Link changes with change in demand in Flow Distribution Networks

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1 2016 International Conference on Information Technology Link changes with change in demand in Flow Distribution Networks Vishal Gupta Department of Computer Science and Information Systems BITS Pilani India Abstract Water Distribution Networks (WDNs) are the foundation and backbone of any city. There maintenance and energy costs demand a significant amount of planning. Judicious scheduling operations can prompt significant savings in energy and prevent disruptions in supply and damages. Water passes through a great deal of system hubs (like pumps, valves, tanks, and so on) while moving from source (reservoir) to individual customers. Such system turn into an uncommon instance of packet switched network, and can be modeled based on networking theory. Similar to bandwidth of each network link, every link in a WDN has a fixed and predefined capacity. Given a source and destination node, there can be a specific estimation of most extreme stream of water which is possible between them. In this paper, our aim is to study the link changes in WDN. Depending upon the change in demand at the destination node, the proposed algorithms recommend optimal modifications in the connecting links of the existing network to increase the maximum possible flow. Keywords Packet Switched Network; Water Distribution Network; maximum flow; flow capacity I. INTRODUCTION Water is one of the most important resources for all human beings and for any infrastructure. It is the primary need for everyone. Therefore uninterrupted and timely supply of water is a highly required aspect of any Water Distribution System (WDN). WDN consist of an interconnected series of pipes, tanks, valves, and other components through which water can flow. Such distribution systems are responsible to transport water from the treatment plant or water-source to the consumer. Typically, the size of water mains, link capacities and, volume of storage reservoir is determined by needs of consumers. A good distribution system should supply water at the destination with adequate pressure, quality, and quantity. The WDN architectural layout should be such that availability is ensured in the case of components wear and tear, and/or breakdown of some parts to a certain extent. This means that there should be a provision of alternate paths for the flow. In addition, maintenance of the distribution system should be easy and economical. There are following four types of distribution networks [2]: 1. Dead End System: In this one main link runs through mainly through the center of the densely populated area and multiple sub links branch off from it. 2. Grid Iron System: It is ideal for places which are laid out as a rectangular plan. Here again main link runs through the center and sub links branch off in perpendicular directions to it. 3. Ring System: Here main link forms a ring around a distribution system. 4. Radial System: Here whole area is divided into multiple sub-areas. Each of the subarea has a central reservoir through which supply is given. A WDN once deployed need to be modified over time since the demand of consumers change. To accommodate this change, we need to replace/modify some intermediate link(s) between the source and the destination. This change can be brought about by changing the valve capacity or the pipe within it. Since this require structural change in the network, it should be optimized because a cost factor is involved. For this, we need to find the path which requires minimum cost to accommodate this change. Such WDNs can be modeled on the existing theory of packet switched distribution networks [1, 7] and Graph Theory. Though packet switched networks can be controlled in distributed fashion, it is more appropriate to control WDN's using centralized approach. In packet switched networks, packets flow from source to destination; and these packets carry information using which they are routed by each intermediate node. Whereas, in WDN's, the flow is of water which do not carry any information. Therefore, a centralized approach is more appropriate to find routes (and thus controlling the network) between the source and destination. United States Environmental Protection Agency (EPA) developed an open source software tool, called EPANET [3-5], to model a WDN. It can be used for running hydraulic and water quality simulation in WDNs. We know that WDN's consist of reservoirs, tanks, pumps, pipes, and nodes. EPANET track the flow of water in each pipe, the pressure at each node, the height of water in each tank, and the concentration of chemical species throughout the network during a simulation period comprised of multiple time steps [4]. In addition to chemical species, water age and source tracing can also be simulated. It has various advantages; like it /16 $ IEEE DOI /ICIT

