# Advanced Graphical–Analytical Method of Pipe Tank Design Integrated with Sensitivity Analysis for Sustainable Stormwater Management in Urbanized Catchments

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Calculation Algorithm Concept

_{i}) reaching the input layer were multiplied by the values of weights w

_{ij}. The obtained sums were transformed using the linear or nonlinear activation function (f) and then were transmitted to the neuron(s) of the output layer. The disadvantage of the neural network method was that the initial values of the weights affected the learning process, which may create problems with finding the global minimum of a function [74].

#### 2.2. Differential Equation of Stormwater Volume Balance

_{or}. For the assumptions mentioned above, the variability in the amount of stormwater flowing into the tanks during the growth phase (t

_{p}) and decline phase (t

_{op}) can be described with the following dependency (see also the graphical presentation in Figure 3):

^{3}s

^{−1}), I

_{m}is the peak flow to the tank (m

^{3}s

^{−1}), t

_{p}is the peak flow time (min), t

_{op}is the time of water lowering, and t is the time (min).

_{z}is the geometric parameter of the tank described as L

_{ret}·D

_{ret}

^{0.60}(where L

_{ret}is the length of the retention chamber (m), and D

_{ret}is the diameter of the retention chamber (m)), B

_{or}is the parameter describing the outlet orifice (expressed as μ·π·4

^{−1}·D

_{out}

^{2}·(2g)

^{0.50}), and V is the tank volume variable in time t (m

^{3}).

**Figure 2.**Scheme of a pipe tank. Where: D

_{ret}is the diameter of the retention chamber, L

_{ret}is the length of the retention chamber (m), D

_{or}is the diameter of the outlet orifice from the pipe chamber (m), D

_{in}is the diameter of the inlet channel (m), D

_{out}is the diameter of the outlet chamber (m), h is the filling level of the retention chamber (m), i is the longitudinal slope of the pipe chamber (-), U

_{av}is the average flow velocity of the stormwater in the retention chamber (ms

^{−1}), I

_{in}is the stormwater discharge flowing into the tank (m

^{3}s

^{−1}), I

_{out}is the stormwater discharge flowing out of the retention chamber (m

^{3}s

^{−1}), 1 is the inlet channel, 2 is the inlet chamber, 3 is the retention chamber, 4 is the safety overflow, and 5 is the outlet chamber.

#### 2.3. Dimensionless Differential Equation of Volume Balance

- -
- Dimensionless time from the start (t*):

- -
- Dimensionless flow rate (I*):

- -
- Dimensionless tank volume (V*):

_{c}is the total inflow hydrograph volume (m

^{3}), V

_{c}= I

_{max}(t

_{p}+ t

_{op})/2.

_{ret}. When the retention chamber is completely filled, the maximal flow rate through the outlet orifice equals:

_{c}= 900 ÷ 45,000 m

^{3}, a peak flow of (I

_{m}= 1.5 ÷ 7.5 m

^{3}s

^{−1}), and peak flow time t

_{p}= 10 ÷ 150 min. For the assumed hydrographs, the obtained values of the ω parameter ranged from 0.15 to 1.00. This assumption taken for the calculation actually meant that the peak time t

_{p}was assumed to be maximal at one-half of the end hydrograph time. That assumption was related to the fact that, to determine the nomogram with the widest possible range of parameters enabling all possible cases were taken into account, even the unlikely ones, to conditions of maximal tank capacity. The length of the retention chamber was calculated assuming β = 0.1–0.9 and the diameter of the outlet orifice D

_{or}= 0.25 ÷ 1.25 m. The differential equation V

_{(h)}= f

_{(t)}was solved using the explicit Runge–Kutta method, in relation to retention chamber length L

_{ret}[79].

#### 2.4. Local Sensitivity Analysis

_{i}which is the product of the standard deviation σ

_{Xi}and the derivative of partial primary function ∂y/∂x

_{i}. The number of sensitivity functions that can be created for one primary function is equal to the number of variables (and parameters) in the primary function.

_{tp}

_{,}S

_{Im}, S

_{λ}, S

_{η}

_{,}where

_{λ}and

_{η}are sensitivity coefficients) were highly sensitive, a local sensitivity analysis was employed to assess the influence of the influent hydrograph parameters (t

_{p}, I

_{m}, λ = t

_{op}·t

_{p}

^{−1}) on the length of the retention chamber. The standard deviation (σ

_{Xi}) of the aforementioned variables in the presented methodology was determined using the Monte Carlo method, by sampling 1000 values from continuous uniform distributions, where the independent variables changed within appropriately assumed ranges. In this case, the sensitivity function described with formula (14) illustrates the change to the retention chamber length in relation to the unit change of parameter x

_{i}. On the basis of the above-mentioned dependencies (see Equation (10)), the relationship describing the length of the chamber was determined and, then, the successive partial derivatives were determined. The following sensitivity functions (S) were developed, enabling us to evaluate the effect of the influent hydrograph parameters and η on the length of the retention chamber:

_{Im}= f(I

_{m}, λ), S

_{tp}= f(t

_{p}, λ), S

_{λ}= f(t

_{p}, λ), and S

_{η}= f(t

_{p}, η).

