# Application of the Monte-Carlo Method to Assess the Operational Reliability of a Household-Constructed Wetland with Vertical Flow: A Case Study in Poland

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## Abstract

**:**

_{5}(biochemical oxygen demand), COD

_{Cr}(chemical oxygen demand), and TSS (total suspended solids), in the 2017–2020 period. Anderson–Darling (A–D) statistics were applied to assess the fit of the theoretical distributions to the empirical distributions of the random variables under study. The selection of the best-fitting statistical distributions was determined using the percentage deviation (PBIAS) criterion. Based on the analyses that were performed, the best-fitting statistical distributions for the pollution indicators of the raw wastewater were the generalised extreme value distribution for BOD

_{5}, the Gaussian distribution for COD

_{Cr}, and the log-normal distribution for TSS. For treated effluent, the log-normal distribution was the best fit for BOD

_{5}and COD

_{Cr}; the semi-normal distribution, for TSS. The new data generated using the Monte-Carlo method allowed the reliability of the VF-CW operation to be assessed by determining the reliability indices, i.e., the average efficiency of the removal of pollutants (η), the technological efficiency index (R), the reliability index (CR), and the risk index of the negative control of the sewage treatment plant operation (Re). The obtained results indicate that only in the case of COD

_{Cr}, the analysed treatment facility may fail to meet the requirements related to the reduction of organic pollutants to the required level, which is evidenced by the values of the indicators CR = 1.10, R = 0.49, and η = 0.82. In addition, the risk index of the negative operation of the facility (Re) assumes a value of 1, which indicates that during the period of its operation, the VF-CW system will not operate with the required efficiency in relation to this indicator. The novelty of this work is the implementation of the indicated mathematical simulation methods for analysing the reliability of the operation of the domestic wastewater treatment facility.

## 1. Introduction

## 2. Materials and Methods

_{5}), chemical oxygen demand (COD

_{Cr}), and total suspended solids (TSS). Raw sewage samples were taken from a septic tank, and treated sewage samples were taken at the inlet to an absorption well. Sampling and transport were carried out in accordance with Polish Standards [28,29]. BOD

_{5}was measured using the dilution method and a WTW OxiTop 538 portable meter PN-EN1899-1:2002 [30]. COD

_{Cr}was determined using a miniaturised method and sealed tubes in accordance with PN-ISO 15705:2005 [31]. The total suspended solids were determined using the weighing method in accordance with PN-EN 872:2002 [32]. The analysis consisted of the following stages: preliminary data analysis, selection of the theoretical function, best-fit empirical distribution, modelling the values of the pollution indicators for sewage, and determining the values of the reliability coefficients for the treatment plant based on the simulation results. Figure 1 illustrates the methodology employed in this study. Based on the current legal standards in Poland [33,34], the following values of analysed pollutant indicators are mandatory for wastewater treatment plants serving up to 2000 Population Equivalents (PEs): BOD

_{5}—40 mgO

_{2}·dm

^{−3}; COD

_{Cr}—150 mgO

_{2}·dm

^{−3}; TSS—50 mg·dm

^{−3}. By law, only two samples may not meet these requirements.

#### 2.1. Description of Study Object

^{3}connected to a single vertical flow bed (VF-CW). The dimensions of the deposit are 1.0 m deep × 12 m

^{2}surface area × 1% slope. The projected capacity of the treatment plant is 0.4 m

^{3}∙d

^{−1}. The hydraulic load of the VF beds is approximately 0.033 m

^{3}∙m

^{2}∙d

^{−1}. To protect the environment from contamination, a 1 mm thick, high-density polyethylene lining was used to seal the bed. The bed was filled with medium sand with a top layer of humus, in which Glyceria maxima was planted. Treated wastewater from the bed was directed to the dry well with a working capacity of 1.9 m

^{3}. The main task of the dry well is to discharge sewage into the ground after pre-treatment in the filtration layer. An overview of the study object is shown in Figure 3.

#### 2.2. Preliminary Data Analysis

_{s}).

#### 2.3. Theoretical Distribution Fitting

^{2}—variance value; μ—mean value.

#### 2.4. Indication of the Best-Fit Theoretical Distributions

#### 2.5. Modelling of Pollutant Indicator Values

_{5}, COD

_{Cr}, and TSS). The new data generated in this way were subjected to statistical analysis. The procedure of this method is based on the generation of 365 random values of the pollution indicators. Each series of simulations was repeated 100 times. From each generated set, a 24 element data set was randomly determined. In each series, the number of simulation observations that did not meet the technological efficiency condition, i.e., whether the simulated values were greater than the acceptable levels, was determined. The control performance score of the WWTP was determined as follows: control indicator (CF) = 1 in the case of a negative assessment result, i.e., if the number of non-compliant samples exceeded the maximum number of these samples; otherwise, CF = 0.

#### 2.6. Determination of Reliability Coefficients for the Operation of the CW System

_{sym}—number of simulations that were carried out; N

_{Xperm}—number of samples from simulations not meeting the limit values for the analysed pollutant indicator.

_{sym}—number of simulations that were carried out.

_{out}—the indicator of the average concentration of pollution in the sewage outflowing from the bed (mg·dm

^{–3}); C

_{in}—the indicator of the average concentration of pollution in the sewage flowing into the bed (mg·dm

^{–3}).

## 3. Results and Discussion

#### 3.1. Preliminary Analysis

_{5}and COD

_{Cr}) and the total suspended solids (TSS) in raw and treated wastewater. Table 1 reports the results of the analysis, and Figure 4 shows the changes in the values of the analysed indicators for the inflow and outflow of sewage in relation to the permissible values.