2 can simulate large sized systems. Frictional head loss along with minor head loss for bends is calculated using Hazen Williams [8], Darcy-Weisbach [9] or Chezy-Manning formula [10]. Finally, the cost of the water supply system setting can be computed. This paper present techniques and a novel means to deal with meeting the demand of a particular destination by applying network flow algorithms. Considering the flow along a pipe and other parameters like diameter of the pipe etc., WDN is modeled as a graph. This graph serves as a model to determine the routing of flow and cost to change the flows of the pipe while meeting the increase in demands of the destination. The techniques presented optimize the link changes with respect to cost required to do the change. II. PROBLEM STATEMENT Suppose a WDN with known sources and destinations is modeled in the form of a network graph with nodes representing reservoirs and/or valves to control the flow and edges representing distribution pipes. The edges are labeled with weights. These weights are an outcome of some function of capacity and cost. Limiting the scope to single source and single destination, the problem is to find optimal change(s) in the links of network when the demand at destination node changes; thus resulting in changed routes of flow. More specifically, by varying demand at the destination node in uniform steps, the problem at hand requires us to optimally suggest modifications in the network so that destination s demand is met. We also need to get the structure of network at each step for analysis. This is a problem which is faced very regularly in real life scenario. Consumer s demand might change or new households need to be added. Also, there may be a breakage at some link because of which we need to divert the flow to some other path. Such situations require us to modify the network. It is required to find cost efficient ways for doing the same. III. EPANET Although, the problem statement given in Section II above can be modeled as a network graph, we need a tool to capture and evaluate water distribution network. EPANET is an open source flow analysis tool for flow distribution networks. It perform simulation of quality behavior within pipe networks. It was developed by the US Environmental Protection Agency s Water Supply and Resources Division. An EPANET network comprise of pipes, nodes, pumps, valves and storage tanks. The input is given in the EPANET input format, i.e..inp format. The output is a binary file which give water quality results at uniform time intervals. EPANET provide various extensions to the programmer to run their own simulations. This helps to customize the functions and working of EPANET according to their own needs and after modifications the programmer can produce dynamic link library (DLL). The Toolkit DLL file is named EPANET2.DLL and is distributed with EPANET. The Toolkit provide a series of functions that allow programmers to customize the use of EPANET s hydraulic and water quality solution engine to their own applications [6]. To implement and test our proposed techniques, the open source code of EPANET was modified with certain additional helper functions. The output we generated include the network description at uniform time intervals and the cost incurred for implementing the change. IV. METHODOLOGY Here, we have designed algorithms to optimize the required change in WDN as a consequence of increase in demand at the destination node. For any given network, there is a maximum flow possible in the existing scenario (given pipes, valves, etc). If there is a change in demand at a node, it might not be possible for the existing network to support that demand. In that case, some of the links should be modified to meet the higher capacity demand. So we need to find out which links to modify. Ford Fulkerson algorithm [11] is used to find out maximum flow through a flow network. This technique is based on residual networks and augmenting paths. We first modified this Ford Fulkerson algorithm to suggest possible change in networks when the demand changes. Also, for the same purpose, another technique is presented and both are compared. Depending upon the algorithm followed, there may be different links possible. We should choose the one with least cost. Various factors are to be kept in mind while deciding the link. Some of these include: Change in capacity Cost of modifying the link Number of links changed Stability and Scalability For modeling the water supply network and implementing our algorithm, we have used the tool EPANET [4, 5]. Two sample networks were taken and the output was generated for them. For each network we found the maximum flow possible, say x. Then we increased the flow in uniform steps from x+y to x+z, where z > y. An algorithm that we devised was applied to find out the change in links to accommodate these changes in flow and the network was modified accordingly. The output consists of modifications introduced in the network at each step and the cost incurred. For cost calculation, it is assumed that the major factors that determine the cost of changing a link depend on: (a) capacity of the new link that is replacing the old one, and (b) its diameter. In fact, capacity and diameter are not necessarily independent of each other. But for simplicity, we have considered both separately in our calculations. Hence, the cost function is as follows: Cost = Diameter * Link Capacity (Eq. 1) The performance is then evaluated and the graphs are plotted for analysis. 100

3 V. PROPOSED TECHNIQUES The input is a network graph with source node s, destination node d, intermediate nodes between s and d, and links connecting these intermediate nodes.. Each link is given a weight, also called as capacity of the link. For evaluating the proposed techniques and comparing them, the demand at destination node is varied; correspondingly the cost of modifying the network is recorded. A. Proposed Technique-1(Algorithm - 1) When the demand increases, the brute force solution is to change all the links as per demand. Such increase is exponential in nature. In this technique (i.e. algorithm-1), we modified the Ford Fulkerson algorithm for finding the maximum flow to meet the changing demand at the destination. Instead of increasing all links flows, this technique selectively increase the like flows so that the cost of changing the links reduces. Figure I show the pseudo code for it. Step 1: Find the Demand of the Destination. Let it be d. Step 2: If the max flow capacity of the network is less than d, then demand can be achieved without modification of any link capacity. Else, follow the steps from 3 to 6. Step 3: Find all the augmenting paths in the network. Step4: Select the smallest Capacity among all augmenting paths. Step First 5: of Increase all, the maximum that smallest flow capacity is calculated by unit using of 1. Ford Step 6: Repeat Step-3 and Step- until the demand of the destination is met. Fig. 1. Algorithm-1 for Proposed Technique-1 B. Proposed Technique-2(Algorithm 2) For this (i.e. algorithm-2), first get the Residual Graph [11] for the network. Now we consider all the links (a, b) in the residual network [11] such that there s a path from source node s to node a; and a path from node b to destination node d. Thus we get a set of possible links that can be changed. It is done using the Dijkstra Shortest Path Algorithm. The bottleneck capacity among such paths is found. Bottleneck capacity refers to the maximum change in capacity possible for that link. The link with the maximum bottleneck is chosen for modification. In case, the network reach a saturated state, when it's not possible to increase the capacity any further by changing one link, we find out the shortest path from s to d and increase the capacities of all the links by some amount. Cost is calculated using the function as specified in Eq 1. It is to be noted that in the implementation of the algorithm in code, there were certain rounding off issues and limitations in calculation of flows, bottlenecks and capacities due to which in some cases, a certain link tended to be selected and changed repeatedly without achieving change in maximum flow. This is an implementation issue and need improvisation to obtain desired results. Thus, it was decided that from the set of possible links that could be changed, a link would be picked at random and changed, and in subsequent iterations all other links would be given a fair chance of being changed by a randomization technique. The pseudo code is as given in Figure 2. Input: A network with nodes V, links E, Capacities of links, Lengths of links. WHILE demand > max_flow total_cost = 0. find all possible links (a, b) along with their bottleneck capacities such that path from s to a and b to d exist. Add this to set P. if (P is empty) find shortest path from src a to dest b. else increase capacity of all links along shortest path by one unit. total_cost = cost (a i,b i), where (a i,b i) is link along shortest path select a link (a i,b i) from P. increase capacity of (a i,b i) by its bottleneck capacity total_cost = cost (a i,b i) calculate max_flow again. Print network details. Return list of total_costs for each demand. Fig. 2. Algorithm-2 for Proposed Technique-2 VI. ANALYSIS AND RESULTS The two algorithms as mentioned in Fig. 1 and Fig. 2 optimally select a link to be modified and calculate the number of links changed. In addition, during each iteration, the algorithms also find the cost of modifying the network. For simulation, we considered two sample networks as shown in Fig. 3 and Fig. 4. The two algorithms are compared for link modifications and cost. Graphs for both of them are drawn to give a clear picture of the working of two algorithms. 101