#### 2.5. Influence of Hydrograph Parameters and Outlet Devices on Calculation Errors

_{ret[mod]}is the tank volume obtained basing on numerical simulations (m

^{3}), V

_{ret[nom]}is the tank volume determined based on the proposed nomograms (m

^{3}), L

_{ret[mod]}is the tank length calculated using the simulations (m), L

_{ret[nom]}is the tank length determined based on the proposed nomogram (see Figure 4) (m), F

_{ret}is the transverse cross-section area of the retention chamber (m

^{2}), and ∆L

_{ret}is the absolute error of the predicted tank length.

_{mod}is the the value of the parameter determined on the basis of a simulation using a hydrodynamic model, i.e., the results from SWMM were taken, and the η was directly substituted and determined here based on data from the hydrograph, the diameter of the retention pipe chamber and the diameter of the outlet orifice from the pipe chamber. The range of parameter changed from 0.0328 to 0.7032.

#### 2.6. Modeling of the Errors in the Predicted Tank Length Using Data Mining Methods

## 3. Results and Discussion

#### 3.1. Dependence η = f(ω, β)

^{2}) in Equation (22) ranged from 0.989 to 0.996 for β = 0.1 ÷ 0.9. The determined curves showed that the values of η increased along with β. The η = f(ω, β = const) curves determined in the form of the nomogram indicated that the greater the value of peak flow reduction and influent hydrograph asymmetry coefficients, the higher the increase in the dispersion of the η parameter values for the analyzed wave asymmetry coefficient (ω = 0.2 ÷ 1.0), and thus the higher the prediction error of the tank volume.

**Figure 4.**Influence of the η parameter and wave asymmetry coefficients (ω) on the peak flow reduction coefficient (β). Points correspond to the values acquired from simulations, whereas curves are related to the values obtained by means of Formula (21).

_{Im}= f(I

_{m}, λ) in the analyzed range. It should be noted that in the range I

_{m}= 0.0 ÷ 3.0 m

^{3}s

^{−1}, the S

_{Im}values for t

_{p}= 500 s and t

_{p}= 1500 s changed in the range from −0.75 × 10

^{6}to −0.25 × 10

^{6}, and from −2.5 × 10

^{6}to −0.30 × 10

^{6}, respectively. In turn, for I

_{m}> 3.0 m

^{3}s

^{−1}the changes to S

_{Im}in the above-mentioned ranges did not exceed 10%. The performed calculations (Figure 5b) indicate that the increase in time from t

_{p}= 500 s to t

_{p}= 2000 s for I

_{m}= 2.0 m

^{3}s

^{−1}did not affect the value of the sensitivity index S

_{p}= f(t

_{p}, λ), while a change to the peak flow at its maximum in the range from I

_{m}= 2.0 m

^{3}s

^{−1}to I

_{m}= 4.0 m

^{3}s

^{−1}reduced the sensitivity index S

_{tp}by 14 × 10

^{6}to 8 × 10

^{6}(43%).

_{p}= 500 ÷ 2000 s (Figure 5c) reduced the sensitivity index S

_{λ}= f(t

_{p}, λ). Nevertheless, it should be noted that in the range of I

_{m}= 0.0 ÷ 3.0 m

^{3}s

^{−1}, the values of S

_{λ}for t

_{p}= 500 s and t

_{p}= 1500 s changed from −4200 to −600 and −11,000 to −600, respectively, whereas for I

_{m}> 3.0 m

^{3}s

^{−1}, the change of S

_{λ}= f(t

_{p}, λ) in the above-mentioned ranges did not exceed 10% and 14%, respectively. The performed calculations (Figure 5d) indicated that, for η ≈ 0 in the analyzed range I

_{max}, there was a sharp change in the sensitivity index S

_{η}, whereas for η > η

_{gr}, the values of the sensitivity index S

_{η}were constant and equaled −0.3 × 10

^{9}.

**Figure 5.**Variability of the sensitivity index (

**a**) S

_{Im}= f(I

_{max}= I

_{m}, t

_{p}), (

**b**) S

_{tp}= f(I

_{max}= I

_{m}, t

_{p}), (

**c**) S

_{λ}= f(λ, t

_{p}), (

**d**) S

_{η}= f(η, I

_{max}= I

_{m}).

