A Probabilistic Model for Predicting the Performance of a Stormwater Overflow Structure as Part of a Stormwater Treatment Plant

Abstract

The purpose of this study was to attempt to develop a stochastic model that describes the operation of the stormwater overflow located in the stormwater sewerage system. The model built for this study makes it possible to simulate the annual volume of the stormwater discharge, the maximum volume of the overflow discharge in a precipitation event, and the share of the latter in the total amount of stormwater conveyed directly, without pre-treatment, to the receiver. The dependence obtained with the linear regression method was employed to identify the occurrence of stormwater discharge. The prediction of the synthetic annual rainfall series was made using the Monte Carlo method. This was performed based on the determined log-normal distribution, the parameters of which were specified using 13-year rainfall series. Additionally, simulation of the stormwater overflow operation was performed with the use of a calibrated hydrodynamic model of the catchment. The model was developed using the Storm Water Management Model (SWMM). The results of the hydrodynamic simulations of the volume and number of discharges were within the scope of the probabilistic solution, which confirms the applicative character of the method presented in this study, intended to assess the operation of stormwater overflow.

1. Introduction

Linear drainage systems collect stormwater from the area surface and convey it to the stormwater treatment plant (STP). The technological facilities of the treatment plant (settling tanks, separators, hydrocyclones, and others) reduce the wastewater pollutant load. In intense precipitation events, when the amount of stormwater flowing from the catchment increases rapidly, exceeding the capacity of stormwater treatment plants, excess wastewater is directly discharged into the river via the stormwater overflow or may be stored for future use. This can lead to the disturbance of the biological and chemical equilibrium in the aquatic environment [1,2,3,4], the washout of the banks and bottom of the watercourse [5,6,7,8], and an increase in the peak flow below the wastewater discharge section [9]. As a result, computational methods have been developed to assess the effect of both the quantity and quality of the stormwater introduced into the watercourse on the conditions prevailing in it [10,11,12,13,14]. In quality-based methods, the amount of dissolved oxygen and concentrations of heavy metals and ammonia in the receiving body are analyzed [15], or the saprobity index is determined [16]. In quantity-based methods, the assessment is based on the number of wastewater discharges, or on the maximum instantaneous flow. The review of the literature [13,17,18] indicates that many studies focus on predictions of the annual number of discharge events through the overflow, and the methods applied can be divided into empirical, deterministic, and stochastic methods [19]. As regards the first category, the methods rely on multiyear measurements of precipitation amounts, and on the multiplicity of the overflow operation. This makes it possible to determine regression dependences or appropriate nomograms [9,17,20]. Deterministic models include the Kuiper method [21,22], in which the calculations of the number of stormwater discharge events are based on the rainfall time series recorded over multiyear periods. However, because the empirical and deterministic methods mentioned above disregard the random character of precipitation events, the dynamics of the event over time, and the catchment retention, the description of the runoff is greatly simplified. Consequently, these methods provide only estimates, and the results may be marred by major errors [19].

For an assessment of the performance of sewerage systems, the hydrodynamic model can be used [9,23,24]. One of the most commonly used is the Storm Water Management Model (SWMM) software [25,26,27], which results from the fact that the source code, written in the C++ programming language, can be modified to fit individual needs. It should be emphasized that to assess the performance of the stormwater overflow, retention reservoir, or other facility in the sewerage system, it is necessary to perform continuous simulations [28,29,30] using rainfall time series collected for many years [31]. The computations [9,32,33] indicate that simulation results can provide the basis to specify the annual number and volume of stormwater discharges through overflows in an assumed return period. However, because of the high costs of taking measurements of precipitation and flows, the development of the model is not always a viable option. Additionally, the complexity of the runoff phenomenon makes it necessary to introduce a number of empirical parameters to describe the catchment and precipitation characteristics, which can only be established by calibration. However, a large number of these and the interactions between the individual parameters [34,35,36] can cause problems related to the unambiguous determination of the numerical values of these parameters, which is a condition for obtaining a correct solution.

