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Forecasting of failures in district heating systems
Engineering Failure Analysis xxx (2015) xxx–xxx
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Engineering Failure Analysis journal homepage: www.elsevier.com/locate/engfailanal
Forecasting of failures in district heating systems _ Bozena Babiarz a,⇑, Katarzyna Chudy-Laskowska b a Rzeszow University of Technology, Faculty of Civil and Environmental Engineering, Department of Heat Engineering and Air Conditioning, 6 ´ ców Warszawy Str., 35-959 Rzeszow, Poland Powstan b ´ ców Warszawy Str., 35-959 Rzeszow, Poland Rzeszow University of Technology, Faculty of Management, Department of Quantitative Methods, 10 Powstan
a r t i c l e
i n f o
Article history: Received 22 September 2014 Received in revised form 17 December 2014 Accepted 18 December 2014 Available online xxxx Keywords: Forecasting Failure Heat supply District heating systems
a b s t r a c t This work presents the method of failures forecasting in district heating systems on the basis of time series analysis. Statistical data concerning the frequency of damage in communal heat distribution networks from a ten-year time period have been used in the article. The data were subjected to statistical analysis. The methods of exponential smoothing, ARIMA, homologous period trend estimation and seasonal indicators were applied as the most adequate for this type of data. Four forecast models of damage in district heating systems were formulated and the best optimal model for the system was selected. After verification, the methods of failure prediction in district heating systems can be applied in the operating process of a heat supply system. These methods give us answers to many questions about planning of repairs, maintenance emergency service, planning of finances and purposefulness of operational control. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction The term ‘forecast’ was introduced to science by Hippocrates. From Greek, ‘forecast’ means knowledge and anticipation. Forecasting makes conclusions about the future reliability state of an element on the basis of a proper mathematical model, based on information regarding the past and future reliability of this element. The aim of forecasting (statistical concluding) is providing information for the effective functioning of the reliability control apparatus in the area of system operation. [4,6]. On the basis of forecasting the gradual and immediate failure of an element included in a given system it is possible to prepare a forecast of reliability indices. Forecasting the failure of an element means creating a forecast of the moment at which the failure occurs and the probability of its existence. It makes it possible to assess the system regarding its specificity and the influence on its reliability [12,16–20,22,23]. A failure may be defined as a partial or full loss of properties of the technical system which may significantly decrease its efficiency or even lead to its complete disappearance. A failure is the result of a great number of factors of a random character both on the micro and macro scales [10,16]. Contemporary knowledge does not fully describe the mechanisms of cooperation of these factors in the aspect of a failure. The process of creating failures may be considered as a ‘‘black box’’. The concept of a ‘‘black box’’ relies on treating the phenomenon or process as unavailable for research and limited only to an analysis, mainly between input and output parameters [9]. This approach allows us to find a model of the creation of failures, which
⇑ Corresponding author. E-mail addresses: [email protected] (B. Babiarz), [email protected] (K. Chudy-Laskowska). http://dx.doi.org/10.1016/j.engfailanal.2014.12.017 1350-6307/Ó 2015 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Babiarz B, Chudy-Laskowska K. Forecasting of failures in district heating systems. Eng Fail Anal (2015), http://dx.doi.org/10.1016/j.engfailanal.2014.12.017
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is a probability model due to its nature. This model, with engineering accuracy, allows the anticipation of failures. The methods presented are applied to the particular case of heat distribution [2,11,13]. The heat supply system includes three subsystems: the subsystem of heat production, constituting the source of heat (heat and power station, heating station), the subsystem of heat supply (SbHS) which constitutes district heating systems along with utilities, and the subsystem of heat distribution, which includes district heating substation and internal central heating installations [3]. Taking into account the structure, specificity and tasks of each subsystem is a condition for their proper assessment with regard to reliability. A district heating system is a repairable structure. However, repair is limited to the exchange of a short section of pipeline or an element of the utilities installed in the network, as a result of which the process of renewal does not significantly influence a change in the reliability properties of the entire system [2,21]. There is a probability of endless failures of a given section, regardless of its length. The duration of failures is short in comparison to the network operation, which allows the assumption that the renewal is immediate. The methods of performing forecasts are properly selected for the needs, with regard to the kind of facility or goods and the time the forecast is being prepared for. The forecasting methods for the needs of reliability are adjusted to the technique of preparing forecasts: they may be calculating or simulating methods [7]. It is proposed to prepare a forecast of SbHS failures on the basis of empirical data on the failures of district heating systems in the real system of heat supply by means of the simulating method with the use of a computer technique (Statistica 10.0). 2. Preliminary analysis of operating data In order to create a model describing the forecast of SbHS failures, data regarding the overall number of failures in the sections of a district heating system within a selected heating centre has been used. The scope of the data regards 79 months of observations of heat network failures. Fig. 1 presents the tendencies of changes in the number of failures within the given period. The forecast regarding the number of failures within the given SbHS included four methods: the exponential smoothing method, ARIMA, homologous period trend estimation method and index analysis. Analysing the operating data regarding the failures in district heating systems (Fig. 1), it must be stated that they indicate a typical seasonal character where the maximum values concern autumn and winter months: September, October, November, December and January when the system is working and is maximally loaded and, moreover, the parameters of operation change while shifting from the summer season into the winter one. On average, most failures are in October (mean x ¼ 290) and November (mean x ¼ 249). In these months there is also a maximum level of this phenomenon. The differentiation is the highest in September std. dev. (r) = 91.6) and November (std. dev. (r) = 94.4), which was established by the standard deviation. In the summer season the number of failures decreases. The lowest average level is noted in June (40) (Table 1). Seasonal fluctuations decrease in amplitude. The character of the entire series is dampened, which means that with time SbHS has fewer and fewer failures. By means of the ANOVA Kruskala–Wallisa test it has been determined whether the differences in the levels of failures, from the monthly perspective, are significant statistically. The test showed that the number of failures for particular months was significantly different p < a (p = 0.0008), (a – level of significance, p – probability test). There are no outliers in the monthly failure numbers (Fig. 2). The model includes seasonal fluctuations, which is why it is purposeful to check autocorrelation, as one of the assumed models requires a stationary process of a series before performing the forecast, which is connected with elimination of autocorrelation before the forecast.
Fig. 1. Changes in the monthly number of failures in the given SbHS.
Please cite this article in press as: Babiarz B, Chudy-Laskowska K. Forecasting of failures in district heating systems. Eng Fail Anal (2015), http://dx.doi.org/10.1016/j.engfailanal.2014.12.017
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B. Babiarz, K. Chudy-Laskowska / Engineering Failure Analysis xxx (2015) xxx–xxx Table 1 Basic descriptive statistics for failures in particular months. Months
I
II
III
IV
V
VI
VII
VIII
IX
X
XI
XII
Number of observation Mean Std. dev. Min Max
7 202.9 74.8 115 309
7 167.0 65.7 69 284
7 148.4 59.7 48 216
7 127.3 49.0 48 196
7 116.7 45.6 54 176
7 107.0 40.4 40 154
7 137 59.0 67 202
6 118.2 39.1 55 162
6 203.3 91.6 113 324
6 290.3 94.4 166 398
6 248.8 77.7 125 327
6 208.5 62.0 123 301
Fig. 2. Monthly failure figures.
