Determining the Pressure Distribution on Water Pipeline Networks Using the Firefly Algorithm

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1 2016 7th International Conference on Intelligent Systems, Modelling and Simulation Determining the Pressure Distribution on Water Pipeline Networks Using the Firefly Algorithm Lala Septem Riza, Jajang Kusnar, Munir Department of Computer Science Education Universitas Pidikan Indonesia Indonesia Riyan Naufal Hays Department of Information Technology Universitas Serang Raya Indonesia Kuntjoro Adji Sidarto Department of Mathematics Institut Teknologi Bandung Indonesia Abstract Along with the increase of population, companies delivering clean water to customers are dealing with some problems on their networks, e.g., flow assurance. It is to guarantee the availability of water on customers or to provide sufficient pressure distribution at any points on the network. This study aims to model and calculate the pressure distribution on the water pipeline networks. Using the analogy of Kirchoff s Law for the electrical current to the flow of water in pipelines, we construct a non-linear equation system for representing the water distribution system. In this research, we consider the Hazen- Williams equation for modeling fluid dynamics. Then, it is solved by using a method of swarm intelligence, namely the firefly algorithm. To provide a better illustration, a case using the Hanoi network is presented. A model and computation determining pressure distribution/head at each point of the network are obtained, along with a comparison with EPANET. Keywords water distribution system; swarm intelligence; model and simulation; pressure distribution I. Introduction It cannot be denied that the human need for clean water is increasing. Since along with the increase of population, the water pipeline networks are growing, the companies are facing many issues related to the networks. One of these problems that should be solved is to ensure the availability of clean water for all consumers, called flow assurance. It is an important task since companies must fulfill the contract with customers. Before analyzing the flow assurance, determining the distribution of the pressure/head at each point is a critical task. Usually, a model of the pressure distribution consists of equations of water flow in the pipes, which are assumed to be steady-state. So, the continuity equation at every point always occurs. According to the assumption, we can construct a non-linear equation system for representing the water distribution system. Here, the system contains non-linear equations since fluid dynamics is expressed in the following equations. For example, we can use Darcy-Weisbach [1], [2], Hazen- Williams [3], Chezy-Manning [3], etc. It is obvious that the more complex distribution pipelinenetworkswehave,thehardersystemofequations to be solved. Several techniques have been carried out by researchers, such as using Jacobian matrix to solve non-linear systems based on the Darcy-Weisbach head-loss models [4]. Genetic Algorithms have been used to solve the equations of water distribution systems [5]. Then, the research using Genetic Algorithms was improved by using Multi Objective Genetic Algorithms [6]. In this study we attempt to implement the firefly algorithm [7] for solving a non-linear equation system constructed on the water distribution system. The firefly algorithm is one of methods included in nature-inspired metaheuristic algorithms based on swarm intelligence. It is inspired by the flashing pattern of tropical fireflies [8]. The remainder of this paper is structured as follows. Section II gives briefly an introduction to fluid dynamics used in this research. Section III presents the firefly algorithm and its pseudocode. In secton IV, we illustrate construction of a model based on the Hazen-William equation. An example of the Hanoi network and its result and discussion are shown in Sections V and VI. Finally, Section VII concludes the paper. II. Fluid Dynamics A. The Bernoulli Equation Therearethreeelementsoffluiddynamicsthatneed to be considered in a pipeline: an elevation (z), a speed (v), and a pressure (p). From these fluid elements, there are three forms of energy as follows: 1) Kinetic or motion energy, which is the energy contained in fluid that has velocity. It can be represented as E K = wv2 2g, (1) where w is a weight of fluid and g is the specific weight. 2) Potential energy, which is the energy that deps on the elevation/position of the object, which is /16 $ IEEE DOI /ISMS

