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Water demand forecasting by trend and harmonic analysis
archives of civil and mechanical engineering 18 (2018) 140–148
Available online at www.sciencedirect.com
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Original Research Article
Water demand forecasting by trend and harmonic analysis Edward Kozłowski a, Beata Kowalska b, Dariusz Kowalski b, Dariusz Mazurkiewicz c,* a Lublin University of Technology, Faculty of Management, Department of Quantitative Methods in Management, Nadbystrzycka 38D, 20-618 Lublin, Poland b Lublin University of Technology, Faculty of Environmental Engineering, Department of Water Supply and Wastewater Disposal, Nadbystrzycka 40B, 20-618 Lublin, Poland c Lublin University of Technology, Faculty of Mechanical Engineering, Department of Production Engineering, Nadbystrzycka 36, 20-618 Lublin, Poland
article info
abstract
Article history:
Water demand forecasting in water supply systems is one of the basic strategic management
Received 6 February 2017
tasks of water supplying companies. This is done using specially designed water consump-
Accepted 24 May 2017
tion models which generate data necessary for planning operational activities. A high
Available online
number of water demand forecasting methods proposed in the literature points to the
Keywords:
companies. However, it must be observed that no universal method applicable to any water
Forecasting methods
supply system has been developed so far. In addition to this, there is no method which could
Water supply system
be considered referential relative to other methods. For this reason, it is necessary to
complexity and significance of the problem for current operation of water supplying
Time series
continue the research on forecasting methods enabling effective forecasts based on suitably
Trend analysis
selected sets of input quantities. This paper proposes a solution for water consumption
Harmonic analysis
forecasting in a water supply system, wherein hourly water consumption is determined by trend analysis and harmonic analysis. Trend analysis consists in estimating parameters of models for individual phases of a cycle, while harmonic analysis is based on the assumption that a time series consists of sine and cosine waves with different frequencies known as harmonics. In addition, relationships between structural parameters of individuals harmonics and ambient temperature are investigated using the least squares method. © 2017 Politechnika Wrocławska. Published by Elsevier Sp. z o.o. All rights reserved.
1.
Introduction
Water demand forecasting is one of the key tasks of water supplying companies to ensure proper standards of service
together with reduced operational costs, including, in particular, that of electric energy required for pumping, usually prevailing in the total economic calculation [1,2]. Water demand forecasting is done using water consumption models specially developed for this purpose. Forecasting data
* Corresponding author. E-mail address: [email protected] (D. Mazurkiewicz). http://dx.doi.org/10.1016/j.acme.2017.05.006 1644-9665/© 2017 Politechnika Wrocławska. Published by Elsevier Sp. z o.o. All rights reserved.
archives of civil and mechanical engineering 18 (2018) 140–148
obtained with such models are then used to manage, control and modernize the existing water supplying infrastructure, as well as to design new elements of the system [3,4]. Water consumption standards are also used to create computersimulated models reflecting hydraulic conditions in waterpipe networks. Such simulations are usually performed using time and space averaged values of these standards. Spatial averages are determined by grouping specific consumers and assigning water consumption to water-pipe network nodes, while time averages are calculated as the averages of temporary values of water consumption at nodes. A correct model for forecasting variations in water consumption will additionally facilitate identification and control of leakages in the water supply system by comparing real and simulated water demand [5–7]. Despite numerous solutions for optimizing water supply infrastructure management reported in the literature, there is a continuous search for verified water demand forecasting models as effective tools for the operational management of a water supplying company. For example, in paper by Loska et al. [8] two subsystems are presented, which create an integrated control and management system for water supply network. The development of forecasting models for the above applications is not an easy task due to the complex, deterministic and random nature of water consumption [9]. Water consumption forecasts can refer to different time intervals depending on its aim [10]. As for current and shortterm forecasts generated one up to several hours in advance, short-term forecasting is usually used exclusively in algorithms for simulating water distribution system operation and developing an optimal strategy for the management of water delivery processes [11,12]. The output value of the model can refer to both daily and hourly flow if the forecasting of pump control and tankage management is planned. Water consumption models predominantly assume hourly time intervals, although there are studies in which shorter intervals (e.g. 15 min) are used [13], the fundamental input parameters of such water demand forecasting models being historical data regarding water consumption, temperature, precipitation [14– 16] and seasonal factor [28] according to which there is a significant difference between water consumption during the summer and the rest of the year. An alternative solution to short-term forecasts are long-term forecasts which are usually applied to plan and design water supply infrastructure, as well as to manage the company's property [15,17]. Given their diversity, water demand forecasting models can be classified according to different criteria, one of them being a division into linear and nonlinear models [18]. Widely used linear models include univariate time series analysis and autoregressive integrated moving average (ARIMA) models [19–22]. These methods are widely used due to their simplicity and practical use in operational activity. The methods for forecasting hourly water consumption time series (ARIMA models and exponential smoothing of time series methods) meet the practical rule of easy availability of output data for forecasting, including predicators. However, they do not omit any external variables and are solely based on chronological sequences of observations from the direct past [23]. Autoregressive integrated moving average (ARIMA) models reflect static and dynamic properties of stationary series and certain
141
classes of non-stationary series interpreted as the result of white noise passing through a discrete, finite-dimensional linear filter. They can be used for current and short-term forecasts of time series regarding water consumption, however the accuracy of their results is often unsatisfactory. Forecast results are usually made more accurate by describing and transforming the raw data by Fourier transform [12]. Widely used nonlinear methods for water demand forecast modelling include: nonlinear regression models, bilinear models, threshold autoregressive models, ANN-based models, fuzzy logic, extended Kalman filter and genetic programming, and model trees [4,14–16,24–26]. Artificial neural networks (ANNs) can be used to analyse many variables simultaneously, hence it is possible to develop a model even if solving the problem is exceptionally complex. The disadvantages of neural networks include difficulty in determining structural parameters, long time of learning, and lack of clear interpretation [1,4,11,14,27]. Among the many new multivariate regression methods, particular attention should be paid to Support Vector Regression (SVR) [29,30]. The algorithm of this method consists in determining a linear function. The nonlinearity of this method results from the fact that observations about a set of learning data are transformed to a new space with a much greater dimension by a non-linear transform. One of the advantages of multivariate regression is the fact that it does not require the user to verify the assumptions concerning the distribution of diagnostic variables. The application of this method enables reduction in forecasting errors regarding observations from the test data set. Moreover, this method is resistant to noise occurrence in the set of learning data. Unfortunately, its operation is automated to a great extent, resembling that of a ‘‘black box’’, and the results can be interpreted only to a small degree. The stages of the learning process and validation required for data tuning are complicated here, too [31,32]. Fontanazza et al. [3] proposed dividing regressive modelbased forecasting methods into 5 categories: regression analysis, time series analysis, computational intelligence approach and stochastic models. Time series models can contain long-term, cyclical and short-term ingredients. Artificial intelligence comprises artificial neural network (ANN) models, fuzzy logic, and agent-based models [3]. In recent years, hybrid models based on earlier developed methods have been effectively used, for instance, the support vector regression method based on Fourier time series proposed by Brentan et al. [33]. In turn, Romano and Kapelan [34] proposed a new method for water demand forecasting based on time series analysis and evolutionary artificial neural networks (EANNs). A survey of methods for modelling water consumption in different time intervals was also given by Qi and Chang [35] as well as House-Peters and Chang [9]. The abundance of water demand forecasting methods points to the complexity and significance of the problem. It must be noted that there is no universal method which could be applied to any water supply system. In addition to this, there is no method which could be regarded as referential in relation to other methods. In light of the above, it is necessary to continue research on forecasting methods, a particular challenge for mathematical forecasting models being here long-term forecasts based on correctly selected sets of input data. This stems from the fact
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that aside from the random nature of water consumption, these methods should also take account of many additional factors such as temperature or precipitation. This paper proposes a solution to short-term forecasting problem. To this end, two methods for water consumption forecasting in a selected real water supply system are presented. Specifically, hourly water consumption is determined by trend and harmonic analyses. Trend analysis assumes that every period (cycle) contains individual phases and consists in determining model parameters for individual phases of the cycle. In contrast, harmonic analysis is based on the assumption that a time series consists of sine and cosine waves with different frequencies which are known as harmonics. For each of the above methods, we examined the relationships between structural parameters of phases of the cycle and individual harmonics versus ambient temperature. Parametric identification was performed by the least squares method. Next, based the forecasts of average ambient temperature, we devised forecasts of hourly water consumption for the next coming month.
2.
