Transactions on Ecology and the Environment vol 19, 1998 WIT Press, ISSN

Transcription

1 Calibration of water distribution network models using genetic algorithms G. Walters, D. Savic, M. Morley, W. de Schaetzen & R. Atkinson Centre for Water Studies, School of Engineering & Computer Science, University of Exeter, Exeter, UK, EX4 4 QF Abstract Calibration of computer models for network analysis is a regular component of the model building process. Re-calibration of existing models is also necessary periodically, to reflect physical changes in the network. The process generally first involves a series of field tests during which pressures and flows are recorded at strategic locations in the system, usually continuously over one or more days. This is followed by a desk exercise during which adjustments are made to the roughness values used in modelling the system until a satisfactory match is obtained between modelled and observed values. If no satisfactory match is obtainable, further site checks are usually made to identify discrepancies between the model and the real system, such as incorrectly modelled valve settings and unrecorded connections. The selection of a satisfactory set of roughness values can be a tedious and timeconsuming business when undertaken by the traditional trial and error approach. The authors show in this paper that the process can be largely automated using an optimisation approach that is well suited to the problem. The method adopted is a variation of the Genetic Algorithm, which is closely coupled with a network solver for maximum efficiency. Experience in using the automatic calibration software is presented using a case study from the UK.

2 132 Hydraulic Engineering Software 1 Introduction For most applications, hydraulic models of water distribution networks must be calibrated so that the predictions they generate may be interpreted with confidence. Model calibration essentially comprises adjusting pipe roughnesses until pressure and flow values predicted by the model are consistent with field measurements. There are vast numbers of potential combinations of pipe roughness values that could be considered in reaching a good calibration Hence there is scope for achieving a calibration much more efficiently and consistently using a computer based numerical optimization technique rather than the traditional trial and error adjustment adopted by most network modelers. At Exeter University, Genetic Algorithm (GA) optimization has been developed for use in a number of different water supply and distribution problems (Savic & Walters', Halhal et al*, Savic & Walters^). In collaboration with Optimal Solutions and Structural Technologies Ltd., Object Oriented GA software has been integrated with the commercial HARP network solver, using STRUMAP as the user interface. This has produced a package called GANET, which is proving a very effective tool for planning the optimal strengthening, rehabilitation and operation of distribution networks (de Schaetzen et al*). GANET has now been enhanced to handle the calibration of network models. In applying GA technology to the model calibration, the objective is to find the solution that has the minimum overall difference between field and model values of flow and pressure. This is achieved by performing a large number of runs of the network model using trial values of pipe roughnesses, which are adjusted throughout the process using the principles of natural evolution. The basics of the GA process are described in the following section. 2 Genetic Algorithms - Overview 2.1 General Evolution Programs, of which the GA is the best-known type, are general artificial evolution search methods based on natural selection and mechanisms of population genetics. They emulate nature's very effective optimization techniques of evolution, which are based on preferential survival and reproduction of thefittestmembers of the population, the maintenance of a population with diverse members, the inheritance of

3 Hydraulic Engineering Software 133 genetic information from parents, and the occasional mutation of genes. These algorithms are best suited to solving combinatorial optimization problems that cannot be solved using more conventional operational research methods. Thus, they can be applied to large, complex problems that are non-linear with multiple local optima. The GA is an algorithmic model of Darwinian evolution that begins with the creation of a set of solutions referred to as a population of individuals. Each individual in a population consists of a set of parameter values that completely describe a solution The computer encoding of candidate solutions takes the form of so-called chromosomes, which are sets of strings of data analogous to the DNA in living things. The probability that the GA will select a chromosome from the original population to produce offspring for the new generation is dependent on its fitness value (a measure of how well a chromosome optimizes the objective function). The preference for more fit solutions produces a selective pressure, which, when applied through many generations, results in an overall trend towards greater fitness in the population of solutions. The fitness is computed by simply decoding the strings of the chromosome into parameter values and substituting them into the objective fiuiction. Then, based on their fitness values, individuals are selected from the population and recombined, producing offspring that will make up the next generation. The term crossover typically denotes the recombination process because of the way that genetic material crosses over from one chromosome to another. Sincefitindividuals have a higher probability of producing offspring than less fit ones, the new generation will have on average a higherfitnessthan the old population. A further mechanism called mutation also plays a role in the reproduction phase, although, contrary to popular belief, it is not the dominant role in the process of evolution. In GA each bit (or gene) is allowed a small probability to randomly mutate. If the probability of mutation is set too high, the GA degenerates into a random search process. This type of performance is undesirable, as a properly tuned GA is not a random search for a solution to a problem. For the simulation of a genetic process a GA uses stochastic mechanisms, but the result is distinctly better than random search.

