Retrofitting low-performance units to abate sewer overflow pollution based on mathematical model and Sobol algorithm

Optimal retrofit of low-performance units (LPUs) is promising to abate overflow pollutant mass loading of sewer systems during wet-weathers. This study presents a combination of mathematical model and Sobol algorithm to help identify LPUs of sewer systems and design retrofitting strategies. Therefore, the solution to minimize the overflow pollutant mass loading from sewers systems can be efficiently obtained. The developed method was demonstrated at a catchment served by one wastewater treatment plant in the Chaohu City, Anhui Province of China, with five pumping stations and a total sewer length of 58.3 km. Within the catchment, there are three rivers and a small lake to receive overflows from the sewer system. Among them, one river that was mostly polluted was selected as the object of overflow pollution abatement during wet weather period. After identifying the LPUs of the sewer system and developing retrofitting strategies using Sobol sequence, the mitigation of overflow pollution during wet weather period was analyzed. Results show that the mass loading of chemical oxygen demand (COD) discharged into the target river could be reduced by 40.6%, by implementing optimal retrofit strategy of LPUs, i.e., increasing the conveyance capacities of two pumping stations by 2.5–3.2 times and augmenting the diameters of 12 sewers by 1.25–1.29 times. To further coordinate the abatement of overflow pollution and retrofit investment, Sobol sensitivity analysis was conducted to screen the dominant LPUs to update the optimal retrofit strategy. By applying the updated strategy, the overflow COD mass loading per overflow event was close to that of non-updated strategy, while the retrofitting length of sewers was reduced by 40%. Therefore, on the basis of the presented method, decision-makers can flexibly develop retrofitting strategies of sewer system to abate overflow pollution during wet weathers in a cost-effective way.


Introduction 
Combined sewer systems that collect sewage and stormwater runoff are often overloaded during wet weathers owing to the insufficient conveyance and interception capacities [1] .As a result, a large amount of untreated sewage overflows into receiving water-sponge city project announced by China [9] .
CSO reduction generally includes volumeoriented and pollution-oriented forms.Specifically, volume-oriented form is often applied to minimize CSO volume and number of occurrences, while pollution-oriented form is designed to minimize the overflow pollutant mass, which is correlated to not only the overflow volume but also the concentration of sewer flow [10][11] .Compared with the latter, most studies are prone to volume-oriented CSO reduction, due to the complex interaction of water quantity and quality dynamics of pollution-oriented form Refs. [12-13].However, pollution-oriented CSO reduction has been proved to be promising in cost-effectively promoting water quality of receiving waterbodies [10][11] .
In practice, three approaches often perform well in managing CSO pollution, which are (1) Reducing stormwater runoff at source, (2) Increasing sewage conveyance and interception efficiencies and (3) Attenuating peak flow entries into sewers, respectively [14][15] .Recently, low impact development facilities have been widely constructed to abate runoff at source [16] , but several factors can affect the performances, such as design parameters, structural layer materials, spatiotemporal variation and maintenance frequency [17] .Enlarging sewer system is often efficient for improving sewage conveyance and interception capacities, but not always a desirable way considering the high cost of long-term construction and disruption to traffic, businesses, and property owners [18] .Activating storage capacity by constructing detention tanks can significantly alleviate peak flows into sewers.However, the required large dedicated areas and high financial investments make it difficult to implement [19] .
After trading off local limits (e.g., budget, space) against performances, municipalities are prone to a relatively cost-effective solution to mitigate CSO pollution.Retrofitting low-performance units (LPUs) of the existing sewer system is promising for meeting their requirements [12] .LPU refers to the local unit that restricts the effective utilization of sewage conveyance and interception capacities of drainage system, such as reducing pipe, lifting pumping station with insufficient delivery capacity and so on.As a result, they are often responsible for hydrodynamic discordance between upstream and downstream sewer segments.
Identification of LPUs is the basis of retrofit, while the process is not straightforward due to the non-visualization and hydraulic complex connectivity of sewer networks.Confronting the challenge, mathematical model was developed to simulate the longitudinal and temporal water level and flow rate within the sewers, and help identify LPUs [20] .For example, Liu et al. [21] established the sewer network model of Hoboken combining sewer failure with groundwater level to identify the LPUs that affected by groundwater infiltration.After identification, retrofitting strategy of LPUs is usually developed by the method of manual "trial and error".However, this kind of method is often subjective and limited, and cannot provide a cost-effective retrofitting strategy of LPUs for CSO pollution mitigation.In the context, simulation-optimizing model that combines urban drainage system model and optimization algorithm has been proposed to help automatically design the retrofitting strategy of LPUs within the entire allowable ranges [22] .On the basis, the coupling of simulation-optimizing model and global sensitivity analysis can further help find out the dominant LPUs that primarily responsible for sewage delivery failure and CSO occurrence, and provide the optimal strategy to coordinate CSO pollution mitigation and retrofit investment.
To address the challenge, the main objectives of this work are to (1) Integrate hydrodynamic and water quality models for simulating the temporal sewage conveyance and overflow of sewer system during wet weather days, and then identifying LPUs, (2) Construct an simulation-optimizing model that combines hydrodynamic-water quality model and optimization algorithm to determinate the optimal retrofitting strategy of LPUs and (3) Introduce global sensitivity analysis to identify the dominant LPUs and update the optimal retrofitting strategy to trade off the reduction effect of CSO pollution against retrofit cost in a real case.