Data for modelling
Observing the behaviour of a technical, physical or economic process, we read or automatically register values which describe this process, thereby creating a database. Currently, as far as water supply systems are concerned, we can observe a significant development of both techniques and methods for measuring elements of such systems, along with solutions which ensure electronic registration of measurement data. Data from automatic cyclical readings are a significant part of the database, and they require suitable processing and inferring with respect to taking decisions concerning optimal management of technical infrastructure [36–38]. Examining values collected in the database, we try to design a mathematical model which could describe as accurately as possible the investigated phenomenon or a technological process. The behaviour of a system is usually modelled by a stochastic process with discrete time, which is known as a time series [39]. The main task here is to identify non-random components and internal disturbances in the series. For this reason, the development of a model for forecasting water demand by a real water supply system is based on the measurement of global water consumption in a town of Puławy. The municipal water supply system supplies water to approximately 50,000 inhabitants. Groundwater is taken by 7 wells with an output of 51, 131, 150, 3 180 and 210 m3/h, respectively. The measurement of water quantity is made using electromagnetic flowmeters. Next, untreated water is directed to two connected parallel and cooperating storage and surge tanks, each with a capacity of 1500 m3. The water from the tanks is shared with the joint conduit to the second stage pumping station, which supplies the water supply network. For the purpose of the present paper, these tanks are conventionally treated as a source of water for a secondary pumping station supplying water directly to the water distribution system. This pumping station is equipped with six KSB-Etanom M80-200 M11 pumps, each with a capacity of 180 m3/h, and it pumps water to a water supply system via two
pipelines providing water to two zones: ‘‘Puławy East’’ and ‘‘Puławy West’’. Zone division was introduced due to varied terrain topography. The Western zone is bigger and gets 1.7 times more water than the Eastern one. There are no significant differences between the zones because the participation of each type of water consumer is similar. The industry present in the city has its own, independent water supply system. The quantity of water pumped by second stage pumping station is measured with two electromagnetic flowmeters, each located on the pipelines supplying water to the above zones. The measurements of flow rate in these pipelines are made and stored with a frequency of 1 min. They serve as a basis for determining global water consumption of both every supply zone and the entire town. The measurement data used in this study encompass a period of two months (1 July 2015–31 August 2015). The data for that period is additionally grouped under two categories: weekdays and weekends. This is because the water consumption depends on both citizens (7 days/week consumption) and industrial sector (only weekdays consumption). The data from the first month is used to indentify models, while the data from the second month is used to test accuracy of the results. The water consumption forecasts for a monthly period ranging from 1 August to 31 August 2015 obtained by modelling are compared with the real measurements for that period. In addition, the computations take account of the effect of air temperature on water consumption, because the results demonstrate that water consumption between 6 am and 11 pm is significantly affected by ambient temperature. This is why for modelling two factors were used: average daily temperature and day category (weekday or weekend).
3.
Trend analysis
Trend analysis consists in identifying periodic variation (all phases of a cycle) occurring in time series. The method for identifying individual phases of a time series is known as the phase trend method [39,40]. The parameters of non-random functions occurring in individual phases of the cycle are predominantly determined by the least squares method or the highest reliability method. The analysis of hourly water consumption is based on the assumption that a cycle (day) contains 24-h phases. Therefore, to perform parametric identification, the investigated time series fxi g1iN is divided into 24 subseries: n o xk1 ¼ x1 ; x25 ; . . .; x24ðT1Þþ1 ; 1kT n o ¼ x2 ; x26 ; . . .; x24ðT1Þþ2 ; xk2 1kT .n. . o xk24 ¼ fx24 ; x48 ; . . .; x24T g;
(1)
1kT
where xkj denotes water consumption from hour j 1 until hour j of day k for 1 j 24 and 1 k T (we assume that 24T = N). The behaviour of each phase of the series is modelled the above, we assume that using first order polynomial.n Given o for the phase 1 j 24 the elements of the series xkj 1k31 satisfy the equation: j
j
xkj ¼ a0 þ a1 tk þ ekj ;
(2)
archives of civil and mechanical engineering 18 (2018) 140–148
where
akj ¼ b0j þ b1j k þ ykj
j
j
a0 and a1 denote the free term and the slope in expression in jth phase, tk – mean daily ambient temperature for kth day, n o ek is a series of independent and identically distribj
1k24
uted random variables with normal distribution N 0; s 2j for
j
j
The values of a0 and a1 were determined by the least n o denote the squares method. The values of the series xkj 1j24
theoretical values (forecasts) of water consumption for kth day: j
j
xkj ¼ a ^0 þ a ^ 1 tk ; j
(3) j
^ 1 , 1 j 24 denote the estimated structural ^ 0 and a where a parameters in model (2). When we will assume that tk = k, then in the model described by Eq. (2) we must investigate a linear dependence between water consumption and time factor. As a result, for each phase we forecast the hourly water consumption with the use of (3) and linear trends.
(6)
and temperature akj ¼ c0j þ c1j tk þ vkj ;
(7)
where tk denotes the mean daily temperature on kth day, n o n o and vkj are the series of independent whereas ykj 1kT
1 j 24.
143
1kT
and identically distributed random variables with the normal distributions N 0; s 2j and N 0; g 2j , respectively. The forecasts (theoretical values) of hourly water consumption are estimated in the following way. For kth day, k > T k ^n 1n24 using the expansion: we calculate the series x ^kn ¼ x
24 X 2pi ð j1Þðn1Þ ; a ^ kj exp 24 j¼1
(8)
^ kj denotes the estimation of jth of this harmonic. where a Harmonics estimators are determined by the equations: 0
1
^ þb ^ k ^ kj ¼ b a j j
(9)
or
4.
^ kj ¼ ^c0j þ ^c1j tk ; a
Harmonic analysis
^0 ; b ^1 , b j j
Harmonic analysis is based on the assumption that a time series consist of sine and cosine waves with different frequencies. Different frequency waves are known as harmonics, while elements of the series are linear combinations of these harmonics (see e.g. [39,41–43]). Periodic variation in the time series fxs g1sN will be identified using Fast Fourier Transformation. Similarly to the trend analysis, we will investigate the dependence of structural parameters of individual harmonics on time factor and ambient temperature. For every day 1 k T, where 24T = N we consider hourly water consumption and analyse the behaviour of elements of def the series xkn 1n24 ¼ fxs g24ðk1Þs24k fxs g1sN . Therefore, k for the series xs 1s24 , 1 k T we take account of 24 harmonics while the elements of the series are expressed as: xkn ¼
24 X 2pi ð j1Þðn1Þ ; akj exp 24 j¼1
(4)
where akj denotes the coefficient for the jth harmonic of kth day, 1 k T, 1 j 24. The values of harmonic coefficients are determined using the inverse Fourier transform: 24 1 X 2pi ð j1Þðn1Þ xkn exp (5) akj ¼ 24 n¼1 24 for 1 k T, 1 j 24. For each day we evaluate the 24 factors of Fourier transform and then we analyse day temperature influence on their values. Obviously, by predicting the n o ^ kj for k > T, we can behaviour of the harmonic series a 1j24 k ^n 1n24 . This study forecast the hourly water consumption x investigates the dependence of each harmonic on time factor:
(10) ^c0j ; ^c1j
where are the values of structural parameter estimators for Eqs. (6) and (7), respectively. As water consumption forecasts we take parts of the real elements of k ^n 1n24 [40,42,43]. the series x
5.
Results
To identify the parametric dependence between hourly water consumption on average ambient temperature in model (2) and the relationships between direct coefficients and average ambient temperature (7) expanded in Fourier series (4), we used data collected for the period between 1 July 2015 and 31 July 2015. In addition to that, the period was divided into weekends and workdays. In compliance with the above, for 48 equations of (2) describing phase trends, we estimated 96 parameters, i.e. we examined hourly water consumption separately for workdays and weekends using Eq. (2) which contains a free term and a directivity coefficient (2 24 2 = 96). Analogically, we determined 96 parameters examining the linear relationships (7) of Fourier transform coefficients (4) and average daily temperature. Using the trend and harmonic analysis methods, we estimated the theoretical values of water consumption by the lines ‘‘Puławy East’’ and ‘‘Puławy West’’ for the period from 1 July 2015 to 31 July 2015 (for k = 1, 2, . . ., 31). For clarity, the results are presented in a graphic form. In Fig. 1 the blue curves illustrate the real values of water consumption xkn 1n24 by ‘‘Puławy East’’ and ‘‘Puławy West’’ during the period from 1 July 2015 to 31 July 2015 (for k = 1, 2, . . ., 31). The red dotted k ^n 1n24 curves represent the theoretical water consumption x estimated by the Fourier Transformation (FT) (8), considering the effect of mean daily ambient temperature on the harmonic components a ^ kj , 1 j 24, 1 k 31. The estimators of the
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Fig. 1 – Empirical and deterministic values obtained by trend analysis and Fourier transform for the period 11 July 2015–17 July 2015.