In view of the above remarks, stochastic models have been developed to assess the performance of stormwater overflow structures. These are based, among others, on the First-Order Reliability Method (FORM), the authors of which [37] accounted for the uncertainty of the estimated parameters in the hydrodynamic model. For the method to be used, it is necessary to employ complex numerical algorithms, which may limit the application of the method in engineering practice. In his hydrodynamic model, which also accounts for parameter uncertainty, Thorndahl [38] used the results of simulations performed on the basis of rainfall time series collected over a 50-year period. In their models, neither Thorndahl [38] nor other researchers [9,29] explained the dynamics of the variation in the discharge volume over the yearly cycle, which is a valuable piece of information with respect to the hydraulic conditions in the rainwater receiver.

The primary objective of this study was to develop a stochastic model to evaluate the performance of a stormwater overflow. The model makes it possible to determine the annual quantity and volume of stormwater discharges, taking into account the stochastic nature of rainfall events. The method developed by the authors also makes it possible to identify extreme events in which the maximum volume of wastewater discharge is found in the precipitation event, and also to compute the share of that volume in the total yearly discharge. For the identification of the event and the computation of the volume of the overflow discharge, a regression dependence was used. It was obtained on the basis of continuous simulations performed by the SWMM program. To predict yearly precipitation data, the Monte Carlo method was applied. Estimation of the statistical distribution parameters describing the dynamics of the rainfall profile in individual precipitation events was performed on the basis of observation data collected in the years 2008–2020.

2. Materials and Methods

2.1. Study Area

This study presents an analysis of the catchment of a Si9 sewer, with a total area of 62 ha, located in the central–eastern part of the city of Kielce in Poland (Figure 1). The main sewer, 1569 m in length, and with diameter ranging from 600 to 1250 mm, collects precipitation wastewater from 17 laterals. The total length of the system is 5583 m. The sewerage system is buried 1.59–3.55 m in the ground. The highest catchment point is located at 271.20 m above the mean sea level (amsl), and the lowest-lying point is at 260.00 m amsl. The building development of the area includes housing estates, institutional buildings, trunk roads, and side streets. In total, roofs constitute 14.3%, pavements 8.4%, roads 17.7%, car parks 11.2%, greenery 47.2%, and school playgrounds 1.3% of the catchment area [39].

Wastewater transported from the catchment by the main sewer, Si9, is delivered to the stormwater treatment plant via the separation chamber (DC in Figure 1). The route along which the wastewater flows through the treatment facility includes an oblong settling tank (SET), coalescence separator (SEP), and control manhole (CS) (Figure 1). When the water level in the chamber (DC) does not exceed 0.42 m, stormwater flows through four ϕ400 mm conduits into the settling tank, and when the level is higher, stormwater is delivered, via the stormwater overflow (OV) (Figure 2), into the discharge (outlet) channel, from where it travels to the receiving water body (i.e., the Silnica River).

Two ultrasonic flow meters (ISCO AV 2150, Teledyne ISCO, Lincoln, NE, USA) were installed in the Si9 collector. The first is located 3.0 m from the inlet to the separation chamber, and the other is at a distance of 5.0 m below the storm overflow. Devices indirectly measure the flow rate based on the direct measurement of the average velocity and the water level (hydrostatic pressure measurement). The water level measurement ranges from 0.01 to 3.0 m, with an accuracy of ± 0.003 m. Furthermore, a probe for measuring the filling, which triggers the automatic rainwater sampling device, was found in the separation chamber (sampler type 6712, Teledyne ISCO, Lincoln, NE, USA). The tipping bucket-type rain gauge RG50 from SEBA Hydrometrie GmbH (Kaufbeuren, Germany) was used to measure the depth of the rain. The measurement frequency ranged from 2 to 5 min, with a resolution of 0.1 mm.

2.2. Precipitation Event and Antecedent Period

The literature review [38,40,41,42,43] indicates that the rainfall definition is unclear because, for an event to be considered relevant, the rainfall depth should range 1–10 mm. As the present study was intended to predict the number of wastewater discharges through the overflow within a one-year cycle, the assumed minimal rainfall depth selected as a basis to show the statistical distribution should make it possible to identify cases when such discharge does not occur. Therefore, the rainfall depth cited in Arbeitsblatt DWA-A 118 [44], equal to P_tot = 10 mm, which defines an intense event, seems to be too high. Conversely, the minimum value of P_tot = 1 mm is not sufficient because, as shown in the investigations conducted by Kotowski et al. [45], it could be marred by a considerable measurement error related to the equipment used. In relation to the above, in this study, a precipitation event was assumed to mean a rainfall depth of a minimum of 2 mm. In many studies [40,41,42,43], this value provides a basis for determining multidimensional distributions of extreme values used to evaluate the operation of the sewerage system. The time interval that separates individual precipitation events was assumed to be 4 h [44].

