Nonlinear Programming Techniques for Operative Planning in Large Drinking Water Networks Jens Burgschweiger1, Bernd Gnädig2 and Marc C

14 The Open Applied Mathematics Journal, 2009, 3, 14-28 Open Access Nonlinear Programming Techniques for Operative Planning in Large Drinking Water Networks Jens Burgschweiger1, Bernd Gnädig2 and Marc C. Steinbach3,* 1Berliner Wasserbetriebe, Abt. NA-G/W, 10864 Berlin, Germany 2Düsseldorf, Germany 3Leibniz Universität Hannover, IfAM, Welfengarten 1, 30167 Hannover, Germany Abstract: Mathematical decision support for operative planning in water supply systems is highly desirable; it leads, however, to very difficult optimization problems. We propose a nonlinear programming approach that yields practically satisfactory operating schedules in acceptable computing time even for large networks. Based on a carefully designed model supporting gradient-based optimization algorithms, this approach employs a special initialization strategy for convergence acceleration, special minimum up and down time constraints together with pump aggregation to handle switching decisions, and several network reduction techniques for further speed-up. Results for selected application scenarios at Berliner Wasserbetriebe demonstrate the success of the approach. Keywords: Water supply, large-scale nonlinear programming, convergence acceleration, discrete decisions, network reduction. INTRODUCTION • experiments with various optimization models and methods [36, 37], Stringent requirements on cost effectiveness and environmental compatibility generate an increased demand • a first nonlinear programming (NLP) model for model-based decision support tools for designing and developed under GAMS [38], operating municipal water supply systems. This paper deals • numerical results for a substantially reduced with the minimum cost operation of drinking water network graph using (under GAMS) the SQP codes networks. Operative planning in water networks is difficult: CONOPT, SNOPT, and the augmented Lagrangian a sound mathematical model leads to nonlinear mixed- code MINOS. integer optimization, which is currently impractical for large water supply networks as in Berlin. Because of the enormous The main goal of the joint work reported here is the complexity of the task, early mathematical approaches development of a decision support tool suitable for routine typically rely on substantially simplified network hydraulics application, to be implemented as an optimization module (by dropping all nonlinearities or addressing the static case, within the new operational control system of Berliner for instance) [1-8], which is often unacceptable in practice. Wasserbetriebe. The approach is restricted to the framework Other authors employ discrete dynamic programming [9-14], sketched above: a pure NLP model (no integer variables), the which is mathematically sound but only applicable to small GAMS modeling environment, and the listed NLP solvers. networks unless specific properties can be exploited to Criteria for applicability are speed (response time), increase efficiency. Optimization methods based on reliability, and practicability. Our mathematical nonlinear models (mostly for the pumps only) are reported in developments toward these goals are based on two internal [15-19]. These approaches employ computationally studies [39, 40] and can be coarsely categorized into expensive meta-heuristics or suffer from inefficient coupling modeling techniques (reported in [41]) and nonlinear of gradient-based optimization with non-smooth simulation programming techniques (reported here). Basic modeling by existing network hydraulics software, such as EPANET techniques include, in particular, a globally smooth and [20]. Other topics in water management include network asymptotically correct approximation of the hydraulic design [21-24], online control [25, 26], state estimation [27], pressure loss in pipes, and suitably aggregated models for and contamination detection [28]. More loosely related collections of pumps that operate in parallel. The NLP recent work addresses modeling and optimization for techniques include, among others, a sequential linear networks of irrigation and sewage canals or for gas programming type initialization procedure for the nonlinear networks, see, e.g., [29-35]. Previous efforts toward mini- iteration, special constraints that ensure minimum up and mum cost operation at Berliner Wasserbetriebe include: down times in pump operation, and various network reduction techniques. Together with pump aggregation, the *Address correspondence to this author at the Leibniz Universität Hannover, up and down time constraints permit the handling of discrete IfAM, Welfengarten 1, 30167 Hannover, Germany; E-mail: decisions (pump switching) without introducing binary [email protected]; http://www.ifam.uni-hannover.de/steinbach variables. Following the NLP-based network-wide 2000 Mathematics Subject Classification. 