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The forecasting of the demand applied to water supply systems has been an important tool to realize time control. The use of the time series to do the forecasting of the demand is the main way that has been used by researchers. By this way, the need of a complete time demand series increases. This work presents two ways to reconstruct the water demand time series synthetically, using the Average Reconstruction Method and Fourier Method. Both the methods were considered interesting to do the synthetic reconstruction and able to complete the time series, but the Fourier Method showed better results and a better fitness to approximation of the water consumption pattern.

The increase of the demand on the urban water supply systems and the need of accuracy on the control of these systems require an increase at the precision of the forecasting of the water demand. According to [

The short term forecasting can be done by the analysis of measured time series of consumption at the sector that is being controlled and managed, and the time demand series can be obtained by the measurement of water flow at consumption points or outflow of reservoirs or tanks for example.

Sometimes, these measurements are done by electronic apparatus which are subject to troubles such as electrical fail or data storage that jeopardize the data consistency and can harm the temporal series analysis.

Several methods to forecasting the water consumption have been proposed by researchers. [

If a time series of water demand is observed, it is possible to see a periodic oscillation on the consumption of water. Furthermore, if the sector that is being managed is a residential sector, the oscillation is more known, because the consumption of water at a residential sector is widely studied. So it is possible to confirm that this oscillation will be repeated with a few variations at the values in a week [

Another observation that is possible to do: the behavior of the demand follows a pattern according the day of the week. By this way, if a value of the demand is known at one time of one weekday, it’s expected a repeat of any value near of the known demand. The deviation of the expected demand can be justified because the value of the demand is affected by the weather factors such as rain, temperature variation, relative humidity [

For this work, the analysis of the water demand was done using a real time series obtained with a measurement of the consumption of water in a residential urban sector at Franca, a city of the state of São Paulo, Brazil. The measurement was done for 26 months every 20 minutes of all days.

The graphic at

At

The method of Averaged Reconstruction was developed based on the methods of synthetic reconstruction of rain time series and considering the demand variation showed above. The method uses the week pattern of consumption and the weekday pattern of consumption. With the measured demand of the two days after the analyzed day and two days next, it is possible to calculate the average consumption for each day and with this value, for each time of measurement, it is possible to determine a factor f_{1}, which means the deviation between the demand of each measurement time and the average value.

where

After, it is possible to calculate the average

After the calculation of the

So is possible to compose a final demand

where D_{i} is the reconstructed water demand for the time i and w is the weight of the demand

With the value of the demand D_{i}, it is possible to calculate the error E according the Equation (3):

where

The value of w is determined by a minimization of E, which shows the factor most important at composition of the final demand using the solver of the MS Excel 2007^{®}.

To validate this method, the work purposes a reconstruction of two days that was full measured and by this way evaluate the final value of E.

Considering the method of the calculation the demand D_{1} (Week average) and D_{2} (Weekday average) the

It is possible to observe the method can do a good reconstruction of the time series but it is not able to reproduce the big variation of the demand all day along as observed at time 35 (

Knowing the repetition of the oscillation on the demand, this second method purposes the use of the Fourier series to reconstruct the time series of the water demand. The method uses a combination of sine and cosine those will be approximate the function to the measured data. The application of the Fourier series was inspired on the work of [

The demand D_{i} can be written as:

The index j corresponds to the number of the terms of Fourier series. At this work, as showed at

Analyzed Day | Error (l/s) |
---|---|

06/05/2012 | 0.19 |

07/28/2012 | 0.16 |

improvements on the capacity of reproduction of the time series when the number of terms is changed of 3 for 4 terms (value of j). The value of x_{i} can be obtained doing a transformation with the value i, the time of measurement, according the Equation (9).

According to [

Analyzed Day | Error (l/s) |
---|---|

06/05/2012 | 0.085 |

07/28/2012 | 0.087 |

where

The Fourier Method was more effective to reconstruct the time series of water demand. Besides the reduction of the squatter error, the graphic shows the less deviation at the shape of the behavior of the demand. The capacity of this method to represent de oscillation of the demand can justify the better results.

The forecasting of demand, mainly the short term forecasting has been more important to definition of the operation at real time on the water-supply systems. The measurement of the demand to generation of a forecasting model, sometimes, has any fails that can give troubles to determine a reliable model. By this way, this work purposed two methods to do a synthetic reconstruction of water demand time series.

At first, the Averaged Reconstruction Method, whose fitness could be considered acceptable, though for some times, it is possible to observe a significant deviation between the measured data and the synthetic data. This deviation is observed because the technique was not able to complete all time series along. By

The second method, Fourier Series Method, was able to determine with more accuracy the behavior of the demand all day along. The advantage of the Fourier method is the reduction of the deviation between the reconstructed demand and the measurement. But also this method was not able to reproduce the extreme variation of the demand some times of the day.

All methods can be applied to compose the study of the time series of the demand. Maybe the Fourier Method can be more efficient to define the unknown demands and can give more reliable forecasting model.