2 expressed by E P = wz. (2) 3) Flow energy, which is the energy required to move the fluid element to other locations: E A = wp g. (3) So the whole energy of the fluid dynamics is the sum of the three energy above as follows: E = E A + E P + E K = wp wv2 + wz + g 2g, (4) where each element has the following units: Newtonmeter (N.m) on SI or foot-pounds (ft lb) on U.S. Customary System. Based on the law of conservation, we can consider the following system of energy on two different points: E 1 = E 2 p 1 g + z 1 + v 2 1 2g = p 2 g + z 2 + v 2 2 2g, (5) which is known as the Bernoulli equation [9]. It should be noted that the unit of force (or weight (w)) are both in the numerator and the denominator so as to mutually eliminated. Thus obtained only unit in meters (m) orfoot(ft) and can be interpreted as the height (height). It is often expressed as head which states height against a reference level. B. The Hazen-William Equation The Hazen-William formula (SI) representing head loss can be expressed by Q =0.2787C k D 0.63 ( hl L ) (6) By considering head loss due to friction (L) on two different points, h L can be expressed by h L = p i p j + z i z j. (7) g Therefore, we have the following equation in U.S. Customary System [3]: ( ( )) Q i j =0.4329C k D 0.63 pi p j + z i z j. L g (8) As in Equation 8, a pipeline connects two nodes: i and j, with length L and diameter D. Then, the rate Q i j declares the water flow rate in the pipeline connecting the node i to j in ft 3 /s while p i and p j denote the pressure at the nodes i and j in lb/ft 2. The elevations, z i and z j,areinft. Moreover, the equation uses the specific weight g, which is equal to 62.4 lb/ft 3 on the temperature of 60 o F. III. The Firefly Algorithm The firefly algorithm is one of swarm-intelligence methods based on nature-inspired metaheuristic inspired by the flashing pattern of tropical fireflies [8]. In particular, although the firefly algorithm has many similarities with other algorithms that are based on swarm intelligence, such as the famous Particle Swarm Optimization [10], Artificial Bee Colony optimization [11], Genetic Algorithms [12], and Bacterial Foraging Algorithm [13], it is very efficient and simple to solve many optimization problems. The firefly algorithm assumes the following three rules [7]: 1) all firelies are unisex so that they can attract each other; 2) attractiveness is proportional to their brightness, which deps on their distances; 3) the objective function determines the brightness of afirefly. According to the assumptions above, we can construct Algorithm 1. There are two aspects including in the input : Objective function f(x), x =(x 1,...,x d ) T output: The best fireflies x Generate initial population of fireflies x i (i =1, 2,...n); Light intensity I i at x i is determined by f(x i ); Define light absorption coefficient γ; while t<maxgeneration do for i =1:n do for j =1:n do if (I j >I i ) then Move firefly i towards j in d-dimension; Calculate attractiveness varies with distance r via exp γr; Calculate evaluate new solutions and update light intensity; Rank the fireflies and find the current best; Algorithm 1: Pseudo code of the firefly algorithm [7]. algorithm above: attractiveness and movement. The first is the function β representing a level of brightness of a firefly to others. It can be expressed as β(r) =β 0 e γr2, (9) where r is a distance of fireflies. Moreover, β 0 is the attractiveness at r = 0 for a given medium with a 32