The autocorrelogram indicates that the analysed series has autocorrelation and is characterised by seasonal fluctuations depicted every 12 months. This system of data imposes the aim of checking several models and selecting the one which is characterised by the lowest forecast errors. 3. Methodology of forecasting 3.1. ARIMA model In order to identify economic phenomena, certain factors may be differentiated, which indicate some delays in the course of certain phenomena in time. In these kinds of situations, autoregression models are applied in which the values of the forecast variable are the function of the value of this variable at the moments or period preceding the studied period and the random element [14]. The ARIMA model was introduced by Box and Jenkins in 1976 [4]. It may be applied for modelling stationary series, i.e. series in which there are only random fluctuations around the average or non-stationary ones, reduced to stationary ones. The structure of the model is based on the autocorrelation phenomenon. The ARIMA model includes two basic processes: autoregression and moving average. They exist together under certain circumstances. The equation of the autoregression model is called the autoregression process of the row p, AR (p) and it is presented by the following formula:
Y t ¼ u0 þ u1 Y t1 þ þu2 Y t2 þ þ up Y tp þ et
ð1Þ
where Yt, Yt1, Ytp – values of dependent variable in time t, t 1, t 2, t p. u0, u1, u2, up – parameters of the model. et – residuals of the model. p – time lag. Please cite this article in press as: Babiarz B, Chudy-Laskowska K. Forecasting of failures in district heating systems. Eng Fail Anal (2015), http://dx.doi.org/10.1016/j.engfailanal.2014.12.017
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Time series consist mainly of observations of mutually dependent values, so it is possible to estimate model coefficients which are described by the following elements of the series on the basis of the preceding elements which are delayed in time. Each observation is the sum of a random element and the linear combination of preceding observations. The AR process will be stable if the parameters belong to a certain range (1, 1). This is the so-called need for stationary series. The form of the average model (MA) is called the process of the moving average of the row q, MA (q)
Y t ¼ h0 þ et h1 et1 h2 et2 hq etq
ð2Þ
where Yt – values of dependent variable in period t. h0, h1, h2, hq – parameters of the model. et, et1, et2, etq – residuals of the model in periods t. . .t q. q – quantity lag. In the process of moving average, each observation consists of a random element and the linear combination of a random component from the past. The term moving average is slightly mistaken as the sum of weights is not equal to one; however, it is used for this model. The equation of moving average may be written in the autoregression form and may be performed when the parameters of the moving average satisfy certain conditions, i.e. the model is reversible. In order to gain higher elasticity in matching the model to the time series, it is necessary to combine both models:
Y t ¼ u0 þ u1 Y t1 þ u2 Y t2 þ . . . þ up Y tp þ et þ h0 h1 et1 h2 et2 hq etq
ð3Þ
This process is called the process of autoregression and moving average with grade (p and q) which is abbreviated to ARMA (p, q) [4]. The model assumes that the value of the forecast variable at the moment and in period t depends on its past values and the difference between past real values of the forecast variable and its values obtained from the model. The ARIMA model includes autoregression parameters and moving average and introduces the differentiation operator. So, there are three parameters: ARIMA (p, d, q): autoregression parameters p, differentiation grade d, moving average parameters q. It is required that the input series used for the ARIMA method is stationary, i.e. it should include a time constant average, variance and lack of autocorrelation. This is why it needs differentiation until it has gained a stationary state. The parameter d expresses how many times the series should be differentiated. Very rarely, the number of parameters p and q must be higher than 2. There are also seasonal models, where additionally three parameters of seasonality are defined: ARIMA (ps, ds, qs) autoregression parameters ps, seasonal differentiation parameters ds and moving average parameters qs [15]. 3.2. Exponential smoothing The essence of exponential smoothing relies on the fact that the time series of the forecast variable is smoothed by means of the moving average and weight established due to the exponential function. The forecast is based on the weighted average expressed in present and previous values of the series [1]. Exponential smoothing may be based on various models that are appropriate to the kind of the components of the analysed time series [8]. In the case of the exponential smoothing models, the key issue is the selection of the values of model parameters. Several ways of selecting parameters are provided in the literature on the subject. If particular components change rapidly, it is assumed that the values of smoothing parameters must be established at a level close to one, or otherwise at a level close to zero. The values of parameters may be also selected by the method of an experiment and by minimising selected mistakes, define three smoothing parameters: – a (alpha) – is the stable parameter of smoothing, indispensible in all models, it assumes values from the range [0, 1]. If a = 0, all smoothed values will equal the preliminary value. If a = 1 each smoothed value (forecast) will equal the previous value. The closer a is to 0, the slower the weights will be decreasing i.e. the effect of preceding observations will decrease at a much lower rate. – d (delta) – is the parameter of seasonal smoothing and it requires defining only in the case of seasonal models. It may assume values from the range [0, 1]. If d = 0, it is assumed that the seasonal component for a given time point is identical to that anticipated for a given moment during the preceding cycle. – c (gamma) and / (fi) – are smoothing parameters of the trend [5]. The parameter c is defined for a linear trend and exponential models as well as for models of a falling trend without seasonal fluctuations. The parameter / is determined for a model of falling trend. Analogous to the seasonal component, when the trend component is included in the process of exponential smoothing, for each time moment, it is calculated as an independent trend component and modified as the function of the forecast error and proper parameter. If the parameter c amounts to 0 (zero), the trend component is stable for all values of the time series. If the parameter equals 1, the trend component is maximally modified in each step by a proper forecast error. If the values of the parameter are in the middle they represent a mixture of these two extreme situations. The parameter c is the parameter of modification of the trend and it defines how strong changes in the trend influence the assessment of the systematic component for the following forecast, i.e. how fast the trend will be declining or rising. The form of a model with a exponential trend and multiplicative fluctuations is presented as follows: Please cite this article in press as: Babiarz B, Chudy-Laskowska K. Forecasting of failures in district heating systems. Eng Fail Anal (2015), http://dx.doi.org/10.1016/j.engfailanal.2014.12.017
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F t1 ¼/ ðyt1 C t1r Þ þ ð1 /ÞðF t2 St2 Þ
ð4Þ
St1 ¼ cðF t1 F t2 Þ þ ð1 cÞSt2 y C t1 ¼ d t1 þ ð1 cÞC t1r F t1
ð5Þ ð6Þ
where Ft1 – equivalent of smoothed value created from a simple exponential smoothing model. St1 – evaluation of trend growth for a moment or the period t 1. Ct1 – evaluation of seasonality index for a moment or the period t 1. r – the length of seasonal cycle – number of phases. a, c, d – the model parameters – values from 0 to 1. 3.3. The model of homologous period trend estimation The method of homologous period trend estimation uses a correlation between the observation from several years for the same period. In the case studied, homologous period variables are months. The procedure relies on estimating the parameters of the analytical function of the trend separately for particular cycle phases. The forecast is determined by means of extrapolation of the estimated trend function for each cycle phase. The application of the method imposes the principles of ‘‘status quo’’, meaning that the observed tendency for each of the cycle phases will be maintained [14]. Each time series referring to a defined cycle phase is described by a linear model in the following form
yij ¼ a0i þ a1i tji þ et
j ¼ 1 . . . k; i ¼ 1 . . . r
ð7Þ
where yij – value of predicted variable, for the i-th phase in the j-th cycle. tij – time variable, tij = i + r(j 1). a0ia1i – structural parameters of the model. et – random component. 3.4. Index analysis This is one of the most commonly used methods in the analysis of seasonal fluctuations. It relies on determining the indexes of seasonal fluctuations for particular cycle phases. When fluctuation amplitudes in analogous cycle phases are approximately the same, it relies on absolutely stable fluctuations. When the amount of fluctuation amplitudes change more or less in the same relation, this is known as relatively stable fluctuations. In the first case, for the description of the shaping of this phenomenon an additive model may be used and in the second case, multiplicative ones may be applied. In the studies on monthly number of failures SbHS, the model of the following equation has been applied:
^ti si þ nt yti ¼ y
ð8Þ
where ytj – real values of forecasted variable. ^ti – smoothed values received form the model trend. y si – pure indicator of seasonality based on i t cycle. nt – random component. The analysis of seasonal fluctuations may include four stages: – – – –
Selection of the trend, elimination of the trend from time series, elimination of accidental fluctuations, calculation of seasonal coefficients.