_{5}, COD

_{Cr}, and TSS in raw sewage were 89%, 58%, and 88%, respectively. In Gawłówek, the composition of the raw sewage discharged from households is typical of that of domestic sewage in southern Poland, as indicated by the results of a study by Kaczor [59]. Additionally, taking into account the values of the coefficient of variation (C

_{s}) for the observational series of indicators of pollution in sewage, the changes in their values were high (BOD

_{5}and TSS) or average (COD

_{Cr}). The size variability of the indicators is a typical feature for domestic wastewater and has been noted and confirmed in the works of other authors, including Jóźwiakowksi et al. [60], Jucherski et al. [61], and Zhang et al. [62]. For the treated sewage, the difference between the extreme values of the BOD

_{5}indicator was 98%, with an average value of 13.3 mgO

_{2}·dm

^{−3}. In the study period, this indicator exceeded the permissible value (40 mgO

_{2}·dm

^{−3}) once. For COD

_{Cr}, the difference between the extreme values was 63%, and the average value was 162 mgO

_{2}·dm

^{−3}. Moreover, 21 observations exceeding the permissible value of 150 mgO

_{2}·dm

^{−3}were recorded in the analysis period. The last analysed indicator was TSS, for which the difference between the extreme values was 98%, with an average value of 21.5 mg∙dm

^{−3}. In the sampling period, three samples did not meet the requirements in relation to the permissible values of the total suspended solids (TSS) in the outflow (40 mg∙dm

^{−3}). A comparison of the mean and median values obtained for the analysed pollution indicators for the raw and treated wastewater suggests that the obtained values are similar. This is because the analysed data series is homogeneous and has no extremes. The exceedance of the permissible values could be determined by important factors affecting the effectiveness of the wastewater treatment, e.g., the volume of the wastewater inflow, the temperature of the wastewater, or the pollutant load in the wastewater flowing into the treatment facility. As reported by Wałęga et al. [63], the quality of wastewater is strongly determined by the characteristics of the farm where the wastewater is generated. Higher concentrations of pollutants in the effluent flowing into the treatment plant may be due to the sparing use of water by residents. Uneven trends in water consumption by inhabitants in households can cause large fluctuations in drains, which can expose systems to unstable operation. The temperature and plant density are the factors that clearly correlate with the pollution removal effect of CW systems. Temperatures above 20 °C are conducive to the multiplication and growth of microorganisms and ensure high microbial activity and a strong ability to absorb and decompose organic matter, which ultimately allows densely grown plants to absorb increasing volumes of organic compounds from wastewater. In addition, plants with a highly developed root system can strongly absorb organic matter [62,64]. The analyses of the pollutant values of the treated wastewater show that higher concentrations were recorded in the autumn–winter and winter–spring periods, when the temperature decreased. The efficiency for removing contaminants will, thus, drop sharply owing to low temperatures, where plants go into a dormant phase and most microbial activity slowly decreases [65,66].

#### 3.2. Fitting the Theoretical Distributions

_{5}and COD

_{Cr}) and total suspended solids in raw sewage: Halfnormal, Pareto, Triangular, and Rayleigh. In the case of the treated wastewater, the results of the A–D test for the observational series showed that the p-values were below 5% for the Triangular and Rayleigh distributions for the indicators from the organic group (BOD

_{5}and COD

_{Cr}) and for the Half-normal and Pareto distributions in the case of the COD

_{Cr}indicator. Therefore, the H

_{0}hypothesis—stating that the analysed theoretical distribution is consistent with the empirical distribution—should be rejected in this case. In the case of the total suspended solids, the analysis of the fit of the theoretical distribution to the empirical distribution showed a lack of consistency in this indicator for the Triangular and Rayleigh functions (p < 0.05). The analysis for fitting the theoretical distribution to the empirical one showed that the Triangular distribution could not be used to describe any of the observational series of the analysed pollution indicators (Table 2). The inapplicability of this distribution is due to the homogeneous nature of this function. As Stain and Keblis [67], Pereira et al. [68], and Młyński et al. [69] have stated in their studies, the Triangular distribution is described by the parameters of the upper and lower limits, i.e., the most probable maximum and minimum values, respectively. In the case of wastewater, the values of the individual quality parameters depend on many factors that vary over time. Thus, the prediction of the most likely minimum and maximum values for a Triangular distribution is problematic. In practice, only a few types of random variables can be described by this distribution.

#### 3.3. Indication of the Best-Fit Probability Distribution

_{5}, the Gaussian distribution for COD

_{Cr}, and the Log-normal distribution for the total suspended solids. For the treated wastewater (Figure 6), Log-normal is the best-fitting distribution for BOD

_{5}and COD

_{Cr}, as indicated by PBIAS values of −0.987 and −0.132, respectively. For the total suspended solids, the Halfnormal distribution turned out to be the best-fitting theoretical distribution.

#### 3.4. Determination of Reliability Coefficients for the Operation of VF-CW

_{e}). The obtained results are presented in Table 4.

