# An Experimental Water Consumption Regression Model for Typical Administrative Buildings in the Czech Republic

^{1}

^{2}

^{*}

## Abstract

**:**

_{3}coefficient, according to the FAVAD concept used for prediction of changes in water consumption related to pressure. The statistical inference is based on the maximum likelihood method. The proposed regression models are tested to evaluate their suitability, particularly, the models are compared using a cross-validation procedure. The significance tests for parameters and model reduction are based on asymptotic properties of the likelihood ratio statistics. Pressure is confirmed in each regression model as a significant variable.

## 1. Introduction

_{3}set coefficients within the meaning of [13] based on real studies. For example the value of the “inside-the-house” consumption coefficient was set at 0.2 for the Johannesburg student campus at the University of Johannesburg in [15]. Also worth mentioning is the principle of minimum pressure, which must be ensured in the water supply network so that the water supply can be realized. If this value is under-stepped, the volume of water supplied is considered zero [9]. In this study, the results of which are presented here, this limit was defined by valid legislation at 0.15 MPa and was not broken during the experiment.

_{3}coefficient used in the FAVAD concept. In the past, a sufficient number of studies used the FAVAD concept, covering the entire spectrum of various types of end users. A new regressive model was not created for this purpose, but an existing model used according to [13].

## 2. Materials and Methods

#### 2.1. Facility Details, Measuring Campaign and Measuring Equipment

#### 2.1.1. Facility Details

#### 2.1.2. Measuring Campaign

#### 2.1.3. Measuring Device

- Change of pressure conditions—a spring-based PRV with the output pressure range of 0.15–0.60 MPa was used. The hydraulic losses caused by the PRV were low even at the maximum hourly flow rate. PRV dimension was chosen with respect to the hydraulic losses and characteristic flow rates through the PRV. The characteristic flow rates in the given building are presented in Table A1 and the PRV head loss diagram is shown in Figure A3.
- Flow volume measurement—The volume of water flowing was measured using a water meter with a pulse generator and the pulse value of 1 liter. The water meter corresponded to the “C” level of precision in the sense of [17]. Nominal flow rate of the water meter in the sense of [17] is 2.5 m
^{3}h^{−1}. In order to sustain the guaranteed precision, undisturbed spacing lengths were maintained both upstream and downstream of the water meter. The value of the pressure was recorded along with the water meter value every 15 s. This was an interrupted measurement with a relatively short time period. - Pressure measurement—The pressure was measured with an integrated pressure sensor with a range of 0.0–1.0 MPa, and measurement accuracy of 0.25% of the range (i.e., 0.0025 MPa).

#### 2.2. Data and Its Verification

^{3}–238.3 m

^{3}–241.1 m

^{3}), while the maximum number of workers per shift continued to be the same.

#### 2.3. Statistical Inference for Pressure

_{3}as a measure of the pressure change to the water consumption. In our case, the coefficients N

_{3}and C

_{0}were considered as unknown regression parameters to be estimated. The pressure P

_{0}is assumed fixed, equal to the average of category-E pressures. The estimates of the regression parameters are obtained by the Maximum Likelihood (ML) method. Hence we obtained the values

#### 2.4. Statistical Inference for Climatological Factors

## 3. Results

#### 3.1. Outlier Identification

#### 3.2. Dependency of Consumption on Pressure

#### 3.3. Dependency of Consumption on Climatological Factors

## 4. Discussion

_{3}coefficient value established by the FAVAD concept, according to [13], for the respective administrative building reaches 0.179. Compared to our expectations, this value is rather high. In administrative buildings, a high percentage of water is used for flushing toilets, while these are often volumetric flushing systems with a pre-stored storage water tank. For example, according to [32], water consumption for toilets in government and private administrative buildings in Singapore presents 37% of all overall consumption, while another 31% is water for cooling systems. Water consumption for cooling was zero in the building where our case study took place, because a cooling system is not installed. This results in a ratio of water used for flushing toilets to be a minimum of 50%, although this ratio was not specifically established during this study. Therefore, at least 50% of the overall volume of consumed water in this building is not dependent on water pressure in the mains.

_{3}coefficient in this study is also rather high in comparison with [15], where the N

_{3}value was established at 0.2. Although this is a higher value, the study was performed at a student campus with a considerable ratio of water consumption for personal hygiene purposes, as well as for gardening and irrigation, which are areas of consumption dependent on pressure.

^{3}year

^{−1}. This represents 12% of the overall annual water consumption in the entire building. The user would save approximately 100 € year

^{−1}.

