Water distribution systems in hilly areas are always divided into several zones due to the undulating terrain. The present approach of dividing water distribution systems lacks an assessment index and is characterized by a low degree of automation. With the building of a mathematical model, this paper introduces two indicators – pressure limitation and pressure variation – to enable the automatic division of the water supply pipe network. It prioritizes economic index as the objective function in the evaluation of the division of water distribution systems in hilly areas, and then selects the optimal division scheme by generic algorithm in a large number of candidates. The SY terrain in YW City China is used for verification. Compared to traditional water supply partition methods, this procedure is easier to operate time-savingly by staff and is more automatic.

## INTRODUCTION

Hilly areas (He 2006) consist of continuous low mountain areas with gentler slopes, an altitude of less than 500 m, with a relative height of no more than 200 m. In China, the total hilly area is approximately 100,000 km^{2}, which accounts for about one-tenth of the country's total area. Overall, the water supply systems in hilly areas are still the weak link in China's water utilities. Usually, hilly areas contain the following landscape features: (1) they are far away from the water source and urban areas, (2) they contain more dispersed water distribution networks, and (3) the terrain elevations in the house group vary greatly. Consequently, the water supply system has many disadvantages, such as the high cost of pipe network construction, imbalance in water pressure distribution and greater difficulty in operation and management in relation to water loss and pipe bursting (Karadirek *et al.* 2011). In hilly areas it is more difficult to divide the water supply system reasonably than it is in flat areas. Many factors, such as the boundaries of administrative divisions, the high and low areas of the terrain, and the water demands of distribution, must be considered. It is also hard to select the control point and minimum service head in pipe network groups Code for Urban Water Supply Engineering Planning (GB 50282-98) suggested that the minimum network service head should meet the requirements of 28 m. The most unfavorable node refers to the node that once its water pressure meets the requirements, the pressure of the entire water distribution system will be able to meet the requirements, too. In China, a water supply system without a booster pump station is called a unified water supply system. It is critical and difficult to select the most unfavorable node and minimum service head when the pump station is integrated with a unified water supply. Improper selection will cause energy waste and low pressure in some consumer nodes.

Many researchers have approached water distribution system division according to the features of natural conditions, the social and economic level, or the requirements of operation stage management (Deuerlein 2008; Izquierdo *et al.* 2009; Herrera *et al.* 2010). Only a few studies have explored methods for the automated creation of district metering areas (DMAs). Tzatchkov *et al.* (2006) and Di Nardo & Di Natale (2010) presented graph theory–based algorithms for water distribution system (WDS) sectorization. Izquierdo *et al.* (2009) and Herrera *et al.* (2010) designed two DMA-partitioning methods based on machine learning, with both graphical and vector information considered. Deuerlein (2008) and Perelman & Ostfeld (2011) developed topological clustering tools for WDS analysis. Diao *et al.* (2013) proposed a methodology that could efficiently decompose real-world all-pipe networks into DMAs on the basis of decomposition theorems for complex systems. However, there is little research systematically on the optimization method of water distribution system division in hilly areas. Lv *et al.* (2010) analyzed the characteristics and problems of water distribution system planning in the Southern Hills, China, and proposed a standard that uses different minimum service head according to the ground elevation in hilly areas.

Overall, more and more approaches, in practice, can be applied to divide water distribution systems in hilly areas on the basis of the development of optimization algorithms. However, there are still some shortcomings. The first is the lack of an assessment index. Many studies have analyzed factors on the division in a real project while failing to establish the quantitative correlation between the influential factors and the optimization objectives. The second is that the existing optimal division method relies more on work experience. In practice, the zoning plans are often determined through empirical method or human–computer interaction to start with, and then validated by establishing models of water distribution systems, rather than being obtained directly from models. As a result, on the one hand this is not conducive to the promotion of optimization methods because of the lack of unified standards; on the other hand, empirical method or human–computer interaction, which requires large amounts of data to be manually inputted is more prone to error, which thus affects the final zoning.

To solve the problems above, this paper intends to develop an evaluation system which has some applicability for the zone planning of water distribution networks, while combining the characteristics of hilly areas. At the same time, a new optimization method of zone planning with a higher degree of automation is put forward to reduce error and improve work efficiency.

## METHOD

This study uses EPANET (Rossman 2002) as the hydraulic analysis software to build up an underlying mathematical model for optimization.