_{5}and TSS indicate the reliable operation of the VF-CW. This is also evidenced by the results of the technological efficiency indicators (R), which are close to one. This suggests that the number of samples that do not meet the requirements for wastewater quality at the outflow from the facility may occur only occasionally during the operation of the VF-CW system. Additionally, for both analysed indicators of contaminants, the value of the risk index of the negative control of the treatment plant operation (R

_{e}) clearly indicates that the number of samples exceeding the permissible values of these indicators at the outlet is lower than the acceptable value for a treatment plant serving a PE below 2000. The high degree of reduction obtained for both indicators confirms the effective distribution of these pollutants at the analysed treatment facility. Only in the case of COD

_{Cr}, the obtained simulation results indicated that the domestic treatment system might not meet the requirements related to the reduction of organic contaminants to the required level. The values of the reliability index (CR), technological efficiency (R), and the average efficiency of the pollutant removal (η) confirmed that the concentration of organic compounds at the outflow may be higher than the permissible concentration set out in the regulation [34]. Moreover, the value of the risk index of the negative operation of the facility (R

_{e}) was one for this indicator, which indicates that the VF-CW system will not operate with the required efficiency in relation to this indicator during the period of the system’s operation. The simulation results suggest that the processes of the biological wastewater treatment in the facility may be disturbed. As studies have shown, the degree of pollutant decomposition described by COD

_{Cr}in VF-CW systems is lower than that described by BOD

_{5}, which is due to the presence of components with low biodegradability in wastewater [70]. Similar results showing the reduced removal efficiency of COD

_{Cr}pollutants in subsurface vertical-flow wastewater systems were obtained in other studies [71,72]. The decomposition of organic compounds determined by BOD

_{5}and COD

_{Cr}occurs by the interaction of microbial and physical mechanisms using dissolved oxygen. The removal of BOD

_{5}occurs through sedimentation and microbial degradation processes carried out by aerobic bacteria residing in the peri-root zone of the CW system. COD

_{Cr}is mainly removed by sedimentation and filtration, and the presence of plants negligibly affects its removal efficiency [73,74]. The simulation results showed that the COD

_{Cr}at the outflow was significantly higher than the values set out in the regulation. The oxygen released from the roots was below the volume needed for the aerobic degradation of the O

_{2}-demanding molecule. Higher concentrations of organic pollutants in the effluent may be caused by higher concentrations of these pollutants at the inlet to the facility [75]. The fats and oils used for frying food, leftover protein- and carbohydrate-rich foods, chemicals, detergents, soaps, and shampoos discharged from households can result in higher concentrations of hard-to-degrade compounds in the effluent and can hinder their decomposition by microorganisms [59]. The temperature is yet another factor that can affect the biological processes taking place in the bed and can affect COD

_{Cr}reduction. A drop in temperature below 5 degrees reduces the efficiency of organic compound decomposition in a vertical flow bed [76]. A possible solution for increasing the COD

_{Cr}removal efficiency in the analysed CW system is the artificial oxygenation of the bed, as demonstrated by [77]. Another way to increase the removal efficiency of hardly biodegradable compounds is to recirculate treated wastewater back into the system, which is supposed to ensure increased aerobic bacterial activity through extensivereactions between pollutants and microorganisms [72,78]. The use of Monte-Carlo simulations as a tool for assessing the reliability of the operation of VF-CW systems is a new concept. To the best of our knowledge, there are currently no published studies that would allow comparison with our results. To date, this method has been used in the work of [25,69], where the authors simulated the values of pollution indicators for collective wastewater treatment plants. In both cited works, the authors stated unequivocally that the Monte-Carlo method is an effective tool for assessing the reliability of the operation of the analysed facilities and that it allows the modelling of wastewater quality parameters at different stages of treatment.

## 4. Conclusions

_{5}, COD

_{Cr}, and TSS) for raw and treated wastewater were characterised based on the acquired data. Then, the fit of the theoretical distributions to the empirical distributions was determined for the random variables. The fit was analysed for 10 probability distributions. The fit of the theoretical distributions to the empirical distributions was assessed using A–D statistics, and the best-fitting distributions were selected using the PBIAS criterion. For the raw wastewater, the theoretical distributions that best fit the empirical distributions are as follows: the generalised distribution of extreme values for BOD

_{5}, the Gaussian distribution for COD

_{Cr}, and the Log-normal distribution for TSS. For the treated wastewater, the Log-normal distribution is best-fitting for BOD

_{5}and COD

_{Cr}, while the Half-normal distribution is best-fitting for TSS. The simulated values of the pollutant indicators were analysed in terms of their likely variations. The results indicate that there may be problems with the reduction of COD