_{max}, λ) on the reservoir area was not analyzed. In turn, the paper by Guo [66] presented an analytical–graphical method supported by several computational variants. This paper concerned two methodologies: a very precise one allowing errors lower than 2% and another, simplified one, for which the difference between the obtained results and the reference data was at the level of 4%.

#### 3.2. Influence of the Influent Hydrograph Parameters and Tank Characteristics on the Errors in the Prediction of Its Volume

_{V}) committed using the developed graph were determined with Formula (22). Table 1 lists only the minimum and maximum values of ε

_{V}.

_{Vmax}) was observed for β = 0.1 and its extreme values ranged from −7.01% to 3.77%. On the contrary, the greatest error noted was for β = 0.9 and varied from −36.33% to 22.13%. This means that the relative error values related to the tank volume prediction were significant, thus confirming the need for their estimation. In the performed work, ε

_{V}was first determined using the multiple stepwise regression method because the variables with negligible effect on the relative error of the retention chamber volume were omitted during the calculations. Thus, the following formula was obtained:

_{m}), the time of the peak flow (t

_{p}), the diameter of the outlet device (D

_{or}), and the diameter of the retention chamber (D

_{ret}). The parameters of the proposed mathematical models are presented in Table 2, while Figure 6 and Figure 7 show a comparison of the ε

_{v}values, both calculated and determined on the basis of the numerical simulations and the developed nomogram η = f(ω, β). The performed calculations (Table 2) indicated a comparable agreement of the calculations and measurements of the relative errors (ε

_{V}) obtained by the means of an artificial neural network with 6 neurons in the hidden and exponential activation functions, and using genetic programming. This was confirmed by the values of the particular fitting parameters (Table 2) since in the case of multiple regressions, ANN, and GP (ω, β), the correlation coefficient and Akaike information criterion amounted to 0.6851 and −3553; 0.9310 and −3010 as well as 0.9010 and −3236, respectively.

**Figure 8.**Comparison of the error values obtained on the basis of the developed nomogram ε

_{V[mod]}, calculated using multiple regression ε

_{V [prog]}.

**Figure 9.**Comparison of the error values obtained on the basis of the developed nomogram ε

_{V[mod]}, calculated using genetic programming ε

_{V [prog]}.

**Figure 10.**Comparison of the error values obtained on the basis of the developed nomogram ε

_{V[mod]}, calculated using ANN ε

_{V [prog]}.

_{V[mod]}< −0.10.

_{max}= 3.34 m

^{3}/s t = 15 min and V = 15,000 m

^{3}i.e., ω = 3.34 × 15 × 60/15,000 = 0.20 (according Equation (11)).

^{2}) × (19.62 × 3.5)

^{0.5}= 0.10 (according Equation (13)).

_{or}= 0.57 × 0.785 × 0.3

^{2}× (19.62

^{0.5}) = 0.313 (according Equation (13)).

_{ret}= [0.313 × 15 × 60/(3.5

^{0.21}× 15,000

^{0.65}× 0.04)]

^{2.857}= 815.3 m (according Equation (10)).

_{ret}= 0.011 was obtained, which gave a final result of L

_{ret}= 823.5 m.

_{ret}calculation error values are relatively small, and amount to 1.1%.