harmonics were determined by Eq. (10). When the harmonic estimators were calculated by Eq. (9), then the water consumption forecasts estimated by the FT (8) were worse (were more different from the empirical values). The green curves represent the theoretical water consumption values n o xkn which depend on the mean daily ambient temper1n24
ature and are estimated by using the Phase Trend Method (PTM) (see Eq. (3)). Fig. 1 reveals that the deterministic values obtained by the trend analysis and Fourier transform (the values of series of (3) and (8)) are practically the same. For the lines ‘‘Puławy East’’ and ‘‘Puławy West’’ we detern o and ekn 1n24 for mined the difference sequences ^ekn 1n24
^kn denote the k = 1, 2, . . ., 31 (consecutive days), where ^ekn ¼ xkn x differences between real water consumption and its theoretical values determined by the Fourier transformation (8), whereas ekn ¼ xkn xkn denote the differences between real water consumption and its theoretical values estimated by trend analysis. Since the theoretical values of water consumption per each line estimated by the phase trend method and Fourier transform are practically the same, then the real part of the sequence
n o ^ekn
1n24
and the values of the sequence ekn 1n24 are similar,
too. For this reason, the subsequent part of this paper will investigate the residues sequence ekn 1n24 . Fig. 2a and b illustrates the values of residues sequences ekn 1n24 , 1 k 31 for ‘‘Puławy East’’ and ‘‘Puławy West’’, respectively, during the period between 1 July 2015 and 31 July 2015. The statistical data describing the water supply models for two regions, Puławy East and Puławy West, in the investigated water supply system (Table 1) demonstrate that the mean hourly water consumption for ‘‘Puławy East’’ is 110.106 m3, for ‘‘Puławy West’’ – 1785.775 m3, whereas the standard deviations for the above lines are 49.914 and 79.004, respectively. To verify the matching of phase trends (2), we investigates the residues sequence ekn 1n24 for k = 1, 2, . . ., 31. The mean values of me of the residues sequence for the above lines are equal to zero, while the standard deviations se are 15.65 and 23.02, respectively. Comparing the values of s and se, we can observe that the application of models (2) and (4) provides more accurate water consumption forecasts than the forecasts based on arithmetic means. In addition to this, the variations in water consumption are explained by models (2) and (4) in 88.02% for ‘‘Puławy East’’ (see Table 1, determination coeffi-
Table 1 – Comparison of statistical data describing the water supply models in the regions Puławy East and Puławy West in the investigated water supply system.
Puławy East Puławy West
m
s
me
se
R2
F
110.106 175.775
49.914 79.004
0 0
15.65 23.02
0.8802 0.8853
5.9415 5.9759
archives of civil and mechanical engineering 18 (2018) 140–148
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Fig. 2 – Values of residues sequences ekn 1n24 and normal distribution matching for lines ‘‘Puławy East’’ and ‘‘Puławy West’’.
cients R2), and in 88.53% for the line ‘‘Puławy West’’. To analyse the significance of the multiple correlation coefficient R, we determined values of the statistics F (see Table 1) which has a Fisher–Snedecor distribution. For the applied significance level a = 0.001 and the number of degrees of freedom (744 96 = 648), the critical value of the statistics F* is 1.5642. Since for every line F > F*, hence as for the significance level a = 0.001 we must reject the working hypotheses about the lack of multiple correlation in models (2) and (4). Given the above, we assume that multiple correlation coefficients significantly differ from zero, and the matching of the models to the empirical data regarding hourly water consumption by ‘‘Puławy East’’ and ‘‘Puławy West’’ is relatively high. Fig. 2c shows the matching of the residues sequence k en 1n24 for k = 1, 2, . . ., 31 to the normal distribution Nð0; 15:65Þ for ‘‘Puławy East’’, while Fig. 2d – to the distribution Nð0; 23:02Þ for ‘‘Puławy West’’ The points corresponding the values of these sequences are not located along the straight line connecting the first and third quantiles. We can observe that the differences with absolute values over 40 m3 diverge from this line; in Fig. 2a and b we can additionally observe the presence of extreme values, the so-called ‘‘outliers.’’ Using the Kolmogorov–Smirnov and Lilliefors tests, it has been found that at a significance level of 0.01 the postulate of normality of
distributions for these residues sequences is not met. Hence, we can predict that the residues sequences ekn 1n24 for k = 1, 2, . . ., 31 have additional internal dynamics, the investigation of which is beyond the scope of this paper. The best way to verify the effectiveness of trend analysis (2) and Fourier transform (4), considering the effect of mean daily ambient temperature on harmonics values (7), is to compare the forecast and empirical results. Based on the mean daily ambient temperatures for the period between 1 August 2015 and 31 August 2015, we forecast water consumption for two lines, ‘‘Puławy East’’ and ‘‘Puławy West’’ Next, the forecast results were compared with the real water consumption on these lines during the period 1/08/2015–31/08/2015. In Fig. 3 the blue curves denote the real values of water consumption k xn 1n24 for the above lines, whereas the red dotted curve 1k7 k ^n 1n24 obtained marks the forecast water consumption x 1k7 with the Fourier transform (8), where the harmonics were determined by Eq. (10). The green curve illustrates the water n o consumption xki 1n24 determined by trend analysis (3). 1k7 We can observe here that the water consumption results per
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Fig. 3 – Empirical and forecast data obtained by phase trend method and Fourier transform for the period 1 August 2015–7 August 2015.
each line estimated by the phase trend method and Fourier transform are practically the same. To perform further data analyses, we determined the residues sequences ekn 1n24 for k = 1, 2, . . ., 31 between the forecast and real water consumption per each line during the period between 1 August 2015 and 31 August 2015. The results obtained are listed in Table 2, offering a comparison of the mean values and standard deviations of the residues sequences ekn 1n24 for k = 1, 2, . . ., 31 for the lines ‘‘Puławy East’’ and ‘‘Puławy West’’. It can therefore be observed that the average hourly underestimation of the forecast versus real consumption is 2.01 m3 for ‘‘Puławy East,’’ and 7.99 m3 for ‘‘Puławy West’’ Standard deviations between the forecast and real water consumption for ‘‘Puławy East’’ and ‘‘Puławy West’’ are 22.19 and 30.71, respectively. In addition, the analysis of normality of distribution of the residues sequences ekn 1n24 by the 1k 31 Kolmogorov–Smirnov and Lilliefors tests reveal that for each of the above lines at a significance level of 0.01, the residues
Table 2 – Comparison of the mean values and standard deviation of difference sequences between the forecast and measured values for the period 1 August 2015–31 August 2015.
Puławy East Puławy West
me
se
2.01 7.99
22.19 30.71
sequences ekn 1n24 do not meet the postulate of normal
1k 31 distribution Nð2:01; 22:19Þ and Nð7:99; 30:71Þ for ‘‘Puławy East’’ and ‘‘Puławy West’’, respectively. Therefore, we can suspect that there are additional factors affecting water consumption or that the residues sequences ekn 1n24 have internal 1k 31 dynamics which should be modelled by ARIMA or ARCH models.
6.
Conclusion
Water consumption depends on a number of factors, including periodicity, time, temperature and randomness. This paper presented two methods for modelling hourly water consumption in a water supply system. The first method involves the use of linear phase trends dependent on ambient temperature. The other consists in using the Fourier transform, wherein the harmonics are linearly dependent on ambient temperature. Structural parameters of the above models were estimated based on the date regarding two lines, ‘‘Puławy East’’ and ‘‘Puławy West,’’ collected between 1 July 2015 and 31 July 2015. The application of the two water demand forecasting models demonstrated that the real part of theoretical values for the Fourier transform and the values of the expected phase trends are similar. Both the matching of these models to the empirical data and the forecasts for the period between 1 August 2015 and 31 August 2015 obtained with these models reflect relatively accurately the variations in daily water consumption.