_{Cr}organic compounds at the studied wastewater treatment facility. The Monte-Carlo simulation method used in this study is a useful tool for modelling the operational reliability of a VF-CW. However, it is important to identify the correct theoretical distribution for the analysed random variables to allow new data to be generated while maintaining the current correlation structures between the random variables. The values generated in this way can be helpful for evaluating the operation of the treatment plant and preparing different scenarios for its operation. The prediction of changes in the quality of wastewater flowing into and effluent leaving the treatment facility allows necessary changes in or reconstruction of existing elements of the treatment plant to be planned, which is related to the rational planning of expenses and work to be incurred by the owner. The presented Monte-Carlo simulation method should be implemented to assess the reliability of the operations of other types of domestic wastewater treatment plants, e.g., horizontal-flow constructed wetlands, hybrid wastewater treatment plants, wastewater treatment plants using activated sludge technology, sequencing batch reactor (SBR) wastewater treatment plants, and wastewater treatment plants with ground filters, in terms of removing pollutants from not only the organic group and total suspended solids but also the total nitrogen and total phosphorus. The obtained results of the analyses would be a valuable source of information for future wastewater treatment facility operators in terms of the reliability of the operation of these plants in different scenarios.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Steer, D.; Fraser, L.; Boddy, B.; Seibert, B. Efficiency of small constructed wetlands for subsurface treatment of single-family domestic effluent. Ecol. Eng.
**2002**, 18, 429–440. [Google Scholar] [CrossRef] - Brix, H.; Arias, C.A. The use of vertical flow constructed wetlands for on-site treatment of domestic wastewater: New Danish guidelines. Ecol. Eng.
**2005**, 25, 491–500. [Google Scholar] [CrossRef] - Seo, D.C.; DeLaune, R.D.; Park, W.Y.; Lim, J.S.; Seo, J.Y.; Lee, J.; Cho, J.S.; Heo, J.S. Evaluation of a hybrid constructed wetland for treating domestic sewage from individual housing units surrounding agricultural villages in South Korea. J. Environ. Monit.
**2009**, 11, 134–144. [Google Scholar] [CrossRef] [PubMed] - Mikosz, J. Wastewater management in small communities in Poland. Desaliation Water Treat.
**2013**, 51, 2461–2466. [Google Scholar] [CrossRef] - Jóźwiakowski, K.; Mucha, Z.; Generowicz, A.; Baran, S.; Bielińska, J.; Wójcik, W. The use of multi-criteria analysis for selection of technology for a household WWTP compatible with sustainable development. Arch. Environ. Prot.
**2015**, 41, 76–82. [Google Scholar] [CrossRef] - Pawełek, J.; Bugajski, P. The development of household wastewater treatment plants in Poland—Advantages and disadvantages. Acta Sci. Pol. Form. Circumiectus
**2017**, 16, 3–14. (In Polish) [Google Scholar] [CrossRef] - GUS. Municipal Infrastructure in 2020; Rocznik Statystyczny Rzeczypospolitej Polskiej; Główny Urząd Statystyczny: Warszawa, Poland, 2021. (In Polish)
- Mucha, Z.; Mikosz, J. Rational application of small wastewater treatment plants according to sustainability criteria. Czas. Tech. Sr.
**2009**, 106, 91–100. (In Polish) [Google Scholar] - Martín, I.; Betancort, J.R.; Pidre, J.R. Contribution of non-conventional technologies for sewage treatment to improve the quality of bathing waters (ICREW project). Desalination
**2007**, 215, 82–89. [Google Scholar] [CrossRef] - Bugajski, P.; Kaczor, G. The assessment of working sewage treatment plant at Wadowice before the modernization. Przemysł Chem.
**2008**, 87, 44–426. (In Polish) [Google Scholar] - Orlik, T.; Jóźwiakowski, K. Evaluation of two BATEX wastewater treatment plants with a filtration outlet. Inżynieria Rol.
**2003**, 3, 109–119. (In Polish) [Google Scholar] - Keffala, C.; Ghrabi, A. Nitrogen and bacterial removal in constructed wetlands treating domestic waste water. Desalination
**2005**, 185, 383–389. [Google Scholar] [CrossRef] - De Filippis, L.F. Role of Phytoremediation in Radioactive Waste Treatment. In Soil Remediation and Plants; Academic Press: San Diego, CA, USA, 2015; pp. 207–254. [Google Scholar]
- Yeh, T.Y.; Chou, C.C.; Pan, C.T. Heavy metal removal within pilot-scale constructed wetlands receiving river water contaminated by confined swine operations. Desalination
**2009**, 249, 368–373. [Google Scholar] [CrossRef] - Sheoran, A.S.; Sheoran, V. Heavy metal removal mechanism of acid mine drainage in wetlands: A critical review. Miner. Eng.
**2006**, 19, 105–116. [Google Scholar] [CrossRef] - Arivoli, A.; Mohanraj, R.; Seenivasan, R. Application of vertical flow constructed wetland in treatment of heavy metals from pulp and paper industry wastewater. Environ. Sci. Pollut. Res.
**2015**, 22, 13336–13343. [Google Scholar] [CrossRef] - Hendy, I.; Zelenakova, M.; Pietrucha-Urbanik, K.; Salama, Y.; Abu-hashim, M. Decentralized Constructed Wetlands for Wastewater Treatment in Rural and Remote Areas of Semi-arid Regions. Water
**2023**, 15, 2281. [Google Scholar] [CrossRef] - Haynes, R.J. Use of industrial wastes as media in constructed wetlands and filter beds—Prospects for removal of phosphate and metals from wastewater streams. Crit. Rev. Environ. Sci. Technol.
**2015**, 45, 1041–1103. [Google Scholar] [CrossRef] - Hua, T.; Haynes, R.J.; Zhou, Y.F.; Boullemant, A.; Chandrawana, I. Potential for use of industrial waste materials as filter media for removal of Al, Mo, As, V and Ga from alkaline drainage in constructed wetlands—Adsorption studies. Water Resour.
**2015**, 71, 32–41. [Google Scholar] [CrossRef] [PubMed] - Młyński, D.; Chmielowski, K.; Młyńska, A. The assesment of the efficency and stability of work sewage treatment plant in Zabajka. Inżynieria Ekol.
**2016**, 47, 123–130. (In Polish) [Google Scholar] [CrossRef] - Olyaei, M.A.; Karamouz, M.; Asce, F.; Farmani, R. Framework for assessing flood reliability and resilience of wastewater treatment plants. J. Environ. Eng.
**2018**, 144, 1–14. [Google Scholar] [CrossRef] - Bugajski, P.; Almeida, M.A.A.; Kurek, K. Reliablity of sewage treatment plants processing sewage from school buildings located in non-urban areas. Infrastruct. Ecol. Rural. Areas
**2016**, 4, 1547–1557. [Google Scholar] - Marzec, M. Reliability of removal of selected pollutants in different technological solutions of household wastewater treatment plants. J. Water Land Dev.
