ABSTRACT 1 INTRODUCTION

Transcription

1 Water distribution network steady state simulation AQUANET model H. Rahal Laboratory of Hydraulics, K. U. Leuven, de Croylaan 2, B-3001 Heverlee, Belgium ABSTRACT This article is devoted to the description of the algorithm upon which the steady state simulation model AQUANET has been developed. This algorithm solves the nonlinear system of the continuity and the loop equations formulated in terms of the link flows, using the Newton-Raphson iterative procedure. To ensure fast convergence, the initial solution is determined using an extended version of the linear method. The state of the art graph theory concepts were used to generate the fundamental circuits (or loops). To take into account, the pipes various types of material and ages, the model allows the use of the most used head loss equation for each pipe link. Among these head loss equations, the Darcy-Colebrook can be used without any trade-off, because of the efficient procedure developed to ensure the fastest convergence. A special algorithm for parallel pipe assembly can be used to reduce the size of the vector space spanned by the steady state equations. Beside the determination of the link discharges and the nodes heads, the algorithm checks the bounding constraints imposed on the link velocity and on the node head pressure. Such feature allows the reinforcement and expansion of existing networks. 1 INTRODUCTION Water distribution steady state simulation has been a major issue for the last half century. The oldest systematic solution was provided by Hardy- Cross [1]. Not only his method addresses the problem correctly, but also is adapted to hand calculations. It suffers however, from slow convergence and instabilities.

2 396 Hydraulic Engineering Software The first general solution of the full set of equations using the Newton- Raphson's numerical algorithm to solve the nodal head pressure formulation is reported to be addressed by Martin and Peters [2] and extended by Shamir and Howard [5] to take into account even variable roughness coefficients, beside flows and heads. The theoretical study achieved by Nielsen [3] proved the numerical superiority of the link flow method on the nodal head method, which lacks stability and fast convergence. The proposed algorithm solves the full steady state equations expressed in terms of the link flow discharges. In the following there is a detailed description of the AQUANET model algorithm and the subsequent issues raised such as: the nonlinear system solution, the use of nonlinear head loss equations, the algorithm convergence, the initial solution generation, the loop generation and the indirect optimization features. 2. BASIC HYDRAULIC BACKGROUND A hydraulic network is composed of n nodes interconnected by p pipe links. The set of nodes is composed of rifh fixed head nodes ( reservoirs,...) and rifd fixed demand ( consumption ) nodes. Any closed chain of links is called a loop. One of the chains of links connecting two fixed head nodes is called a fundamental pseudo-loop. The graph theory cyclomatic number stipulates that the number p of links is equal to the sum of the m? real loops, the rrip pseudo-loops and rifd fixed demand nodes. p = rn,. +?7%p + rt/d (1) The steady state governing equations major difficulty is due to the nonlinearity of the head loss equations. The most used equations can be put in the resistance form: Ay = -^- 9" = r?" (2) K D Where hj is the friction head loss, L is the pipe length, D is the pipe diameter, q is the pipe discharge, K is the roughness coefficient, r is the pipe resistance,and a, ra and n are various constants and exponents depending on the chosen head loss formula, such as Hazen-Williams, Manning- S trickier, or Darcy-Wiesbach. In contrast to the formulas of Hazen-Williams and Manning-S trickier which are explicit, the Darcy equation coupled with Colebrook formula form an implicit system of equations, which needs an iterative procedure. The proposed procedure to solve such system will be elaborated in Section 5.2. Beside the friction head loss occurring along the pipe, there are singular head losses due to various localfittingssuch as: valves, elbows, contractions, junctions,... Losses in fittings have been found, in general, to vary as: hs = ksq^, where k$ is the singularity head loss coefficients.

3 Hydraulic Engineering Software 397 In contrast to all apparatus inducing energy losses, the pumps provide boosting energy excess, according to a given characteristic curve. This curve expresses the pumping head hp versus the discharge q. The most common fitting curve is the quadratic expression: hp = a + bq + cq*. The total head loss h along a pipe is the sum of the friction and local head losses from which the eventual boosting head is subtracted. Its expression versus the pipe discharge is as follows: /,,(9) - /4W = r<t + 6,g2 - (a + 6% + c^) (3) 3. THE MODEL GOVERNING EQUATIONS The retained link flow formulation expresses the problem in terms of the link flow discharges vector Q = [^i, f/2,..., </p]*, which consists of a nonlinear system formed by the continuity and the loop equations. The governing system is formed by rifd fixed demand nodes continuity equations, rrip pseudo-loop equations and ra^ real loops equations. 3.1 The Continuity equations The continuity equation, with respect to any fixed demand node j, stipulates that the algebraic sum of discharges of the nl links connected to this node, is equal to the node demand discharge: dj. Where Sij is the link ij direction sign, with respect to node j. The conventional direction s^ = 1, when node j corresponds to the link downstream node. On the contrary, Sij 1, when the node j corresponds to the pipe upstream node. The choice of the upstream and downstream nodes of a pipe link is arbitrary. 3.2 The loop equations Beside the classical loop, consisting of a closed circuit of pipes, the pseudoloop ( or hydraulic path) is the chain of pipes connecting two fixed head nodes (reservoirs). The choice between the different paths connecting two fixed head nodes is irrelevant. The pseudo-loop equation stipulates, that the algebraic sum of the head losses along the path connecting the two fixed head end-nodes, is equal to the difference between the z heads of the path upstream and downstream nodes. For pseudo-loop?, the head equation is: k=l (5)