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Moreover, the differences between the theoretical and empirical data on water consumption indicate that there exist some additional factors affecting water consumption. In presented water consumption models the seasonality factor was not taken into account directly, because the seasonality is strong correlated with the ambient temperature. Assumed weekends and weekdays partition together with considered the ambient temperature allows us to explain the variability of hourly water consumption in 88%. The advantage of presented model is its ability to construct forecasts based on a short history of data. The results confirm that the above models can effectively be used for water demand forecasting (coefficient of determination above 0.88 and multiple correlation coefficient significantly different from zero) and designing controllers for water supply pumps in order to optimize the costs of maintaining water surface at the desired level. Due to the fact that the differences between theoretical and empirical data on water demand do not have a normal distribution, further studies will focus on determining additional factors which affect water consumption (e.g. waterworks malfunctions, swimming pools water replacement, precipitation quantity, inner dynamics identification using ARIMA or ARCH models).
references
[1] S. Zhou, T.A. McMahon, A. Walton, J. Lewis, Forecasting operational demand for an urban water supply zone, Journal of Hydrology 259 (1–4) (2002) 189–202. [2] T.G. Mamo, J. Ilan, S. Isam, Urban water demand forecasting using the stochastic nature of short term historical water demand and supply pattern, Journal of Water Resource and Hydraulic Engineering 2 (2013) 92–103. [3] C.M. Fontanazza, V. Notaro, V. Puleo, G. Freni, Multivariate statistical analysis for water demand modeling, Procedia Engineering 89 (2014) 901–908. [4] M. Tiwari, J. Adamowski, Medium term urban water demand forecasting with limited data using an ensemble waveletbootstrap machine-learning approach, Journal of Water Resources Planning and Management 141 (2) (2015) 04014053. [5] M. Iwanek, B. Kowalska, E. Hawryluk, K. Kondraciuk, Distance and time of water effluence on soil surface after failure of buried water pipe. Laboratory investigations and statistical analysis, Eksploatacja i Niezawodnosc (Maintenance and Reliability), 18 (2) (2016) 278–284. , http://dx.doi.org/10.17531/ ein.2016.2.16. [6] R. Klempous, J. Kotowski, J. Nikodem, J. Ułasiewicz, Optimization algorithms of operative control in water distribution systems, Journal of Computational and Applied Mathematics 84 (1) (1997) 81–99. [7] F. Odan, L. Reis, Hybrid water demand forecasting model associating artificial neural network with Fourier series, Journal of Water Resources Planning and Management 138 (3) (2012) 245–256. [8] A. Loska, W. Moczulski, R. Wyczółkowski, A. Dąbrowski, Integrated system of control and management of exploitation of water supply system, Diagnostyka 17 (1) (2016) 65–74. [9] L.A. House-Peters, H. Chang, Urban water demand modeling. Review of concepts, methods, and organizing principles, Water Resources Research 47 (2011) W05401. [10] M. Bakker, K. Ven Schagen, J. Timmer, Flow control by prediction of water demand, Journal of Water Supply: Research and Technology 52 (2003) 417–424.
147
[11] M. Herrera, L. Torgo, J. Izquierdo, R. Perez-Garcia, Predictive models for forecasting hourly urban water demand, Journal of Hydrology 387 (1–2) (2010) 141–150. [12] Z. Siwoń, W. Cieżak, J. Cieżak, Stochastic models of water demand in a chosen water-supply system, Ochrona Środowiska 27 (2005) 7–13. [13] M. Bakker, J.H.G. Vreeburg, L.J. Palmen, V. Sperber, G. Bakker, L.C. Rietveld, Better water quality and higher energy efficiency by using model predictive flow control at water supply system, Journal of Water Supply: Research and Technology – AQUA 62 (2013) 1–13. [14] J. Adamowski, H.F. Chan, S.O. Prasher, B. Ozga-Zielinski, A. Sliusarieva, Comparison of multiple linear and nonlinear regression, autoregressive integrated moving average, artificial neural network, and wavelet artificial neural network methods for urban demand forecasting in Montreal, Canada, Water Resources Research 48 (1) (2012) W01528. [15] J. Boguadis, K. Adamowski, R. Diduch, Short-term municipal water demand forecasting, Hydrological Processes 19 (1) (2005) 137–148. [16] A. Jain, A. Varshney, U. Joshi, Short-term water demand forecast modeling at IIT Kanpur using artificial neural networks, Water Resources Management 15 (5) (2001) 299– 321. [17] S. Behboudian, M. Tabesh, M. Falahnezhad, F.H. Ghavanini, A long-term prediction of domestic water demand using preprocessing in artificial neural network, Journal of Water Supply: Research and Technology 63 (1) (2014) 31–42. [18] G. Zhang, An investigation of neural networks for linear time-series forecasting, Computers and Operations Research 28 (12) (2001) 1183–1202. [19] J. Alhumoud, Freshwater consumption in Kuwait; analysis and forecasting, Journal of Water Supply: Research and Technology – AQUA 57 4 (2008) 279–288. [20] T. Hughes, Peak period design standards for small western U. S., Water Supply 16 (4) (1980) 661–667. [21] D. Maidment, S. Miaou, M. Crawford, Transfer function models of daily urban water use, Water Resources Research 21 (4) (1985) 425–432. [22] S. Zhou, T.A. McMahon, A. Walton, J. Lewis, Forecasting daily urban water demand: a case study of Melbourne, Journal of Hydrology 236 (3–4) (2000) 153–164. [23] O. Voitcu, Y.S. Wong, On the construction of a nonlinear recursive predictor, Journal of Computational and Applied Mathematics 190 (1) (2006) 393–407. [24] C. Bunett, R.A. Stewart, C.D. Beal, ANN-based residential water end-use demand forecasting model, Expert Systems and Applications 40 (4) (2013) 1014–1023. [25] P. Cutore, A. Campisano, Z. Kapelan, C. Modica, D. Savic, Probabilistic prediction of urban water consumption using the SCEM-UA algorithm, Urban Water Journal 5 (2) (2008) 125– 132. [26] M. Nasseri, A. Moeini, M. Tabesh, Forecasting monthly urban water demand using extended Kalman filter and genetic programming, Expert Systems and Applications 38 (6) (2011) 7387–7395. [27] C. Benetti, R.A. Stewart, C.D. Beal, ANN based residential water end-use demand forecasting model, Expert Systems and Applications 40 (4) (2013) 1014–1023. [28] A. Loska, Scenario modeling exploitation decision-making process in technical network systems, Eksploatacja i Niezawodnosc – Maintenance and Reliability 19 (2) (2017) 268–278. , http://dx.doi.org/10.17531/ein.2017.2.15. [29] I.S. Msiza, F.V. Nelvamondo, T. Marwala, Artificial neural networks and support vector machines for water demand time series forecasting, in: IEEE International Conference on System, Man and Cybernetics, IEEE, 2007 638–643.
148
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[30] M. Trzęsiok, Symulacyjna ocena jakości zagregowanych modeli zbudowanych metodą wektorów nośnych. W: Zastosowania metod matematycznych w ekonomii i zarządzaniu, Wydawnictwo Uniwersytetu Ekonomicznego w Katowicach, Katowice, 2013. [31] I. Banicescu, R.L. Cariño, J.I. Harvill, J.P. Lestrade, Investigating asymptotic properties of vector nonlinear time series models, Journal of Computational and Applied Mathematics 236 (3) (2011) 411–421. [32] C.L.Z. Tu-Qiao, Hourly water demand forecast model based on Bayesian least squares support vector machine, Journal of Tianjin University 9 (2006) 005. [33] B.M. Brentan, E. Luvizotto Jr., M. Herrera, J. Izquierdo, R. Perez-Garcia, Hybrid regression model for near teal-time urban water demand forecasting, Journal of Computational and Applied Mathematics (2016), http://dx.doi.org/10.1016/j. cam.2016.02.009 (in press). [34] M. Romano, Z. Kapelan, Adaptive water demand forecasting for near real-time management of smart water distribution systems, Environmental Modelling and Software 60 (2014) 265–276. [35] C. Qi, N. Chang, System dynamics modeling for municipal water demand estimation in the urban region under uncertain economic impacts, Journal of Environment Management 92 (2011) 1628–1641.
[36] B. Hazen, C.H. Boone, J. Ezell, L.A. Jones-Farmer, Data quality for data science, predictive analytics, and big data in supply chain management: An introduction to the problem and suggestions for research and applications, International Journal of Production Economics 154 (4) (2014) 72–80. [37] E. Kosicka, E. Kozłowski, D. Mazurkiewicz, The use of stationary tests for analysis of monitored residual processes, Eksploatacja i Niezawodnosc (Maintenance and Reliability), 17 (4) (2015) 604–609. , http://dx.doi.org/10.17531/ ein.2015.4.17. [38] D. Mazurkiewicz, Computer-aided maintenance and reliability management systems for conveyor belts, Eksploatacja i Niezawodnosc (Maintenance and reliability), 16 (3) (2014) 377–382. [39] E. Kozłowski, Analiza i identyfikacja szeregów czasowych, Wyd. Politechnika Lubelska, Lublin, 2015. [40] A. Zeliaś, B. Pawełek, S. Wanat, Prognozowanie ekonomiczne, Teoria, przykłady, zadania, PWN, Warszawa, 2004. [41] G.E.P. Box, G.M. Jenkins, Analiza szeregów czasowych, PWN, Warszawa, 1983. [42] G.C. Chow, Ekonometria, PWN, Warszawa, 1995. [43] J.D. Hamilton, Time Series Analysis, Princeton University Press, Princeton, 1994.
