Global Gradient Algorithm Extension to Distributed Pressure Driven Pipe Demand Model
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It has been proved that the standard representation of water demand in a Water Distribution Network (WDN) leads to pipe head loss errors as well that the fully satisfied demand regardless water pressure assumption is misleading. This follows that different algorithms have been developed in order to overcome these two drawbacks although separately and independently. Consequently, this paper introduces an alternative formulation of the Global Gradient Algorithm (GGA), referred to as UD-PD, which is able to solve uniformly distributed pressure driven demands along the pipes of a WDN in extended period simulations. In addition, this new scheme is tested against reference solutions and its performances are compared with the standard WDN models. Finally, the UD-PD is applied to a real WDN under pressure deficit conditions. Numerical results show that the hydraulic heads computed with the UD-PD result higher than those simulated with standard demand driven models and that the UD-PD is able both to capture the non linear behavior of the hydraulic head along the network and to correctly compute the flow inversion even in pressure driven conditions.
KeywordsWater distribution network modelling Uniformly distributed demand Pressure driven demand Global Gradient Algorithm
Nowadays, the water scarcity scenarios (Jakob and Steckel 2016) and the effects of climate change on water resources (Oki and Kanae 2006) drive the development of hydraulic tools able to correctly simulate Water Distribution Networks (WDNs). It can be argued that these are mandatory steps in order to manage and prevent urban water crisis as recently recognized by the European Environmental Agency and underlined by the Water Framework Directive (Conte et al. 2012). In particular, more accurate models of pressures and flow rates in WDNs are intent on properly support water system designs, rehabilitation plans and water demands management strategies.
Since early 1930s, many network hydraulics solvers have been developed. Firstly, Cross (1936) designed a scheme based on iterative solution of flow equations in looped pipes. Afterwards, different authors applied the Newton-Raphson technique to calculate water flows and hydraulic heads in both opened and closed networks (Epp and Fowler 1970; Boulos and Altman 1991; Murty 1972). It is only with Todini and Pilati (1988)’s works that the researchers’ efforts arrived to a breakthrough when the so called Global-Gradient-Algorithm (GGA) was developed. This scheme combines energy loss equations and mass balance equations providing a simultaneous solutions for nodal head and pipe flows. This method has become de-facto the standard for WDN solvers and was adopted in the widespread and open-source program EPANET (Rossman 2000). Some examples of EPANET diffusion in the scientific community as well in the WDN design and management sectors can be found in Avesani et al. (2012), where EPANET is extended in order to simulate unsteady flow in water distribution networks with variable head tanks, in Ingeduld et al. (2006), Ingeduld (2007), which uses EPANET to model both water intermittent water flows and real-time demands forecasting in WDN. Specifically, a complete and historical review about WDN solver methods can be found in Ormsbee (2006) and a survey on EPANET in Rossman (2000).
Despite the utility and feasibility of the above schemes, they presents two limits: firstly, water demands are modelled with lumped withdrawals at network nodes and not distributed along pipes. This approximation does not preserve the energy balance along pipes and leads to significant head loss errors, as reported in Berardi et al. (2010), Giustolisi and Todini (2009). Secondly, standard hydraulic network solvers are not able to represent WDNs affected by pressure deficit conditions (Giustolisi and Todini 2009).
Consequently, in the past decade, the researchers’ efforts have been focused on the development of proper extensions and alternative formulations of standard WDNs models in order to overcome those two drawbacks. Starting with models able to correctly taking into account actual demand along pipes, initially Giustolisi and Todini (2009) formulated a modification of the GGA able to simulate uniformly water demand along pipes in the so called EGGA scheme. Afterward, Berardi et al. (2010) extended the EGGA to any empirical head loss monomial formula and Menapace et al. (2018) developed an EPANET extension to consider both node and uniformly distributed demand driven demands.
A second category of schemes involve instead models able to correctly consider pressure demand driven although still at the individual node level. For example, Ang and Jowitt (2006) presents a novel algorithm based on artificial reservoirs; Rossman (2007) introduces pressure dependent demands using proper emitters and Kalungi and Tanyimboh (2003)’s scheme recognizes the pressure dependency of water consumption in the solution procedure by adjusting nodal flows. Other nodal pressure driven approaches instead are based on head-flow relationship embedded in hydraulic network equations. Some examples can be found in Giustolisi and Todini (2009), Tanyimboh and Templeman (2010) and Siew and Tanyimboh (2012) in which a constitutive head-flow equation allows to compute the actual withdrawn water in a node according to the nodal demand and pressure.