**2017**, 35, 141–148. [Google Scholar] [CrossRef] - Jóźwiakowski, K.; Bugajski, P.; Mucha, Z.; Wójcik, W.; Jucherski, A.; Nastwany, M.; Siwiec, T.; Mazur, A.; Obroślak, R.; Gajewska, M. Reliability and efficiency of pollution removal during long-term operation of a one-stage constructed wetland system with horizontal flow. Sep. Purif. Technol.
**2017**, 187, 60–66. [Google Scholar] [CrossRef] - Młyński, D.; Bugajski, P.; Młyńska, A. Application of the mathematical simulation methods for the assessment of the wastewater treatment plant operation work reliability. Water
**2019**, 11, 873. [Google Scholar] [CrossRef] - Mitrenga, D. Methodological Basis of the Monte Carlo Stochastic Simulation. Stud. Ekon. Uniw. Ekon. W Katowicach
**2014**, 204, 164–180. (In Polish) [Google Scholar] - Ferson, S. What Monte Carlo methods cannot do. Hum. Ecol. Risk Assess. Int. J.
**1996**, 4, 990–1007. [Google Scholar] [CrossRef] - PN-C-04620-02:1974; Water and Sewage—Sampling—General Provision and Scope of the Standard. Polski Komitet Normalizacji, Miar i Jakości: Warszawa, Poland, 1975. (In Polish)
- PN-EN 25667-2:1999; Water Quality—Sampling—Guidance on Sampling Techniques. Polski Komitet Normalizacyjny: Warszawa, Poland, 1999. (In Polish)
- Water Quality—Determination of Biochemical Oxygen Demand after n Days (BOD)—Part 1: Dilution and Vaccination Method with the Addition of Allythiourea; Polski Komitet Normalizacyjny: Warszawa, Poland, 2002. (In Polish)
- Water Quality—Determination of the Chemical Oxygen Demand Index (ST-COD)—Small-Scale Sealed-Tube Method; Polski Komitet Normalizacyjny: Warszawa, Poland, 2005. (In Polish)
- Water Quality—Determination of Suspended Solids—Method by Filtration trough Filters; Polski Komitet Normalizacyjny: Warszawa, Poland, 2007. (In Polish)
- Regulation of the Minister of Environment of November 18, 2014 Laying down Conditions for the Introduction of Sewage into Water or Soil and Substances Particularly Harmful to the Aquatic Environments (No 2014 Item 1800). Available online: https://isap.sejm.gov.pl/isap.nsf/download.xsp/WDU20140001800/O/D20141800.pdf (accessed on 10 September 2023). (In Polish)
- Rozporządzenie Ministra Gospodarki Morskiej i Żeglugi Śródlądowej z dnia 12 lipca 2019 r.w Sprawie Substancji Szczególnie Szkodliwych dla Środowiska Wodnego Oraz Warunków, Jakie Należy Spełnić przy Wprowadzaniu do wód lub do Ziemi ścieków, a Także przy Odprowadzaniu wód Opadowych lub Roztopowych do wód lub do Urządzeń Wodnych. Regulation of the Minister of Maritime Economy and Inland Navigation of 12 July 2019 on Substances Particularly Harmful to the Aquatic Environment and Conditions to Be Met during Sewage Discharge into the Water or into the Ground and during Rainwater or Snowmelt Discharge into the Water or into the Water Devices. Available online: https://isap.sejm.gov.pl/isap.nsf/download.xsp/WDU20190001311/O/D20191311.pdf (accessed on 10 September 2023). (In Polish)
- Jagiełło, R.; Beker, C.; Jagodziński, A.M. Goodness of fit evaluation of the breast height diameter distributions of beech stands differing in age with selected theoretical distributions. Sylwan
**2016**, 160, 107–119. (In Polish) [Google Scholar] - Glickman, T.S.; Xu, F. The distribution of the product of two random variables. Stat. Probab. Lett.
**2008**, 78, 2821–2826. [Google Scholar] [CrossRef] - Wang, X. Vehicle Noise and Vibration Refinement; Woodhead Publishing: Cambridge, UK, 2010. [Google Scholar]
- Zoran, D.; Weiss, Y. From learning models of natural image patches to whole image restoration. In Proceedings of the 2011 International Conference on Computer Vision, Barcelona, Spain, 6–13 November 2011; pp. 479–486. [Google Scholar]
- Yu, G.; Sapiro, G.; Mallat, S. Solving inverse problems with piecewise linear estimators: From gaussian mixture models to structured sparsity. IEEE Trans. Image Process.
**2012**, 21, 2481–2499. [Google Scholar] - Wałęga, A.; Rutkowska, A.; Policht-Latawiec, A. Sensitivity of beta and Weibull synthetic unit hydrographs to input parameter changes. Pol. J. Environ. Stud.
**2014**, 1, 221–229. [Google Scholar] - Alam, M.A.; Emuro, K.; Farnham, C.; Yuan, J. Best-fit probability distributions and return periods for maximum monthly rainfall in Bangladesh. Climate
**2018**, 6, 9–25. [Google Scholar] [CrossRef] - Zeng, X.; Wang, D.; Wu, J. Evaluating the three methods of goodness of fit test for frequency analysis. J. Risk Anal. Crisis Response
**2015**, 5, 178–187. [Google Scholar] [CrossRef] - Evans, D.L.; Drew, J.H.; Leemis, L.M. The distribution of the Kolmogorov–Smirnov, Cramer–von Mises, and Anderson–Darling test statistics for exponential populations with estimated parameters. Comput. Probab. Appl.
**2017**, 247, 165–190. [Google Scholar] - Jäntschi, L.; Bolboacă, S.D. Computation of probability associated with Anderson–Darling statistic. Mathematics
**2018**, 6, 88–104. [Google Scholar] [CrossRef] - Ajmal, M.; Waseem, M.; Kim, D.; Kim, T.W. A Pragmatic Slope-Adjusted Curve Number Model to Reduce Uncertainty in Predicting Flood Runoff from Steep Watersheds. Water
**2020**, 12, 1469. [Google Scholar] [CrossRef] - Gupta, H.V.; Sorooshian, S.; Yapo, P.O. Status of automatic calibration for hydrologic models: Comparison with multilevel expert calibration. J. Hydrol. Eng.
**1999**, 4, 135–143. [Google Scholar] [CrossRef] - Moriasi, D.N.; Arnold, J.G.; Van Liew, M.W.; Bingner, R.L.; Harmel, R.D.; Veith, T.L. Model evaluation guidelines for systematic quantification of accuracy in watershed simulations. Trans. ASABE
**2007**, 50, 885–900. [Google Scholar] [CrossRef] - Yuan, Y.; Nie, J.; McCutcheon, S.C.; Taguas, E.V. Initial abstraction and curve numbers for semiarid watersheds in south eastern Arizona. Hydrol. Process.
**2014**, 28, 774–783. [Google Scholar] [CrossRef] - Archibald, J.A.; Buchanan, B.; Fuka, D.R.; Georgakakos, C.B.; Lyon, S.W.; Walter, M.T. A simple, regionally parameterized model for predicting nonpoint source areas in the northeastern US. J. Hydrol. Reg. Stud.
**2014**, 1, 74–91. [Google Scholar] [CrossRef] - Donigian, A.S.; Imhoff, J.C.; Bicknell, B.R. Predicting water quality resulting from agricultural nonpoint-source pollution via simulation: HSPF. In Agricultural Management and Water Quality; Iowa State University Press: Ames, IA, USA, 1983; pp. 200–249. [Google Scholar]
- Singh, J.; Knapp, H.V.; Arnald, J.G.; Demissie, M. Hydrologic modeling of the Iroquois River watershed using HSPF and SWAT. J. Am. Water Resour. Assoc.