3 Figure 1. A simple network containing 6 nodes fixed light absorption coefficient γ. The movement of the firefly i to the firefly j is defined by with x i = x i + β 0 e γrij 2 (x j x i )+α(rand 1 2 ) (10) r ij = x i x j = d (x i,k x j,k ) 2, (11) k=1 α representing the randomization parameter, and the random number generator rand being uniformly distributed in [0, 1]. IV. The Model Development A. Water Distribution Networks Water distribution networks consist of a number of pipelines that connect nodes. A node represents a branching point or the inlet/outlet of the water flow to/from the system. The water flows through a pipeline due to differences of head on both s of the pipeline. In some nodes, amount of pressures are known, while the rest will be calculated. For simplicity, here it is assumed the system was in a steady state, isothermal conditions, and no pumps and control valves on the network. Using the analogy of Kirchoff s Law for the electrical current in an electric circuit on the water flow in pipeline networks, we can state that the sum of the amount of water into and out of a node is equal to zero. For example, we have a simple network as shown in Figure 1. By considering Kirchoff s Law, we obtain a non-linear system consisting of six equations (i.e., f) corresponding tosixnodes,asfollows: f 1 = Q N1 Q 1 2 =0, f 2 = Q 1 2 Q 2 3 Q 2 5 Q 2 6 Q 2N =0, f 3 = Q 2 3 Q 3 4 Q 3 5 Q 3N =0, f 4 = Q 3 4 Q 4N =0, f 5 = Q Q 3 5 Q 5N =0, f 6 = Q 2 6 Q 6N =0, (12) where Q N1 is a flow rate of reservoir/supplier while Q 2N, Q 3N, Q 4N, Q 5N,andQ 6N are given flow rates on the outlets2,3,4,5,and6.moreover,q i j,substituted by Equation 8, is a flow rate on the pipeline connecting the following nodes: i and j. Inthiscase,weneedto determine pressure/head on each node (i.e., p) thatgives the values f closed to 0. B. A Solution Using the Firefly Algorithm As we mentioned, in this research we attempt to solve the non-linear equation system by performing the firefly algorithm for determining pressure distribution on each node. Basically, three following steps need to be considered to do this task: defining the fitness function, initializing the firefly parameters, and following the algorithm. First, we can set the fitness function as F (x) = 1 1+f(x) with f(x) = f 1 (x) +f 2 (x) f N (x). Therefore, we transform the problem to be an optimization task by maximizing the value F, which is closed to 1. Then, the parameters required to assign are β 0, γ, and α representing the initial attractiveness, the fixed light absorption coefficient, and the randomization parameter, respectively. Lastly, we just follow the steps as shown in Algorithm 1. V. Experimental Study: Hanoi Network Hanoi Network [14] is a model that is often used by many researchers as a study case and their benchmarking. For example, Tabu search algorihtm was used for water network optimization on Hanoi network [15]. In the study [16], by using Hanoi network, a genetic algorithm was conducted for design of water distribution networks. Optimal design using shuffled complex evolution with the study case Hanoi network has been studied as well [17]. A. Data Gathering Hanoi Network has one reservoir, 31 customers, and no elevation on each node as shown in Figure 2. In this experiment, the following data need to be assigned: pressure/head on reservoir, flow rates on each node, diameter, and length. While the flow rate and pressure on reservoir (i.e., Node 1) are ft 3 /s and lb/ft 2, rates on other nodes are illustrated in Table I. It should be noted that negative means the rate goes out of the system. Table II shows sizes of diameter and length of each pipeline. B. Firefly Parameters Table III illustrates parameter values of the firefly algorithm used on this experiment. It can be seen that we perform 10 simulations with variation on firefly populations and maximum iterations, in order to do sensitivity analysis and to obtain the best result. 33