Available online at www.sciencedirect.com
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Original Research Article
Water demand forecasting by trend and harmonic analysis Edward Kozłowski a, Beata Kowalska b, Dariusz Kowalski b, Dariusz Mazurkiewicz c,* a Lublin University of Technology, Faculty of Management, Department of Quantitative Methods in Management, Nadbystrzycka 38D, 20-618 Lublin, Poland b Lublin University of Technology, Faculty of Environmental Engineering, Department of Water Supply and Wastewater Disposal, Nadbystrzycka 40B, 20-618 Lublin, Poland c Lublin University of Technology, Faculty of Mechanical Engineering, Department of Production Engineering, Nadbystrzycka 36, 20-618 Lublin, Poland
article info
abstract
Article history:
Water demand forecasting in water supply systems is one of the basic strategic management
Received 6 February 2017
tasks of water supplying companies. This is done using specially designed water consump-
Accepted 24 May 2017
tion models which generate data necessary for planning operational activities. A high
Available online
number of water demand forecasting methods proposed in the literature points to the
Keywords:
companies. However, it must be observed that no universal method applicable to any water
Forecasting methods
supply system has been developed so far. In addition to this, there is no method which could
Water supply system
be considered referential relative to other methods. For this reason, it is necessary to
complexity and significance of the problem for current operation of water supplying
Time series
continue the research on forecasting methods enabling effective forecasts based on suitably
Trend analysis
selected sets of input quantities. This paper proposes a solution for water consumption
Harmonic analysis
forecasting in a water supply system, wherein hourly water consumption is determined by trend analysis and harmonic analysis. Trend analysis consists in estimating parameters of models for individual phases of a cycle, while harmonic analysis is based on the assumption that a time series consists of sine and cosine waves with different frequencies known as harmonics. In addition, relationships between structural parameters of individuals harmonics and ambient temperature are investigated using the least squares method. © 2017 Politechnika Wrocławska. Published by Elsevier Sp. z o.o. All rights reserved.
1.
Introduction
Water demand forecasting is one of the key tasks of water supplying companies to ensure proper standards of service
together with reduced operational costs, including, in particular, that of electric energy required for pumping, usually prevailing in the total economic calculation [1,2]. Water demand forecasting is done using water consumption models specially developed for this purpose. Forecasting data
* Corresponding author. E-mail address: [email protected] (D. Mazurkiewicz). http://dx.doi.org/10.1016/j.acme.2017.05.006 1644-9665/© 2017 Politechnika Wrocławska. Published by Elsevier Sp. z o.o. All rights reserved.
archives of civil and mechanical engineering 18 (2018) 140–148
obtained with such models are then used to manage, control and modernize the existing water supplying infrastructure, as well as to design new elements of the system [3,4]. Water consumption standards are also used to create computersimulated models reflecting hydraulic conditions in waterpipe networks. Such simulations are usually performed using time and space averaged values of these standards. Spatial averages are determined by grouping specific consumers and assigning water consumption to water-pipe network nodes, while time averages are calculated as the averages of temporary values of water consumption at nodes. A correct model for forecasting variations in water consumption will additionally facilitate identification and control of leakages in the water supply system by comparing real and simulated water demand [5–7]. Despite numerous solutions for optimizing water supply infrastructure management reported in the literature, there is a continuous search for verified water demand forecasting models as effective tools for the operational management of a water supplying company. For example, in paper by Loska et al. [8] two subsystems are presented, which create an integrated control and management system for water supply network. The development of forecasting models for the above applications is not an easy task due to the complex, deterministic and random nature of water consumption [9]. Water consumption forecasts can refer to different time intervals depending on its aim [10]. As for current and shortterm forecasts generated one up to several hours in advance, short-term forecasting is usually used exclusively in algorithms for simulating water distribution system operation and developing an optimal strategy for the management of water delivery processes [11,12]. The output value of the model can refer to both daily and hourly flow if the forecasting of pump control and tankage management is planned. Water consumption models predominantly assume hourly time intervals, although there are studies in which shorter intervals (e.g. 15 min) are used [13], the fundamental input parameters of such water demand forecasting models being historical data regarding water consumption, temperature, precipitation [14– 16] and seasonal factor [28] according to which there is a significant difference between water consumption during the summer and the rest of the year. An alternative solution to short-term forecasts are long-term forecasts which are usually applied to plan and design water supply infrastructure, as well as to manage the company's property [15,17]. Given their diversity, water demand forecasting models can be classified according to different criteria, one of them being a division into linear and nonlinear models [18]. Widely used linear models include univariate time series analysis and autoregressive integrated moving average (ARIMA) models [19–22]. These methods are widely used due to their simplicity and practical use in operational activity. The methods for forecasting hourly water consumption time series (ARIMA models and exponential smoothing of time series methods) meet the practical rule of easy availability of output data for forecasting, including predicators. However, they do not omit any external variables and are solely based on chronological sequences of observations from the direct past [23]. Autoregressive integrated moving average (ARIMA) models reflect static and dynamic properties of stationary series and certain
141
classes of non-stationary series interpreted as the result of white noise passing through a discrete, finite-dimensional linear filter. They can be used for current and short-term forecasts of time series regarding water consumption, however the accuracy of their results is often unsatisfactory. Forecast results are usually made more accurate by describing and transforming the raw data by Fourier transform [12]. Widely used nonlinear methods for water demand forecast modelling include: nonlinear regression models, bilinear models, threshold autoregressive models, ANN-based models, fuzzy logic, extended Kalman filter and genetic programming, and model trees [4,14–16,24–26]. Artificial neural networks (ANNs) can be used to analyse many variables simultaneously, hence it is possible to develop a model even if solving the problem is exceptionally complex. The disadvantages of neural networks include difficulty in determining structural parameters, long time of learning, and lack of clear interpretation [1,4,11,14,27]. Among the many new multivariate regression methods, particular attention should be paid to Support Vector Regression (SVR) [29,30]. The algorithm of this method consists in determining a linear function. The nonlinearity of this method results from the fact that observations about a set of learning data are transformed to a new space with a much greater dimension by a non-linear transform. One of the advantages of multivariate regression is the fact that it does not require the user to verify the assumptions concerning the distribution of diagnostic variables. The application of this method enables reduction in forecasting errors regarding observations from the test data set. Moreover, this method is resistant to noise occurrence in the set of learning data. Unfortunately, its operation is automated to a great extent, resembling that of a ‘‘black box’’, and the results can be interpreted only to a small degree. The stages of the learning process and validation required for data tuning are complicated here, too [31,32]. Fontanazza et al. [3] proposed dividing regressive modelbased forecasting methods into 5 categories: regression analysis, time series analysis, computational intelligence approach and stochastic models. Time series models can contain long-term, cyclical and short-term ingredients. Artificial intelligence comprises artificial neural network (ANN) models, fuzzy logic, and agent-based models [3]. In recent years, hybrid models based on earlier developed methods have been effectively used, for instance, the support vector regression method based on Fourier time series proposed by Brentan et al. [33]. In turn, Romano and Kapelan [34] proposed a new method for water demand forecasting based on time series analysis and evolutionary artificial neural networks (EANNs). A survey of methods for modelling water consumption in different time intervals was also given by Qi and Chang [35] as well as House-Peters and Chang [9]. The abundance of water demand forecasting methods points to the complexity and significance of the problem. It must be noted that there is no universal method which could be applied to any water supply system. In addition to this, there is no method which could be regarded as referential in relation to other methods. In light of the above, it is necessary to continue research on forecasting methods, a particular challenge for mathematical forecasting models being here long-term forecasts based on correctly selected sets of input data. This stems from the fact
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that aside from the random nature of water consumption, these methods should also take account of many additional factors such as temperature or precipitation. This paper proposes a solution to short-term forecasting problem. To this end, two methods for water consumption forecasting in a selected real water supply system are presented. Specifically, hourly water consumption is determined by trend and harmonic analyses. Trend analysis assumes that every period (cycle) contains individual phases and consists in determining model parameters for individual phases of the cycle. In contrast, harmonic analysis is based on the assumption that a time series consists of sine and cosine waves with different frequencies which are known as harmonics. For each of the above methods, we examined the relationships between structural parameters of phases of the cycle and individual harmonics versus ambient temperature. Parametric identification was performed by the least squares method. Next, based the forecasts of average ambient temperature, we devised forecasts of hourly water consumption for the next coming month.
2.
