Use of a neural network to apply the M uskingum model to sewer networks
Certain towns and cities frequently suffer from failures of their sewer networks, especially in rainy weather. Pollution of the host environment, as the direct consequence of occasionally untimely spills, is not appreciated by the natural environment or the human population. Improving the quality of the natural environment therefore involves an increasingly sophisticated control of the hydraulics and the pollutant load in drainage systems, and especially in sewer networks. Real-time management of sewer networks can provide a solution for the protection of the natural environment. In this case, control strategies are provided for the sluices and pumps of the sewer network during a rainy event to minimize the urban effluent. Moreover, a better understanding and modeling of the transport of pollution in the mains is required. To that end, not only must the hydraulic operation of the mains be correctly modeled (shape of the hydrograph, value and temporal position of the peak flow), but this numerical model must also be stable and converge towards the solution, irrespective of the initial conditions for modeling of the pollution, and the computer time must be compatible with the requirements of real-time management. The most representative model of unidimensional flows is that of Barré de Saint-Venant (1871). The non-linearity of the model, resulting in difficulties in solving these equations, together with the computer time required, are such that not all the criteria for a real-time application can be met. The conceptual equations model of Muskingum is another model that can be used. In the case of round sewerage mains with a slope ranging from practically nil to a few per-thousandths and a few kilometers long, the K and α coefficients traditionally used do not yield correct results with respect to the benchmark model of Barré de Saint-Venant. To keep the advantages of the simplification of the Muskingum equations, and to avoid having to solve the Barré de Saint Venant system, we propose new parameters for the Muskingum equations and we use optimization and correlation calculation techniques using neural networks. In modeling the mains of a sewer network, the discretization of their length, within the usual limits [50 m; 1000 m] is chosen empirically. This discretization plays an essential part in the propagation of the wave in a main. To take this effect into account, the round main of length L is discretized into N sections, and K is expressed on the basis of the maximum speed of the flow Vmax. The model setting parameters are now N and α, and will be calibrated for a wide range of slopes, lengths and flow rates for round mains with a constant roughness. The calculation procedure is as follows: - Setting of the optimal values of N and alpha giving results close to those calculated by Barré de Saint Venant; - Determination of correlations of the parameters N and alpha according to the slope, length and diameter; - Validation of the Muskingum model in relation to that of Barré de Saint-Venant. The parameters alpha and N are set by minimizing an objective function giving the agreement between the results of the hydraulic simulations by Barré de Saint-Venant and the simulations of the proposed model. The objective function is defined by the sum of the relative quadratic deviations of the values and times of maximum flow rates. The maximum errors are in fact reduced from 90% to 10% on peak flows and from 30% to 10% at a given point in time during the peak flow. The mean error is reduced forty-fold for peak flow, and five-fold in the temporal position, with a reduction of the same order for the standard deviations. Correlations of alpha and N are sought according to the slope, length and diameter of the mains modeled. As linear type relations failed to provide satisfactory results, the multi-layer Perceptron type (artificial) neural network model was used. The model includes 3 inputs and 2 outputs. The first, essential stage consists of finding the optimal number of neurons in the masked layer. It is important to mention that despite maximum errors of 40% and 20% on the prediction of time and peak flow rate, mean errors of only 3% and 4% are observed. Given this result, 4 neurons were chosen in the masked layer. This model therefore includes 3 inputs, 4 neurons in the masked layer, and 2 outputs. Following the learning phase with the results of the optimization phase, the so-called prediction phase was then performed. This consists of using the neural network with data with intermediary values with respect to those used in the learning phase. The neural network is used solely to predict values within the minimum and maximum limits of the learning phase. The prediction (or validation) phase revealed that the mean errors are in the order of 2.7% for the peak flow value and 5.5% for the instant of the same flow. The choice of 4 neurons in the masked layer during the prediction phase gives results with the same order of magnitude as in the learning phase, thus validating the structure of the neural network chosen. Subsequently, the proximity of the value and of the time position of the maximum flow rate for the propagation of rectangular hydrograms was studied. The performance of the model proposed is now verified by studying the propagation of a hydrogram of any given shape. Use of this model, validated on a hydrogram of any given shape and presenting several peaks of different intensities, yields a satisfactory reproduction of the output hydrogram and is a distinct improvement on the classic Muskingum model.