4 398 Hydraulic Engineering Software Where 5/^ = 1 when the kth link direction corresponds to the loop chain's direction, on the contrary s^ = 1. /is the number of links forming the pseudo-loop. The loops equations are identical to those of the pseudo-loops, except that the head difference between the end nodes is Az% = The resulting nonlinear system assembly In terms of the link flow discharges vector, The steady state nonlinear system of p equations can be written as: /i(9n 92?' ',%) = 0 /or = 0 4. THE ALGORITHM SOLUTION 4.1 Linearization The solution of a nonlinear system goes through the use of a Taylor's series expansion around a known vector Q^ = [q[, q%, -,<??]* f the ith iteration. The Newton- Raphson procedure assumes that the second order term of the expansion is negligible and fk(q^^) ~ 0. Therefore, the retained first order linearization turns to be: The above equation can be expressed in terms of Ag^ q^ q '. It can be extended, as well, to the whole system using the jacobian matrix J, as: J(Q(*'))AQ(''+i) = -F(Q^) (7) A linear system solver ( Gauss elimination ) determines at each iteration, the vector AQ^+*\ which allows the determination of the new discharge vector: ' i) - Q(*') (8)

5 Hydraulic Engineering Software The algorithm convergence Mathematically, the convergence is reached, when the nonlinear system F(Q) = 0. However, numerically the convergence is considered reached, when the residual error: e[f(q')] < e, where e is a fixed precision. The residual error is estimated using the Euclidean norm: To guarantee the best solution, the Newton-Raphson algorithm has been damped. In fact, at each iteration the residual error is computed. If the new residual error is higher (worse) than the subsequent one, the step size AQ(*+^ is divided by 2, as many times as possible, until the new iteration gives better residual error. The convergence rate depends on the closeness of the initial solution. The generation of this initial solution, not only grantees the convergence, but also allows its fastest rate. This has been confirmed by the determination of the jacobian matrix condition, which turns to be, virtually, never ill-conditioned. k=l 5. THE ALGORITHM OTHER FEATURES 5.1 The initial solution generation The overall Newton-Raphson algorithm convergence rate depends particularly on the closeness of the initial solution guess to the exact one. In this respect, an algorithm has been developed to approximate as good as possible the initial guess estimate. This algorithm is based on the linear theory method proposed by Wood [6]. It consists of assembling a linear system of p link flow equations. The n/d continuity equations are already linear. However, the loop equations are nonlinear, making their linearization necessary. The proposed linearization is based on the fact that practically the order of magnitude of the pipe velocity and its Darcy-Weisbach friction factor are respectively of about 1 m/s and Thus, the loop equation becomes: > - f E^ g^* = Az, (10) 5.2 Darcy-Colebrook formula use The Darcy-Weisbach head loss formula is the most indicated one, particularly, when it is coupled with the Colebrook- White formula for determining the friction coefficient. Such system of equations has the advantage of taking into account the fluid viscosity v and of covering a wide range of flow

6 400 Hydraulic Engineering Software regimes. However, the two equations are implicit and require an iterative procedure. The author solved the combined nonlinear function expressed in terms of the friction head loss: = 0 (11) The proposed algorithm uses the Net won- Raphson iterative procedure, which requires the determination of f(hf) and it's derivative f'(hf). The rate covergence depends upon the closeness of the initial guess. The best initial guess is achieved by the use of Zigrang-Sylvester [7] explicit friction coefficient formula. The implementation of such algorithm showed outstanding convergence. In fact, not more than two iterations are necessary to converge to even a very high fixed precision. 5.3 The loop generation Determining manually the set of loops of a large size network turns to be very constraining and some times a source of errors. Therefore, developing a routine that can generate automatically the loops is extremely important, particularly if a pipe break has to be simulated, since the loops might change at each simulation run. The AQUANET model loop generator routine starts by generating a short path tree (instead of a spanning tree, which is less efficient) to identify the tree branches from the co-tree chords. For each chord, there is an associated fundamental circuit. This chord forms thefirstlink of the circuit, the rest of the circuit links form a unique set of branches. The short path ( or skim ) tree is generated using Disjkstra's procedure combined with a binary search strategy. The circuit branches are determined by a minimum path routine. The pseudo-loops are generated the same way as for the real loops, except that the chord is replaced by the pseudo-chord joining the two fixed head nodes. 5.4 Parallel pipe assembly Existing water supply system end up, in most cases, with several types of pipe material of different ages, necessitating the use of different head loss formulas. In addition, to enhance the efficiency of the system to meet increasing demand, adding parallel pipes to reinforce the system is a common practice. An algorithm to assemble parallel pipes using different head loss formula developed by Rahal and Berlamont [6] was implemented and added to the AQUANET model to reduce the size of the nonlinear system of as many equations as the number of parallel links.