Site description
The study region is an urban catchment served by a WWTP in the Chaohu City, Anhui Province of China (Fig. 1), with a total sewer length of 58.3 km and a total coverage area of 22.4 km 2 where approximately 140 000 inhabitants reside.The impervious area accounts for 72.4% of the total coverage area.On the basis of the sewage discharge survey results, 606 sewage sources were confirmed in this area, including 203 sources from residential communities and 403 sources from public services such as hotels, restaurants, commercial and office buildings, schools, hospitals, shops, and so on.Correspondingly, the cumulitive daily sewage volume generated in the study area was estimated 28 000 m 3 .
As seen in Fig. 1, the subcatchments of the sewer system are connected by five pumping stations, including two stations with both stormwater and sewage lifting pumps (pumping stations I, II), two stations with sewage lifting pumps (pumping stations III, IV), and one terminal station for lifting WWTP inflow (pumping station V).The detailed information concerning the five pumping stations is provided in our previous study [1] .Operational records of the terminal influent pumping system show that the daily WWTP-treated volume in the dry weather period was approximately 47 000 m 3 , while the maximum daily treatment capacity of WWTP during wet weather periods was about 61 000 m 3 (i.e., 1.3 times to dry weather days' capacity).However, during wet weathers, the total amount of sewage and stormwater runoff would exceed this maximum capacity, leading untreated CSOs delivered into Donghuancheng River, Xier pool, Xihuancheng River and Tianhe River through outlets 1-4, outlet 5, outlet 6 and outlet 7.Among these outlets, outlets 1, 2, 5 and 7 are pumping outlets, and outlets 3, 4 and 6 are gravity outlets.Additionally, as depicted in Fig. 1, most of the outlets (outlets 1-4) are distributed at the bank of Donghuancheng River, which explains the worst water quality of the river among the four waterbodies.

Construction of the simulation-optimizing model
for optimal retrofitting strategy exploration Design of the methodology for establishing a simulation-optimizing model for LPU retrofit is presented in Fig. 2.

Objective function
From the perspective of water quality restoration, the designed retrofit objective is to minimize the pollutant mass of CSOs during wet-weather periods.In mathematical form, the objective function is expressed where f is the total wet-weather overflow pollutant loading of outlet 1, , K  , K is the number of sewer overflow outlets, k W is the overflow pollutant of outlet k released into the receiving environment.Overflow pollutant at each outlet is dynamically simulated through hydrodynamic and water quality models of the sewer system.

Constraint condition
The inflow to WWTP is capped by the maximum treatment capacity on wet-weather days, which determines the overall performance of the retrofitted system, i.e.
influent WWTP where influent V is the daily WWTP influent volume, WWTP V is the maximum daily WWTP treatment capacity during wet-weather periods.