### Partition method

The case of a water supply system with a single water plant has been adopted in this study. It is assumed that the main design of the pipe network has been finished, and that the position and pressure of the most unfavorable node which is the lowest pressure node in the zone has been discovered. Based on these assumptions, this paper introduces the ‘pressure limitation’ and the ‘pressure variation’: two concepts to conduct the topological division of the WDSs. The ‘pressure limitation’ is the maximum or minimum pressure value set according to the pump head of the hydraulic model. It is used to determine whether a specific water node is within the boundary of the initial designated zone. As long as the ‘pressure variation’ is given, the number and size of the divisions would be determined. The ‘pressure variation’ is used to determine whether the surrounding nodes should be incorporated into the same division.

The method introduced in this study contains the following procedures:

Offering the initial pumping head of the station in the water plant, conducting the water distribution network hydraulic calculation, and then searching for the most unfavorable node in the water distribution network.

Determining the most unfavorable node, shown as ‘1’ in Zone

*I*(Figure 1). If its pressure is less than the ‘pressure limitation’*A*of the unified water supply system, there is no need for the districted water supply, and hence division ends. Otherwise, incorporating the node in division*i*(with the same value as*I*), and searching the adjacent nodes simultaneously.Searching the node adjacent to the most unfavorable node, shown as ‘2’ in Figure 1. Checking its pressure variation compared with the most unfavorable node. If the pressure difference between node 2 and the most unfavorable node is higher than ‘pressure variation’

*B*, it suggests that node 2 should be incorporated into the remaining Zone*I**+*1, rather than in the same division with the most unfavorable node. Otherwise, it should be incorporated in division*I*; then move to the next step to search the other adjacent nodes.Checking out all the adjacent nodes according to the condition in step 3, shown as ‘3’ in Figure 1, and then continuing to search on the basis of these satisfactory nodes, shown as yellow dots in Figure 1. The detection of the partition community boundary is complete when there is no longer an adjacent node according to the condition and thus the final sub-partition community

*i*is identified.Return to ‘step 2’ and start to identify Zone

*i**+*1, or finish.

The method applied to locate the booster pump station of each division is described as follows: from the point of view of water supply security, set the booster pump station at the node with the connection of largest diameter. If the connection diameters of multiple nodes are equal, choose the node with the highest elevation, so that the water supply can take full advantage of gravity to lower the booster pump head and reduce energy consumption.

It is not difficult to understand that when ‘pressure limitation’ *A* and ‘pressure variation’ *B* are fixed; the topological division of the network is unique. When the ‘pressure limitation’ *A* and ‘pressure variation’ *B* change in value, various division schemes can be obtained. According to the characteristics of this method, we can obtain a large number of candidate zoning plan schemes within a real project.

### Optimization model

This paper focuses on the method of setting the division scheme and the location of the booster pump reasonably on the premise that water treatment plant and pipeline layout has been determined. Therefore, the main consideration within the economic calculation of the different schemes includes three parts: the power cost of the pump station, the power cost of the booster pump station and its capital cost (Perelman & Ostfeld 2011; Diao *et al.* 2013).

*m*

^{th}pump station, L/s. = efficiency of pump station, %;

*S*= total number of pump station; = connections set on a line from the pump station to its controlled node; = head loss of the

*k*

^{th}pipe of the connections on the line from the

*m*

^{th}pump station to its controlled node,

*m*; = required free head of the controlled node in the network,

*m*; = elevation difference between the controlled node in the network and the water well of the pump station,

*m*.

*i*

^{th}booster pump station,

*L/s*; = head of the booster pump station,

*m*; = efficiency of booster pump station, %;

*V*= total number of booster pump station.

*T*= the operational life of the booster pump station, year;

*I*= the annual interest rate, %; = annual depreciation and overhaul coefficient of the booster pump station, calculated as a percentage of the capital cost of the pump station; = equipment cost of pump station per kilowatt, yuan/KW, used in the calculation of equipment investment; = reserve factor of booster pump unit; = efficiency of booster pump station and power transmission line,%; = capital cost coefficient of the pump station and its auxiliary structures, yuan/m

^{3}, used to calculate the capital cost of the booster pump station. = the scale of water supply, m

^{3}/s, which determines the total capital cost.

*f*are same as above mentioned.

_{1}, f_{2}, f_{3}### Constraints

*m*;, = the minimum and maximum head of the pump, respectively,

*m*.

*i, m*; , = the lower and upper limit of free node at node

*I*, respectively,

*m*.

### Model solution

According to the above network topology partitioning method, the values of ‘pressure limitation’ and ‘pressure variation’ can be randomly defined, so there are a wide range of preliminary results of network topology partitions. Assuming there are 100 possible partitioned sections, and each partition has two choices: zone water supply with booster stations, and unified water supply with pumping stations, so the total schemes number will reach 2^{100}. It is difficult and extremely time-consuming to find the optimal solution through the enumeration method. So it requires an intelligent optimization algorithm to quickly search the optimal solution (or approximate optimal solution). In this study, the genetic algorithm is adopted to solve the model.