Data for modelling
Observing the behaviour of a technical, physical or economic process, we read or automatically register values which describe this process, thereby creating a database. Currently, as far as water supply systems are concerned, we can observe a significant development of both techniques and methods for measuring elements of such systems, along with solutions which ensure electronic registration of measurement data. Data from automatic cyclical readings are a significant part of the database, and they require suitable processing and inferring with respect to taking decisions concerning optimal management of technical infrastructure [36–38]. Examining values collected in the database, we try to design a mathematical model which could describe as accurately as possible the investigated phenomenon or a technological process. The behaviour of a system is usually modelled by a stochastic process with discrete time, which is known as a time series [39]. The main task here is to identify non-random components and internal disturbances in the series. For this reason, the development of a model for forecasting water demand by a real water supply system is based on the measurement of global water consumption in a town of Puławy. The municipal water supply system supplies water to approximately 50,000 inhabitants. Groundwater is taken by 7 wells with an output of 51, 131, 150, 3 180 and 210 m3/h, respectively. The measurement of water quantity is made using electromagnetic flowmeters. Next, untreated water is directed to two connected parallel and cooperating storage and surge tanks, each with a capacity of 1500 m3. The water from the tanks is shared with the joint conduit to the second stage pumping station, which supplies the water supply network. For the purpose of the present paper, these tanks are conventionally treated as a source of water for a secondary pumping station supplying water directly to the water distribution system. This pumping station is equipped with six KSB-Etanom M80-200 M11 pumps, each with a capacity of 180 m3/h, and it pumps water to a water supply system via two
pipelines providing water to two zones: ‘‘Puławy East’’ and ‘‘Puławy West’’. Zone division was introduced due to varied terrain topography. The Western zone is bigger and gets 1.7 times more water than the Eastern one. There are no significant differences between the zones because the participation of each type of water consumer is similar. The industry present in the city has its own, independent water supply system. The quantity of water pumped by second stage pumping station is measured with two electromagnetic flowmeters, each located on the pipelines supplying water to the above zones. The measurements of flow rate in these pipelines are made and stored with a frequency of 1 min. They serve as a basis for determining global water consumption of both every supply zone and the entire town. The measurement data used in this study encompass a period of two months (1 July 2015–31 August 2015). The data for that period is additionally grouped under two categories: weekdays and weekends. This is because the water consumption depends on both citizens (7 days/week consumption) and industrial sector (only weekdays consumption). The data from the first month is used to indentify models, while the data from the second month is used to test accuracy of the results. The water consumption forecasts for a monthly period ranging from 1 August to 31 August 2015 obtained by modelling are compared with the real measurements for that period. In addition, the computations take account of the effect of air temperature on water consumption, because the results demonstrate that water consumption between 6 am and 11 pm is significantly affected by ambient temperature. This is why for modelling two factors were used: average daily temperature and day category (weekday or weekend).
3.
Trend analysis
Trend analysis consists in identifying periodic variation (all phases of a cycle) occurring in time series. The method for identifying individual phases of a time series is known as the phase trend method [39,40]. The parameters of non-random functions occurring in individual phases of the cycle are predominantly determined by the least squares method or the highest reliability method. The analysis of hourly water consumption is based on the assumption that a cycle (day) contains 24-h phases. Therefore, to perform parametric identification, the investigated time series fxi g1iN is divided into 24 subseries: n o xk1 ¼ x1 ; x25 ; . . .; x24ðT1Þþ1 ; 1kT n o ¼ x2 ; x26 ; . . .; x24ðT1Þþ2 ; xk2 1kT .n. . o xk24 ¼ fx24 ; x48 ; . . .; x24T g;
(1)
1kT
where xkj denotes water consumption from hour j 1 until hour j of day k for 1 j 24 and 1 k T (we assume that 24T = N). The behaviour of each phase of the series is modelled the above, we assume that using first order polynomial.n Given o for the phase 1 j 24 the elements of the series xkj 1k31 satisfy the equation: j
j
xkj ¼ a0 þ a1 tk þ ekj ;
(2)
archives of civil and mechanical engineering 18 (2018) 140–148
where
akj ¼ b0j þ b1j k þ ykj
j
j
a0 and a1 denote the free term and the slope in expression in jth phase, tk – mean daily ambient temperature for kth day, n o ek is a series of independent and identically distribj
1k24
uted random variables with normal distribution N 0; s 2j for
j
j
The values of a0 and a1 were determined by the least n o denote the squares method. The values of the series xkj 1j24
theoretical values (forecasts) of water consumption for kth day: j
j
xkj ¼ a ^0 þ a ^ 1 tk ; j
(3) j
^ 1 , 1 j 24 denote the estimated structural ^ 0 and a where a parameters in model (2). When we will assume that tk = k, then in the model described by Eq. (2) we must investigate a linear dependence between water consumption and time factor. As a result, for each phase we forecast the hourly water consumption with the use of (3) and linear trends.
(6)
and temperature akj ¼ c0j þ c1j tk þ vkj ;
(7)
where tk denotes the mean daily temperature on kth day, n o n o and vkj are the series of independent whereas ykj 1kT
1 j 24.
143
1kT
and identically distributed random variables with the normal distributions N 0; s 2j and N 0; g 2j , respectively. The forecasts (theoretical values) of hourly water consumption are estimated in the following way. For kth day, k > T k ^n 1n24 using the expansion: we calculate the series x ^kn ¼ x
24 X 2pi ð j1Þðn1Þ ; a ^ kj exp 24 j¼1
(8)
^ kj denotes the estimation of jth of this harmonic. where a Harmonics estimators are determined by the equations: 0
1
^ þb ^ k ^ kj ¼ b a j j
(9)
or
4.
^ kj ¼ ^c0j þ ^c1j tk ; a
Harmonic analysis
^0 ; b ^1 , b j j
Harmonic analysis is based on the assumption that a time series consist of sine and cosine waves with different frequencies. Different frequency waves are known as harmonics, while elements of the series are linear combinations of these harmonics (see e.g. [39,41–43]). Periodic variation in the time series fxs g1sN will be identified using Fast Fourier Transformation. Similarly to the trend analysis, we will investigate the dependence of structural parameters of individual harmonics on time factor and ambient temperature. For every day 1 k T, where 24T = N we consider hourly water consumption and analyse the behaviour of elements of def the series xkn 1n24 ¼ fxs g24ðk1Þs24k fxs g1sN . Therefore, k for the series xs 1s24 , 1 k T we take account of 24 harmonics while the elements of the series are expressed as: xkn ¼
24 X 2pi ð j1Þðn1Þ ; akj exp 24 j¼1
(4)
where akj denotes the coefficient for the jth harmonic of kth day, 1 k T, 1 j 24. The values of harmonic coefficients are determined using the inverse Fourier transform: 24 1 X 2pi ð j1Þðn1Þ xkn exp (5) akj ¼ 24 n¼1 24 for 1 k T, 1 j 24. For each day we evaluate the 24 factors of Fourier transform and then we analyse day temperature influence on their values. Obviously, by predicting the n o ^ kj for k > T, we can behaviour of the harmonic series a 1j24 k ^n 1n24 . This study forecast the hourly water consumption x investigates the dependence of each harmonic on time factor:
(10) ^c0j ; ^c1j
where are the values of structural parameter estimators for Eqs. (6) and (7), respectively. As water consumption forecasts we take parts of the real elements of k ^n 1n24 [40,42,43]. the series x
5.
Results
To identify the parametric dependence between hourly water consumption on average ambient temperature in model (2) and the relationships between direct coefficients and average ambient temperature (7) expanded in Fourier series (4), we used data collected for the period between 1 July 2015 and 31 July 2015. In addition to that, the period was divided into weekends and workdays. In compliance with the above, for 48 equations of (2) describing phase trends, we estimated 96 parameters, i.e. we examined hourly water consumption separately for workdays and weekends using Eq. (2) which contains a free term and a directivity coefficient (2 24 2 = 96). Analogically, we determined 96 parameters examining the linear relationships (7) of Fourier transform coefficients (4) and average daily temperature. Using the trend and harmonic analysis methods, we estimated the theoretical values of water consumption by the lines ‘‘Puławy East’’ and ‘‘Puławy West’’ for the period from 1 July 2015 to 31 July 2015 (for k = 1, 2, . . ., 31). For clarity, the results are presented in a graphic form. In Fig. 1 the blue curves illustrate the real values of water consumption xkn 1n24 by ‘‘Puławy East’’ and ‘‘Puławy West’’ during the period from 1 July 2015 to 31 July 2015 (for k = 1, 2, . . ., 31). The red dotted k ^n 1n24 curves represent the theoretical water consumption x estimated by the Fourier Transformation (FT) (8), considering the effect of mean daily ambient temperature on the harmonic components a ^ kj , 1 j 24, 1 k 31. The estimators of the
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Fig. 1 – Empirical and deterministic values obtained by trend analysis and Fourier transform for the period 11 July 2015–17 July 2015.