65K10, 90C06, 90C11, 90C30, optimization, nonlinear mixed-integer models are solved 90C35, 90C90. locally at each waterworks. 1874-1142/09 2009 Bentham Open Operative Planning in Large Drinking Water Networks The Open Applied Mathematics Journal, 2009, Volume 3 15 Table 1. Notation Symbol Explanation Value Unit Q Volumetric flow rate in arcs m3/s D Demand flow rate at junctions m3/s H Pressure potential at nodes (head) m H Pressure increase at pumps, decrease at valves m L Pipe length m d Pipe diameter (bore) m k Pipe roughness m A Pipe cross-sectional area m2 Pipe friction coefficient – r Pipe hydraulic loss coefficient s2/m5 Water density 1000 kg/m3 2 g Gravity constant 9. 81 m/s We start by summarizing the component models of all 1.2. Optimization Horizon and Dynamic Variables the network elements in Section 1, followed by the overall We consider a planning period of length T in discrete NLP model. In Section 2, the smoothing and SLP time, t = 1, 2, ,T, with initial conditions at t = 0. The initialization are discussed along with further convergence … enhancement techniques. Section 3 is devoted to subinterval (t 1, t) will be referred to as period t and has the physical length t. At Berliner Wasserbetriebe, the combinatorial issues, particularly the prevention of undesired planning period represents the following day in 24 hourly pump switching. Several network reduction strategies are intervals. then developed in Section 4 with special emphasis on suitable smoothing of the hydraulic friction loss. Finally, We measure the pressure by the head H: the sum of the Section 5 presents selected application scenarios at Berliner geodetic elevation and of the elevation gain corresponding to Wasserbetriebe in order to demonstrate the success of our the hydraulic pressure. Pressure variables Hjt are associated approach. with every node j N and time period t. Volumetric flow rates Qat are associated with every arc a A. Control 1. OPTIMIZATION MODEL variables are the pressure increase across pumps and decrease across valves, Hat, a Apu Avl. All variables We will first summarize the nonlinear programming have simple bounds. model developed in [41] in order to keep the paper self- contained. This model covers the physical and technical 1.3. Network Dynamics and Model Constraints network behavior. Later on we will add further constraints and develop graph reduction techniques to achieve a desired The network hydraulics include over-all flow balances solution behavior and to enable efficient treatment by the and pressure relations, with one equation per network selected standard NLP solvers. The basic notation used in element: our model is given in Table 1. 0 cflow : Q Q D ,j, (1) = jt = ijt jkt jt N jc i:ijA k:jkA 1.1. Network Topology 0 = cflow := Q Q E (H ,H ), j N , The network model is based on a directed graph G = (N, jt ijt jkt jt j,t1 jt tk (2) i:ij k:jk A) whose node set represents junctions, reservoirs, and A A tanks, and whose arc set represents pipes, pumps, and gate head 0 = c jt := H jt H j,jN rs , valves, loss N = Njc Nrs Ntk, 0 = c := H H + H (Q ), a = ij A , (3) a t jt it a at pi A = Api Apu Avl. 0 cdiff : H H H , ij , = a t = jt it at a = Apu The set of pumps consists of raw water pumps and pure water pumps, Apu = Apr App. We denote arcs as a A or, 0 = cdiff := H H + H , a = ij A , (4) with tail and head i, j N, as ijA. A flow from i to j is a t jt it at vl positive, from j to i negative. 0 csign := H Q , a = ij A , (5) Fig. (1) illustrates the main network of Berliner a t at at vl Wasserbetriebe, with 1481 nodes and 1935 arcs. Earlier Externally given data include the predicted junction investigations were based on a small test configuration with demands Djt, constant reservoir heads H j, tank inflow 144 nodes and 192 arcs; cf. [41]. 16 The Open Applied Mathematics Journal, 2009, Volume 3 Burgschweiger et al. Fig. (1). Main distribution network of Berliner Wasserbetriebe. characteristics Ejt(Hj,t1, Hjt), and pipe pressure loss Nontrivial inequalities include bounds on: the flow characteristics Ha(Qat). The sign condition (5) ensures gradients at the raw water pumps, the daily discharge of the consistency of the valve pressure decrease with the unknown waterworks outlets, a Apo, and the total power consumption direction of flow. of raw water and pure water pumps a Apr(w) App(w) at each waterworks or pumping station, w : The hydraulic pressure loss in pipes is usually expressed W in terms of a loss coefficient ra(Qat) depending on the pipe cgrad : Q Q [ Q , Q+ ], , (8) at = at a,t-1 at at a Apr length and diameter, T 8L day + a c := tQ[Q ,Q ], A , Ha(Qat) = ra(Qat)Qat|Qat|, ra(Qat) = 2 5 a(Qat), (6) a at at at

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