**2004**, 41, 343–360. [Google Scholar] [CrossRef] - Van Liew, M.W.; Arnold, J.G.; Garbrecht, J.D. Hydrologic simulation on agricultural watersheds: Choosing between two models. Trans. ASAE
**2003**, 46, 1539–15512. [Google Scholar] [CrossRef] - Halton, J.H. A retrospective and prospective survey of the Monte Carlo method. SIAM Rev.
**1970**, 12, 1–63. [Google Scholar] [CrossRef] - Schauberger, G.; Piringer, M.; Baumann-Stanzer, K.; Knauder, W.; Petz, E. Use of a Monte Carlo technique to complete a fragment set of H2S emission rates from a wastewater treatment plant. J. Hazard. Mater.
**2013**, 263, 694–701. [Google Scholar] [CrossRef] [PubMed] - Hendren, C.O.; Badireddy, A.R.; Casman, E.; Wiesner, M.R. Modeling nanomaterial fate in wastewater treatment: Monte Carlo simulation of silver nanoparticles (nano-Ag). Sci. Total Environ.
**2013**, 449, 418–425. [Google Scholar] [CrossRef] [PubMed] - Barton, L.E.; Auffan, M.; Durenkamp, M.; McGrath, S.; Bottero, J.; Wiesner, M.R. Monte Carlo simulations of the transformation and removal of Ag, TiO
_{2}, and ZnO nanoparticles in wastewater treatment and land application of biosolids. Sci. Total Environ.**2015**, 511, 535–543. [Google Scholar] [CrossRef] [PubMed] - Andraka, D.; Dzienis, L. Modeling of risk in the operation of wastewater treatment plants. Rocz. Ochr. Sr.
**2013**, 15, 1111–1125. [Google Scholar] - Kurek, K.; Bygajski, P.; Operacz, A.; Młyński, D.; Wałęga, A. Technological reliability of sewage treatment plant with the Pomiltek Mann type bioreactor. J. Water Land Dev.
**2020**, 46, 146–152. [Google Scholar] - Kaczor, G. Concentrations of the pollutants in the sewage drained from the rural sewerage systems in lesser Poland voivodeship. Infrastrukt. Ekol. Teren. Wiej.
**2009**, 9, 97–104. (In Polish) [Google Scholar] - Jóźwiakowski, K.; Bugajski, P.; Kurek, K.; Carvalho, F.; Almeida, A.; Siwiec, T.; Borowski, G.; Czekała, W.; Dach, J.; Gajewska, M. The efficiency and technological reliability of biogenic compounds removal during long-term operation of a one-stage subsurface horizontal flow constructed wetland. Sep. Purif. Technol.
**2018**, 202, 216–226. [Google Scholar] [CrossRef] - Jucherski, A.; Nastawny, M.; Walczowski, A.; Jóźwiakowski, K.; Gajewska, M. Assessment of the technological reliability of a hybrid constructed wetland for wastewater treatment in a mountain eco-tourist farm in Poland. Water Sci. Technol.
**2017**, 75, 2649–2658. [Google Scholar] [CrossRef] - Zhang, L.; Zhang, L.; Liu, Y.D.; Shen, Y.W.; Liu, H.; Xiong, Y. Effect of limited artificial aeration on constructed wetland treatment of domestic wastewater. Desalination
**2010**, 250, 915–920. [Google Scholar] [CrossRef] - Wałęga, A.; Chmielowski, K.; Młyński, D. Influence of the Hybrid Sewage Treatment Plant’s Exploitation on Its Operation Effectiveness in Rural Areas. Sustainability
**2018**, 10, 2689. [Google Scholar] [CrossRef] - Lu, S.; Pei, L.; Bai, X. Study on method of domestic wastewater treatment through new-type multi-layer artificial wetland. Int. J. Hydrogen Energy
**2015**, 40, 11207–11214. [Google Scholar] [CrossRef] - Kuschk, P.; Wiener, A.; Kappelmeyer, U.; Weissbrodt, E.; Kästner, M.; Stottmeister, U. Annual cycle of nitrogen removal by a pilot-scale subsurface horizontal flow in a constructed wetland under moderate climate. Water Resour.
**2003**, 37, 4236–4242. [Google Scholar] [CrossRef] [PubMed] - Kadlec, R.H.; Reddy, K. Temperature Effects in Treatment Wetlands. Water Environ. Res.
**2001**, 73, 543–557. [Google Scholar] [CrossRef] - Stein, W.; Keblis, M. A new method to simulate the triangular distribution. Math. Comput. Model.
**2009**, 49, 1143–1147. [Google Scholar] [CrossRef] - Pereira, E.J.S.; Pinho, J.T.; Galhardo, M.A.B.; Macêdo, W.M. Methodology of risk analysis by Monte Carlo Method applied to power generation with renewable energy. Renew. Energy
**2014**, 69, 347–355. [Google Scholar] [CrossRef] - Młyński, D.; Młyńska, A.; Chmielowski, K.; Pawełek, J. Investigation of the Wastewater Treatment Plant Processes Efficiency Using Statistical Tools. Sustainability
**2020**, 12, 10522. [Google Scholar] [CrossRef] - Vymazal, J. The use of sub-surface constructed wetlands for wastewater treatment in the Czech Republic: 10 years experience. Ecol. Eng.
**2002**, 18, 633–646. [Google Scholar] [CrossRef] - Abdelhakeem, S.G.; Aboulroos, S.A.; Kamel, M.M. Performance of a vertical subsurface flow constructed wetland under different operational conditions. J. Adv. Res.
**2016**, 7, 803–814. [Google Scholar] [CrossRef] - Ilyas, H.; Masih, I. The performance of the intensified constructed wetlands for organic matter and nitrogen removal: A review. J. Environ. Manag.
**2017**, 198, 372–383. [Google Scholar] [CrossRef] - Sehar, S.; Sumera, S.; Naeem, I.; Perveen, N.; Ahmed, S. A comparative study of macrophytes influences on wastewater treatment through subsurface flow hybrid constructed wetland. Ecol. Eng.
**2015**, 81, 62–69. [Google Scholar] [CrossRef] - Zhu, D.; Suna, C.; Zhang, H.; Wub, Z.; Jia, B.; Zhang, Y. Roles of vegetation, flow type and filled depth on livestock wastewater treatment through multi-level mineralized refuse-based constructed wetlands. Ecol. Eng.
**2012**, 39, 7–15. [Google Scholar] [CrossRef] - Gajewska, M. Złoża hydrofitowe z pionowym przepływem ścieków charakterystyka procesów i zastosowań. Wydaw. Pol. Akad. Nauk.
**2019**, 150, 1–309. Available online: https://kis.pan.pl/images/stories/pliki/pdf/Monografie/Monografia-M-Gajewska-vol-150.pdf (accessed on 10 September 2023). (In Polish). - Soroko, M. Annual Treatment Of Wastewater From Small Fruit And Vegetable Processing Plant In The Reed Bed System with Vertical Flow. Woda Sr. Obsz. Wiej.
**2011**, 11, 289–298. [Google Scholar] - Wu, H.; Fan, J.; Zhang, J.; Ngo, H.H.; Guo, W.; Hu, Z.; Liang, S. Decentralized domestic wastewater treatment using intermittently aerated vertical flow constructed wetlands: Impact of influent strengths. Bioresour. Technol.
**2015**, 176, 163–168. [Google Scholar] [CrossRef] - Wu, S.; Kuschk, P.; Brix, H.; Vymazal, J.; Dong, R. Development of constructed wetlands in performance intensifications for wastewater treatment: A nitrogen and organic matter targeted review. Water Resour.
**2014**, 57, 40–55. [Google Scholar] [CrossRef]