2.3. Precipitation Data

Precipitation data were obtained from precipitation stations located within the study catchment area and about 2.0 km from its borders. These posts have been continuously collecting atmospheric data since 2008, and precipitation is continuously monitored with a resolution of 1–5 min. Data from 2008–2020 were used for the analysis, with the total precipitation (P_tot) ranging from 2.0 to 45.2 mm during the period. The highest observed daily precipitation totals were in 2010, when days with more than 20 mm of precipitation occurred eight times, including two days during which 43.8 mm and 45.2 mm fell, which, using the formula given by Suligowski [46], classifies them as precipitation with a probability of occurrence close to p = 5% (once in 20 years). The number of precipitation events analyzed during a given year ranged from 52 to 75. Finally, 480 rainfall events were used to determine the synthetic rainfall generator for the P_t10, P_t15 and P_t30 forecasts. The duration of rainfall of the observed events (t_d) varied in the range of 20–2366 min, and the rain-free period (t_ap) was 0.17–60 days. The maximum 10, 15, and 30 min values (P_t10, P_t15, P_t30, respectively) were adopted, according to the literature [47,48], as authoritative for describing the intensity of the course of the considered phenomenon in a rainfall event. In the analyzed catchment, they are P_t10 = 0.2–14.4 mm, P_t15 = 0.2–19.3 mm, and P_t30 = 0.3–21.2 mm.

For the observed precipitation events described by means of t_ap, P_tot, P_t10, P_t15, and P_t30, empirical statistical distributions were stated, to which one of the following distributions, namely, Weibull, chi-square, exponential, GEV, Fisher–Tippett, gamma, log-normal, or beta, was fitted. The congruence between the empirical and theoretical distributions was made based on the results of the Kolmogorov–Smirnov and chi-square tests.

2.4. Hydrodynamic Model

The hydrodynamic model of the catchment was developed using SWMM 5.0 software. The model consisted of 92 subcatchments, with an area of 0.12–2.10 ha, 200 manholes, and 72 sections of the sewers. Stormwater from the roofs of buildings is discharged directly into the sewer system and surface runoff is discharged into street inlets. The Manning roughness coefficients for impervious and pervious areas of the catchment were 0.015 and 0.15 m^−1/3∙s, respectively, and the roughness coefficient of sewer walls was 0.015 m^−1/3∙s. The amounts of ground retention of the impervious areas (52.8% of the catchment) and pervious ones (47.2%) were 1.5 and 6.0 mm, respectively, and the weighted average retention (d_avg,ret) determined based on the share of each type of development was equal to d_avg,ret = 0.528∙1.5 + 0.472∙6.0 = 3.62 mm.

The mathematical model of the STP was developed based on field measurements and the design documentation of the facility [49]. Details of the construction and calibration of the model can be found in the works of Szeląg and Bąk [50]. To compute the amount of wastewater flowing into the treatment plant and that of the wastewater discharged through the stormwater overflow (V_d), simulations performed with the SWMM 5.0 program [51,52,53] were used. To calibrate the developed model of the separation chamber with the stormwater overflow, flow measurement sequences from 2008 to 2020 were used.

2.5. Calibration of the Hydrodynamic Model

The following parameters were used to assess the consistency of the measured and simulated fillings of the separation chamber:

-

The ratio of the maximum measured to simulated filling:

$$\mathsf{\delta }={\displaystyle \frac{{\mathrm{h}}_{\mathrm{K}\mathrm{R}\left(\mathrm{m}\right)}}{{\mathrm{h}}_{\mathrm{K}\mathrm{R}\left(\mathrm{s}\mathrm{i}\mathrm{m}\right)}}}$$

(1)

where h_KR(m) is the maximum measured filling of the separation chamber (m), and h_KR(sim) is the maximum calculated filling of the separation chamber in the SWMM program (m);

