optimization method, is established. In special, based on the analyses of the water supply system containing variable-speed pumps, a software tool has been developed successfully. The application of this model to the city of Shenyang (China) is compared to experiential strategy. The results of this study show that the developed model is a very promising optimization method to control the large-scale water supply systems. 展开更多
关键词 optimal dispatching water supply system water networks implicit model hierarchical optimization
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Comparative study on identification methods of pipe roughness coefficients in water networks 被引量:2
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作者 刘永鑫 邹平华 《Journal of Harbin Institute of Technology(New Series)》 EI CAS 2011年第6期133-138,共6页
In this paper,we improve object functions and constraint conditions of genetic algorithms (GAs) applied in PRCs identification of water networks.This identification method can increase calculation efficiency,but can n... In this paper,we improve object functions and constraint conditions of genetic algorithms (GAs) applied in PRCs identification of water networks.This identification method can increase calculation efficiency,but can not solve an identification problem with infinitely many solutions well.Then we propose PRCs identification based on the minimal norm method,which satisfies observability conditions and has advantages of high computing efficiency and short time consumption.The two identification methods are applied in a water network,and their identification results are compared under the same conditions.From the results,we know that PRCs identification based on the minimal norm method has advantages of higher computing efficiency,shorter time consumption and higher precision. 展开更多
关键词 PRCs water networks IDENTIFICATION improved GA minimal norm method
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