^{−1}·day

^{−1}, today this value is approximately at half.

_{3}coefficient (upon using the (7) model), which reaches the value of 0.091 for temperature and 0.178 for humidity.

## 5. Conclusions

_{3}coefficient for an administrative building was established at 0.179. Stipulating this value is important to establish realistic values for multiple types of users, using individual values of the N

_{3}coefficient for the optimization of pressure conditions in water distribution networks, including water consumption as an optimization criterion according to (3).

_{3}coefficient values.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

Day (Working Hours) | Minimal Consumption (Liters) | Mean Consumption (Liters) | Maximal Consumption (Liters) |
---|---|---|---|

Monday + Wednesday (7–17) | 608 | 892 | 1307 |

Tuesday + Thursday (7–15) | 481 | 710 | 1252 |

Friday (7–13) | 307 | 623 | 927 |

All days (7–13) | 307 | 545 | 982 |

Hourly consumption—all days | 10 | 91 | 381 |

No. | Date | Day of Week | Problem Description | Explanation of Problem |
---|---|---|---|---|

1 | 26.09.2016 | Monday | Zero consumption during 8–11 | Technical problems with measuring device |

2 | 03.10.2016 | Monday | Unknown consumption during 9–11 | |

3 | 16.11.2016 | Wednesday | Very low consumption per person | Failure of water supply—water mains breakage—failures duration 6 h |

4 | 21.12.2016 | Wednesday | Very high consumption per person | Very low number of workers—Christmas holidays + 7 visitors over the course of approximately 4 h |

5 | 22.12.2016 | Thursday | Very low number of workers—Christmas holidays + 15 visitors over the course of approximately 2 h | |

6 | 27.12.2016 | Tuesday | Very low number of workers—only accountants present + creation of an ice rink | |

7 | 28.12.2016 | Wednesday | ||

8 | 29.12.2016 | Thursday | Very low number of workers—only accountants present + some workers outside the evidence | |

9 | 30.12.2016 | Friday | ||

10 | 10.01.2017 | Tuesday | Excluded on the basis of Grubbs’ test results | |

11 | 26.01.2017 | Thursday | Low consumption per person | 8 left the building for an unknown number of hours |

12 | 20.04.2017 | Thursday | 21 workers left the building for 5 h | |

13 | 10.07.2017 | Monday | Negligible consumption per person | Very low number of workers—summer holidays—only workers in the reception—smaller number of workers in the complex than in evidence |

14 | 11.07.2017 | Tuesday | ||

15 | 12.07.2017 | Wednesday | ||

16 | 13.07.2017 | Thursday | ||

17 | 14.07.2017 | Friday | ||

18 | 17.07.2017 | Monday | ||

19 | 18.07.2017 | Tuesday | ||

20 | 19.07.2017 | Wednesday | ||

21 | 20.07.2017 | Thursday | ||

22 | 30.08.2017 | Wednesday | Very low consumption per person | 19 workers left the building for 3.5 h |

23 | 31.08.2017 | Thursday | 23 workers left the building for 4.5 h |

**Figure A4.**Water consumption in CZ and Zlin region for years 2000–2016 [19].

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**Figure 1.**Time series of observed consumption; only working days plotted. Values labelled as invalid are highlighted by red circles.

**Figure 3.**Standardized residuals of the fit by regression curve (3). Horizontal lines show the residual mean and interval of half-width equal to 3 standard deviations.

**Figure 4.**Histogram of frequencies in pressure categories; (

**a**) boxplots of consumption observations within the categories; (

**b**) whiskers show 1.5 inter-quantile range.

**Figure 6.**Pairwise plot of covariates: (

**a**) pressure against humidity; (

**b**) pressure against temperature; and (

**c**) temperature against humidity.

**Figure 7.**Plot of observed consumption against temperature (

**a**,

**c**) and humidity (

**b**,

**d**) within pressure categories B (

**a**,

**b**) and C (

**c**,

**d**).

Statistical Criterion | Category of Pressure | ||||
---|---|---|---|---|---|

A | B | C | D | E | |

${\chi}^{2}$ test | 0.0084 | 0.5049 | 0.6747 | 0.4959 | 0.3735 |

Lilliefors test | 0.0018 | 0.2961 | 0.4133 | 0.5000 | 0.3215 |

A-D test | 0.0011 | 0.1239 | 0.1991 | 0.4207 | 0.3382 |

**Table 2.**Results of one-way ANOVA applied to consumption under pressure categorization. Abbreviations: sum of squares (SS), degrees of freedom (df), mean square (MS), F-statistics (F).