Decision-making variables
In the study, the retrofit of urban sewer system is associated with sewage lifting pumps and sewers.The corresponding decision-making variables of these facilities are the capacities of sewage pumps and sewer diameters, which can exert significant effects on the efficiency of sewer system for sewage conveyance and interception.Additionally, these decisionmaking variables conform to the following constraints: are the minimum and maximum allowable retrofit diameters of sewer n and N is the number of sewers to be retrofitted.

Realization of the simulation-optimizing model and coupling with global sensitivity analysis
The architecture to realize the simulationoptimizing model and then coupling with global sensitivity analysis are provided in Fig. 3, which follows four steps as described below.
Step 1: The sewer network model of the studied catchment is constructed to simulate the dynamic evolution process of sewage yield, conveyance, and overflow under current operational situation.On the basis, LPUs of the sewer system and the corresponding decision-making variables are determined.
Step 2: Sobol sequence method was used to generate retrofitting strategies 1, , L  for LPUs based on their decision-making variables (see Eq. ( 5)).The generation was achieved by "QMCPy" [23] , a Python library with the function of computing elements of Sobol quasi random sequence, which is a type of quasi random sequence with lower disparity than the pseudorandom sequence [24] .Using Sobol sequence method for random number generation will lead to a global exploitation of the search space.
, , , , where , i sp l Q is the capacity of lifting sewage pump i in strategy l , I is the number of lifting sewage pumps to be retrofitted, L is the number of strategies, , j s l D is the diameter of sewer j in strategy l and J is the number of sewers to be retrofitted.
Step 3: Retrofitting strategies 1, , L  of LPUs are embedded into the sewer network model in turn to automatically simulate the corresponding overflow pollutant mass loadings 1 , , L f f  based on PySWMM [25] , a Python language-based package for the creation and manipulation of the structure, dynamics, and function of USEPA stormwater management model.After the comparison of 1 , , L f f  , the minimum overflow pollutant mass loading and the corresponding optimal retrofitting strategy for LPUs were attained.
Step 4: Retrofitting strategies of LPUs and the corresponding overflow pollutant mass loadings are integrated to calculate the Sobol sensitivity index (SSI) of decision-making variables to help identify the dominant LPUs and update the optimal retrofitting strategy.
To explain SSI, the following generic model description is used [26] where is the set of p decisionmaking variables, Y is the model output representing the overflow pollutant mass loading in the study.The function f can be further decomposed into summands of increasing dimensionality.
If the input variables are independent and each term of the equation is chosen with zero average and is square integrable, 0 f is a constant and equal to the expectation value of the output, and the summands are mutually orthogonal.Additionally, this decomposition is unique.
The total unconditional variance can be defined as where p  represents the p-dimensional unit hyperspace (i.e., variable ranges are scaled between 0, 1).The partial variances, which are the components of the total variance decomposition, are calculated from each of the terms in Eq. ( 7) 1 With the assumption that the variables are mutually orthogonal, this results in Eq. (10) for the variance decomposition.
In this way, the variance contributions to the total output variance of individual variables and variable interactions can be determined.These contributions are characterized by the ratio of the partial variance to the total variance, that is SSI: Second-order SSI = ( ) The first-order SSI is a measure for the variance contribution of individual variable i X to the total model variance.The partial variance i V in Eq. ( 11) is given by the variance of the conditional expectation = ( ( | )) and is also called the main effect of variable i X on model output Y .The impact on the model output variance of the interaction between variable i X , j X is given by ij S , and i T S is the result of the main effect of i X and all its interactions with the other variables (up to the th p order).
The calculation of i T S can be based on the variance i V  that results from the variation of all variables, except i X = 1 ( ) For additive models and under the assumption of orthogonal input factors, S is greater than 1, while the sum of i S is less than 1.By analyzing the difference between i T S , i S , the impact of the interactions between variable i X and the other variables can be determined.Previous study shows that variables with first-order SSI larger than the sensitivity thresholds 0.01 and 0.10 exert weak and strong dominant impacts on model output [27][28][29][30][31] .