The selection of the trend relies on smoothing the time series by means of the r-word central or non-central moving average or the analytical function. The aim of data aggregation is to ascertain the time series in which seasonal fluctuations appear. It is performed by summing data in the periods assumed in the studies into data adequate for the periods that equal the length of the seasonal cycle. The elimination of the trend in the case of a time series with additive fluctuations is performed while calculating the difference between real values of the forecast variable and smoothed values obtained from the trend model. In the case of multiplicative fluctuations, the quotient of real values of the forecast variable is determined by means of the adequate smoothed values.
zti ¼
yti ^ti y
ð9Þ
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where zti – real difference between variable values and smoothed values received from the trend model. yti – the real value. ^ti – empirical value. y The calculated values take into account seasonal fluctuations. Elimination of the random component is made by calculating so-called raw seasonal fluctuation indices. They constitute average numbers determined on the basis of the numbers zti, regarding the same phase of fluctuations. The seasonal coefficients (pure) are determined according to formula (10) and the forecast according to formula (11).
si ¼
zi q
ð10Þ
where: si – pure indicator of seasonality. q – the mean values based on zti. r – the number of cycle phases. ðtÞ
ðtÞ
yti ¼ yti si ;
ð11Þ
yti – the forecast for the period ðtÞ yti – preliminary forecasts for
in the i-th phase of the cycle. the period in the i-th phase of the cycle. ci – pure indicator of seasonality in the i-th phase of the cycle.
4. Results and discussion 4.1. ARIMA method The assumptions of the method require that before estimating parameters the series must be led to a stationary state, i.e. it must have a time stable average, variance and should not possess autocorrelation. Analysing the data taken for studies, none of the assumptions are fulfilled. The series is declining, so the average is decreasing in time, and the amplitude of fluctuations is also decreasing, which is why variance is falling in time (Fig. 1). The data are seasonal, so autocorrelation exists (Fig. 3). In order to lead the series to a stationary state, it must undergo determination of the logarithm and differentiation. After such procedures, the series should be stationary. After finding the logarithm and differentiation, the series is stationary, which is why it is possible to shift to the estimation of model parameters. On the basis of the studies, the best model was the ARIMA model (1, 1, 1) (1, 0, 0) which has one autoregression parameter p and one seasonal autoregression parameter ps, one moving average parameter q and
Fig. 3. Autocorrelation function.
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one parameter of differentiation d. Having structured the model it is necessary to check whether the distribution of residuals is a normal distribution or its series no longer has autocorrelation. The following hypothesis has been assumed: H0: residuals have a normal distribution and H1: no residuals in the normal distribution. The studies assumed a = 0.05. The analysis concludes that p > a (p = 0.64197), so there is no basis for rejecting H0. It is most probable that the distribution of model residuals is a normal distribution (Fig 4). No autocorrelation was found in the properly structured model, as is shown in Fig. 5. It is clearly visible in the correlogram that residuals are within the confidence level and there is no autocorrelation. A forecast was obtained for failures of the district heating systems in the studied period for such a structured model as in Fig. 6. 4.2. Exponential smoothing In choosing the model, the tendency of the series must be determined as well as the kind of periodical fluctuations, length and the seasonal delay. Several different models of smoothing have been checked and the best turned out to be the model
Fig. 4. Normal distribution of residuals – ARIMA model (1, 1, 1)(1, 0, 0).
Fig. 5. Autocorrelation function of residuals in ARIMA (1, 1, 1)(1, 0, 0) model.
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Fig. 6. Monthly failure numbers forecast based on ARIMA model.
with a linear tolerance and multiplicative fluctuations with the parameters: a = 0.3, d = 0.1 and c = 0.1. This model was characterised with the smallest errors, which is why it was accepted as the final one. For the given model, the distribution of residuals was checked assuming H0, that the residuals have a normal distribution. The probability found, p = 0.46738, gives no basis to reject H0, so the distribution of residuals is most likely normal (Fig. 7). A forecast was obtained for failures of the heat network in the studied period and the results are presented graphically in Fig. 8. 4.3. The model of homologous period trend estimation Another model which can be used in the case of monthly data which is characterised by seasonal fluctuations and a tendency, in this case a decreasing one, is the homologous periods model. For previously prepared data, monthly trends were determined by means of simple linear models based on a time line (Table 2). The model parameters were estimated by the method of least squares. As the analysed data have a monthly character, 12 linear models were estimated, which allowed forecasting of other values. The forecast performed on the basis of the estimated models is presented in Fig. 9. 4.4. Index method The final method presented is based on the analysis of seasonal indices calculated for particular cycle phases. The indices are used for structuring the forecast. Firstly, the linear function of the trend is estimated – the overall tendency contained in
Fig. 7. Normal distribution of residuals in exponential smoothing.
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Fig. 8. Monthly failure numbers forecast based on exponential smoothing.
Table 2 The results of structural parameters for the models of homologous period trend estimation. Number of months
Equation of the model
Number of months
Equation of the model
January (7) February (7) March (7) April (7) May (7) June (7)
y = 32.92t + 334.5 y = 28t + 281.4 y = 24t + 233 y = 18.85t + 202.7 y = 19.25t + 193.7 y = 13.64t + 161.5
July (7) August (6) September (6) October (6) November (6) December (6)
y = 25.32t + 238.2 y = 16.6t + 176.2 y = 44.85t + 360.3 y = 43.42t + 442.3 y = 39.28t + 386.3 y = 24.6t + 294.6
Fig. 9. Monthly failure numbers forecast based on homologous period trend (TM).
the data. It assumes the following formula for failures in the heat network: Y = 2.22t + 259.24. Next, the values were determined (zti) as quotients of real and empirical data. However, they include the effect of seasonal as well as accidental fluctuations, so this is corrected by calculating raw indices of seasonal processes (zi), i.e. the average of indices which are adequate to homologous phases. Pure indices of the seasonal processes allowing us to structure the forecast are determined as quotients of raw seasonal indices zi and q. For example, in January the pure index of seasonal processes amounted to c1 = 1.131, (113.1%) which means that in the January months the number of failures SbHS was higher, on average, by 13.1% than the one resulting from the trend line (Table 3). Please cite this article in press as: Babiarz B, Chudy-Laskowska K. Forecasting of failures in district heating systems. Eng Fail Anal (2015), http://dx.doi.org/10.1016/j.engfailanal.2014.12.017
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Table 3 Raw and pure indices of seasonal processes. Forecasting based on the index method. Months
zi
ci
t
Forecasting
January (z1) February (z2) March (z3) April (z4) May (z5) June (z6) July (z7) August (z8) September (z9) October (z10) November (z11) December (z12)
1.142 0.942 0.852 0.742 0.693 0.655 0.823 0.675 1.144 1.705 1.470 1.274 12.116 1.010
1.131 0.933 0.843 0.735 0.686 0.649 0.815 0.668 1.133 1.688 1.456 1.262
89 90 91 92 93 94 95 96 97 98 99 100
69.730 55.473 48.259 40.424 36.209 32.801 39.419 30.818 49.737 70.370 57.461 46.981
R q
Fig. 10. Monthly failure numbers – forecast based on the index method.