Nevertheless, to the best of authors knowledge, none mathematical solution has been yet formulated capable of representing distributed water demands and pressure deficit conditions simultaneously. As a result, the aim of this paper is to developed a proper mathematical formulation in order to model distributed pressure driven demand in a WDN. The paper is organized as follows: Section 2 presents the mathematical model for the uniformly distributed pression driven head loss equation; Section 3 describes the new numerical model in order to introduce the distributed pressure driven demand into the GGA scheme; in Section 4 the new scheme is compared with reference solutions and applied to real test cases; and finally, Section 5 summaries the main finding and outcomes of the present paper.
2 Mathematical Model
It has to be noticed that from a mathematical point of view both Eqs. 1 and 3 are discontinuous functions. Specifically, Eq. 3 is a step function of general validity, which is able to properly taken into account the flow direction along each pipes even when a flow inversion occurs and Eq. 1 is a sampling function. It is obvious that this topological representation, which is provided by Eqs. 1 and 3 and shown in Fig. 1, is correct and accurate although it presents two drawbacks. First, the discontinuity in the flow and in the withdrawal force to split the problem in a number of joint pipe equal to the number of connections. This increase the complexity in the topological representation of the WDN reducing the computational efficiency. Second, in real applications and network analysis there is the assumption of uniformly distributed water demands along the pipe due to the lack of information about the actual connections and demands.
Nevertheless, differently from the original (1), the wij function results now continuous and differentiable. In particular it could be approximated within an iterative scheme like those characterizing WDN hydraulic solvers. Therefore, the novel idea presented in this paper is to developed a strategy in order to compute an approximate wij function which still allows a correct simulation of the head loss in a pipe. In particular, the core of the scheme is borrowed from high order numerical Finite Element Methods (FEM) where high order polynomials approximate the exact solution in each mesh element from the values computed in the nodes of the mesh by the FEM solver.
3 The New Scheme for Distributed Pressure Driven Demand
As in the FEM-based schemes, WDN are made by nodes linked not with multidimensional elements but with one dimensional element, i.e. the pipes, where an arbitrary number of additional nodes can be added in order to reconstruct the water withdrawal function, known the pressure head in each nodes, as shown in Fig. 2b and d. It follows that, starting from the classical Global Gradient Algorithm, each iterative step could provide the solution of the hydraulic head both in water supply network nodes and in the pipes additional nodes. After that, known the hydraulic head and chosen a proper head-flow relationship such as those described in nodal pressure demand schemes, the withdrawal function along each pipe of the network can be approximated.
where nij,l is the number of withdrawal polynomial points, nt is the total number of nodes and nn is the total number of junction, i.e., the number of nodes where the piezometric head is unknown. According to the standard formulation, Eq. 16a describes the head losses along each pipe of the network and Eq. 16b represents the mass balance at each of the unknown head nodes. On the other hand, differently from the origial formulation, the Eq. 16c has been also introduced in order to properly compute the unknown pressure dependent withdrawal function along each pipe. This follows that the System (16) is a non linear system of equations which, as for the standard GGA scheme, can be linearized using the Taylor series expansion.
which can be directly cast inside the GGA procedure where only one additional step in the iterative procedure has to be considered. In order to facilitate the developed and the explanation of the new scheme, we limit ourself to a general discussion of the method. However, Appendix B reports the Equations in case of second order approximation for water withdrawals, flow rate and head loss in a generic pipe. The coefficients for Taylor series expansion are also derived.
4 Test Cases
4.1 Reference Test Case
The aim of this test case is to verified the present new UD-PD scheme with a reference solution. In particular, the test consists in simulating the pressure and the flow rate in a single dead-end pipe connecting two nodes, named i and j, where the hydraulic pressure in the node i and the water demand are the only known variables of the problem. In order to compare the solution obtained with the different methods the following reference solution is considered: the ij pipe is divided with 100 nodes equally distributed which have a water demand of 0.1 l/s. This follows that it is possible to solve the standard head loss equation, where the flow rate is constant, for each section of the pipe and to apply the mass balance in each nodes sequentially starting from the initial node i where the pressure is known. In particular, it is possible to calculate in each node the actual withdrawal, which depends locally on the pressure as described by the Eq. 23, and through an appropriate mass balance, know the water passing through the next node.