Model construction
Prior 1 to the coupling of Sobol algorithm, a mathematical model for the studied sewer network was established.First, a catchment runoff module was incorporated with a one-dimensional hydrodynamic model to simulate the longitudinal and temporal water level and flow rate within the sewer system.With simulated in-sewer hydrodynamic patterns, a onedimensional water quality model, the catchment pollutant flushing module, was further employed to simulate longitudinal and temporal profiles of the pollutant concentrations in the sewer system.In the study, chemical oxygen demand (COD) was used as the water quality indicator because it is the priority control pollutant for the receiving waterbodies [32] .
For this study site, the developed model consists of 12 subcatchments, 212 sewage pipes, 5 pumping stations and 7 overflow outlets.Information for each divided sub-catchment and background discharges connecting to the sewer network under dry-weather period are presented in our previous study [1] .

Model calibration
The established model was calibrated with the measured flows and COD concentrations within the sewer system on June 5, 2017, when a precipitation event of 38 mm mainly occurred during 14:00-16:00 (as shown in Fig. 4(a)).Specifically, the time-series flow at WWTP inlet (i.e., site f1) was obtained by pumping operation records, and the in-sewer sewage at measurement site c1 was continuously collected at 10 min interval using polyethylene bottles, resulting in total of 13 samples.The collected sewage samples were transported back to our laboratory for immediate processing and COD concentration analysis.After that, the sewer flow model was calibrated with measured time-series data at f1 monitoring site, and the sewer water quality model was calibrated with measured time-variable COD concentration at c1 sampling site.The modeling calibration results are shown in Figs.  1.

Model validation
The calibrated model was further validated with the measured time-variable flow and COD concentration within the sewer system on July 1, 2017, when a precipitation event of 46 mm mainly occurred during Cleaning efficiency for green area/% 0 Wash-off coefficient for green area 0.01 Wash-off exponent for green area

Current situation of sewage conveyance and pollution overflow
According to the historical rainfalls recorded from the local meteorological department, 93% of the rainfalls were below the level of a one-year return period, which corresponded to approximately 42 mm within one precipitation duration [32] .Therefore, the design scenario of a one-year return period was employed to study the retrofit of LPUs and CSO pollution reduction.The Chicago approach was used to generate a synthetic hyetograph of a one-year event based on the local stormwater intensity formula [1] .Accordingly, the designed instantaneous rainfall intensity under a one-year return period is provided in Fig. 6(a).Under the rainfall scenario, the LPUs of the sewer system were identified and the time-series COD concentrations of runoff and overflows were analyzed.

Identification of LPUs
Before the rainfall (00:00-9:00), sewage pumps in every pumping station operated below their maximum flow rates, and there were no overflows in the whole system.About 30 min after the start of rainfall (see Fig. 6(b)), sewage lifting pumps of pumping station III upstream of Xier Pool reached their maximum capacities, and three upstream pipes of the station were filled-up, while the downstream pipes still possessed sufficient conveying capacity (the fullness of the downstream pipes was less than 0.4).This result indicates that pumping station III and the three pipes are the LPUs of the sewer system.Then, the fullness of eight pipes along Xihuancheng River was close to 1.0 but the upstream and downstream pipes still have sufficient spaces for sewage conveyance (as presented in Fig. 6(c)), reflecting the blocking effects of the eight LPUs on sewage delivery.Meanwhile, all According to the above analysis, the insufficient conveyance capacities of LPUs (mainly pumping stations and pipes) are primarily responsible for CSO pollution discharges of the sewer system during wet-weathers.

Time-series COD concentrations of runoff and overflow
Based on the validated sewer network model and the designed storm hyetograph under a one-year return period, the simulated time-series COD concentrations in sub-catchment runoff and outlet overflow are presented in Fig. 7. Figure 7 shows that high COD concentration (> 200 mg / L) mainly occurred at 9:30-10:00, 9:45-10:40 in sub-catchment runoff and overflow.With the consideration of the time from runoff generation to inflow into drainage system (about 15 min) [33] , the sub-catchment runoff with high COD concentration would enter into the sewer system at the period of 9:45-10:15, which was partly consistent with the overflow time of sewage with high COD concentration.As depicted in Figs.6(c)-6(e), during 9:45-10:15, the sewer network was severely blocked by LPUs.It indicates that the occurrence of local sewage conveyance failure caused by LPUs may be the most important factor to induce the runoff with high COD concentration directly discharging into receiving waterbodies.