## 4. Summary

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Arfanuzzaman, M.; Atiq Rahman, A. Sustainable water demand management in the face of rapid urbanization and ground water depletion for social–ecological resilience building. Glob. Ecol. Conserv.
**2017**, 10, 9–22. [Google Scholar] [CrossRef] - Nagypál, V.; Mikó, E.; Hodúr, C. Sustainable Water Use Considering Three Hungarian Dairy Farms. Sustainability
**2020**, 12, 3145. [Google Scholar] [CrossRef] [Green Version] - Özerol, G.; Dolman, N.; Bormann, H.; Bressers, H.; Lulofs, K.; Böge, M. Urban water management and climate change adaptation: A self-assessment study by seven midsize cities in the North Sea Region. Sustain. Cities Soc.
**2020**, 55, 102066. [Google Scholar] [CrossRef] - Kärrman, E. Strategies towards sustainable wastewater management. Urban. Water
**2001**, 3, 63–72. [Google Scholar] [CrossRef] - Garrick, D.; Iseman, T.; Gilson, G.; Brozovic, N.; O’Donnell, E.; Matthews, N.; Miralles-Wilhelm, F.; Wight, C.; Young, W. Scalable solutions to freshwater scarcity: Advancing theories of change to incentivise sustainable water use. Water Secur.
**2020**, 9, 100055. [Google Scholar] [CrossRef] - Willuweit, L.; O’Sullivan, J.J. A decision support tool for sustainable planning of urban water systems: Presenting the Dynamic Urban Water Simulation Model. Water Res.
**2013**, 47, 7206–7220. [Google Scholar] [CrossRef] - Paithankar, D.N.; Taji, S.G. Investigating the hydrological performance of green roofs using storm water management model. Mater. Today Proc.
**2020**, 32, 943–950. [Google Scholar] [CrossRef] - Simon, D. Our Common Future: Report of the World Commission on Environment and Development (Book Review). Third World Plan. Rev.
**1987**, 9, 285. [Google Scholar] [CrossRef] - Demuzere, M.; Orru, K.; Heidrich, O.; Olazabal, E.; Geneletti, D.; Orru, H.; Bhave, A.G.; Mittal, N.; Feliu, E.; Faehnle, M. Mitigating and adapting to climate change: Multi-functional and multi-scale assessment of green urban infrastructure. J. Environ. Manag.
**2014**, 146, 107–115. [Google Scholar] [CrossRef] - La Rosa, D.; Pappalardo, V. Planning for spatial equity - A performance based approach for sustainable urban drainage systems. Sustain. Cities Soc.
**2020**, 53, 101885. [Google Scholar] [CrossRef] - Wakode, H.B.; Baier, K.; Jha, R.; Azzam, R. Impact of urbanization on groundwater recharge and urban water balance for the city of Hyderabad, India. Int. Soil Water Conserv. Res.
**2018**, 6, 51–62. [Google Scholar] [CrossRef] - Tapia Silva, F.O.; Wehrmann, A.; Henze, H.-J.; Model, N. Ability of plant-based surface technology to improve urban water cycle and mesoclimate. Urban. For. Urban. Green.
**2006**, 4, 145–158. [Google Scholar] [CrossRef] - Paul, M.J.; Meyer, J.L. Streams in the Urban Landscape. Annu. Rev. Ecol. Syst.
**2001**, 32, 333–365. [Google Scholar] [CrossRef] - Mielby, S.; Eriksson, I.; Campbell, D.; De Beer, H.; Bonsor, H.; Le Guern, C.; van der Krogt, R.; Lawrence, D.; Ryzynski, G.; Schokker, J. Opening up the Subsurface for the Cities of Tomorrow. Considering the Access to Subsurface Knowledge-Evaluation of Practices and Techniques; COST Action TU1206 Sub-Urban Report; TU1206-WG2-001; TNO Publications; 2016, p. 119. Available online: http://sub-urban.squarespace.com/s/TU1206-WG2-001-Opening-up-the-subsurface-for-the-cities-of-tomorrow_Summary-Report.pdf (accessed on 13 September 2020).
- Timm, A.; Kluge, B.; Wessolek, G. Hydrological balance of paved surfaces in moist mid-latitude climate—A review. Landsc. Urban. Plan.
**2018**, 175, 80–91. [Google Scholar] [CrossRef] - Wang, S.; Wang, H. Extending the Rational Method for assessing and developing sustainable urban drainage systems. Water Res.
**2018**, 144, 112–125. [Google Scholar] [CrossRef] - Cheng, M.-S.; Coffman, L.S.; Clar, M.L. Low-Impact Development Hydrologic Analysis. In Urban Drainage Modeling, Proceedings of the Specialty Symposium of the World Water and Environmental Resources Congress, Orlando, FL, USA, 20–24 May 2001; American Society of Civil Engineers: Reston, VA, USA, 2001; pp. 659–681. [Google Scholar]
- Dietz, M.E. Low Impact Development Practices: A Review of Current Research and Recommendations for Future Directions. Water Air Soil Pollut.
