J. Cent. South Univ. Technol. (2007)03-0436-06

DOI: 10.1007/s11771-007-0085-1牋 牋牋牋牋牋牋

Optimal operation of water distribution networks under local pipe failures

TIAN Yi-mei(田一梅)¹, G. Y. FU², CHI Hai-yan(迟海燕)¹, LIU Ye(刘 烨)¹

(1. School of Environmental Science and Engineering, Tianjin University, Tianjin 300072, China;

2. Department of Civil Engineering, Faculty of Applied Science, University of British Columbia,

Vancouver, BC, V6T 1Z4, Canada)

Abstract: The optimal operation of water distribution networks under local pipe failures, such as water main breaks, was proposed. Based on a hydraulic analysis and a simulation of water distribution networks, a macroscopic model for a network under a local pipe failure was established by the statistical regression. After the operation objectives under a local pipe failure were determined, the optimal operation model was developed and solved by the genetic algorithm. The program was developed and examined by a city distribution network. The optimal operation alternative shows that the electricity cost is saved approximately 11%, the income of the water corporation is increased approximately 5%, and the pressure in the water distribution network is distributed evenly to ensure the network safe operation. Therefore, the proposed method for optimal operation under local pipe failure is feasible and cost-effective.

Key words: water distribution network; local pipe failure; macroscopic model; optimal operation

1 Introduction

Local pipe failures in water distribution networks, such as water main breaks, take place frequently in water corporations in China. They normally cause water shutting down and traffic block in an accidental area. During repair, they also result in some surrounding industries shutting down, no water supplying for the residents and water pressure decreasing in the downstream area of the pipe failure. The losses are costly^[1]. Although water corporations have adopted all kinds of preventive measures, sometimes water main breaks still occur due to a variety of factors including pipe materials, temperature, pressure, traffic loading and construction. Therefore, not only will water corporations repair broken water mains immediately when local pipe failures happen, but also they will need to adjust operations of the whole water distribution network properly in order to minimize the affected area. The latter part can be achieved by establishing optimal operations using computer analysis of water distribution network under local pipe failures.

At present, there are three aspects of research on local pipe failures of a water distribution network. The first one focuses on structure and hydraulic reliabilities^[2-4]. The second one deals with a fault diagnosis^[5-6], in which a neural network-based inverse analysis method was established to detect the location and extent of the fault and the affected area^[7]. The third one emphasizes operation of water distribution networks under pipe failures^[8], in which a simple method was presented for estimating the impact of component failures on overall network, based on a calculation of microflow distribution. However, the study on how to optimize operation of a water distribution network under local pipe failures has not been reported.

?/span>If a city has good network operating conditions, such as an adequate capacity of water supply, a sufficient regulating capacity of reservoirs and a reasonable arrangement of pump stations, not only will the operation of its water distribution network under local pipe failures be easy and flexible but also it will be reliable. However, it is a challenge if the construction fund is inadequate, water supply facilities are incomplete and the network is too old and without proper maintenance. Currently, operation of a distribution network under local pipe failures is completely dependent upon operators?experience. Thus it is difficult to estimate and judge the impact of each pipe failure and whether the operation alternative is proper. Therefore, it is necessary for water suppliers in China to use scientific methods to achieve a computer-aid optimal operation of water distribution network under local pipe failures.

The objectives of the optimal operation of wate distribution network under local pipe failures are to meet customer’s demands for quantity and pressure as much as possible, minimize the affected area, supply water safely, and reduce energy consumption through reasonably managing water supply facilities in a water distribution network. It is necessary to address that the local pipe failures in a water distribution network in this paper mean serious pipe failures significantly impact water use for customers and operation alternative on the overall network must be adjusted.

The computer-aid optimal operation of water distribution networks under local pipe failures includes two processes as follows. Firstly, the pipe failure condition is simulated.Secondly, according to optimal objects and system conditions, an optimal operation model is established and solved to provide an optimal operation alternative.