**Figure 2.**The location of Gawłówek village against the background of the Lesser Poland Voivodeship in Poland.

**Figure 4.**The values of the pollutant indicators for raw and treated sewage and corresponding permissible levels (

**a**) BOD

_{5}; (

**b**) COD

_{Cr}; (

**c**) TSS.

**Figure 5.**Q–Q plots of best-fit theoretical distributions of the random-variable distributions for indicators for raw sewage (

**a**) BOD

_{5}(GEV distribution); (

**b**) COD

_{Cr}(Gaussian distribution); (

**c**) TSS (Log-normal distribution).

**Figure 6.**Q–Q plot of best-fit statistical distributions of indicators for treated wastewater (

**a**) BOD

_{5}(Log-normal distribution); (

**b**) COD

_{Cr}(Log-normal distribution); (

**c**) TSS (Half-normal distribution).

**Table 1.**Results of preliminary data analysis for pollutant indicators BOD

_{5}, COD

_{Cr}, and TSS for inflow and outflow of sewage.

Type of Indicator | Unit | Descriptive Statistics | |||||
---|---|---|---|---|---|---|---|

Raw Sewage | |||||||

Min | Max | Mean | Median | s | C_{s} | ||

BOD_{5} | mgO_{2}∙dm^{−3} | 172.0 | 1700.0 | 760.7 | 780.0 | 10.6 | 0.79 |

COD_{Cr} | mgO_{2}∙dm^{−3} | 848.0 | 2041.0 | 1314.1 | 1257.0 | 245.9 | 0.19 |

TSS | mg∙dm^{−3} | 125.0 | 1084.0 | 393.2 | 343.0 | 184.2 | 0.47 |

Treated Sewage | |||||||

BOD_{5} | mgO_{2}∙dm^{−3} | 1.0 | 48.0 | 13.3 | 10.0 | 10.6 | 0.79 |

COD_{Cr} | mgO_{2}∙dm^{−3} | 114.0 | 304.0 | 162.0 | 150.0 | 41.2 | 0.25 |

TSS | mg∙dm^{−3} | 2.4 | 96.0 | 21.5 | 16.0 | 18.8 | 0.87 |

_{s}—coefficient of variation.