**2007**, 186, 351–363. [Google Scholar] [CrossRef] - Ahiablame, L.M.; Engel, B.A.; Chaubey, I. Effectiveness of Low Impact Development Practices: Literature Review and Suggestions for Future Research. Water Air Soil Pollut.
**2012**, 223, 4253–4273. [Google Scholar] [CrossRef] - Kaykhosravi, S.; Khan, U.; Jadidi, A. A Comprehensive Review of Low Impact Development Models for Research, Conceptual, Preliminary and Detailed Design Applications. Water
**2018**, 10, 1541. [Google Scholar] [CrossRef] [Green Version] - Khan, U.; Valeo, C.; Chu, A.; He, J. A Data Driven Approach to Bioretention Cell Performance: Prediction and Design. Water
**2013**, 5, 13–28. [Google Scholar] [CrossRef] - Elliott, A.; Trowsdale, S. A review of models for low impact urban stormwater drainage. Environ. Model. Softw.
**2007**, 22, 394–405. [Google Scholar] [CrossRef] - Fletcher, T.D.; Shuster, W.; Hunt, W.F.; Ashley, R.; Butler, D.; Arthur, S.; Trowsdale, S.; Barraud, S.; Semadeni-Davies, A.; Bertrand-Krajewski, J.-L.; et al. SUDS, LID, BMPs, WSUD and more—The evolution and application of terminology surrounding urban drainage. Urban. Water J.
**2015**, 12, 525–542. [Google Scholar] [CrossRef] - Ishaq, S.; Hewage, K.; Farooq, S.; Sadiq, R. State of provincial regulations and guidelines to promote low impact development (LID) alternatives across Canada: Content analysis and comparative assessment. J. Environ. Manag.
**2019**, 235, 389–402. [Google Scholar] [CrossRef] [PubMed] - Ghodsi, S.H.; Zahmatkesh, Z.; Goharian, E.; Kerachian, R.; Zhu, Z. Optimal design of low impact development practices in response to climate change. J. Hydrol.
**2020**, 580, 124266. [Google Scholar] [CrossRef] - Gaffin, S.; Khanbilvardi, R.; Rosenzweig, C. Development of a Green Roof Environmental Monitoring and Meteorological Network in New York City. Sensors
**2009**, 9, 2647–2660. [Google Scholar] [CrossRef] [PubMed] - Abdollahian, S.; Kazemi, H.; Rockaway, T.; Gullapalli, V. Stormwater Quality Benefits of Permeable Pavement Systems with Deep Aggregate Layers. Environments
**2018**, 5, 68. [Google Scholar] [CrossRef] [Green Version] - Šijanec Zavrl, M.; Tanac Zeren, M. Sustainability of Urban Infrastructures. Sustainability
**2010**, 2, 2950–2964. [Google Scholar] [CrossRef] [Green Version] - Sartipi, M.; Sartipi, F. Stormwater retention using pervious concrete pavement: Great Western Sydney case study. Case Stud. Constr. Mater.
**2019**, 11, e00274. [Google Scholar] [CrossRef] - Boogaard, F.; Lucke, T. Long-Term Infiltration Performance Evaluation of Dutch Permeable Pavements Using the Full-Scale Infiltration Method. Water
**2019**, 11, 320. [Google Scholar] [CrossRef] [Green Version] - Saadeh, S.; Ralla, A.; Al-Zubi, Y.; Wu, R.; Harvey, J. Application of fully permeable pavements as a sustainable approach for mitigation of stormwater runoff. Int. J. Transp. Sci. Technol.
**2019**, 8, 338–350. [Google Scholar] [CrossRef] - Zhu, H.; Yu, M.; Zhu, J.; Lu, H.; Cao, R. Simulation study on effect of permeable pavement on reducing flood risk of urban runoff. Int. J. Transp. Sci. Technol.
**2019**, 8, 373–382. [Google Scholar] [CrossRef] - Nordbo, A.; Järvi, L.; Haapanala, S.; Wood, C.R.; Vesala, T. Fraction of natural area as main predictor of net CO 2 emissions from cities. Geophys. Res. Lett.
**2012**, 39, 2012GL053087. [Google Scholar] [CrossRef] - Morakinyo, T.E.; Lam, Y.F.; Hao, S. Evaluating the role of green infrastructures on near-road pollutant dispersion and removal: Modelling and measurement. J. Environ. Manage.
**2016**, 182, 595–605. [Google Scholar] [CrossRef] [PubMed] - Capotorti, G.; Alós Ortí, M.M.; Copiz, R.; Fusaro, L.; Mollo, B.; Salvatori, E.; Zavattero, L. Biodiversity and ecosystem services in urban green infrastructure planning: A case study from the metropolitan area of Rome (Italy). Urban. For. Urban. Green.
**2019**, 37, 87–96. [Google Scholar] [CrossRef] - Bressy, A.; Gromaire, M.-C.; Lorgeoux, C.; Saad, M.; Leroy, F.; Chebbo, G. Efficiency of source control systems for reducing runoff pollutant loads: Feedback on experimental catchments within Paris conurbation. Water Res.
**2014**, 57, 234–246. [Google Scholar] [CrossRef] [Green Version] - Palermo, S.A.; Talarico, V.C.; Turco, M. On the LID systems effectiveness for urban stormwater management: Case study in Southern Italy. IOP Conf. Ser. Earth Environ. Sci.
**2020**, 410, 012012. [Google Scholar] [CrossRef] - Dabas, R.; Kumar, S.; Kumar, M. Applications of Low Impact Development for Managing the Storm Water Surface Runoff in Urban Areas; Springer: Singapore, 2021; pp. 275–283. [Google Scholar]
- Kevern, J.T. Green Building and Sustainable Infrastructure: Sustainability Education for Civil Engineers. J. Prof. Issues Eng. Educ. Pract.