4.2 Two Tank Test Case
In order to discuss the implications of the new UD-PD scheme, a simplify WDN is considered. It consists in two tanks, named T01 and T02, connected with three pipes (see layout and detailed report of the network in Fig. 8 and Table 1 of Electronic Supplementary Material). In order to cover both demand driven and pressure deficit conditions, four scenarios are simulated. In the first case, case(i), the water levels is set at 80 m for the tank T01 and 70 m for the tank T02 while in the second case, case(ii), the water levels are set at 30 m and 20 m respectively for the tank T01 and the tank T02 (see Table 1 in Supplementary Material). In both cases two water total demands are taken into account: Dtot,AB = 5 l/s and Dtot,AB = 20 l/s. These configurations have been specifically chosen because they allow to compare the presented UD-PD scheme, where the total water demand is uniformly distributed along the pipe AB and is pressure driven, with both the standard uniformly distributed and demand driven schemes. In the N-DD, the total demand equally distributed in both node A and B.
Moving to case (ii), the pressure drops and the two tanks do not ensure enough pressure to satisfy the water demand. This implies that the differences between the N-DD, UD-DD and UD-PD models increase, as reported in Fig. 4c and d. Focusing on Dtot,AB = 5 l/s demand, Fig. 4c shows that both distributed models simulate a non linear behaviour in the hydraulic pressure along the pipe AB the pressure differs both along the pipes and in the nodes. On the contrary, the pressure computed in the node A and B with the demand driven models results similar in the nodes and differs from the one computed with the UD-PD. Similar to case(i), both distributed models are able to capture the flow inversion even in case of Dtot,AB = 20 l/s although the pressure in the UD-PD results higher due the fact less water is withdrawn in case of pressure driven demand. It worth to notice that this results are in agreement with the conclusions presented in Section 4.1 and further underline the differences between the UD-DD and the demand driven models in pressure scarcity conditions.
4.3 Case Study
This test case has two purposes: firstly, to show the applicability of the new UD-PD scheme to a real network; and secondly, to demonstrate that the hydraulic head distribution could considerable change when the distributed pressure driven demand is considered. As a consequence, the new UD-PD method has been applied to a real WDN, known in scientific literature as Apulian Network test case (Giustolisi et al. 2008) (see layout and detailed report of the network in Fig. 9 and Table 2 of Electronic Supplementary Material). It has to be underlined that the original nodal water demand presented in Giustolisi et al. (2008)’s work are herein distributed along each pipe to enforce the differences between pressure and demand driven simulations.
This paper presents a new formulation of the GGA scheme in order to properly model uniformly pressure driven demand along pipes in a WDN. In particular, a new approximation of the head loss equation, based on solving high order polynomial equation, is formulated as well an additional step in the iterative Newton-Rapson scheme is introduced. In addition, the new uniformly pressure driven scheme is tested with reference solution to prove its performance and accuracy. Furthermore applications to two WDNs illustrate some comparisons between the uniformly pressure driven scheme and the standard uniformly demand driven scheme. These simulations underline that these two schemes, pressure and demand driven, lead to the same results in case of sufficient pressure in a WDN. On the contrary, the uniformly pressure driven scheme results into a higher nodal pressure and simultaneously a lower withdrawn water where the pressure is not sufficient to deliver all the demand water.
The presented results make the new distributed pressure driven demand head-loss equation formulation within the GGA scheme an essential tool to properly simulate flow and pressure in a WDN in pressure deficit conditions.
This work has been partially carried out within the Research project AI-ALPEN, CUP:B26J16000300003 funded by the PAB (Autonomy Provence of Bozen-Bolzano,Italy) for University Research-2014. The EFRE-FESR project Thermo Fluid Dynamics, infrastructures for applied research (ERDF 2014- 2020, CUP: I52F16000850005) support is also acknowledged. The authors thank Professor Righetti Maurizio for his constant guidance and the encouragement in water researches.
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Conflict of interests
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