The fitness function should be convex (i.e. the existence of local best value point) in the genetic algorithm. The greater the number of partitions, the more easy it is to control the pressure in a reasonable range, but it will increase the construction and maintenance costs of the pumping station. The fewer the partitions, the less the pumping station construction and maintenance costs will be, but it will increase the operating costs of pumping station. Therefore, the cost function can be constructed in line with the requirements of the genetic algorithm.

In this study, with a known network topology partition, the genetic optimization algorithm is adopted to find the real water partitioning scheme, following these steps:

Basic parameter setting. Select the population size

*N*, crossover probability*P*_{c}, mutation probability*P*_{m}and the maximum iteration algebra*G*_{max}, and termination conditions. For a smaller-scale pipe network, population size is 100 which is between 20 and 100 empirically, crossover probability is about 0.8 which is between 0.4 and 0.9 empirically, mutation probability can be about 0.05 which is between 0.0001 and 0.1 empirically, and the maximum iteration algebra is 1,000.Initialization. A random initial population of water partitioning scheme group

*[A]*is generated by random function*Random ()*. In this paper, the code ‘1’ represents setting booster pump station with partitioning water supply, and ‘0’ represents unified water supply with pumping station. If all the codes are ‘0’ in all partitions, it means there is no partition. If they are all ‘1’, it means that all candidates have set up a water booster station to supply water partition.Individual fitness calculation. Individual fitness calculation is to calculate the economic costs of different partitioning schemes, and the objective function is to minimize the fitness function. In this study, when a partition is coded as ‘1’, the construction and operation expenses of the booster pump station in this partition would be calculated; when it is coded as ‘0’, we only consider the influence of the most unfavorable node in this partition to the head of pumping station. As the pumping station construction investments and operating costs of each partition can be calculated in advance, the constraint variable of fitness function is the water head value of the pumping station under a certain partition scheme.

Termination condition and optimization criteria. If the current fitness value meets the accuracy requirements, then the end of the calculation, and results output,

*[Ax]*, is the final solution of the water partition scheme. If the current fitness value does not satisfy the requirements of the termination conditions, then the generation of individual will be proceeded to selection, crossover and mutation, and generates the next generation of individual populations, return to step 3, until reaching the termination condition.When the termination condition is reached, the individual with the largest fitness can be output as the optimal solution in the evolutionary process, and the calculation terminated.

### Case study

SY is located in the southern zone of YW city, with a ground elevation of 51–91 m. There is a topographic map (Figure 2) illustrating the elevation of all nodes in the system. The water system of the two towns in SY is decentralized. In 2020, the water demand plan of SY will be approximately 79,000 m^{3}/d, which cannot be met by current water supply capacity. Therefore, YW is planning to build a water plant with a treatment scale of 100,000 m^{3}/d in the north of SY.

As the maximum interior elevation difference in SY is up to 35 m, it is appropriate to adopt zone water supply. This research conducts the validation of the automatic approach for dividing water supply systems in hilly areas on the basis of all information regarding SY.

### Hydraulic model and parameters

^{3}. Daily variation coefficient and hourly variation coefficient are specified as 1.2 and 1.4, respectively, and hence the highest daily maximum hourly water consumption is approximately 4,600 m

^{3}/h.

The adjustment calculation shows that the water head of the water plant should be set to 31.3 m to guarantee 28 m free head for the most unfavorable node through the unified water supply. Although the water head is not too high, the pressure distribution is extremely non-uniform because of the large elevation difference. As shown in Figure 3, the highest interior pressure is up to 59.5 m, and the pressure is higher than 50 m in large areas (yellow nodes). Excessive free head not only results in a waste of energy, but also increases the probability of damaging the water appliances. Hence, it is necessary to carry out reasonable division for the projected water network, so as to achieve a more uniform pressure distribution and lower energy consumption.

There are some requirements for water pressure, as follows:

Minimum service head. According to the recommendation of the Specification of Urban Water Supply (GB 50282-98), this research specifies 28 m as the minimum service head.

Upper limit of the maximum pressure is specified to be 60 m.