2 Operation simulation under local pipe failures

In order to achieve computer-aid optimal operation, it is necessary to evaluate the working conditions of a water distribution network under local pipe failures. Based on the actual operating data, the macroscopic model is generally established to simulate water distribution networks^[9-10]. However, the whole network configuration will change when some pipelines or local networks are shut down during pipe failures, which causes the macroscopic model changing. Therefore, the macroscopic model needs to be re-established when simulating a network under pipe failures. However, the macroscopic model for one pipe failure should be established with a large amount of operating data under the pipe failure. On the other hand, the operating data under one pipe failure are not enough since the chance of the same pipe failure is low. Therefore, the macroscopic model for one pipe failure can be established not only based on operating data, but based on mathematical simulation. That抯 to say, the network is simulated by computer according to the hydraulic theories. Firstly, various conditions under pipe failures on the simulated network will be obtained with a computer by shutting down valves in certain pipelines. Secondly, characteristic data of the network, such as node flows, node pressures, and water flow and pressure supplied by each pump station, are obtained by conducting a hydraulic analysis under pipe failures. Finally, through analyzing hydraulic calculation results and selecting water flow and pressure supplied by each pump station as well as some node pressures (monitoring point pressures), the macroscopic model under pipe failures is established using statistical regression, which is expressed as follows:

+

(1)

where-Q_t is the total water consumption of the network; Q_i is the water flow supplied by the pump station i; HPP_j and HPM_k are predicted pressure of the pump station j and monitoring point k, respectively; HP_j is the pressure supplied by pump station j; HM_k is the pressure at the monitoring point k; t and (t-1) are the tth time period and (t-1)th time period in the network under pipe failures, respectively; m is the number of monitoring points; n is the number of pump stations; α is a coefficient, 1.85- 2.00；A1_ij, A2_ij, B1_ij, ?/span>D1_ij and D2_ij are regression coefficients determined by the stepwise regression, respectively.

The macroscopic model under pipe failures is to establish the relationship between state parameters and network configuration under local pipe failures. It predicts the pressure at each monitoring point during the tth time period by the water flow supplied in the network under pipe failures during the tth time period and the pressure of each monitoring point during the (t-1)th time period. This macroscopic model also indicates that different water supply alternatives for network under pipe failures lead to different pressure distribution and different affected extent of a pipe failure. Therefore, the most reasonable alternative can be selected from all kinds of possible alternatives when combining with the objectives of optimal operation.

3 Mathematic model for optimal operation of water distribution networks under local pipe failures

3.1 Objective function

Optimal operation of a water distribution network under local pipe failures is based on analysis and balance of variety of urgent operation alternatives. The most reasonable operation alternative can be obtained under the conditions of assuring the quality of water supply service as much as possible, minimizing the impact of a pipe failure, supplying water safely and reducing energy and cost. Therefore, the objectives of the optimal operation of water distribution networks under pipe failures are given as follows.

1) The network water supply is to meet quantity demands from consumers as much as possible and thus minimize the affected area. That is, the differences between water supply and demand are minimal as follows:

minF₁= (2)

2) The improper operation alternatives should be avoided. They lead to excessively high pressures and cause a higher potential of pipe failures near the pump station while they may result in excessively low pressures that cannot meet the demand of the consumers in the downstream area of the pipe failure.

The pressure at every monitoring point in a network must be close to the pressure demand in this point, that is, the square sum of differences between supply and demand pressures of every monitoring point in a network is minimal as follows:

 (3)

where-HM_k is the predicted pressure of the monitoring point k based on the macroscopic model under pipe failures; HM_k,f is the pressure demand of the monitoring point k under pipe failures.

The determination of HM_k,f is dependent upon pipe failure locations. Different pipe failure locations result in different pressure demands in every monitoring point in a network. These different pressure demands can be determined by computer simulations and analyses for all different kinds of pipe failures.

3) Optimal operation of water distribution networks under local pipe failures should also achieve economical operation. Net income of the systems is maximized as follows:

     (4)

where i refers to the i th pump station/treatment plant; j refers to the j th pump; S_1i?is the unit income of water supply; S_2i is the unit cost of water production; S_3i?is the unit cost of electricity energy; Q_ij is the flow of the pump；H(Q_ij) is the pump head of the pump；η(Q_ij) is the efficiency of the pump; δ_ij is the mechanical and transmission efficiency of the pump; N_ij is the number of the pumps running; NP_i is the number of pump types; r is the conversion coefficient.

The first item is the income of water supply, the second is the cost of water production, and the third is the electricity expenses of water supply.

  3.2 Constraint condition

The optimal operation of water distribution networks under pipe failures should meet the constraint conditions as follows.