Table 4 Forecast list of failures in heat distribution network generated on the basis of four models and ex post errors of the given forecast.
VIII IX X XI XII I II III IV V VI VII The annual total failures number
Monthly failures numbers
ARIMA
Error ex post (%)
Exponential smoothing
Error ex post (%)
Method of univariate periods
Error ex post
Seasonal indicators method
Error ex post (%)
34 77 153 83 92 84 70 60 56 45 32 39 825
64 91 109 93 92 88 67 56 56 59 67 71 913
89 18 29 12 0 5 4 7 0 32 109 83
53 88 132 111 95 83 64 59 52 45 43 50 875
55 15 14 34 4 2 8 2 7 0 34 29
60 46 138 111 122 71 57 41 52 40 52 36 828
76 40 10 34 33 15 18 32 7 12 64 9
31 50 70 57 47 70 55 48 40 36 33 39 578
9 35 54 31 49 17 21 20 28 20 3 1
The bold values indicates the closest value of annual failures number.
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Fig. 11. Comparison of forecasting obtained by four methods.
A forecast was performed on the basis of the index method numerically presented in Table 3 and graphically in Fig. 10. The forecast shows that the index method dampened seasonal fluctuation and allowed the obtaining of the lowest values for monthly forecasts of failures in the SbHS network in comparison to other models. 4.5. Comparison of applied methods The results of SbHS failures obtained with the use of four methods were presented in Table 4. For each forecast, made by means of four models, the error was calculated ex post so as to check how much the value of forecasting was different from the factual state of failure in the heat network. Fig. 11 presents a graphical image of the obtained forecasting. On the basis of the error analysis and according to the graph presenting the comparison of forecasting methods, it may be claimed that the models ARIMA and exponential smoothing show the smallest errors in the most critical moments, when the network operates at maximum load. A higher number of failures are observed in October and November. This is caused by the changing parameters connected with the beginning of the heating season. The methods, however, re-estimate the numbers of failures due to large errors in other periods. The best method seems to be the homologous period trend estimation, where the number of failures for the entire studied period (a year) gives a value that can be compared with real operational data. 5. Conclusions The performed analysis of failures in district heating systems allowed the development of a method of forecasting failures in the heat network, on the basis of the analysis of time series. Four methods were used: ARIMA, exponential smoothing, homologous period trend estimation and index methods, as the most adequate for this type of data. The forecast of failures was developed by selecting the proper models. The most adequate turned out to be the homologous period trend estimation method. The forecast describes the reliability state of SbHS in the given time interval. This state may be defined by the value of one or several reliability indices. The proper forecast of the reliability index may be obtained only with reliable and complete input data. The makes necessary the diligent collection of reliability data of the studied facilities. The method, after verification, gives the possibility of preparing for operating stand-by of repair services. As a result, it is possible to answer many questions that come from the operator of the heat network regarding the plans of repairs, maintenance of repair services, planning finance or the desirability of installing a monitoring system. It may also be used as a tool in the assessment of the risk of lack of heat supply to recipients. References [1] Aczel Amir D. Statistics in management. Warsaw: PWN; 2000. s. 642. [2] Babiarz B. An introduction to the assessment of reliability of the heat supply systems. Int J Press Vessels Pip 2006;83(4):230–5. [3] Babiarz B. Heat supply system reliability management. Safety and reliability: Methodology and Applications. In: Nowakowski et al., editor. Proc. of the European safety and reliability conference, ESREL 2014, Wroclaw, Poland, 14–18 September 2014. 2015 Taylor & Francis Group, London. p. 1501–1506.
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12 [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
B. Babiarz, K. Chudy-Laskowska / Engineering Failure Analysis xxx (2015) xxx–xxx Box GEP, Jenkins GM, Reinsel GC. Time series analysis: forecasting and control. 4th ed. Wiley; 2008. Cies´lak M. Economic forecasting, methods and applications. Warsaw: PWN; 1997. Dittmann P. Sales forecasting methods in the enterprise. UE Wroclaw; 1998. Dolinay V, Vasek L. Simulation and experiments on the secondary heat distribution network system. Int J Math Comput Simul 2012;6(5):465–72. Grubb H, Mason A. Long lead-time forecasting of UK air passengers by Holt 0 Winters methods with damped trend. Int J Forecast 2001;17:71–82. International Electrotechnical Commission: analysis techniques for system reliability – procedure for failure mode and effect analysis. IEC Standard, Publication 812; 1990. Ma B, Shuai J, Liu D, Xu K. Assessment on failure pressure of high strength pipeline with corrosion defects. Eng Failure Anal 2013;32:209–19. Macdonald KA, Cosham A. Best practice for the assessment of defects in pipelines – gouges and dents. Eng Failure Anal 2005;12(5):720–45. Miszta-Kruk K. Reliability analysis of vacuum sewerage systems using the total probability theorem. Advances in safety, reliability and risk management. London, UK: Taylor & Francis Group; 2011. p. 2072–9. Pinna T, Caporali R, Cambi G, Burgazzi L, Poucet A, Porfiri MT. Failure mode and effect analysis on ITER heat transfer systems. Eng Failure Anal 2012;25:156–68. Radzikowska B. Methods of forecasting. UE Wroclaw; 2004. Samagaio A, Wolters M. Comparative analysis of government forecast for the Lisbon Airport. J Air Transp Manage 2010;16:213–7. Song Y, Ma L, Morris J. A practical approach for reliability prediction of pipeline systems. Eur J Oper Res 2009;198:210–4. Szybka J, Broniec Z, Pilch R. Forecasting the failure of a thermal pipeline on the basis of risk assessment and exploitation analysis. Maint Reliab 2011;4:5–10. Teixeira AP, Guedes Soares C, Netto TA, Estefen SF. Reliability of pipelines with corrosion defects. Int J Press Vessels Pip 2008;85(4):228–37. Valis D, Zak L, Pokora O. On approaches for non-direct determination of system deterioration. Eksploatacja i Niezawodnosc—Maint Reliab 2014;14(1):33–41. Werbinska-Wojciechowska S. Time resource problem in logistics systems dependability modeling. Eksploatacja i Niezawodnosc—Maint Reliab 2013:427–33. Wojdyga K. An influence of weather conditions on heat demand in district heating systems. Energy Buildings 2008;40:2009–14. Xu LY, Cheng YF. Reliability and failure pressure prediction of various grades of pipeline steel in the presence of corrosion defects and pre-strain. Int J Press Vessels Pip 2012;89:75–84. Zorc B, Kosec B, Kosec L, Nagode A. Analysis of hot water pipeline system leakage. Eng Failure Anal 2013;28:78–81.