harmonics were determined by Eq. (10). When the harmonic estimators were calculated by Eq. (9), then the water consumption forecasts estimated by the FT (8) were worse (were more different from the empirical values). The green curves represent the theoretical water consumption values n o xkn which depend on the mean daily ambient temper1n24
ature and are estimated by using the Phase Trend Method (PTM) (see Eq. (3)). Fig. 1 reveals that the deterministic values obtained by the trend analysis and Fourier transform (the values of series of (3) and (8)) are practically the same. For the lines ‘‘Puławy East’’ and ‘‘Puławy West’’ we detern o and ekn 1n24 for mined the difference sequences ^ekn 1n24
^kn denote the k = 1, 2, . . ., 31 (consecutive days), where ^ekn ¼ xkn x differences between real water consumption and its theoretical values determined by the Fourier transformation (8), whereas ekn ¼ xkn xkn denote the differences between real water consumption and its theoretical values estimated by trend analysis. Since the theoretical values of water consumption per each line estimated by the phase trend method and Fourier transform are practically the same, then the real part of the sequence
n o ^ekn
1n24
and the values of the sequence ekn 1n24 are similar,
too. For this reason, the subsequent part of this paper will investigate the residues sequence ekn 1n24 . Fig. 2a and b illustrates the values of residues sequences ekn 1n24 , 1 k 31 for ‘‘Puławy East’’ and ‘‘Puławy West’’, respectively, during the period between 1 July 2015 and 31 July 2015. The statistical data describing the water supply models for two regions, Puławy East and Puławy West, in the investigated water supply system (Table 1) demonstrate that the mean hourly water consumption for ‘‘Puławy East’’ is 110.106 m3, for ‘‘Puławy West’’ – 1785.775 m3, whereas the standard deviations for the above lines are 49.914 and 79.004, respectively. To verify the matching of phase trends (2), we investigates the residues sequence ekn 1n24 for k = 1, 2, . . ., 31. The mean values of me of the residues sequence for the above lines are equal to zero, while the standard deviations se are 15.65 and 23.02, respectively. Comparing the values of s and se, we can observe that the application of models (2) and (4) provides more accurate water consumption forecasts than the forecasts based on arithmetic means. In addition to this, the variations in water consumption are explained by models (2) and (4) in 88.02% for ‘‘Puławy East’’ (see Table 1, determination coeffi-
Table 1 – Comparison of statistical data describing the water supply models in the regions Puławy East and Puławy West in the investigated water supply system.
Puławy East Puławy West
m
s
me
se
R2
F
110.106 175.775
49.914 79.004
0 0
15.65 23.02
0.8802 0.8853
5.9415 5.9759
archives of civil and mechanical engineering 18 (2018) 140–148
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Fig. 2 – Values of residues sequences ekn 1n24 and normal distribution matching for lines ‘‘Puławy East’’ and ‘‘Puławy West’’.
cients R2), and in 88.53% for the line ‘‘Puławy West’’. To analyse the significance of the multiple correlation coefficient R, we determined values of the statistics F (see Table 1) which has a Fisher–Snedecor distribution. For the applied significance level a = 0.001 and the number of degrees of freedom (744 96 = 648), the critical value of the statistics F* is 1.5642. Since for every line F > F*, hence as for the significance level a = 0.001 we must reject the working hypotheses about the lack of multiple correlation in models (2) and (4). Given the above, we assume that multiple correlation coefficients significantly differ from zero, and the matching of the models to the empirical data regarding hourly water consumption by ‘‘Puławy East’’ and ‘‘Puławy West’’ is relatively high. Fig. 2c shows the matching of the residues sequence k en 1n24 for k = 1, 2, . . ., 31 to the normal distribution Nð0; 15:65Þ for ‘‘Puławy East’’, while Fig. 2d – to the distribution Nð0; 23:02Þ for ‘‘Puławy West’’ The points corresponding the values of these sequences are not located along the straight line connecting the first and third quantiles. We can observe that the differences with absolute values over 40 m3 diverge from this line; in Fig. 2a and b we can additionally observe the presence of extreme values, the so-called ‘‘outliers.’’ Using the Kolmogorov–Smirnov and Lilliefors tests, it has been found that at a significance level of 0.01 the postulate of normality of
distributions for these residues sequences is not met. Hence, we can predict that the residues sequences ekn 1n24 for k = 1, 2, . . ., 31 have additional internal dynamics, the investigation of which is beyond the scope of this paper. The best way to verify the effectiveness of trend analysis (2) and Fourier transform (4), considering the effect of mean daily ambient temperature on harmonics values (7), is to compare the forecast and empirical results. Based on the mean daily ambient temperatures for the period between 1 August 2015 and 31 August 2015, we forecast water consumption for two lines, ‘‘Puławy East’’ and ‘‘Puławy West’’ Next, the forecast results were compared with the real water consumption on these lines during the period 1/08/2015–31/08/2015. In Fig. 3 the blue curves denote the real values of water consumption k xn 1n24 for the above lines, whereas the red dotted curve 1k7 k ^n 1n24 obtained marks the forecast water consumption x 1k7 with the Fourier transform (8), where the harmonics were determined by Eq. (10). The green curve illustrates the water n o consumption xki 1n24 determined by trend analysis (3). 1k7 We can observe here that the water consumption results per
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Fig. 3 – Empirical and forecast data obtained by phase trend method and Fourier transform for the period 1 August 2015–7 August 2015.
each line estimated by the phase trend method and Fourier transform are practically the same. To perform further data analyses, we determined the residues sequences ekn 1n24 for k = 1, 2, . . ., 31 between the forecast and real water consumption per each line during the period between 1 August 2015 and 31 August 2015. The results obtained are listed in Table 2, offering a comparison of the mean values and standard deviations of the residues sequences ekn 1n24 for k = 1, 2, . . ., 31 for the lines ‘‘Puławy East’’ and ‘‘Puławy West’’. It can therefore be observed that the average hourly underestimation of the forecast versus real consumption is 2.01 m3 for ‘‘Puławy East,’’ and 7.99 m3 for ‘‘Puławy West’’ Standard deviations between the forecast and real water consumption for ‘‘Puławy East’’ and ‘‘Puławy West’’ are 22.19 and 30.71, respectively. In addition, the analysis of normality of distribution of the residues sequences ekn 1n24 by the 1k 31 Kolmogorov–Smirnov and Lilliefors tests reveal that for each of the above lines at a significance level of 0.01, the residues
Table 2 – Comparison of the mean values and standard deviation of difference sequences between the forecast and measured values for the period 1 August 2015–31 August 2015.
Puławy East Puławy West
me
se
2.01 7.99
22.19 30.71
sequences ekn 1n24 do not meet the postulate of normal
1k 31 distribution Nð2:01; 22:19Þ and Nð7:99; 30:71Þ for ‘‘Puławy East’’ and ‘‘Puławy West’’, respectively. Therefore, we can suspect that there are additional factors affecting water consumption or that the residues sequences ekn 1n24 have internal 1k 31 dynamics which should be modelled by ARIMA or ARCH models.
6.
Conclusion
Water consumption depends on a number of factors, including periodicity, time, temperature and randomness. This paper presented two methods for modelling hourly water consumption in a water supply system. The first method involves the use of linear phase trends dependent on ambient temperature. The other consists in using the Fourier transform, wherein the harmonics are linearly dependent on ambient temperature. Structural parameters of the above models were estimated based on the date regarding two lines, ‘‘Puławy East’’ and ‘‘Puławy West,’’ collected between 1 July 2015 and 31 July 2015. The application of the two water demand forecasting models demonstrated that the real part of theoretical values for the Fourier transform and the values of the expected phase trends are similar. Both the matching of these models to the empirical data and the forecasts for the period between 1 August 2015 and 31 August 2015 obtained with these models reflect relatively accurately the variations in daily water consumption.