-

The correlation coefficient (R):

$$\mathrm{R}={\displaystyle \frac{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}({\mathrm{h}}_{\mathrm{o}\left(\mathrm{i}\right)}-{\mathrm{h}}_{\mathrm{s}\mathrm{r},\mathrm{o}})·({\mathrm{h}}_{\mathrm{s}\left(\mathrm{i}\right)}-{\mathrm{h}}_{\mathrm{s}\mathrm{r},\mathrm{s}})}{\sqrt{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{({\mathrm{h}}_{\mathrm{o}\left(\mathrm{i}\right)}-{\mathrm{h}}_{\mathrm{s}\mathrm{r},\mathrm{o}})}^2·{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{({\mathrm{h}}_{\mathrm{s}\left(\mathrm{i}\right)}-{\mathrm{h}}_{\mathrm{s}\mathrm{r},\mathrm{s}})}^2}}}$$

(2)

where h_o(i) is the rainwater distribution chamber fill values observed with a one-minute time step (m); h_s(i) is the distribution of the chamber fill values obtained from simulations with a one-minute time step (m); h_sr,o is the average measured distribution chamber fill (m); h_sr,s is the average distribution chamber fill calculated by numerical simulations (m); n is the total number of observations.

2.6. Analysis of the Effect of the Precipitation Characteristics on the Overflow Performance

Due to the fact that hydrodynamic simulations are time-consuming, and it is not always possible to calibrate the model in the optimal way, for practical purposes, only those measurement results are used on the basis of which the regression dependences are derived. The later application of these dependences is much simpler and faster than that of the hydrodynamic model. The results obtained using these dependences are close to those produced through numerical simulations. To determine the discharge volume or the size of the pollutant load, multiple regression or the least-squares method is usually used [38,54,55].

To determine the effect of the individual parameters describing the non-uniformity of the rainfall depth during the event (P_tot, P_t10, P_t15, P_t30), the event duration (t_d), and time interval separating individual events (t_ap) on the volume of the overflow discharge (V_d), a Pearson correlation matrix was generated. Also, the relations that were statistically relevant at the adopted confidence level of p = 0.05 were found. Using computation results to avoid the necessity of applying the hydrodynamic model to determine the number and volume of discharges, the multiple regression equation was employed:

$${\mathrm{V}}_{\mathrm{d}}={\sum }_{\mathrm{i}=1}^{\mathrm{j}}{\mathsf{\alpha }}_{\mathrm{i}}·{\mathrm{P}}_{\mathrm{i}}+\mathsf{\beta }·{\mathrm{t}}_{\mathrm{d}}+\mathsf{\chi }·{\mathrm{t}}_{\mathrm{a}\mathrm{p}}-\mathsf{\theta }$$

(3)

where the characteristics of precipitation events during which wastewater discharge through overflow occurred were identified on the basis of the dependence:

$${\sum }_{\mathrm{i}=1}^{\mathrm{j}}{\mathsf{\alpha }}_{\mathrm{i}}·{\mathrm{P}}_{\mathrm{i}}+\mathsf{\beta }·{\mathrm{t}}_{\mathrm{d}}+\mathsf{\chi }·{\mathrm{t}}_{\mathrm{a}\mathrm{p}}=\mathsf{\theta }$$

(4)

where α_i, β, χ, and θ are the empirical parameters found with the least-squares method, and P_i, t_ap, and t_d are the parameters that describe the precipitation.

Equation (4) describes the multidimensional space of precipitation events during which stormwater discharge through the overflow will not occur. After appropriate transformations, dependence (4) offers the possibility of determining the limit value of the rainfall depth, the exceeding of which leads to discharge.

2.7. Prediction of the Annual Number and Volume of Discharge Events

Because of the relatively short period of precipitation observations, it was assumed in the computations that the number of rainy days was the same as that in individual years, which is a fairly common assumption made by researchers [56]. Computational analyses performed by Szeląg et al. [47], which took into account a uniform distribution of the number of rainfall days per year, showed that the average numbers of overflow discharges obtained in the variant taking this distribution into account and in the variant excluding it are practically identical. In the probabilistic approach, to determine an annual series (n) of precipitation events, described by means of the t_d, t_d(t−1), t_ap, P_i, and P_i(t−1), it is necessary to run a simulation of a single series at least 1000 times using the Monte Carlo method [56,57].

Using the multiple regression model developed, the annual number and volume of discharges through the stormwater overflow were obtained for the generated precipitation series. On this basis, it was possible to find empirical distribution functions that describe the probability that, in the assumed time period, the discharge number and volume will not exceed a certain value. Additionally, the occurrence of events with the maximum stormwater discharge through the overflow was analyzed. Also, the share of such events in the overall annual discharge of wastewater into the receiving water body was defined.