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Contents lists available at ScienceDirect
Engineering Failure Analysis journal homepage: www.elsevier.com/locate/engfailanal
Forecasting of failures in district heating systems _ Bozena Babiarz a,⇑, Katarzyna Chudy-Laskowska b a Rzeszow University of Technology, Faculty of Civil and Environmental Engineering, Department of Heat Engineering and Air Conditioning, 6 ´ ców Warszawy Str., 35-959 Rzeszow, Poland Powstan b ´ ców Warszawy Str., 35-959 Rzeszow, Poland Rzeszow University of Technology, Faculty of Management, Department of Quantitative Methods, 10 Powstan
a r t i c l e
i n f o
Article history: Received 22 September 2014 Received in revised form 17 December 2014 Accepted 18 December 2014 Available online xxxx Keywords: Forecasting Failure Heat supply District heating systems
a b s t r a c t This work presents the method of failures forecasting in district heating systems on the basis of time series analysis. Statistical data concerning the frequency of damage in communal heat distribution networks from a ten-year time period have been used in the article. The data were subjected to statistical analysis. The methods of exponential smoothing, ARIMA, homologous period trend estimation and seasonal indicators were applied as the most adequate for this type of data. Four forecast models of damage in district heating systems were formulated and the best optimal model for the system was selected. After verification, the methods of failure prediction in district heating systems can be applied in the operating process of a heat supply system. These methods give us answers to many questions about planning of repairs, maintenance emergency service, planning of finances and purposefulness of operational control. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction The term ‘forecast’ was introduced to science by Hippocrates. From Greek, ‘forecast’ means knowledge and anticipation. Forecasting makes conclusions about the future reliability state of an element on the basis of a proper mathematical model, based on information regarding the past and future reliability of this element. The aim of forecasting (statistical concluding) is providing information for the effective functioning of the reliability control apparatus in the area of system operation. [4,6]. On the basis of forecasting the gradual and immediate failure of an element included in a given system it is possible to prepare a forecast of reliability indices. Forecasting the failure of an element means creating a forecast of the moment at which the failure occurs and the probability of its existence. It makes it possible to assess the system regarding its specificity and the influence on its reliability [12,16–20,22,23]. A failure may be defined as a partial or full loss of properties of the technical system which may significantly decrease its efficiency or even lead to its complete disappearance. A failure is the result of a great number of factors of a random character both on the micro and macro scales [10,16]. Contemporary knowledge does not fully describe the mechanisms of cooperation of these factors in the aspect of a failure. The process of creating failures may be considered as a ‘‘black box’’. The concept of a ‘‘black box’’ relies on treating the phenomenon or process as unavailable for research and limited only to an analysis, mainly between input and output parameters [9]. This approach allows us to find a model of the creation of failures, which
⇑ Corresponding author. E-mail addresses: [email protected] (B. Babiarz), [email protected] (K. Chudy-Laskowska). http://dx.doi.org/10.1016/j.engfailanal.2014.12.017 1350-6307/Ó 2015 Elsevier Ltd. All rights reserved.
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is a probability model due to its nature. This model, with engineering accuracy, allows the anticipation of failures. The methods presented are applied to the particular case of heat distribution [2,11,13]. The heat supply system includes three subsystems: the subsystem of heat production, constituting the source of heat (heat and power station, heating station), the subsystem of heat supply (SbHS) which constitutes district heating systems along with utilities, and the subsystem of heat distribution, which includes district heating substation and internal central heating installations [3]. Taking into account the structure, specificity and tasks of each subsystem is a condition for their proper assessment with regard to reliability. A district heating system is a repairable structure. However, repair is limited to the exchange of a short section of pipeline or an element of the utilities installed in the network, as a result of which the process of renewal does not significantly influence a change in the reliability properties of the entire system [2,21]. There is a probability of endless failures of a given section, regardless of its length. The duration of failures is short in comparison to the network operation, which allows the assumption that the renewal is immediate. The methods of performing forecasts are properly selected for the needs, with regard to the kind of facility or goods and the time the forecast is being prepared for. The forecasting methods for the needs of reliability are adjusted to the technique of preparing forecasts: they may be calculating or simulating methods [7]. It is proposed to prepare a forecast of SbHS failures on the basis of empirical data on the failures of district heating systems in the real system of heat supply by means of the simulating method with the use of a computer technique (Statistica 10.0). 2. Preliminary analysis of operating data In order to create a model describing the forecast of SbHS failures, data regarding the overall number of failures in the sections of a district heating system within a selected heating centre has been used. The scope of the data regards 79 months of observations of heat network failures. Fig. 1 presents the tendencies of changes in the number of failures within the given period. The forecast regarding the number of failures within the given SbHS included four methods: the exponential smoothing method, ARIMA, homologous period trend estimation method and index analysis. Analysing the operating data regarding the failures in district heating systems (Fig. 1), it must be stated that they indicate a typical seasonal character where the maximum values concern autumn and winter months: September, October, November, December and January when the system is working and is maximally loaded and, moreover, the parameters of operation change while shifting from the summer season into the winter one. On average, most failures are in October (mean x ¼ 290) and November (mean x ¼ 249). In these months there is also a maximum level of this phenomenon. The differentiation is the highest in September std. dev. (r) = 91.6) and November (std. dev. (r) = 94.4), which was established by the standard deviation. In the summer season the number of failures decreases. The lowest average level is noted in June (40) (Table 1). Seasonal fluctuations decrease in amplitude. The character of the entire series is dampened, which means that with time SbHS has fewer and fewer failures. By means of the ANOVA Kruskala–Wallisa test it has been determined whether the differences in the levels of failures, from the monthly perspective, are significant statistically. The test showed that the number of failures for particular months was significantly different p < a (p = 0.0008), (a – level of significance, p – probability test). There are no outliers in the monthly failure numbers (Fig. 2). The model includes seasonal fluctuations, which is why it is purposeful to check autocorrelation, as one of the assumed models requires a stationary process of a series before performing the forecast, which is connected with elimination of autocorrelation before the forecast.
Fig. 1. Changes in the monthly number of failures in the given SbHS.
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B. Babiarz, K. Chudy-Laskowska / Engineering Failure Analysis xxx (2015) xxx–xxx Table 1 Basic descriptive statistics for failures in particular months. Months
I
II
III
IV
V
VI
VII
VIII
IX
X
XI
XII
Number of observation Mean Std. dev. Min Max
7 202.9 74.8 115 309
7 167.0 65.7 69 284
7 148.4 59.7 48 216
7 127.3 49.0 48 196
7 116.7 45.6 54 176
7 107.0 40.4 40 154
7 137 59.0 67 202
6 118.2 39.1 55 162
6 203.3 91.6 113 324
6 290.3 94.4 166 398
6 248.8 77.7 125 327
6 208.5 62.0 123 301
Fig. 2. Monthly failure figures.