**2011**, 137, 107–112. [Google Scholar] [CrossRef] - Winz, I.; Brierley, G.; Trowsdale, S. Dominant perspectives and the shape of urban stormwater futures. Urban. Water, J.
**2011**, 8, 337–349. [Google Scholar] [CrossRef] - Russo, A.; Escobedo, F.J.; Cirella, G.T.; Zerbe, S. Edible green infrastructure: An approach and review of provisioning ecosystem services and disservices in urban environments. Agric. Ecosyst. Environ.
**2017**, 242, 53–66. [Google Scholar] [CrossRef] - Starzec, M.; Dziopak, J. A Case Study of the Retention Efficiency of a Traditional and Innovative Drainage System. Resources
**2020**, 9, 108. [Google Scholar] [CrossRef] - Piro, P.; Carbone, M.; Morimanno, F.; Palermo, S.A. Simple flowmeter device for LID systems: From laboratory procedure to full-scale implementation. Flow Meas. Instrum.
**2019**, 65, 240–249. [Google Scholar] [CrossRef] - Sepehri, M.; Malekinezhad, H.; Ilderomi, A.R.; Talebi, A.; Hosseini, S.Z. Studying the effect of rain water harvesting from roof surfaces on runoff and household consumption reduction. Sustain. Cities Soc.
**2018**, 43, 317–324. [Google Scholar] [CrossRef] - Soonthornnonda, P.; Christensen, E.R. Source apportionment of pollutants and flows of combined sewer wastewater. Water Res.
**2008**, 42, 1989–1998. [Google Scholar] [CrossRef] - Park, M.-H.; Swamikannu, X.; Stenstrom, M.K. Accuracy and precision of the volume–concentration method for urban stormwater modeling. Water Res.
**2009**, 43, 2773–2786. [Google Scholar] [CrossRef] - Oviedo-Ocaña, E.R.; Dominguez, I.; Ward, S.; Rivera-Sanchez, M.L.; Zaraza-Peña, J.M. Financial feasibility of end-user designed rainwater harvesting and greywater reuse systems for high water use households. Environ. Sci. Pollut. Res.
**2018**, 25, 19200–19216. [Google Scholar] [CrossRef] [Green Version] - Burns, M.J.; Fletcher, T.D.; Duncan, H.P.; Hatt, B.E.; Ladson, A.R.; Walsh, C.J. The performance of rainwater tanks for stormwater retention and water supply at the household scale: An empirical study. Hydrol. Process.
**2015**, 29, 152–160. [Google Scholar] [CrossRef] - Xu, W.; Fletcher, T.; Duncan, H.; Bergmann, D.; Breman, J.; Burns, M. Improving the Multi-Objective Performance of Rainwater Harvesting Systems Using Real-Time Control Technology. Water
**2018**, 10, 147. [Google Scholar] [CrossRef] [Green Version] - Musz-Pomorska, A.; Widomski, M.K.; Gołębiowska, J. Financial Sustainability of Selected Rain Water Harvesting Systems for Single-Family House under Conditions of Eastern Poland. Sustainability
**2020**, 12, 4853. [Google Scholar] [CrossRef] - Suleiman, L.; Olofsson, B.; Saurí, D.; Palau-Rof, L. A breakthrough in urban rain-harvesting schemes through planning for urban greening: Case studies from Stockholm and Barcelona. Urban. For. Urban. Green.
**2020**, 51, 126678. [Google Scholar] [CrossRef] - Melville-Shreeve, P.; Ward, S.; Butler, D. Rainwater Harvesting Typologies for UK Houses: A Multi Criteria Analysis of System Configurations. Water
**2016**, 8, 129. [Google Scholar] [CrossRef] [Green Version] - Lazarova, V.; Hills, S.; Birks, R. Using recycled water for non-potable, urban uses: A review with particular reference to toilet flushing. Water Supply
**2003**, 3, 69–77. [Google Scholar] [CrossRef] - GhaffarianHoseini, A.; Tookey, J.; GhaffarianHoseini, A.; Yusoff, S.M.; Hassan, N.B. State of the art of rainwater harvesting systems towards promoting green built environments: A review. Desalin. Water Treat.
**2015**, 1–10. [Google Scholar] [CrossRef] - Oberascher, M.; Zischg, J.; Palermo, S.A.; Kinzel, C.; Rauch, W.; Sitzenfrei, R. Smart Rain Barrels: Advanced LID Management through Measurement and Control; Springer: Cham, Switzerland, 2019; pp. 777–782. [Google Scholar]
- Palermo, S.A.; Talarico, V.C.; Pirouz, B. Optimizing Rainwater Harvesting Systems for Non-potable Water Uses and Surface Runoff Mitigation; Springer: Cham, Switzerland, 2020; pp. 570–582. [Google Scholar]
- Taji, S.G.; Saraf, V.R.; Regulwar, D.G. Smart Rain Water Harvesting for Smart Cities, In Studies in Systems, Decision and Control.; Springer: Cham, Switzerland, 2021; Volume 308, pp. 91–116. [Google Scholar]
- Campisano, A.; Creaco, E.; Modica, C. Application of Real-Time Control Techniques to Reduce Water Volume Discharges from Quality-Oriented CSO Devices. J. Environ. Eng.
**2016**, 142, 04015049. [Google Scholar] [CrossRef] - Akan, A.O.; Houghtalen, R.J. Urban Hydrology, Hydraulics, and Stormwater Quality: Engineering Applications and Computer Modeling; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2003; ISBN 0471431583. [Google Scholar]
- Mrowiec, M. The Effective Dimensioning and Dynamic Regulation Sewage Reservoirs; Wydawnictwo Politechniki Częstochowskiej: Częstochowa, Poland, 2009; ISBN 9788371934247. [Google Scholar]
- De Paola, F.; De Martino, F. Stormwater Tank Performance: Design and Management Criteria for Capture Tanks Using a Continuous Simulation and a Semi-Probabilistic Analytical Approach. Water