3. Results and Discussion

On the basis of the computations performed using SWMM software and the results of the measurements of the total rainfall depth (P_tot), rainfall duration (t_d), and stormwater flow rate, the stormwater overflow parameters were stated by means of calibration. This made it possible to compute the criteria of the fitting of the water levels in the separation chamber for seven selected precipitation events (

Table 1. Characteristics of precipitation events and parameters of fitting of the separation chamber water level, measured and obtained from simulations.

No.	Hydrograph of the Flow from the Catchment			Separation Chamber
	Ptot mm	td min	Vtot m3	δ -	R -
1	8.6	60	1733	0.88	0.90
2	9.2	286	2221	0.95	0.94
3	12.5	107	1908	0.97	0.91
4	16.5	270	3415	0.94	0.86
5	4.2	26	2133	0.96	0.88
6	5.4	56	684	0.91	0.87
7	3.6	92	327	0.92	0.90

Note: P_tot—total rainfall depth in the precipitation event; t_d—rainfall duration; V_tot—volume of hydrograph of flow from catchment; δ—ratio of the maximum measured water level to the maximum simulated water level; R—correlation coefficient.

). Events 1–5 involved intense rainfall, during which overflow discharge occurred. Conversely, during events 6–7, such discharge was not noted.

The analysis of the results of the fitting of the stormwater overflow model (δ) shows that the maximum instantaneous flows discharged through the overflow, determined with the SWMM model, are underestimated by, maximally, 12%. As part of the calculations, 20 rainfall events were selected from each year, for which calibration was performed (10 each with and without discharge). Table 1 lists seven sample precipitation events for which the fill of the separation chamber was determined and compared with the results of measurements determining the δ and R, varying in the ranges of 0.88–0.97 and 0.86–0.94, respectively.

The values of the correlation coefficients indicate the satisfactory fitting of the measurement results and hydrodynamic simulations with respect to variation in the water level of the separation chamber. In view of the above, it can be stated that the hydraulic model of the separation chamber with the overflow, developed by the authors, describes the chamber operation with satisfactory accuracy and can provide a basis for further analyses. Figure 3 shows an exemplary comparison of the filling level of the separation chamber, measured and obtained from simulations for precipitation event No. 4 (Table 1). The patterns of both curves show large similarities.

The shift in the theoretical curve relative to the curve obtained from measurements may be due to the lower average value of the catchment retention for real conditions than what was assumed in the simulation, which results in a later response of the drainage system to surface runoff. In the model, rainfall events were separated out when performing the simulation, ignoring the current state of the catchment retention between the rainfall events, which affects the results obtained. In addition, there are many factors, such as the amount of sediment in the sewers, the condition of the pipes, and sections with contraflows, that are difficult to determine precisely without very thorough field surveys.

The computations showed that the coefficient of the overflow is 0.36, and the values of the coefficients of the local resistances at the inlets and outlets of the channels conveying stormwater to the settling tank were in the ranges of 0.48–0.53 and 0.95–1.10, respectively. Simulations carried out to model the stormwater overflow performance made it possible to establish a likely number and volume of the overflow discharges in individual years (2008–2020). The values are as follows: 20 (discharge events) and 9234 m³ in 2008; 17 and 7911 m³ in 2009; 15 and 6274 m³ in 2010; 18 and 14,219 m³ in 2011; 21 and 13,259 m³ in 2012; 22 and 13,749 m³ in 2013; 30 and 29,553 m³ in 2014; 26 and 23,456 m³ in 2015; 24 and 21,432 m³ in 2016; 28 and 32,126 m³ in 2017; 19 and 18,754 m³ in 2018; 17 and 11,652 m³ in 2019; 14 and 9873 m³ in 2020, which gives 271 discharges altogether over 13 years.

Based on computations of the discharge volumes and measured rainfall parameters (P_tot, P_t10, P_t15, P_t30, t_d, t_ap), the correlation matrix was determined (

Table 2. Values of the correlation coefficients between precipitation parameters and discharge volume.