The autocorrelogram indicates that the analysed series has autocorrelation and is characterised by seasonal fluctuations depicted every 12 months. This system of data imposes the aim of checking several models and selecting the one which is characterised by the lowest forecast errors. 3. Methodology of forecasting 3.1. ARIMA model In order to identify economic phenomena, certain factors may be differentiated, which indicate some delays in the course of certain phenomena in time. In these kinds of situations, autoregression models are applied in which the values of the forecast variable are the function of the value of this variable at the moments or period preceding the studied period and the random element [14]. The ARIMA model was introduced by Box and Jenkins in 1976 [4]. It may be applied for modelling stationary series, i.e. series in which there are only random fluctuations around the average or non-stationary ones, reduced to stationary ones. The structure of the model is based on the autocorrelation phenomenon. The ARIMA model includes two basic processes: autoregression and moving average. They exist together under certain circumstances. The equation of the autoregression model is called the autoregression process of the row p, AR (p) and it is presented by the following formula:
Y t ¼ u0 þ u1 Y t1 þ þu2 Y t2 þ þ up Y tp þ et
ð1Þ
where Yt, Yt1, Ytp – values of dependent variable in time t, t 1, t 2, t p. u0, u1, u2, up – parameters of the model. et – residuals of the model. p – time lag. Please cite this article in press as: Babiarz B, Chudy-Laskowska K. Forecasting of failures in district heating systems. Eng Fail Anal (2015), http://dx.doi.org/10.1016/j.engfailanal.2014.12.017
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Time series consist mainly of observations of mutually dependent values, so it is possible to estimate model coefficients which are described by the following elements of the series on the basis of the preceding elements which are delayed in time. Each observation is the sum of a random element and the linear combination of preceding observations. The AR process will be stable if the parameters belong to a certain range (1, 1). This is the so-called need for stationary series. The form of the average model (MA) is called the process of the moving average of the row q, MA (q)
Y t ¼ h0 þ et h1 et1 h2 et2 hq etq
ð2Þ
where Yt – values of dependent variable in period t. h0, h1, h2, hq – parameters of the model. et, et1, et2, etq – residuals of the model in periods t. . .t q. q – quantity lag. In the process of moving average, each observation consists of a random element and the linear combination of a random component from the past. The term moving average is slightly mistaken as the sum of weights is not equal to one; however, it is used for this model. The equation of moving average may be written in the autoregression form and may be performed when the parameters of the moving average satisfy certain conditions, i.e. the model is reversible. In order to gain higher elasticity in matching the model to the time series, it is necessary to combine both models:
Y t ¼ u0 þ u1 Y t1 þ u2 Y t2 þ . . . þ up Y tp þ et þ h0 h1 et1 h2 et2 hq etq
ð3Þ
This process is called the process of autoregression and moving average with grade (p and q) which is abbreviated to ARMA (p, q) [4]. The model assumes that the value of the forecast variable at the moment and in period t depends on its past values and the difference between past real values of the forecast variable and its values obtained from the model. The ARIMA model includes autoregression parameters and moving average and introduces the differentiation operator. So, there are three parameters: ARIMA (p, d, q): autoregression parameters p, differentiation grade d, moving average parameters q. It is required that the input series used for the ARIMA method is stationary, i.e. it should include a time constant average, variance and lack of autocorrelation. This is why it needs differentiation until it has gained a stationary state. The parameter d expresses how many times the series should be differentiated. Very rarely, the number of parameters p and q must be higher than 2. There are also seasonal models, where additionally three parameters of seasonality are defined: ARIMA (ps, ds, qs) autoregression parameters ps, seasonal differentiation parameters ds and moving average parameters qs [15]. 3.2. Exponential smoothing The essence of exponential smoothing relies on the fact that the time series of the forecast variable is smoothed by means of the moving average and weight established due to the exponential function. The forecast is based on the weighted average expressed in present and previous values of the series [1]. Exponential smoothing may be based on various models that are appropriate to the kind of the components of the analysed time series [8]. In the case of the exponential smoothing models, the key issue is the selection of the values of model parameters. Several ways of selecting parameters are provided in the literature on the subject. If particular components change rapidly, it is assumed that the values of smoothing parameters must be established at a level close to one, or otherwise at a level close to zero. The values of parameters may be also selected by the method of an experiment and by minimising selected mistakes, define three smoothing parameters: – a (alpha) – is the stable parameter of smoothing, indispensible in all models, it assumes values from the range [0, 1]. If a = 0, all smoothed values will equal the preliminary value. If a = 1 each smoothed value (forecast) will equal the previous value. The closer a is to 0, the slower the weights will be decreasing i.e. the effect of preceding observations will decrease at a much lower rate. – d (delta) – is the parameter of seasonal smoothing and it requires defining only in the case of seasonal models. It may assume values from the range [0, 1]. If d = 0, it is assumed that the seasonal component for a given time point is identical to that anticipated for a given moment during the preceding cycle. – c (gamma) and / (fi) – are smoothing parameters of the trend [5]. The parameter c is defined for a linear trend and exponential models as well as for models of a falling trend without seasonal fluctuations. The parameter / is determined for a model of falling trend. Analogous to the seasonal component, when the trend component is included in the process of exponential smoothing, for each time moment, it is calculated as an independent trend component and modified as the function of the forecast error and proper parameter. If the parameter c amounts to 0 (zero), the trend component is stable for all values of the time series. If the parameter equals 1, the trend component is maximally modified in each step by a proper forecast error. If the values of the parameter are in the middle they represent a mixture of these two extreme situations. The parameter c is the parameter of modification of the trend and it defines how strong changes in the trend influence the assessment of the systematic component for the following forecast, i.e. how fast the trend will be declining or rising. The form of a model with a exponential trend and multiplicative fluctuations is presented as follows: Please cite this article in press as: Babiarz B, Chudy-Laskowska K. Forecasting of failures in district heating systems. Eng Fail Anal (2015), http://dx.doi.org/10.1016/j.engfailanal.2014.12.017
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F t1 ¼/ ðyt1 C t1r Þ þ ð1 /ÞðF t2 St2 Þ
ð4Þ
St1 ¼ cðF t1 F t2 Þ þ ð1 cÞSt2 y C t1 ¼ d t1 þ ð1 cÞC t1r F t1
ð5Þ ð6Þ
where Ft1 – equivalent of smoothed value created from a simple exponential smoothing model. St1 – evaluation of trend growth for a moment or the period t 1. Ct1 – evaluation of seasonality index for a moment or the period t 1. r – the length of seasonal cycle – number of phases. a, c, d – the model parameters – values from 0 to 1. 3.3. The model of homologous period trend estimation The method of homologous period trend estimation uses a correlation between the observation from several years for the same period. In the case studied, homologous period variables are months. The procedure relies on estimating the parameters of the analytical function of the trend separately for particular cycle phases. The forecast is determined by means of extrapolation of the estimated trend function for each cycle phase. The application of the method imposes the principles of ‘‘status quo’’, meaning that the observed tendency for each of the cycle phases will be maintained [14]. Each time series referring to a defined cycle phase is described by a linear model in the following form
yij ¼ a0i þ a1i tji þ et
j ¼ 1 . . . k; i ¼ 1 . . . r
ð7Þ
where yij – value of predicted variable, for the i-th phase in the j-th cycle. tij – time variable, tij = i + r(j 1). a0ia1i – structural parameters of the model. et – random component. 3.4. Index analysis This is one of the most commonly used methods in the analysis of seasonal fluctuations. It relies on determining the indexes of seasonal fluctuations for particular cycle phases. When fluctuation amplitudes in analogous cycle phases are approximately the same, it relies on absolutely stable fluctuations. When the amount of fluctuation amplitudes change more or less in the same relation, this is known as relatively stable fluctuations. In the first case, for the description of the shaping of this phenomenon an additive model may be used and in the second case, multiplicative ones may be applied. In the studies on monthly number of failures SbHS, the model of the following equation has been applied:
^ti si þ nt yti ¼ y
ð8Þ
where ytj – real values of forecasted variable. ^ti – smoothed values received form the model trend. y si – pure indicator of seasonality based on i t cycle. nt – random component. The analysis of seasonal fluctuations may include four stages: – – – –
Selection of the trend, elimination of the trend from time series, elimination of accidental fluctuations, calculation of seasonal coefficients.