**2013**, 5, 1699–1711. [Google Scholar] [CrossRef] [Green Version] - Starzec, M.; Dziopak, J.; Słyś, D.; Pochwat, K.; Kordana, S. Dimensioning of Required Volumes of Interconnected Detention Tanks Taking into Account the Direction and Speed of Rain Movement. Water
**2018**, 10, 1826. [Google Scholar] [CrossRef] [Green Version] - Szeląg, B.; Kiczko, A.; Dąbek, L. Stormwater Reservoir Sizing in Respect of Uncertainty. Water
**2019**, 11, 321. [Google Scholar] [CrossRef] [Green Version] - Kisiel, A.; Kisiel, J.; Malmur, R.; Mrowiec, M. Retention tanks as key elements of modern drainage systems. Tech. J. Environ.
**2008**, 105, 41–63. [Google Scholar] - Mrowiec, M. A tubular tank for stormwater storage in the stormwater collection system. Gas. Water Sanit. Eng.
**2002**, 7, 236–239. [Google Scholar] - Guo, J.C.Y. Detention Storage Volume for Small Urban Catchments. J. Water Resour. Plan. Manag.
**1999**, 125, 380–382. [Google Scholar] [CrossRef] - Hong, Y.-M.; Yeh, N.; Chen, J.-Y. The simplified methods of evaluating detention storage volume for small catchment. Ecol. Eng.
**2006**, 26, 355–364. [Google Scholar] [CrossRef] - Hong, Y.-M. Graphical estimation of detention pond volume for rainfall of short duration. J. Hydro-environ. Res.
**2008**, 2, 109–117. [Google Scholar] [CrossRef] - Szeląg, B.; Kiczko, A. The graphic method of sizing pipe reservoir for short, high-intensity rainfalls. Ann. Warsaw Univ. Life Sci. L. Reclam.
**2014**, 46, 221–232. [Google Scholar] [CrossRef] [Green Version] - Rutkowska, D.; Piliński, M.; Rutkowski, L. Neural Networks, Genetic Algorithms and Fuzzy Systems; Wydaw. Naukowe PWN: Warszawa; Łódź, Poland, 1997; ISBN 978-83-01-12304-8.
- Koza, J.R. Genetic programming as a means for programming computers by natural selection. Stat. Comput.
**1994**, 4, 87–112. [Google Scholar] [CrossRef] - Szelag, B.; Barbusinski, K.; Studzinski, J. Activated sludge process modelling using selected machine learning techniques. Desalin. WATER Treat.
**2018**, 117, 78–87. [Google Scholar] [CrossRef] - Capodaglio, A. Sludge bulking analysis and forecasting: Application of system identification and artificial neural computing technologies. Water Res.
**1991**, 25, 1217–1224. [Google Scholar] [CrossRef] - Rutkowski, L. Arificial Intellignce Methods and Techniques; Wydawnictwo Naukowe PWN: Warszawa, Poland, 2006. [Google Scholar]
- Szeląg, B.; Mrowiec, M. The methods of evaluating storage volume for single-chamber reservoir in urban catchments. Arch. Environ. Prot.
**2016**, 42, 20–25. [Google Scholar] [CrossRef] [Green Version] - Froehlich, D.C. Graphical Sizing of Small Single-Outlet Detention Basins in the Semiarid Southwest. J. Irrig. Drain. Eng.
**2009**, 135, 779–790. [Google Scholar] [CrossRef] - Guo, J.C.Y. Hydrology-Based Approach to Storm Water Detention Basin Design Using New Routing Schemes. J. Hydrol. Eng.
**2004**, 9, 333–336. [Google Scholar] [CrossRef] - McEnroe, B.M. Preliminary Sizing of Detention Reservoirs to Reduce Peak Discharges. J. Hydraul. Eng.
**1992**, 118, 1540–1549. [Google Scholar] [CrossRef] - Starzec, M.; Dziopak, J.; Słyś, D. An Analysis of Stormwater Management Variants in Urban Catchments. Resources
**2020**, 9, 19. [Google Scholar] [CrossRef] [Green Version] - Pianosi, F.; Beven, K.; Freer, J.; Hall, J.W.; Rougier, J.; Stephenson, D.B.; Wagener, T. Sensitivity analysis of environmental models: A systematic review with practical workflow. Environ. Model. Softw.
**2016**, 79, 214–232. [Google Scholar] [CrossRef] - Kiczko, A.; Szeląg, B.; Kozioł, A.P.; Krukowski, M.; Kubrak, E.; Kubrak, J.; Romanowicz, R.J. Optimal Capacity of a Stormwater Reservoir for Flood Peak Reduction. J. Hydrol. Eng.
**2018**, 23, 4018008. [Google Scholar] [CrossRef] - Rossman, L.A. Storm Water Management Model. User’s Manual Version 5.1; National Risk Management Research Laboratory Office of Research and Development, U.S. Environmental Protection Agency: Cincinnati, OH, USA, 2015; Volume 352. [Google Scholar]
- Zhang, S.; Guo, Y. SWMM Simulation of the Storm Water Volume Control Performance of Permeable Pavement Systems. J. Hydrol. Eng.
**2015**, 20, 06014010. [Google Scholar] [CrossRef] - El-Sharif, A.; Hansen, D. Application of SWMM to the Flooding Problem in Truro, Nova Scotia. Can. Water Resour. J.
**2001**, 26, 439–459. [Google Scholar] [CrossRef] - Sañudo, E.; Cea, L.; Puertas, J. Modelling Pluvial Flooding in Urban Areas Coupling the Models Iber and SWMM. Water
**2020**, 12, 2647. [Google Scholar] [CrossRef] - TIBCO Software Inc. Statistica (Automated Neural Network); Version 10; TIBCO Software Inc.: Palo Alto, CA, USA, 2011. [Google Scholar]
- Paik, K. Analytical derivation of reservoir routing and hydrological risk evaluation of detention basins. J. Hydrol.
**2008**, 352, 191–201. [Google Scholar] [CrossRef]