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Moreover, the differences between the theoretical and empirical data on water consumption indicate that there exist some additional factors affecting water consumption. In presented water consumption models the seasonality factor was not taken into account directly, because the seasonality is strong correlated with the ambient temperature. Assumed weekends and weekdays partition together with considered the ambient temperature allows us to explain the variability of hourly water consumption in 88%. The advantage of presented model is its ability to construct forecasts based on a short history of data. The results confirm that the above models can effectively be used for water demand forecasting (coefficient of determination above 0.88 and multiple correlation coefficient significantly different from zero) and designing controllers for water supply pumps in order to optimize the costs of maintaining water surface at the desired level. Due to the fact that the differences between theoretical and empirical data on water demand do not have a normal distribution, further studies will focus on determining additional factors which affect water consumption (e.g. waterworks malfunctions, swimming pools water replacement, precipitation quantity, inner dynamics identification using ARIMA or ARCH models).
references
[1] S. Zhou, T.A. McMahon, A. Walton, J. Lewis, Forecasting operational demand for an urban water supply zone, Journal of Hydrology 259 (1–4) (2002) 189–202. [2] T.G. Mamo, J. Ilan, S. Isam, Urban water demand forecasting using the stochastic nature of short term historical water demand and supply pattern, Journal of Water Resource and Hydraulic Engineering 2 (2013) 92–103. [3] C.M. Fontanazza, V. Notaro, V. Puleo, G. Freni, Multivariate statistical analysis for water demand modeling, Procedia Engineering 89 (2014) 901–908. [4] M. Tiwari, J. Adamowski, Medium term urban water demand forecasting with limited data using an ensemble waveletbootstrap machine-learning approach, Journal of Water Resources Planning and Management 141 (2) (2015) 04014053. [5] M. Iwanek, B. Kowalska, E. Hawryluk, K. Kondraciuk, Distance and time of water effluence on soil surface after failure of buried water pipe. Laboratory investigations and statistical analysis, Eksploatacja i Niezawodnosc (Maintenance and Reliability), 18 (2) (2016) 278–284. , http://dx.doi.org/10.17531/ ein.2016.2.16. [6] R. Klempous, J. Kotowski, J. Nikodem, J. Ułasiewicz, Optimization algorithms of operative control in water distribution systems, Journal of Computational and Applied Mathematics 84 (1) (1997) 81–99. [7] F. Odan, L. Reis, Hybrid water demand forecasting model associating artificial neural network with Fourier series, Journal of Water Resources Planning and Management 138 (3) (2012) 245–256. [8] A. Loska, W. Moczulski, R. Wyczółkowski, A. Dąbrowski, Integrated system of control and management of exploitation of water supply system, Diagnostyka 17 (1) (2016) 65–74. [9] L.A. House-Peters, H. Chang, Urban water demand modeling. Review of concepts, methods, and organizing principles, Water Resources Research 47 (2011) W05401. [10] M. Bakker, K. Ven Schagen, J. Timmer, Flow control by prediction of water demand, Journal of Water Supply: Research and Technology 52 (2003) 417–424.
147
[11] M. Herrera, L. Torgo, J. Izquierdo, R. Perez-Garcia, Predictive models for forecasting hourly urban water demand, Journal of Hydrology 387 (1–2) (2010) 141–150. [12] Z. Siwoń, W. Cieżak, J. Cieżak, Stochastic models of water demand in a chosen water-supply system, Ochrona Środowiska 27 (2005) 7–13. [13] M. Bakker, J.H.G. Vreeburg, L.J. Palmen, V. Sperber, G. Bakker, L.C. Rietveld, Better water quality and higher energy efficiency by using model predictive flow control at water supply system, Journal of Water Supply: Research and Technology – AQUA 62 (2013) 1–13. [14] J. Adamowski, H.F. Chan, S.O. Prasher, B. Ozga-Zielinski, A. Sliusarieva, Comparison of multiple linear and nonlinear regression, autoregressive integrated moving average, artificial neural network, and wavelet artificial neural network methods for urban demand forecasting in Montreal, Canada, Water Resources Research 48 (1) (2012) W01528. [15] J. Boguadis, K. Adamowski, R. Diduch, Short-term municipal water demand forecasting, Hydrological Processes 19 (1) (2005) 137–148. [16] A. Jain, A. Varshney, U. Joshi, Short-term water demand forecast modeling at IIT Kanpur using artificial neural networks, Water Resources Management 15 (5) (2001) 299– 321. [17] S. Behboudian, M. Tabesh, M. Falahnezhad, F.H. Ghavanini, A long-term prediction of domestic water demand using preprocessing in artificial neural network, Journal of Water Supply: Research and Technology 63 (1) (2014) 31–42. [18] G. Zhang, An investigation of neural networks for linear time-series forecasting, Computers and Operations Research 28 (12) (2001) 1183–1202. [19] J. Alhumoud, Freshwater consumption in Kuwait; analysis and forecasting, Journal of Water Supply: Research and Technology – AQUA 57 4 (2008) 279–288. [20] T. Hughes, Peak period design standards for small western U. S., Water Supply 16 (4) (1980) 661–667. [21] D. Maidment, S. Miaou, M. Crawford, Transfer function models of daily urban water use, Water Resources Research 21 (4) (1985) 425–432. [22] S. Zhou, T.A. McMahon, A. Walton, J. Lewis, Forecasting daily urban water demand: a case study of Melbourne, Journal of Hydrology 236 (3–4) (2000) 153–164. [23] O. Voitcu, Y.S. Wong, On the construction of a nonlinear recursive predictor, Journal of Computational and Applied Mathematics 190 (1) (2006) 393–407. [24] C. Bunett, R.A. Stewart, C.D. Beal, ANN-based residential water end-use demand forecasting model, Expert Systems and Applications 40 (4) (2013) 1014–1023. [25] P. Cutore, A. Campisano, Z. Kapelan, C. Modica, D. Savic, Probabilistic prediction of urban water consumption using the SCEM-UA algorithm, Urban Water Journal 5 (2) (2008) 125– 132. [26] M. Nasseri, A. Moeini, M. Tabesh, Forecasting monthly urban water demand using extended Kalman filter and genetic programming, Expert Systems and Applications 38 (6) (2011) 7387–7395. [27] C. Benetti, R.A. Stewart, C.D. Beal, ANN based residential water end-use demand forecasting model, Expert Systems and Applications 40 (4) (2013) 1014–1023. [28] A. Loska, Scenario modeling exploitation decision-making process in technical network systems, Eksploatacja i Niezawodnosc – Maintenance and Reliability 19 (2) (2017) 268–278. , http://dx.doi.org/10.17531/ein.2017.2.15. [29] I.S. Msiza, F.V. Nelvamondo, T. Marwala, Artificial neural networks and support vector machines for water demand time series forecasting, in: IEEE International Conference on System, Man and Cybernetics, IEEE, 2007 638–643.
148
archives of civil and mechanical engineering 18 (2018) 140–148
[30] M. Trzęsiok, Symulacyjna ocena jakości zagregowanych modeli zbudowanych metodą wektorów nośnych. W: Zastosowania metod matematycznych w ekonomii i zarządzaniu, Wydawnictwo Uniwersytetu Ekonomicznego w Katowicach, Katowice, 2013. [31] I. Banicescu, R.L. Cariño, J.I. Harvill, J.P. Lestrade, Investigating asymptotic properties of vector nonlinear time series models, Journal of Computational and Applied Mathematics 236 (3) (2011) 411–421. [32] C.L.Z. Tu-Qiao, Hourly water demand forecast model based on Bayesian least squares support vector machine, Journal of Tianjin University 9 (2006) 005. [33] B.M. Brentan, E. Luvizotto Jr., M. Herrera, J. Izquierdo, R. Perez-Garcia, Hybrid regression model for near teal-time urban water demand forecasting, Journal of Computational and Applied Mathematics (2016), http://dx.doi.org/10.1016/j. cam.2016.02.009 (in press). [34] M. Romano, Z. Kapelan, Adaptive water demand forecasting for near real-time management of smart water distribution systems, Environmental Modelling and Software 60 (2014) 265–276. [35] C. Qi, N. Chang, System dynamics modeling for municipal water demand estimation in the urban region under uncertain economic impacts, Journal of Environment Management 92 (2011) 1628–1641.
[36] B. Hazen, C.H. Boone, J. Ezell, L.A. Jones-Farmer, Data quality for data science, predictive analytics, and big data in supply chain management: An introduction to the problem and suggestions for research and applications, International Journal of Production Economics 154 (4) (2014) 72–80. [37] E. Kosicka, E. Kozłowski, D. Mazurkiewicz, The use of stationary tests for analysis of monitored residual processes, Eksploatacja i Niezawodnosc (Maintenance and Reliability), 17 (4) (2015) 604–609. , http://dx.doi.org/10.17531/ ein.2015.4.17. [38] D. Mazurkiewicz, Computer-aided maintenance and reliability management systems for conveyor belts, Eksploatacja i Niezawodnosc (Maintenance and reliability), 16 (3) (2014) 377–382. [39] E. Kozłowski, Analiza i identyfikacja szeregów czasowych, Wyd. Politechnika Lubelska, Lublin, 2015. [40] A. Zeliaś, B. Pawełek, S. Wanat, Prognozowanie ekonomiczne, Teoria, przykłady, zadania, PWN, Warszawa, 2004. [41] G.E.P. Box, G.M. Jenkins, Analiza szeregów czasowych, PWN, Warszawa, 1983. [42] G.C. Chow, Ekonometria, PWN, Warszawa, 1995. [43] J.D. Hamilton, Time Series Analysis, Princeton University Press, Princeton, 1994.