Pressure limitation

*A*and pressure variation*B*. To reflect the randomness of the division and select the most optimal solution from a large amount of alternative schemes, pressure limitation*A*integrated with the initial water head of the pump station (*H*_{0}) should be set in the model, as the values of*A*and*B*are in wide ranges. In this study, pressure limitation*A*is equal to 28 m which is higher than the initial head of the pump station and the nodes with eligible service head. The heads of the water plants' terminal pump stations in YW city, range from 20 to 40 m and the maximum value is 50 m, which is the head of the water plant in YJS. Considering these data as the reference, the calculation is conducted with the value of*B*from 2 to 22 m. (22 equals 50 minus 28).

### Objective function

The local electricity price for water supply in YW city is 0.84 yuan/(kW·h). The energy variation coefficient is equal to 0.69 through calculations based on the data provided by the water supply enterprises. The efficiency of the pump station and the booster pump station are both specified to be 0.7. The annual interest rate is 3.5% and the annual depreciation and overhaul coefficient of the booster pump station is 6%. The reserve factor of the booster pump unit is 1.5.

*i*

^{th}booster pump station,

*L/s*; = the head of the booster pump station, m;

*V*= number set of the possible booster pump station.

*T*= the operational life of the booster pump station, year; the definition of , , is same as for Equation (8).

### Analysis of the optimization scheme

This approach to the optimization of water supply system division was applied in different year intervals. Choosing 5, 10, 20, 30 years as operational life, respectively, optimization schemes for water supply partition in different conditions can be obtained. Presented here are several representative outcomes corresponding to different pressure differences, being 6 m, 8 m, 10 m, 14 m, respectively. The details are shown in Table 1.

Scheme ID | Pressure difference (m) | Community number | Total cost of the scheme (thousand yuan) | ||||||
---|---|---|---|---|---|---|---|---|---|

5 y | 10 y | 20 y | 30 y | 5 y | 10 y | 20 y | 30 y | ||

1 | 6 | 1 | 1 | 1 | 2 | 9,062 | 19,572 | 46,881 | 85,285 |

2 | 8 | 0 | 0 | 3 | 3 | 9,108 | 19,926 | 47,811 | 85,395 |

3 | 10 | 0 | 0 | 0 | 0 | 9,108 | 19,926 | 48,035 | 87,684 |

4 | 14 | 0 | 0 | 0 | 0 | 9,108 | 19,926 | 48,035 | 87,684 |

Scheme ID | Pressure difference (m) | Community number | Total cost of the scheme (thousand yuan) | ||||||
---|---|---|---|---|---|---|---|---|---|

5 y | 10 y | 20 y | 30 y | 5 y | 10 y | 20 y | 30 y | ||

1 | 6 | 1 | 1 | 1 | 2 | 9,062 | 19,572 | 46,881 | 85,285 |

2 | 8 | 0 | 0 | 3 | 3 | 9,108 | 19,926 | 47,811 | 85,395 |

3 | 10 | 0 | 0 | 0 | 0 | 9,108 | 19,926 | 48,035 | 87,684 |

4 | 14 | 0 | 0 | 0 | 0 | 9,108 | 19,926 | 48,035 | 87,684 |

It can be concluded from the analysis of the operational results that among several conditions, a relatively better performance will be conducted when the pressure difference is 6 m and it is showed clearly that the total cost is a minimum in scheme 1 (shown in Table 1). Compared with unified water supply, the longer the operational years, the more obvious the economic advantages of the division of the water supply will be.

The result of the optimization of water supply partition in SY shows that when the pressure limitation is 28 m and the pressure difference is 6 m, a relatively better partition scheme can be obtained (total cost is minimum). A more detailed introduction to the specific optimization scheme obtained under such conditions with 30 years' operational time is given as follows, as well as a comparison with the unified water supply scheme.

1. Results of optimization scheme. Two partition communities are obtained from the optimization scheme. Water flows are 82.0 L/s and 104.0 L/s, respectively. Heads of the booster pump stations are 32.4 m and 28.0 m, respectively. Disposable investment costs are 213 thousand yuan and 258 thousand yuan, respectively. Depreciation and overhaul costs are 10.5 thousand yuan and 12.9 thousand yuan, respectively. Annual operating costs are 112.6 thousand yuan and 123.4 thousand yuan, respectively. The details are shown in Tables 2–4.