**Figure 1.**A schematic presentation of the algorithm for establishing the parameters of the pipe tank.

**Table 1.**Maximum values of relative errors (ε

_{Vmax}) for the peak flow reduction coefficients β = 0.1 ÷ 0.9.

β | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 |
---|---|---|---|---|---|---|---|---|---|

ε_{Vmax} | 3.77 | 5.75 | 6.16 | 6.31 | 9.45 | 11.19 | 12.92 | 18.44 | 22.13 |

ε_{Vmin} | −7.01 | −7.28 | −9.79 | −12.07 | −18.08 | −20.51 | −22.93 | −25.03 | −36.33 |

**Table 2.**Fitting measures of the obtained models predicting the relative error of tank volume (ε

_{v}).

Method | RRE | MPE | r | AIC |
---|---|---|---|---|

Regression | 0.0442 | 47.51 | 0.6851 | −3553 |

ANN | 0.0272 | 5.60 | 0.9310 | −3010 |

GP(ω, η) | 0.0345 | 8.02 | 0.8872 | −3321 |

GP(ω, β) | 0.0312 | 7.89 | 0.8953 | −3299 |

GP(ω, β) | 0.0296 | 7.21 | 0.9010 | −3236 |

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**MDPI and ACS Style**

Szeląg, B.; Kiczko, A.; Musz-Pomorska, A.; Widomski, M.K.; Zaburko, J.; Łagód, G.; Stránský, D.; Sokáč, M.
Advanced Graphical–Analytical Method of Pipe Tank Design Integrated with Sensitivity Analysis for Sustainable Stormwater Management in Urbanized Catchments. *Water* **2021**, *13*, 1035.
https://doi.org/10.3390/w13081035

**AMA Style**

Szeląg B, Kiczko A, Musz-Pomorska A, Widomski MK, Zaburko J, Łagód G, Stránský D, Sokáč M.
Advanced Graphical–Analytical Method of Pipe Tank Design Integrated with Sensitivity Analysis for Sustainable Stormwater Management in Urbanized Catchments. *Water*. 2021; 13(8):1035.
https://doi.org/10.3390/w13081035

**Chicago/Turabian Style**

Szeląg, Bartosz, Adam Kiczko, Anna Musz-Pomorska, Marcin K. Widomski, Jacek Zaburko, Grzegorz Łagód, David Stránský, and Marek Sokáč.
2021. "Advanced Graphical–Analytical Method of Pipe Tank Design Integrated with Sensitivity Analysis for Sustainable Stormwater Management in Urbanized Catchments" *Water* 13, no. 8: 1035.
https://doi.org/10.3390/w13081035