1) Conservation of energy: pressure head of pumps should be equal to the pressure that water distribution networks demand

H(Q_ij)-Z_ij-h_ij=HPP_i(5)

where HPP_i is the pressure demand of water distribution networks, which is determined by the macroscopic model for water distribution networks under pipe failures; Z_ij is the height from suction reservoir water level to pump axis; h_ij is the head loss of suction pipe of the pump.

2) Limitation of water supply capacity in the pump station:

Q_i,min≤Q_i≤Q_i,max(6)

where Q_i,min and Q_i,max are the minimum and maximum limits of the flow supplied by the pump station, respectively.

3) Limitation of flow supplied by each pump: the pumps need to be operated in the high efficient section:

Q_ij,min≤Q_ij≤Q_ij,max(7)

where Q_ij,min and Q_ij,max are the minimum and maximum flow limits of the pump in the high efficient section, respectively.

4) Limitation of the number of pumps running:

0≤N_ij≤N_ij,max (8)

where N_ij,max is maximum number of the pumps j running in the pump station i.

5) Each monitoring point pressure in the network should be controlled in the permitted range:

HPM_k,min≤HPM_k≤HPM_k,max(9)

where HPM_k,min and HPM_k,max are the minimum and maximum limits of pressure demand at the monitoring point k under pipe failures, respectively.

3.3 Mathematic model and solution for optimal

operation under local pipe failures

Based on the above discussion, the optimal operation of water distribution network under local pipe failures is a multi-objective problem. However, these objectives relate to each other. In resolving this model, we convert the multi-objective problem into single-objective problem using the multiplication and division method. After standardizing each objective (F_i→f_i, f_i means the sub-objective i by standardization), the objectives in solving the minimum and maximum are taken as numerator and denominator, respectively. Then the single-objective problem pursues the minimum. The mathematic model for the optimal operation of water distribution networks under pipe failures is given as follows:

minF=(10)

   (11)

　The above model pursues the goals of the reasonable pressure distribution and the maximum water supply net income in a network under local pipe failures to meet flow and pressure demands from the consumers as much as possible.

　There are continuous variables (Q_ij) and discrete variables（N_ij）in the model. So it is a mixed integer non-linear programming problem. Generally, the model can be solved by a mathematical programming. However, it is difficult to solve the model for a more complex distribution network with many types of pumps, which results in too many variables^[11-12]. At present, genetic algorithms (GAs) are developing gradually, in particular, GA can effectively solve the complex function optimization problems, which cannot be solved using conventional optimization methods. More GAs are being used to solve various optimization problems of science research and real project in many fields such as the optimal design and operation of water distribution networks^[13-15]. In this study, the GA was used to solve the model.

  4 Analysis of example

The example was obtained from a city in north China. Average water supply is 2.0?/span>10⁵ m³/d in the water distribution network. The diameter of the entire pipes is in the range of 100-1 000 mm. In the city, there are 2 water treatment plants with 2 pump stations in which 16 pumps belong to 5 types, and 7 pressure-monitoring points in the distribution network. In this paper a DN800 mm pipe failure was used as an example to explain the optimal operation alternative solving process.

4.1 Simulation of distribution networks under local pipe failures

　　Firstly, the total water consumption in the distribution network is predicted after the accidental valve segments(valve segment means when a pipeline breaks, some valves located near the failure pipeline will be shut down to form a closed area to cut off the relation between the failure pipeline and the nearby pipelines) are shut down under pipe failures. Data of 24 h from several consecutive days of the example in July 2004 were collected. The data included total water consumption and the corresponding water consumption in the accidental valve segments. The data in 7 d (168 h) were selected and the difference between the total water consumption and the water consumption in the accidental valve segment was used as the sample data under the failure condition. The water consumption in the 8th day (24 h) was predicted using the back-propagation (BP) artificial neural network. The predicted water consumption was 1.045 5?/span>10⁴ m³/h in the operating time period.