4 Fig. 3. Network-1 map in EPANET Fig. 5. Changes in Links with change in Demand for Network-1 Fig. 4. Network-2 map in EPANET Using Ford-Fulkerson algorithm, for Network-1 the maximum flow originally possible was 290 units and for Network-2 the value was 110 units. We increased the demand at destination by 10 units at each iteration and observed the change/modifications required in corresponding network. Figure-5 and Figure-6 shows the number of links required to be changed in the two networks respectively with this increase in demand at destination. For Algorithm-2, it is observed that the number of links changed varies from 0 to 4 and 0 to 7 in Network-1 and Network-2 respectively. After the demand reached 380 units in network-1, the number of links changed became constant at 3. While in Network-2, it oscillates between 6 and 7 for demands above 160. Similarly, Figure-5 and Figure-6 shows the performance of Algorithm-1. Comparing the results of the two algorithms, we can see that as the demand at destination increases the number of links to be modified by Algorithm-2 is almost double than that of Algorithm-1. This suggests that the efficiency of Algorithm 2 is almost double as compared to that of Algorithm 1; although experiments on more dense networks are further required to validate this statement. Fig. 6. Changes in Links with change in Demand for Network-2 Another analysis is done for the cost of changing the networks with increase in demand. Figure-7 and Figure-8 illustrate the same. For both the proposed algorithms on Network-1 and Network-2, it is a common observation that the cost is low initially but as we increase the demand, it tends to increase linearly. Thus the performance of both the proposed algorithms is better than the brute force solution; in which case the cost will increase exponentially with increase in demand. Performance of Algorithm-1 shows that the cost calculated by it keeps on increasing with a higher slope right from the start. While in case of Algorithm-2, cost is negligible for initial iterations and later it increases linearly with a lesser slope than the first one. 102

5 in demand at the destination. The algorithms developed works pretty well for the networks under consideration. Results illustrate that increase in cost for modifying the network with increase in demand at destination is linear. It is much better than exponential increase. Future work could include extending the proposed algorithms to work for multiple destinations. Acknowledgment Thanks to Ruthwik Masina, Samarth Dixit, and Shalaka Somani for helping in the implementation of the proposed techniques and critically analyzing the techniques. Fig. 7. Cost versus change in Demand for Network-1 References Fig. 8. Cost versus change in Demand for Network-2 VII. CONCLUSION With the increase in demand at the destination(s) of an already deployed WDN it often becomes a necessity to modify it. We need to find ways to change such WDNs more efficiently and economically. Since these changes require a huge amount of input in the form of money as well as time, we should make sure such changes are as less as possible and incur minimum cost. This paper aims at finding out such algorithms of modifying an existing WDN as per the increase [1] Lee, Young, and James M. Tien. "Static and dynamic approaches to modeling end-to-end routing in circuit-switched networks." Networking, IEEE/ACM Transactions on 10.5 (2002): [2] A.C. Panchdhari. " Water Supply and Sanitory applications", New Age International, Jan [3] L. A. Rossman, EPANET Manual, EPA United States Environmental Protection Agency, Aug [4] EPANET Programmer s Toolkit, EPA United States Environmental Protection Agency, Aug [5] E. Salomons, Waternetgen- EPANET- extension pipe sizing, Aug [Online]. Available: [6] Cheung, P. B., J. E. Van Zyl, and L. F. R. Reis. "Extension of EPANET for pressure driven demand modeling in water distribution system." Computing and Control for the Water Industry 1 (2005): [7] Arora, Sanjeev, Tom Leighton, and Bruce Maggs. "On-line algorithms for path selection in a nonblocking network." Proceedings of the twentysecond annual ACM symposium on Theory of computing. ACM, [8] Liou, Chyr Pyng. "Limitations and proper use of the Hazen-Williams equation." Journal of Hydraulic Engineering (1998). [9] Brown, Glenn O. "The history of the Darcy-Weisbach equation for pipe flow resistance." Environmental and Water Resources History 38.7 (2002): [10] Migliaccio, Kati W., and P. Srivastava. "Hydrologic components of watershed-scale models." Transactions of the ASABE 50.5 (2007): [11] Ford, Lester R., and Delbert R. Fulkerson. "Maximal flow through a network." Canadian journal of Mathematics 8.3 (1956):