The selection of the trend relies on smoothing the time series by means of the r-word central or non-central moving average or the analytical function. The aim of data aggregation is to ascertain the time series in which seasonal fluctuations appear. It is performed by summing data in the periods assumed in the studies into data adequate for the periods that equal the length of the seasonal cycle. The elimination of the trend in the case of a time series with additive fluctuations is performed while calculating the difference between real values of the forecast variable and smoothed values obtained from the trend model. In the case of multiplicative fluctuations, the quotient of real values of the forecast variable is determined by means of the adequate smoothed values.
zti ¼
yti ^ti y
ð9Þ
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where zti – real difference between variable values and smoothed values received from the trend model. yti – the real value. ^ti – empirical value. y The calculated values take into account seasonal fluctuations. Elimination of the random component is made by calculating so-called raw seasonal fluctuation indices. They constitute average numbers determined on the basis of the numbers zti, regarding the same phase of fluctuations. The seasonal coefficients (pure) are determined according to formula (10) and the forecast according to formula (11).
si ¼
zi q
ð10Þ
where: si – pure indicator of seasonality. q – the mean values based on zti. r – the number of cycle phases. ðtÞ
ðtÞ
yti ¼ yti si ;
ð11Þ
yti – the forecast for the period ðtÞ yti – preliminary forecasts for
in the i-th phase of the cycle. the period in the i-th phase of the cycle. ci – pure indicator of seasonality in the i-th phase of the cycle.
4. Results and discussion 4.1. ARIMA method The assumptions of the method require that before estimating parameters the series must be led to a stationary state, i.e. it must have a time stable average, variance and should not possess autocorrelation. Analysing the data taken for studies, none of the assumptions are fulfilled. The series is declining, so the average is decreasing in time, and the amplitude of fluctuations is also decreasing, which is why variance is falling in time (Fig. 1). The data are seasonal, so autocorrelation exists (Fig. 3). In order to lead the series to a stationary state, it must undergo determination of the logarithm and differentiation. After such procedures, the series should be stationary. After finding the logarithm and differentiation, the series is stationary, which is why it is possible to shift to the estimation of model parameters. On the basis of the studies, the best model was the ARIMA model (1, 1, 1) (1, 0, 0) which has one autoregression parameter p and one seasonal autoregression parameter ps, one moving average parameter q and
Fig. 3. Autocorrelation function.
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one parameter of differentiation d. Having structured the model it is necessary to check whether the distribution of residuals is a normal distribution or its series no longer has autocorrelation. The following hypothesis has been assumed: H0: residuals have a normal distribution and H1: no residuals in the normal distribution. The studies assumed a = 0.05. The analysis concludes that p > a (p = 0.64197), so there is no basis for rejecting H0. It is most probable that the distribution of model residuals is a normal distribution (Fig 4). No autocorrelation was found in the properly structured model, as is shown in Fig. 5. It is clearly visible in the correlogram that residuals are within the confidence level and there is no autocorrelation. A forecast was obtained for failures of the district heating systems in the studied period for such a structured model as in Fig. 6. 4.2. Exponential smoothing In choosing the model, the tendency of the series must be determined as well as the kind of periodical fluctuations, length and the seasonal delay. Several different models of smoothing have been checked and the best turned out to be the model
Fig. 4. Normal distribution of residuals – ARIMA model (1, 1, 1)(1, 0, 0).
Fig. 5. Autocorrelation function of residuals in ARIMA (1, 1, 1)(1, 0, 0) model.
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Fig. 6. Monthly failure numbers forecast based on ARIMA model.
with a linear tolerance and multiplicative fluctuations with the parameters: a = 0.3, d = 0.1 and c = 0.1. This model was characterised with the smallest errors, which is why it was accepted as the final one. For the given model, the distribution of residuals was checked assuming H0, that the residuals have a normal distribution. The probability found, p = 0.46738, gives no basis to reject H0, so the distribution of residuals is most likely normal (Fig. 7). A forecast was obtained for failures of the heat network in the studied period and the results are presented graphically in Fig. 8. 4.3. The model of homologous period trend estimation Another model which can be used in the case of monthly data which is characterised by seasonal fluctuations and a tendency, in this case a decreasing one, is the homologous periods model. For previously prepared data, monthly trends were determined by means of simple linear models based on a time line (Table 2). The model parameters were estimated by the method of least squares. As the analysed data have a monthly character, 12 linear models were estimated, which allowed forecasting of other values. The forecast performed on the basis of the estimated models is presented in Fig. 9. 4.4. Index method The final method presented is based on the analysis of seasonal indices calculated for particular cycle phases. The indices are used for structuring the forecast. Firstly, the linear function of the trend is estimated – the overall tendency contained in
Fig. 7. Normal distribution of residuals in exponential smoothing.
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Fig. 8. Monthly failure numbers forecast based on exponential smoothing.
Table 2 The results of structural parameters for the models of homologous period trend estimation. Number of months
Equation of the model
Number of months
Equation of the model
January (7) February (7) March (7) April (7) May (7) June (7)
y = 32.92t + 334.5 y = 28t + 281.4 y = 24t + 233 y = 18.85t + 202.7 y = 19.25t + 193.7 y = 13.64t + 161.5
July (7) August (6) September (6) October (6) November (6) December (6)
y = 25.32t + 238.2 y = 16.6t + 176.2 y = 44.85t + 360.3 y = 43.42t + 442.3 y = 39.28t + 386.3 y = 24.6t + 294.6
Fig. 9. Monthly failure numbers forecast based on homologous period trend (TM).
the data. It assumes the following formula for failures in the heat network: Y = 2.22t + 259.24. Next, the values were determined (zti) as quotients of real and empirical data. However, they include the effect of seasonal as well as accidental fluctuations, so this is corrected by calculating raw indices of seasonal processes (zi), i.e. the average of indices which are adequate to homologous phases. Pure indices of the seasonal processes allowing us to structure the forecast are determined as quotients of raw seasonal indices zi and q. For example, in January the pure index of seasonal processes amounted to c1 = 1.131, (113.1%) which means that in the January months the number of failures SbHS was higher, on average, by 13.1% than the one resulting from the trend line (Table 3). Please cite this article in press as: Babiarz B, Chudy-Laskowska K. Forecasting of failures in district heating systems. Eng Fail Anal (2015), http://dx.doi.org/10.1016/j.engfailanal.2014.12.017
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Table 3 Raw and pure indices of seasonal processes. Forecasting based on the index method. Months
zi
ci
t
Forecasting
January (z1) February (z2) March (z3) April (z4) May (z5) June (z6) July (z7) August (z8) September (z9) October (z10) November (z11) December (z12)
1.142 0.942 0.852 0.742 0.693 0.655 0.823 0.675 1.144 1.705 1.470 1.274 12.116 1.010
1.131 0.933 0.843 0.735 0.686 0.649 0.815 0.668 1.133 1.688 1.456 1.262
89 90 91 92 93 94 95 96 97 98 99 100
69.730 55.473 48.259 40.424 36.209 32.801 39.419 30.818 49.737 70.370 57.461 46.981
R q
Fig. 10. Monthly failure numbers – forecast based on the index method.