**Table 2.**Results for fitting theoretical distributions to empirical distributions of random variables.

Distribution | BOD_{5} | COD_{Cr} | TSS | |||
---|---|---|---|---|---|---|

A–D | p | A–D | p | A–D | p | |

Raw sewage | ||||||

Johnson SB | 0.621 | 0.628 | 0.315 | 0.927 | N/A | N/A |

Weibull | 1.045 | 0.334 | 0.701 | 0.557 | 1.939 | 0.099 |

Gaussian | 0.344 | 0.901 | 0.313 | 0.928 | 0.233 | 0.979 |

GEV | 0.945 | 0.387 | 0.322 | 0.921 | 0.353 | 0.893 |

Half-normal | 7.914 | 0.000 | 13.698 | 0.000 | 5.948 | 0.001 |

Log-normal | 1.692 | 0.137 | 0.322 | 0.921 | 0.584 | 0.662 |

Pareto | 5.689 | 0.001 | 11.054 | 0.000 | 5.295 | 0.002 |

Normal | 0.859 | 0.440 | 0.427 | 0.821 | 2.483 | 0.051 |

Triangular | 4.399 | 0.006 | 2.863 | 0.032 | 6.563 | 0.001 |

Rayleigh | 3.117 | 0.024 | 8.236 | 0.000 | 2.153 | 0.076 |

Treated sewage | ||||||

Johnson SB | 0.185 | 0.994 | 0.106 | 1.000 | 0.802 | 0.479 |

Weibull | 0.223 | 0.982 | 1.612 | 0.152 | 0.411 | 0.837 |

Gaussian | 0.287 | 0.947 | 0.195 | 0.992 | 0.433 | 0.814 |

GEV | 0.296 | 0.941 | 0.209 | 0.988 | 0.208 | 0.988 |

Half-normal | 0.388 | 0.860 | 11.495 | 0.000 | 0.934 | 0.393 |

Log-normal | 0.394 | 0.854 | 0.649 | 0.602 | 0.203 | 0.990 |

Pareto | 0.473 | 0.773 | 7.641 | 0.000 | 0.860 | 0.439 |

Normal | 1.541 | 0.167 | 1.350 | 0.217 | 2.314 | 0.062 |

Triangular | 4.667 | 0.004 | 6.915 | 0.000 | 9.779 | 0.000 |

Rayleigh | 4.821 | 0.003 | 6.021 | 0.001 | 6.603 | 0.001 |

Distribution | PBIAS [%] | |||||
---|---|---|---|---|---|---|

BOD_{5} | COD_{Cr} | TSS | ||||

Raw | Treated | Raw | Treated | Raw | Treated | |

Johnson SB | −1.276 | −0.185 | −0.026 | 0.184 | - | −2.551 |

Weibull | 0.521 | −0.506 | 0.783 | 1.341 | 1.922 | 1.014 |

Gaussian | 0.648 | 4.318 | −0.089 * | 0.045 | −0.197 | 8.392 |

GEV | −1.648 * | −0.629 | −0.083 | 0.218 | −0.269 | −0.550 |

Half-normal | - | 1.106 | - | 27.211 | - | −4.032 * |

Log-normal | −1.097 | −0.987 * | −0.038 | −0.132 * | −0.359 * | −0.422 |

Pareto | - | 8.085 | - | - | - | 11.007 |

Normal | −0.048 | 38.029 | −0.036 | 0.038 | 2.575 | 42.708 |

Rayleigh | - | - | - | - | 6.071 | - |

**Table 4.**Values of pollutant removal efficiency indicators for a VF-CW system according to Monte-Carlo simulations.

Reliability Coefficient | Pollutant Indicator | ||
---|---|---|---|

BOD_{5} | COD_{Cr} | TSS | |

CR | 0.33 | 1.10 | 0.46 |

R | 0.98 | 0.49 | 0.91 |

R_{e} | 0.00 | 1.00 | 0.00 |

η | 0.98 | 0.82 | 0.93 |

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## Share and Cite

**MDPI and ACS Style**

Migdał, K.; Jóźwiakowski, K.; Czekała, W.; Śliz, P.; Tavares, J.M.R.; Almeida, A.
Application of the Monte-Carlo Method to Assess the Operational Reliability of a Household-Constructed Wetland with Vertical Flow: A Case Study in Poland. *Water* **2023**, *15*, 3693.
https://doi.org/10.3390/w15203693

**AMA Style**

Migdał K, Jóźwiakowski K, Czekała W, Śliz P, Tavares JMR, Almeida A.
Application of the Monte-Carlo Method to Assess the Operational Reliability of a Household-Constructed Wetland with Vertical Flow: A Case Study in Poland. *Water*. 2023; 15(20):3693.
https://doi.org/10.3390/w15203693

**Chicago/Turabian Style**

Migdał, Karolina, Krzysztof Jóźwiakowski, Wojciech Czekała, Paulina Śliz, Jorge Manuel Rodrigues Tavares, and Adelaide Almeida.
2023. "Application of the Monte-Carlo Method to Assess the Operational Reliability of a Household-Constructed Wetland with Vertical Flow: A Case Study in Poland" *Water* 15, no. 20: 3693.
https://doi.org/10.3390/w15203693