4 TABLE I Input data: Rate on customers. Node Rate (ft 3 /s) Node Rate (ft 3 /s) Figure 2. The Hanoi network TABLE II Input data: Diameter and length on Hanoi Network. Pipe Name From To D (ft) L (ft) TABLE III Parameters of the firefly algorithm. Parameters Values Number of simulation 10 Pairs of population and iteration {40,4000}, {50,250}, {100,500}, {200,1000}, {40,200}, {500,5000}, {50,5000}, {100,1000}, {200,2000}, {1000,10000} The initial attractiveness 1 (β 0 ) The fixed light absorption 1 coefficient (γ) The randomization parameter 0.2 (α) VI. Results and Discussion As mentioned on Section IV-A, the non-linear equation system is constructed according to the networks. Since we have 32 nodes in Hanoi network, we obtain a system consisting of 32 non-linear equations as shown in Table IV. It should be noted that the equations in Table IV are automatically generated by the firefly algorithm with considering the fitness function. Moreover, since the Hanoi network contains looping pipelines, there are many options of flow directions. So, before finishing a simulation, we do not know which flow direction is correct and met with the equations. It means signs of Q i j can be positive of negative. Again, the firefly algorithm determines the correct one, which is represented by signs on the equation shown in Table IV. Besides constructing the model, we also compare the results with EPANET, which a software used for developing model of water distribution system. It was 34

5 TABLE IV The Model of Hanoi Network. Node Equations 1 f 1 = Q N1 Q 1 2 =0 2 f 2 = Q 1 2 Q 2 3 Q 2N =0 3 f 3 = Q 2 3 Q 3 4 Q 3 19 Q 3 20 Q 3N =0 4 f 4 = Q 3 4 Q 4 5 Q 4N =0 5 f 5 = Q 4 5 Q 5 6 Q 5N =0 6 f 6 = Q 5 6 Q 6 7 Q 6N =0 7 f 7 = Q 6 7 Q 7 8 Q 7N =0 8 f 8 = Q 7 8 Q 8 9 Q 8N =0 9 f 9 = Q 8 9 Q 9 10 Q 9N =0 10 f 10 = Q 9 10 Q Q Q 10N =0 11 f 11 = Q Q Q 11N =0 12 f 12 = Q Q Q 12N =0 13 f 13 = Q Q 13N =0 14 f 14 = Q Q Q 14N =0 15 f 15 = Q Q Q 15N =0 16 f 16 = Q Q Q Q 16N =0 17 f 17 = Q Q Q 18N =0 18 f 18 = Q Q Q 18N =0 19 f 19 = Q 3 19 Q Q 19N =0 20 f 20 = Q 3 20 Q Q Q 20N =0 21 f 21 = Q Q Q 21N =0 22 f 22 = Q Q 22N =0 23 f 23 = Q Q Q Q 23N =0 24 f 24 = Q Q Q 24N =0 25 f 25 = Q Q Q Q 25N =0 26 f 26 = Q Q Q 26N =0 27 f 27 = Q Q Q 27N =0 28 f 28 = Q Q Q 28N =0 29 f 29 = Q Q Q 29N =0 30 f 30 = Q Q Q 30N =0 31 f 31 = Q Q Q 31N =0 32 f 32 = Q Q Q 32N =0 TABLE V Result and Comparison with EPANET Node Head (m) EPANET s head (m) Margin (%) developed by the United States Environmental Protection agency s (EPA) Water Supply and Water Resources Division in 1993 [18]. After performing 10 simulation, we obtain the best pressure distribution if it is compared with EPANET as illustrated in Table V. We only compare with EPANET because the actual pressures/heads obtained from a measurement are not available. According to Table V, the biggest margin, which is 2.87, is on Node 9 and 17 while the most similar head is on Node 25. Furthermore, we provide the average margin of 10 simulations compared with EPANET as depicted on Table VI. In general speaking, all simulations show consistent results even though higher values of population and iteration do not mean providing better results since the firefly algorithm is a stochastic method. VII. Conclusion and Future Work The main constribution on this research is to provide a model for pressure distributions on water pipeline networks. The model expressed in a non-linear equation system is solved by the firefly algorithm. The pressure distribution obtained from 10 simulations show consistent and reasonable results. Therefore, the firefly algorithm can be used as an alternative method to perfom TABLE VI The average margin of 10 simulations compared with EPANET. No Population of Fireflies Max Iteration The Average Margin compared with EPANET (%) optimization problems, such as determining pressure distribution of water pipeline networks. As future work, we plan to implement other algorithms, such as Particle Swarm Optimization and Genetic Algorithm, for the same task. Moreover, determining optimal design of networks considering behavior pressure on nodes and pipelines is interesting to investigate. In this case, we might transform the optimization into regression task, and then solve it by using several machine-learning methods, such as fuzzy rule based sys- 35

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