Table 4 Forecast list of failures in heat distribution network generated on the basis of four models and ex post errors of the given forecast.
VIII IX X XI XII I II III IV V VI VII The annual total failures number
Monthly failures numbers
ARIMA
Error ex post (%)
Exponential smoothing
Error ex post (%)
Method of univariate periods
Error ex post
Seasonal indicators method
Error ex post (%)
34 77 153 83 92 84 70 60 56 45 32 39 825
64 91 109 93 92 88 67 56 56 59 67 71 913
89 18 29 12 0 5 4 7 0 32 109 83
53 88 132 111 95 83 64 59 52 45 43 50 875
55 15 14 34 4 2 8 2 7 0 34 29
60 46 138 111 122 71 57 41 52 40 52 36 828
76 40 10 34 33 15 18 32 7 12 64 9
31 50 70 57 47 70 55 48 40 36 33 39 578
9 35 54 31 49 17 21 20 28 20 3 1
The bold values indicates the closest value of annual failures number.
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Fig. 11. Comparison of forecasting obtained by four methods.
A forecast was performed on the basis of the index method numerically presented in Table 3 and graphically in Fig. 10. The forecast shows that the index method dampened seasonal fluctuation and allowed the obtaining of the lowest values for monthly forecasts of failures in the SbHS network in comparison to other models. 4.5. Comparison of applied methods The results of SbHS failures obtained with the use of four methods were presented in Table 4. For each forecast, made by means of four models, the error was calculated ex post so as to check how much the value of forecasting was different from the factual state of failure in the heat network. Fig. 11 presents a graphical image of the obtained forecasting. On the basis of the error analysis and according to the graph presenting the comparison of forecasting methods, it may be claimed that the models ARIMA and exponential smoothing show the smallest errors in the most critical moments, when the network operates at maximum load. A higher number of failures are observed in October and November. This is caused by the changing parameters connected with the beginning of the heating season. The methods, however, re-estimate the numbers of failures due to large errors in other periods. The best method seems to be the homologous period trend estimation, where the number of failures for the entire studied period (a year) gives a value that can be compared with real operational data. 5. Conclusions The performed analysis of failures in district heating systems allowed the development of a method of forecasting failures in the heat network, on the basis of the analysis of time series. Four methods were used: ARIMA, exponential smoothing, homologous period trend estimation and index methods, as the most adequate for this type of data. The forecast of failures was developed by selecting the proper models. The most adequate turned out to be the homologous period trend estimation method. The forecast describes the reliability state of SbHS in the given time interval. This state may be defined by the value of one or several reliability indices. The proper forecast of the reliability index may be obtained only with reliable and complete input data. The makes necessary the diligent collection of reliability data of the studied facilities. The method, after verification, gives the possibility of preparing for operating stand-by of repair services. As a result, it is possible to answer many questions that come from the operator of the heat network regarding the plans of repairs, maintenance of repair services, planning finance or the desirability of installing a monitoring system. It may also be used as a tool in the assessment of the risk of lack of heat supply to recipients. References [1] Aczel Amir D. Statistics in management. Warsaw: PWN; 2000. s. 642. [2] Babiarz B. An introduction to the assessment of reliability of the heat supply systems. Int J Press Vessels Pip 2006;83(4):230–5. [3] Babiarz B. Heat supply system reliability management. Safety and reliability: Methodology and Applications. In: Nowakowski et al., editor. Proc. of the European safety and reliability conference, ESREL 2014, Wroclaw, Poland, 14–18 September 2014. 2015 Taylor & Francis Group, London. p. 1501–1506.
Please cite this article in press as: Babiarz B, Chudy-Laskowska K. Forecasting of failures in district heating systems. Eng Fail Anal (2015), http://dx.doi.org/10.1016/j.engfailanal.2014.12.017
12 [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
B. Babiarz, K. Chudy-Laskowska / Engineering Failure Analysis xxx (2015) xxx–xxx Box GEP, Jenkins GM, Reinsel GC. Time series analysis: forecasting and control. 4th ed. Wiley; 2008. Cies´lak M. Economic forecasting, methods and applications. Warsaw: PWN; 1997. Dittmann P. Sales forecasting methods in the enterprise. UE Wroclaw; 1998. Dolinay V, Vasek L. Simulation and experiments on the secondary heat distribution network system. Int J Math Comput Simul 2012;6(5):465–72. Grubb H, Mason A. Long lead-time forecasting of UK air passengers by Holt 0 Winters methods with damped trend. Int J Forecast 2001;17:71–82. International Electrotechnical Commission: analysis techniques for system reliability – procedure for failure mode and effect analysis. IEC Standard, Publication 812; 1990. Ma B, Shuai J, Liu D, Xu K. Assessment on failure pressure of high strength pipeline with corrosion defects. Eng Failure Anal 2013;32:209–19. Macdonald KA, Cosham A. Best practice for the assessment of defects in pipelines – gouges and dents. Eng Failure Anal 2005;12(5):720–45. Miszta-Kruk K. Reliability analysis of vacuum sewerage systems using the total probability theorem. Advances in safety, reliability and risk management. London, UK: Taylor & Francis Group; 2011. p. 2072–9. Pinna T, Caporali R, Cambi G, Burgazzi L, Poucet A, Porfiri MT. Failure mode and effect analysis on ITER heat transfer systems. Eng Failure Anal 2012;25:156–68. Radzikowska B. Methods of forecasting. UE Wroclaw; 2004. Samagaio A, Wolters M. Comparative analysis of government forecast for the Lisbon Airport. J Air Transp Manage 2010;16:213–7. Song Y, Ma L, Morris J. A practical approach for reliability prediction of pipeline systems. Eur J Oper Res 2009;198:210–4. Szybka J, Broniec Z, Pilch R. Forecasting the failure of a thermal pipeline on the basis of risk assessment and exploitation analysis. Maint Reliab 2011;4:5–10. Teixeira AP, Guedes Soares C, Netto TA, Estefen SF. Reliability of pipelines with corrosion defects. Int J Press Vessels Pip 2008;85(4):228–37. Valis D, Zak L, Pokora O. On approaches for non-direct determination of system deterioration. Eksploatacja i Niezawodnosc—Maint Reliab 2014;14(1):33–41. Werbinska-Wojciechowska S. Time resource problem in logistics systems dependability modeling. Eksploatacja i Niezawodnosc—Maint Reliab 2013:427–33. Wojdyga K. An influence of weather conditions on heat demand in district heating systems. Energy Buildings 2008;40:2009–14. Xu LY, Cheng YF. Reliability and failure pressure prediction of various grades of pipeline steel in the presence of corrosion defects and pre-strain. Int J Press Vessels Pip 2012;89:75–84. Zorc B, Kosec B, Kosec L, Nagode A. Analysis of hot water pipeline system leakage. Eng Failure Anal 2013;28:78–81.
Please cite this article in press as: Babiarz B, Chudy-Laskowska K. Forecasting of failures in district heating systems. Eng Fail Anal (2015), http://dx.doi.org/10.1016/j.engfailanal.2014.12.017