In order to rapidly simulate the operation conditions of a network and find the optimal operation alternative under a pipe failure, the macroscopic models for each pipe failure were established in advance according to the methods in section 2 in this paper. A hydraulic analysis under pipe failures was conducted by the computer simulation and 96 groups of data were obtained. The macroscopic model under pipe failures was established with the former 48 groups of data and was verified with the latter 48 groups of data. The verification results of the macroscopic model at every monitoring point are given in Table 1. The average relative error is in the range of 5%-10% when the pressure is predicted by the macroscopic model at every monitoring point under pipe failures. The precision is within an acceptable range for a purpose of practical applications.

Table 1 Verification results of macroscopic model at

every monitoring point

No.1 and No.2 represent monitoring points in the pump station and No.3-No.9 represent monitoring points in the networks，respectively. R means correlation coefficient of macroscopic model, M_D means average values of the actual monitoring data, while F_D means average values of the predicted data, R_E means an average relative error.

4.2 Establishment of mathematic model and solution

for optimal operation under local pipe failure

There are 5 types of pumps in the 2 pump stations of the distribution network.?So the number of variables is 10 including 5 discrete variables and 5 continuous variables, and the number of constraint is 25. The mathematic models for the optimal operation in a pipe failure were established according to Eqns.(10)-(11) and they were solved by GA. The optimal operation alternative was obtained，which is given in Tables 2 and 3.

4.3 Analysis and discussion

Based on the hydraulic analysis, the operation records for the real distribution networks and the simulation for empirical operation methods, the pumps in the example were operated twice by manual experience in order to investigate the social and economic benefits from the optimal operation under a pipe failure. Technical and economical parameters for each operation alternative are given in Table 4.

　Based on the analysis of each operation alternative, it is learnt that the first empirical operation alternative is to meet the consumers?demand for flow and pressure. Because the operation alternatives cannot be discriminated one by one, the pressure heads supplied in the first and second pump station are increased by牋 69.1 kPa and 45.5 kPa comparing with the pressure demands, respectively. As a result, the pressures are excessively high in a part of the distribution network. The excessively high pressure results in the leakage in the distribution network and causes more energy cost in water supply such as approximately 11% electricity cost higher than the optimal operation. This increases the chance of potential problems in some weak parts of the distribution network and adds risks to the safe operation in the water distribution network. Meanwhile, the second empirical operation alternative is expected to ensure the safe water supply but the decreased pressure and flow cause approximately 8% less supply than demand. Moreover, the pressures at the two monitoring points in the distribution network are decreased by 60.2 kPa and 54.9 kPa, respectively, which leads to pressure decrease of 50-100 kPa in surrounding areas. As a result, the impact of the pipe failure is expanded without intention and the quality of water supply service is deteriorated. On the other hand, not only does the performance of the optimal operation meet the water demand of the consumers, but also it provides the consumers with sufficient pressures at the end of the distribution network and avoids excessively high pressures in some regions. The optimal operation alternative is selected among a lot of pump sets operation alternatives in order to make the pressure distribution evenly in the water distribution network, while it also increases income of the water supply corporation approximately 5%. Therefore, the

Table 2 Optimal operation alternative-pump operation alternative in each pump station

Table 3 Optimal operation alternative-pressure of every monitoring point in network

H_D represents pressure demand in every monitoring point; H_O represents optimal pressure in every monitoring point.

Table 4 Technical and economic parameters for each operation alternative

optimal operation under local pipe failure can enhance the social and economic benefits from water supply.

5 Conclusions

1) A method for the optimal operation of water distribution networks under local pipe failures was presented, which included a macroscopic model for simulating a water distribution network under local pipe failures, an optimal operation model with the multi-objectives, and their solutions.

2) The case study presented shows that using the optimal operation for a water distribution network under local pipe failures has several advantages. The first is that operators can make better estimation and judgment for the affecting extent of each pipe failure and whether each operation alternative is reasonable. The second is to make the operation decision more scientific, which is to ensure the network safety and enhance the social and economical benefits for water supply corporations.

3) The methods proposed can also be applied for network maintenance, such as choosing maintenance alternative of main pipes or main valves, improving the operation techniques for water supply corporations.

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(Edited by LI Xiang-qun)

Foundation item: Project(50278062) supported by the National Natural Science Foundation of China; Project(003611611）supported by the Natural Science Foundation of Tianjin, China

Received date: 2006-06-28; Accepted date: 2006-10-22

Corresponding author: TIAN Yi-mei, Associate professor; Tel: +86-22-27408298; E-mail: ymtian_2000@yahoo.com.cn