4 134 Hydraulic Engineering Software 2.2 Benefits of GAs The benefits of GAs stem from their ability to converge rapidly on an optimal or near-optimal solution, having analyzed only a tiny fraction of the number of possible solutions available. For large problems such as those typically associated with networks, the exhaustive analysis of all options is unlikely ever to be feasible. In a network expansion and reinforcement problem recently analyzed by Optimal Solutions using GA, there were 721 pipe links each of which could adopt any of 15 diameters (including 'no pipe'). This presents a problem size of 15 or 9.16 x 10*". Even if one billion design evaluations (i.e. network simulation runs) could be performed in a second, the time needed to evaluate all possible schemes would be much greater than the age of the earth (estimated at 4.6 billion years). The GA optimization search was able to converge on the lowest cost solutions by carrying out several hundred thousand evaluations only. 2.3 Limitations GA optimization is a very powerful approach with a proven ability to identify near-optimal solutions. However the correct operation of a GA optimization depends on careful setting up and parameter tuning, requiring appropriate skills and experience. Inappropriate penalty levels will distort the results away from the end user's perception of 'optimum', and too high or low a rate of genetic exchange ('crossover' or 'mutation') result in degeneration to a random search, or failure to converge on the global optimum, respectively. Because of the complexity involved in setting up a GA to operate effectively, this form of optimization is not well suited to 'trivial' problems, i.e. those for which the number of possible solutions is small 3 Application of GA to model calibration To apply GA to the network calibration problem, the decision variables are generally defined to be the roughness values for each pipe in the system. These can be expressed as the "k" value in the Colebrook-White formulation, or as the Hazen-Williams "C" coefficients. All pipes can have individually variable roughness values, or groups of pipes can be pre-selected to have a common variable roughness, the selection being based on diameter, age, material, location or a combination of these factors. Variables other than pipe roughness could, in principle, be used in the calibration, although this is not common practice. For instance,

5 Hydraulic Engineering Software 135 neither demands nor pipe diameters may be known precisely, and hence it could be justifiable to allow them to vary to some extent to achieve a good modeling fit. Each potential solution consists of a set of pipe roughness values, and must be evaluated to obtain itsfitness.calculation of the fitness requires the network model to be run using the set of assumed roughnesses, to obtain the simulated heads and flows. Several snapshot simulations are used, generally at maximum and minimum demand times, and at an average time during a typical 24-hour cycle. At each measurement point and for each snapshot, the difference between simulated and observed data (head and/or flow) is calculated, and an overall error value for the whole network is calculated. Several different forms of error function can be used, but normally a root mean square formulation is adopted. Different weightings between head and flow measurements can also be incorporated into the error expression. As this simulation is performed, many thousands of times during the evolution of the best calibration, it is important that the GA and the analysis modules are fully integrated. Usually, there are anomalies in the data or in the basic model representing the physical network, which need resolution. Initial data inspection and model analysis, followed by several preliminary runs of the GA are generally required to eliminate misleading data originating from faulty loggers and to correct model features such as wrongly closed valves. Without these corrections an accurate calibration would be impossible. 4 Example 4.1 The Saltash Model The software was used by Optimal Solutions to calibrate an existing model of the Saltash water distribution system for South West Water, Saltash being located in the South West of the United Kingdom. There are 1043 pipes in the model, with pressures logged at 92 points and flows logged at 15 points over a two day period. The model calibration exercise used an objective function which calculated the root mean square of the absolute errors of all flow and pressure measurements at three snapshot times during a 24 hour period There were some obvious anomalies in the field data provided, due to apparent logger failures, identified either by head readings constant over the 24 hour period, or by heads which were lower than the elevation of

6 136 Hydraulic Engineering Software the corresponding monitoring site. Evidence of obvious anomalies led to the exclusion of 19 of the 92 original pressure data sets. The Colebrook-White "k" values for pipe roughnesses were used as variables. To ensure that the GA solution contained pipe roughnesses that generally lay within a credible range, a maximum value of 30mm was set. In some locations where initial results suggested a system anomaly (for example, a partially closed valve) this value was exceeded by manual adjustment to simulate the apparent extra resistance. 4,2 Initial Model Analysis To identify any further problems with the data and structure of the model prior to the optimization, an initial solution was obtained by running the model with default roughness values of 0.3 mm for all significant pipes. In most circumstances, a substantial head error at the 'minimum hour' condition would indicate a data anomaly. In theory, the flows should be very low at minimum hour with correspondingly low head losses. Hence monitored and modeled heads should correspond approximately, unless incorrect elevation data has been used. Several pressure sets were thereby identified as suspect and removed from the calibration data as a result. 4.3 Preliminary GA Solution A preliminary optimization was performed using the GA, with the purpose of highlighting any remaining problems. As a result of the initial analysis and the preliminary GA runs, nine data sets were excluded for various reasons including suspected wrong elevations, suspected wrong Table 1: Summary of Pipe Roughness Values for Optimum Solution g 350 f 300 % 250 H \ 0 ]0.1;5] ]5;10] ]10;20J ]20;30J k»100 'k' values Interval (mm)