Source | SS | df | MS | F | p-Value |
---|---|---|---|---|---|

Groups | 3.9132 | 4 | 0.97807 | 6.87 | <0.0001 |

Error | 31.1841 | 219 | 0.14239 | ||

Total | 35.0964 | 223 |

**Table 3.**Results of the 10-fold cross-validation procedure for power regression (3) and linear regression (2).

Regression Type | Mean Squared Error | ||
---|---|---|---|

Mean | Median | Standard Deviation | |

Power regression | 0.1413 | 0.1413 | 0.0073 |

Linear regression | 0.1501 | 0.1379 | 0.0698 |

**Table 4.**The estimated regression parameters relative to the model (3): point estimates, 95% confidence interval for the parameters obtained from the profile likelihood, value of likelihood ratio statistics and corresponding p-value of parameter significance.

Regression Parameter | Estimates | |||
---|---|---|---|---|

Value | 95% Conf. Interval | LR Statistics | Significance p-Value | |

${C}_{0}$ | 2.8841 | [2.834, 2.935] | - | - |

${N}_{3}$ | 0.1788 | [0.116, 0.244] | 22.644 | <0.0001 |

**Table 5.**Spearman’s rank correlation coefficient between covariates and the observed p-values of significance tests.

Pair of Covariates | Corr. Coefficient | Significance p-Value | |
---|---|---|---|

pressure | temperature | −0.0536 | 0.4247 |

pressure | humidity | 0.0508 | 0.4497 |

temperature | humidity | −0.2865 | <0.0001 |

**Table 6.**Estimated regression parameters and p-values of slope significance test—for models significantly different from constant at level 0.05, p-values are highlighted with an underline. Both temperature and humidity models are shown.

Pressure Category | Temperature | Humidity | ||||
---|---|---|---|---|---|---|

Intercept | Slope | p-Value | Intercept | Slope | p-Value | |

A | 2.8386 | −0.0169 | 0.025 | 2.3898 | 0.0029 | 0.614 |

B | 3.0592 | −0.0157 | 0.002 | 2.2866 | 0.0087 | 0.005 |

C | 2.9733 | −0.0105 | 0.117 | 3.0246 | −0.0025 | 0.518 |

D | 2.9708 | −0.0216 | 0.125 | 1.9333 | 0.0123 | 0.0002 |

E | 3.0383 | −0.0140 | 0.008 | 3.2202 | −0.0031 | 0.492 |

**Table 7.**ML estimates of the parameters of model (7) and their 95% confidence intervals obtained from profile likelihood. Likelihood ratio statistics and corresponding p-value shown for parameter of climatological covariates.

Included Covariate and Regression Parameters | Estimates | ||||
---|---|---|---|---|---|

Value | 95% Conf. Interval | LR Statistics | Significance p-Value | ||

Temperature | ${C}_{0}$ | 3.0000 | [2.953, 3.047] | - | - |

${N}_{3}$ | 0.0906 | [0.035, 0.148] | - | - | |

$\beta $ | −0.0137 | [−0.017, −0.011] | 23.218 | <0.0001 | |

Humidity | ${C}_{0}$ | 2.6183 | [2.568, 2.668] | - | - |

${N}_{3}$ | 0.1782 | [0.109, 0.249] | - | - | |

$\beta $ | 0.0029 | [0.0032, 0.0046] | 4.714 | 0.0299 |

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## Share and Cite

**MDPI and ACS Style**

Rucka, J.; Holesovsky, J.; Suchacek, T.; Tuhovcak, L.
An Experimental Water Consumption Regression Model for Typical Administrative Buildings in the Czech Republic. *Water* **2018**, *10*, 424.
https://doi.org/10.3390/w10040424

**AMA Style**

Rucka J, Holesovsky J, Suchacek T, Tuhovcak L.
An Experimental Water Consumption Regression Model for Typical Administrative Buildings in the Czech Republic. *Water*. 2018; 10(4):424.
https://doi.org/10.3390/w10040424

**Chicago/Turabian Style**

Rucka, Jan, Jan Holesovsky, Tomas Suchacek, and Ladislav Tuhovcak.
2018. "An Experimental Water Consumption Regression Model for Typical Administrative Buildings in the Czech Republic" *Water* 10, no. 4: 424.
https://doi.org/10.3390/w10040424