	Ptot	Pt30	Pt15	Pt10	td	tap	Vd
Ptot	1.00	0.41	0.29	0.25	0.59	−0.38	0.62
Pt30	-	1.00	0.95	0.75	−0.26	−0.13	0.92
Pt15	-	-	1.00	0.87	−0.39	−0.05	0.85
Pt10	-	-	-	1.00	−0.43	−0.02	0.83
td	-	-	-	-	1.00	−0.27	−0.05
tap	-	-	-	-	-	1.00	−0.21
Vd	-	-	-	-	-	-	1.00

Note: P_tot—total rainfall depth in the precipitation event; P_t30—maximum 30 min rainfall depth; P_t15—maximum 15 min rainfall depth; P_t10—maximum 10 min rainfall depth; t_d—rainfall duration; t_ap—antecedent period duration; V_d—volume of overflow discharge.

). The matrix provided the basis for deriving, with the use of multiple regression, the dependence for the determination of the volume of the overflow discharge.

In Table 2, a very strong correlation (r = 0.92) can be observed between the volume of the overflow discharge (V_d) and the maximum 30 min rainfall depth (P_t30), and a strong correlation (r = 0.85 and r = 0.83) holds for the 10 and 15 min rainfall depths. A medium-strength correlation (r = 0.62) exists between the variables V_d − P_tot, whereas for the pair of variables V_d − t_ap, a weak correlation is shown.

Taking into account the results presented in Table 2, and also the strong correlations between the parameters considered, to determine the regression dependence, the method of multiple stepwise regression backwards was used. On the basis of the computations, the following dependence was obtained:

$${\mathrm{V}}_{\mathrm{d}}=243.02\left(\pm 18.86\right)·\left({\mathrm{P}}_{\mathrm{t}30}-1.85\right) (\mathrm{r}=0.920)$$

(5)

where the value ± 18.86 is the standard deviation.

The dependence (5) describes the amount of wastewater discharged through the overflow during a single precipitation event, which is confirmed by the correlation coefficient r = 0.920. It follows from Equation (5) that stormwater overflow discharge will occur when P_t30 exceeds the rainfall amount of 1.85 mm, whereas the weighted average value of the catchment retention (d_avg,ret) is much higher and amounts to 3.62 mm.

For predictions of the annual number and volume of overflow discharges, the statistical distribution of the maximum 30 min rainfall depth in the precipitation event was determined. For this purpose, values of the probability (p) of the Kolomogorov–Smirnov and chi-square tests were determined for the following distributions: Weibull, chi-square, exponential, GEV, Fisher–Tippett, log-normal, and beta. The analysis of the results of the computations (

Table 3. Results of computations (p) of Kolmogorov–Smirnov and chi-square tests of the fitting of the empirical distribution function to the theoretical distribution function obtained with the use of statistical distributions of concern.

Distribution	Kolmogorov–Smirnov Test	Chi-Square Test
Beta	0.0001	0.0002
Chi-square	0.0002	0.0003
Weibull	0.0253	0.0312
Exponential	0.0007	0.0005
GEV	0.0002	0.0003
Fisher–Tippett	0.0003	0.0004
Log-normal	0.3130	0.2526

, Figure 4) indicates that for the measurement data from the years 2008–2020, the log-normal distribution is the best fitted one (Kolmogorov–Smirnov test: p = 0.3130; and chi-square: p = 0.2526 for α = 5%), having a standard deviation of σ = 0.970 mm and a mean value of μ = 0.403 mm.

Simulations conducted using the Monte Carlo method generated, 1000 times, 58 precipitation events in a year. The results of the computations of the number of overflow discharges, based on the generated series, were illustrated in the form of a cumulative probability density distribution (Figure 5). Next, from Formula (5), discharge volumes for 30 min synthetic rainfall depths were determined. This allowed for the determination of the annual volume (Figure 6) and the identification of extreme events that occurred in the year and during which the maximum discharge through the overflow occurred (Figure 7). Additionally, on the basis of the simulation results, the share of the stormwater volume in a single precipitation event during which the maximum wastewater discharge took place in the total yearly discharge was determined (Figure 8).

The graph in Figure 5 shows that the expected value (p = 0.50) of the annual number of discharges through the stormwater overflow in the urbanized catchment of concern is 22, whereas the percentile values p = 0.05 and p = 0.95 equal 17 and 28, respectively. The yearly number of the discharges obtained with the SWMM program ranges from 15 to 29, which corresponds to percentile values of p = 0.01 and p = 0.97.