Community ID | Nodes number/percentage | Water flow (L/s)/percentage | ||
---|---|---|---|---|

1 | 7 | 4.12% | 82.0 | 6.41% |

2 | 10 | 5.88% | 104.0 | 8.12% |

Community ID | Nodes number/percentage | Water flow (L/s)/percentage | ||
---|---|---|---|---|

1 | 7 | 4.12% | 82.0 | 6.41% |

2 | 10 | 5.88% | 104.0 | 8.12% |

Community ID | Node index of pump station | Most unfavorable node index | Pressure difference/m | Recommended pump head/m |
---|---|---|---|---|

1 | 91 | 92 | 4.45 | 32.4 |

2 | 63 | 63 | 0 | 28.0 |

Community ID | Node index of pump station | Most unfavorable node index | Pressure difference/m | Recommended pump head/m |
---|---|---|---|---|

1 | 91 | 92 | 4.45 | 32.4 |

2 | 63 | 63 | 0 | 28.0 |

Community ID | Disposable investment (thousand yuan) | Overhaul and depreciation expenses (thousand yuan) | Annual operating cost (thousand yuan) |
---|---|---|---|

1 | 213 | 10.5 | 112.6 |

2 | 258 | 12.9 | 123.4 |

Community ID | Disposable investment (thousand yuan) | Overhaul and depreciation expenses (thousand yuan) | Annual operating cost (thousand yuan) |
---|---|---|---|

1 | 213 | 10.5 | 112.6 |

2 | 258 | 12.9 | 123.4 |

The details of the different operational years are shown in Table 5.

Scheme | Head (m) | Construction investment (thousand yuan) | Operating cost (thousand yuan) | Operational years | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 5 | 10 | 15 | 20 | 25 | 30 | ||||

Unified | 313 | 0.0 | 1,698 | 1,698 | 3,455 | 9,105 | 19,920 | 32,760 | 48,020 | 66,130 | 87,650 |

Regional | 252 | 472 | 1,626 | 2,114 | 3,815 | 9,282 | 19,750 | 32,170 | 46,930 | 64,460 | 85,290 |

D-value (thousand yuan) | 417 | 360 | 1,770 | −1,700 | −590 | −1,080 | −1,670 | −2,370 |

Scheme | Head (m) | Construction investment (thousand yuan) | Operating cost (thousand yuan) | Operational years | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 5 | 10 | 15 | 20 | 25 | 30 | ||||

Unified | 313 | 0.0 | 1,698 | 1,698 | 3,455 | 9,105 | 19,920 | 32,760 | 48,020 | 66,130 | 87,650 |

Regional | 252 | 472 | 1,626 | 2,114 | 3,815 | 9,282 | 19,750 | 32,170 | 46,930 | 64,460 | 85,290 |

D-value (thousand yuan) | 417 | 360 | 1,770 | −1,700 | −590 | −1,080 | −1,670 | −2,370 |

## CONCLUSIONS

The methodology proposed in this paper aims to make automatic optimization division of water distribution systems in hilly areas. The biggest advantage of this method is that it is easy to operate and accomplish the division plan automatically. It just needs the water supply network model and hydraulic parameters to complete the partition. Compared with the traditional method, the new method can assess the whole water supply area, and give detailed results about area, pressure, water quantity and boost water pump of every division, and supply guidelines to the water engineers.

It proposes two indicators, pressure limitation and pressure variation, to realize the automatic partition of water supply pipe networks by using EPANET with C# programming language. It chooses water supply pressure, free head of the node and so on, as safety indicators, and the operational cost of the pump stations and the construction investment of booster pump stations as the objective function. Finally, the optimal partition scheme can be selected by generic algorithm among a large number of candidates. An application program for automatic water supply system partition is applied to the real project of SY terrain in YW City. It is tested on the condition that operational life is, respectively, set to be 5, 10, 20 and 30 years, the pressure limitation is 28 m and the pressure variation is 2 to 22 m. The results show that among different conditions, total costs of the scheme are lowest when the pressure variation equals 6 m, and that two booster pump stations should be set up in the area of corresponding peak nodes. With a construction investment of 472 thousand yuan and taking into account the annual overhaul and depreciation expenses, the total costs of the two plans are basically the same in the 7th year. Furthermore, the regional water supply scheme increases economic advantage after the 8th year. In the 30th year, the regional water supply scheme can save an economic cost of 2,370 thousand yuan in comparison to the unified water supply scheme.

The values of pressure limitation and pressure variation will definitely affect the division results, but it is difficult to determine what value is appropriate because of the different water distribution network topology and water supply requirements, so the range of these values should be researched in the future.

## ACKNOWLEDGEMENTS

The data for this paper are available in the report ‘The special planning of water supply pipe network in Yiwu City’, edited by Yiwu Urban Planning and Design Institute in February 2009, and this work was supported by the Major Science and Technology Project – Water Pollution Control and Treatment (No. 2012ZX07403-004) and National Natural Science Foundation of China (Grant Nos. 51478326 and 51378374).