Keywords: modelling, water distribution system, calibration, roughness, demand, optimization.

The aim of the paper is to demonstrate that error dynamics may be used for the calibration of water distribution system (WDS) models. Our suggestions are: to use an optimization procedure to obtain proportions of leakages and unaccounted demands; to rank pressure measurements according to their closeness of the measurement to the water source; to analyze error dynamics to control whether roughness or demand are over or underestimated.

Any WDS contains leakages and unaccounted demands. Therefore the amount of water inflow into system is often higher than accounted demand. Water balance in a WDS must be analysed before calibration. An optimization procedure will be useful for this analysis.

The data for the optimization problem are:

1. Dynamics of measured water input into the system;
2. Measured water demand with different patterns;
3. Measured pressures at some points in the system to obtain leakages dynamics.

It is proposed to rank pressure measurements according to the closeness of the measurement to water source. The measurement of the first order means the measurement located nearest to the water source. The measurement of the second order means that between that measurement and the water source there is the measurement of first order. The third order means that there are the first and second order measurements between the measurement and the water source (and so on). Ranking of the measurements according to their closeness to water source is necessary because, for example, errors in modelling of a WDS area, that is located close to the water source, influence modelling results in the remaining area. Therefore, error dynamics of the first order measurements are analysed first.

Dependencies of errors on water flow are created for each junction with the first order measurements. Then the matrix of first derivatives of these dependencies is analysed. The analysis may lead to different cases:

1. First derivatives equal to zero for these dependencies for all junctions with the first order measurements.
2. First derivatives have the same sign for all junctions with the first order measurements.
3. First derivatives for these dependencies are positive for some junctions and negative for others.
4. First derivatives for these dependencies change sign for some junctions.

The next step of the analysis will depend on the specific case. In case 3, for example, the attempt is accomplished to rearrange unaccounted demands or leakages. The task is formulated in the following way. Find the demand or leakage changes which minimize the sum of squares of slopes of error dependencies on the water flow. The sum of positive and negative changes must be equal to zero and changes may not be higher than unaccounted demands or leakages. The task is solved by optimization.

The application of the proposed methodology for the operational WDS showed that

• The analysis of water balance of a WDS by an optimization procedure allows assessing of unaccounted demands and obtaining more real values for leakages.
• The analysis of dynamics of pressure errors gives additional information for calibration and for location of unaccounted demand and leakages.
• The analysis proposed has been successfully tested on operational WDS.

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