7 Hydraulic Engineering Software 401 This algorithm determines the equivalent pipe diameter of a set of p parallel pipes, using each one a particular head loss formula. The expression of the equivalent diameter is: A i=p AT'^r (12) Where the subscripts e and i refer respectively to the equivalent pipe and to the ith parallel pipe. The equivalent length is chosen as the weighted ( on the diameters) average length of the parallel pipes system. The parallel pipes coefficients and formulas are known. However, a decision has to made upon the choice of the formula for the equivalent system. The algorithm overcomes this difficulty by choosing the formula which has the highest sum of pseudo-resistance CY]- L^/Df). 5.5 Pseudo-optimization Beside computing the links flows and nodes head pressures, the AQUANET model allows an "indirect optimization". In fact, it provides the system total cost for a part or the whole system, constituted of new pipes, and indicates if there is any violated constraints. The imposed constraints are those of the bounded link velocities: Vmin < V < Vmax, and of the bounded node heads: z^,^ < z < z^^. This indirect optimization is remarkably useful, particularly for existing distribution systems, which needs to be expanded or reinforced. In most cases, only few links ought to be added or reinforced. Therefore, only few computer runs are necessary to optimize the system, without going through sophisticated optimization algorithms. 5.6 Final remark In this paper, the author omitted intentionally to discuss the dimensionality problems posed by the jacobian matrix for large size networks. In fact, a great research effort is still underway, to find the appropriate solutions for this issue. The author is studying whether, it is appropriate to add a sparse matrix routine solver or to reformulate the system of nonlinear equations through graph theory and various substitutions. 6. CONCLUSIONS The AQUANET model is developed, essentially to simulate the steady state in a water distribution network. This model has been under development

8 402 Hydraulic Engineering Software for the last few years during which the author tried to include as many efficient procedure as possible. The global algorithm convergence has been carefully studied by damping the Newton-Raphson procedure in a way the best solution is retained. Furthermore, this convergence is accelerated by generating an outstanding initial solution estimate based on an improvement of the linear method. The AQUANET model has not shown for the last few years, any cases of nonconvergence, except those due to input data errors. In addition the Jacobian matrix condition determination did not show any ill-conditioned cases. Finally, the author come up with the same conclusions as Nielsen [3] indicating that the best convergence is reached using the linear method for the first iteration and the Newton-Raphson for the subsequent iterations. Beside the algorithm convergence issue, many procedures have been developed with great success such as: the starting initial solution estimate, the use of comprehensive head loss formulas, the loops generation, the parallel pipes assembly and design constraints verifications which determines whether or not a specific design is feasible with respect to the link velocity and nodal head bound constraints. 7. REFERENCES [1]. Hardy Cross (1936). "Analysis of flow in networks of conduits or conductors." BWWm <?##, f&e C/muerszZ?/ o////nzozg Em/. EzperzmemZ ^Wzom, Urbana, Illinois. [2]. Martin, D. W., and Peters, G. (1963). "The application of Newton Raphson's method of network analysis by digital computers.",journal of the Institute of Water Engineers, 17, [3]. Nielsen, H. B. (1989). "Methods for analyzing pipe networks." Journal o/ffz/drwz'c #%#., 115 (2), [4]. Rahal, H. and Berlamont J.(1993). "Algorithm for parallel pipe assembly ",/%2e<7raW Compiler App/zmZzoms m MWer J)%pp/%/, l/b/.); Me2/to& &W Procedures /or ^z/sfems j^muwzmz &?W CWZro/, edited by Bryan Coulbek, Research Studies Press Lts and John Wiley Sons Inc. pp [5]. Shamir, U., and Howard, C. D. D. (1968). "Water distribution systems analysis." JowrW o/f/te Tf^raWzcs Dzm^orn, ASCE, 94 (HY1), [6]. Wood, D. J. and Charles, C.O.A (1972)." Hydraulic network analysis using linear theory", Jo%rW o/#%/<fr. DzmszW, ASCE, 98 (HY7), [7]. Zigrang, D.J. and Sylvester, N.D. (1982)." Explicit approximation to the solution of Colebrook's friction factor.", Journal of Amer. Inst. of 28 (3),