7 Hydraulic Engineering Software 137 location relative to a Pressure Relief Valve, inconsistency between adjacent loggers and suspect demand allocations 4.4 Best GA Solution Once these anomalies had been resolved, a series of runs were performed using the GA to obtain a range of solutions. The resulting *k' values as determined by the best GA solution are summarized in Table 1. Of particular interest was the high friction needed in a particular run of pipes, suggesting the possible presence of a partially closed valve on a highly strategic link. Consistently high k values were obtained for these particular pipes in all GA optimization runs. Table 2 summarizes the head errors for the maximum hour snapshot, this being generally the worst of the three snapshot times. It can be seen that most results lie within 1m of the monitored pressure, with a small number of discrepancies remaining. Table 2: Summary of Head Errors at Maximum Hour (m) I.*J n - r-i n [0;0.2] ]0.2;0.4] ]0.4; 0.6] ]0.6; 0.8] ]0.8;1.0] ]1.0; 1.5] ]1.5;2.0[ error interval [m] Table 3 summarizes the flow results. These are presented as absolute error values (in 1/s), together with the computed flows at each point to give an indication of the relative scale of each error. The flow errors are generally small, particularly at the largest flow points, suggesting that the strategic representation of the calibrated model is good

8 138 Hydraulic Engineering Software Node Absolute Flow Error and Computed Flow [1/s] hours 08:00 hours \l 5:00 hours Error Flow Error Flow Error Flow Table 3: Summary of 'Best GA Solution' Results - Flows. 24 hours total head variations " observed profile calculated profile 96 Time [h] Figure 1: Example of a typical 24 hours simulation comparison.

9 Hydraulic Engineering Software 139 The overall quality of a simulation model may also be judged by comparing the continuous field data profile with the model predictions over a 24-hour period. This comparison was carried out at regular stages during the calibration process and demonstrated that a good profile match was consistently achieved where the error at all three optimization snapshots was small. This confirms that the use of only three well defined loading conditions (which represent the minimum, maximum and average water demand) in the optimization is sufficiently robust to obtain a good match for the whole simulation period. An example of a chart comparing field and model profiles is presented in Figure Suspected Anomalies identified during calibration During the course of the calibration, interim solutions were assessed manually at various stages by analyzing the cause of errors between field and predicted values. Initial attention focused on flow errors to ensure that these were reasonable, resulting in the identification of an incorrectly specified reservoir level: a bottom water level at 102m had been used instead of 100.2m giving rise to large inconsistencies in flow Pressure errors in excess of 1m post calibration were then examined, with the particular objective of identifying apparently anomalous circumstances which could bias the solution by forcing the GA to improve inappropriate matches at the possible expense of legitimate ones. Several probable anomalies were identified where it was reasonable to make changes, either to the model or to the field data, to effect an improvement in the results. These included the following: Increased friction factors (from the maximum limit of 20mm to 100mm) in two pipes to simulate a probable significant hydraulic resistance, perhaps a partially closed valve, on a strategic link; Correction of two suspected incorrect elevations where minimum pressure errors and inconsistencies with nearby adjacent measurement points were apparent Correction of two pressure data sets where there was an apparent one-hour lag in the field data compared to other data points and model predictions. 4.6 Discussion of Results The calibration has produced a good model of the system, although a few discrepancies remain as yet unresolved. When originally received the model contained 9 "artificial" throttle valves, apparently inserted by the original model builders in an attempt to manually calibrate the system.

10 140 Hydraulic Engineering Software The final GA calibrated version removed the necessity for their inclusion, except for the one pipe run mentioned, in which some real pipe throttling is suspected. The distribution of pipe roughness values in Table 1, giving a preponderance of high and low values, may be indicative of some distortion in the calibration caused, for example, by inaccurate demand allocations or pipe diameters. 5 Discussion and Conclusions The GA approach to model calibration is in its infancy, but has already demonstrated its ability to assist the modeler in the task of deriving a good network model. As well as removing most of the routine and tedious aspects of the job, it will generally achieve better fits to the available data than can be achieved manually, and, very importantly, will adopt a consistent and unbiased approach which is more likely to highlight real inconsistencies and problems. However, the input of experienced modeling staff is still required to validate data and interpret results within an essentially iterative process. 6 Acknowledgments This work is partially supported by Teaching Company Programme 2286 between Exeter University and Ewan Associates Ltd. The cooperation of South West Water and Optimal Solutions is gratefully acknowledged. 7 References 1 Savic, DA and Walters, G A Genetic Algorithms for Least-Cost Design of Water Distribution Networks, ASCE Journal of Water Resources Planning and Management, 123 (2), pp , Halhal, D., Walters, G.A., Ouazar, D. and Savic, D.A. Water Network Rehabilitation with a Structured Messy Genetic Algorithm, ASCE J. Water Resources Planning and Man., 123 (3), pp , Savic, D.A. and Walters, G A Evolving Sustainable Water Networks, Hydrological Sciences J., 42 (4), pp , de Schaetzen, W., Randall-Smith, M.J., Savic, D and Walters, G. A. A Genetic Algorithm Approach for Rehabilitation in Water Supply Systems, Proceedings Int. Conference on Rehabilitation Technology for the Water industry, Lille, March 1998.