The yearly discharge volume required for the percentile of p = 0.50 is 12450 m³, and for p = 0.05 and p = 0.95, the values are 7303 m³ and 19,903 m³, respectively (Figure 6). According to the simulations carried out using the hydrodynamic model, the lowest annual discharge volume is 7911 m³, which corresponds to p = 0.07, whereas the highest is 29,953 m³ (for p = 0.99).

On the basis of the computations performed using the stochastic model shown in Figure 7, it can be stated that the expected value of the maximum volume of the overflow discharge in a single precipitation event equals 2341 m³, and values of the percentiles p = 0.05 and p = 0.95 correspond to discharge volumes of 1184 m³ and 5840 m³.

The comparison of the results of the computations obtained using the hydrodynamic model with the stochastic solution shown in the graph (Figure 7) indicates that the maximum volume of the discharge in a single precipitation event based on the SWMM simulations amounts to 6120 m³. This value is close to that of the determined percentile p = 0.95 in the probabilistic model (empirical distribution function). In contrast, the minimal volume of discharge in the precipitation event, computed with the mathematical model of the catchment, is 2120 m³ and corresponds to p = 0.41.

The developed probabilistic model offers the possibility of determining the expected value that describes the share of the maximum volume of the stormwater discharge in a single precipitation event in the total annual discharge (Figure 8). This share amounts to 0.18 for the percentile p = 0.50, whereas the extreme values, i.e., for p = 0.05 and p = 0.95, are equal to 0.12 and 0.36, respectively.

The shares of the maximum discharges in the overall annual discharge, computed using the SWMM software, correspond to a percentile range of p = 0.07–0.97. This indicates considerable variation in the individual discharge values in the years 2008–2020 relative to the total annual discharge.

The results of the computations of the number and volume of overflow discharges obtained using the calibrated hydrodynamic model in the SWMM program and the stochastic model show high variation in the parameters describing the performance of the stormwater overflow. The annual number and volume of discharges obtained, and also the discharge volume in a single precipitation event obtained with the SWMM model, remain in the range of the probabilistic solution. This confirms the application capacity of the method presented in this study. In addition, a wide range of variation in the share of maximum discharge volumes in a single event in the total yearly discharge indicates substantial differences as regards the wastewater volumes in individual precipitation events, in which the wastewater is directly conveyed to the receiving water body.

To order to expand the applicability of the developed model, it is recommended to extend the study to other urban catchments with different land use patterns and different topologies of stormwater sewer systems. This approach can be implemented in parallel with computational experiments performed for the study catchment. Using experiment planning techniques, it is possible to design virtual catchments with different characteristics, for which simulations of the overflow performance can be performed using the developed SWMM model. The statistical models obtained in this way can be compared with predictions for other urban catchments.

4. Conclusions

On the basis of the computations, it can be stated that the stochastic model presented to describe the performance of the stormwater overflow can be applied to determine the number and volume of overflow discharges, and also to identify the extreme events during which the maximum wastewater discharge occurred during the year. In the analyses presented above, the results of the numerical computations performed with the SWMM program allowed for establishing the regression dependence between the P_t30 and the volume of the overflow discharge. Additionally, as regards research paths and the number of parameters describing the catchment and the sewerage system characteristics in the SWMM model, and to account for the interactions between these parameters, it is advisable to conduct an analysis of the uncertainties of the estimated parameters in the hydrodynamic model. This makes it possible to identify reliable values of the parameters by means of suitable statistical methods, which allows for the physical interpretation of the parameters obtained in Equation (5).

The results of the hydrodynamic simulations of the volume and number of discharges were within the scope of the probabilistic solution, which confirms the applicative character of the method presented in this study, intended to evaluate the performance of stormwater overflows.

It is necessary to validate the parameters of the statistical distribution describing the variation in the maximum P_t30, and to account for the varied number of precipitation events in the yearly cycle by developing a statistical distribution to describe this parameter.

The method described in this paper can be applied to other small urban catchments for stormwater drainage networks with simple topologies, with the prerequisite of continuous high-resolution rainfall measurement series, as well as the identification of a regression relationship between the overflow discharge volume and the parameters describing the rainfall dynamics during individual rainfall events.

The adopted calibration method ignores interactions between the identified SWMM parameters. Further research directions should be focused on developing models for the V_d, t_d, and number of overflow discharge predictions that take into account the uncertainty of the SWMM parameters describing the catchment retention.