Modelling daily water consumption through potential curves. Disaggregating apparent and real losses

ABSTRACT

This paper presents a model, based on potential curves, that describes the behaviour of the inverse of the daily cumulated frequency of the flows provided to a District Metered Area (DMA). The model has two terms, the first corresponds to the variable consumption due to the aggregation of demand patterns of consumers. The evolution of this term presents periodic behaviours with annual and weekly frequency. An extreme drought episode that affected Catalunya, reduced this parameter 19%. A second term presents exponential behaviour in its evolution and includes the real leakage. The leakage disaggregation together with the billing information allows the estimation of the apparent loses, 14.89% in the case study. The difficulty of estimating the parameters in a potential model, a complex problem of optimization, is simplified by applying mathematical moments. Hence, daily parameters become a linear relation of the daily moments that allows their algebraic operation.

KEYWORDS:

• Water demand
• potential curves
• model calibration
• real losses
• apparent losses
• mathematical moment

Acknowledgements

This work has been partially funded by the Spanish State Research Agency (AEI) and the European Regional Development Fund (ERFD) through the projects SCAV (ref. MINECO DPI2017-88403-R) and DEOCS (ref. MINECO DPI2016-76493) and AGAUR ACCIO RIS3CAT UTILITIES 4.0 – P4 MODEM. Data have been gently provided by the company Aigües de Manresa.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notation

The following symbols are used in this paper:

DMA District Metered Area

$MSE$ Mean squared error

$MAE$ Mean absolute error

MAPE Mean Absolute Percentage Error

RRMSE Relative Root Mean Square Error

NSE Nash–Sutcliffe Model Efficiency

$Y$ Series of work data, a single series per day and flowmeter, with n_d values $\left\{Y_1\dots \dots Y_{\mathrm{i}}\dots ..Y_{\mathrm{n}\mathrm{d}}\right\}$

$Y_{sum}$ Series of work daily data adding the flows of m flowmeters, with n_d values $\left\{{Y_{sum}}_1\dots \dots {Y_{sum}}_{\mathrm{i}}\dots ..{Y_{sum}}_{\mathrm{n}\mathrm{d}}\right\}$

$f_i$ Frequency associated with the order number of $Y_{\mathrm{i}}$ in $Y$

$Y_{\mathrm{n}\mathrm{d}}$ Minimal daily flow measured

${Y_{\mathrm{n}\mathrm{d}}}_{sum}$ Minimal daily flow measured adding the flows of m flowmeters

$\bar{Y}$Arithmetic mean of work daily data $Y_{\mathrm{i}}$

$\overline{\stackrel{‾}{Y}}$ Arithmetic mean of the daily arithmetic mean

$\hat{Y}$ Potential adjust data of Y series

${\hat{Y}}_{\mu }$ Potential adjust data of Y series calculated through moments

${\hat{Y}}_{\mu sum}$ Potential adjust data of $Y_{sum}$ series calculated through moments

$\overline{\stackrel{ˆ}{Y}}$Arithmetic mean of the daily potential adjust data

$\overline{\stackrel{ˆ}{Y}}$ Arithmetic mean of the daily arithmetic mean

$Y^{\mathbf{s}\mathbf{y}\mathbf{m}}$ Series with the order of the inverted data, a single series per day and flowmeter, with n_d values $\left\{Y_{\mathrm{n}\mathrm{d}}\dots \dots Y_{\mathrm{i}}\dots ..Y_1\right\}$

${\hat{Y}}^{sym}$ Potential adjust data of $Y^{\mathbf{s}\mathbf{y}\mathbf{m}}$ series

$V$ Daily flow volume calculated by Y series

$V_{sum}$ Sum m daily flow volume $V$

$V_T$ Trimestral volume measured

$\bar{V}$Arithmetic mean of $V$

$\hat{V}$ Daily flow volume, calculated with $\hat{Y}$

$\overline{\stackrel{ˆ}{V}}$Arithmetic mean of $\hat{V}$

$V_{\mu }$ Daily flow volume, calculated with the adjust potential daily data through moments

$V_{\mu s}$ Daily flow volume $V_{\mu }$ associated with the consumer’s demand

$V_{\mu l}$ Daily flow volume $V_{\mu }$ associated with real leakage

$V_{\mu sum}$ Sum m daily flow volume $V_{\mu }$

$a$ Exponent of the potential function, daily calculated

$a^{\prime }$ Exponent of the potential function, daily calculated

$k$ Parameter of the potential function, daily calculated

$k_{\mu }$ Parameter of the potential function, daily calculated through moments

${ }_{\mu ear}$ Annual daily mean $k_{\mu }$

$k_{\mu sum}$ Sum m daily parameters $k_{\mu }$

$N$ Offset of the potential function, daily calculated

$N_{\mu }$ Offset of the potential function, daily calculated through moments

$N_{\mu s}$ Fraction of $N_{\mu }$, associated to consumers

$N_{\mu l}$ Fraction of $N_{\mu }$, associated to real leaks

$S_{\mu }$ Coefficient characteristic of the sector, to calculate ${\mathrm{N}}_{\mu l}$

$N_{\mu sum}$ Sum m daily parameters $N_{\mu }$

${\mu }_n$ Moment of degree n respect to f=0, calculated with the work daily data

${\mu }_n^{sym}$ Moment of degree n respect to f=1, calculated with the work daily data

${\hat{\mu }}_n$ Moment of degree n respect to f=0, calculated with the adjust potential daily data

${\hat{\mu }}_{nsum}$ Sum m daily moments ${\hat{\mu }}_n$

$Er{r}_{com}$ Combined error

$Er{r}_p$ Point error

$Er{r}_s$ Shape error

mc Metered Consumption

al Apparent Losses

al_µ Apparent Losses calculated by adjust potential data through moments

$P()$ Normalized probability density function