Performance of the optimal retrofitting strategy
Based on the above analysis, LPUs of the studied sewer system were determined, and their contributions to the occurrence of CSOs was also proved.In order to effectively achieve the target of overflow pollution mitigation, the LPUs of sewer system should be given high priorities to be retrofitted.Among the four receiving water bodies, the sewer system along Donghuancheng River is the focus of retrofit due to the repetitive occurrences of black-foul problems in the river.Under the current operational situation of the sewer system, COD mass loading overflowed into Donghuancheng River through outlets 1-4 was about 1 749 kg.To abate the overflow pollution, this study designed a series of retrofitting strategies for LPUs using Sobol sequence.As mentioned above, the LPUs along Donghuancheng River consisted of three sewage lifting pumps of pumping stations I, II and 12 pipes (P s1 -P s12 ) upstream and downstream of these stations.The total length of these pipes was 437 m. Figure 8 lists the current values and allowable ranges of the decision-making variables of these LPUs.
After introducing retrofitting strategies for the LPUs, the overflow COD mass loading during the optimal process is illustrated in Fig. 9.In this process, the number of retrofitting strategies generated by the Sobol sequence was 2 000.By implementing the optimal retrofitting strategy, the minimum overflow COD mass loading per overflow event could be 1 039 kg, which is 40.6% lower than that of before optimization.The optimal retrofitting strategy includes the following measures: (1) Increasing the conveyance capacities of the two sewage lifting pumps of pumping station I from 0.10 m³/s to 0.25 m³/s, and the sewage lifting pump of pumping station II from 0.10 m³/s to 0.32 m³/s, (2) Augmenting the diameters of pipes P s1 -P s12 by 1.25-1.29 times.

Identification of the dominant LPUs and the performance of the updated optimal retrofitting strategy
To further identify the dominant LPUs, Sobol sensitivity analysis was performed.Result shows that the sum of the first-order SSIs of the LPUs was 0.89 less than that of total-order SSIs (1.04), which indicates that the interactions between LPUs may explain an important part of the CSO pollution discharge.The first-order SSIs of pipes P s1 , P s5 and P s9 were larger than 0.1, implying that these LPUs belong to the strong dominant LPUs and will induce significant impacts on the reduction of overflow pollution after being retrofitted.Meanwhile, sewage pumps I, II, pipes P s2 , P s4 , P s6 , P s7 and P s10 with the first-order SSIs in the range of 0.01-0.10were identified as the weak dominant LPUs, in other words, they exert weaker impacts than those of pipes P s1 , P s5 and P s9 .
Additionally, the first-order SSIs of pipes P s3 , P s8 , P s11 and P s12 were too small (< 0.01) to be excluded from the dominant LPUs.According to the topology structure of the sewer system along Donghuancheng River (see Fig. 1), the retrofits of reducing pipes and pipes directly connected with outlets (i.e., P s1 , P s5 and P s9 ) can explain the most significant impacts on overflow pollution abatement.By contrast, the pipes (i.e., P s2 , P s4 , P s6 , P s7 and P s10 ) that directly connected to the above three strong dominant LPUs only affect the abasement of overflow pollution indirectly.Also, the delivery capacities of sewage lifting pumps both in pumping stations I, II are important factors affecting overflow pollution abatement, which conforms that pumping stations working as hydrodynamic regulators can significantly influence the interception efficiency of sewer system.The non-dominant LPUs (i.e., pipes P s3 , P s8 , P s11 and P s12 ) are mainly located in the middle or downstream of those weak dominant LPUs, and passively accepted the upstream shock loading or downstream blockage.In other words, these units do not have to be primarily responsible for sewage delivery failure and CSO occurrence.Based on the screening for the dominant LPUs, the optimal retrofit strategy was updated to only retrofit the dominant LPUs to explore the abatement efficiency for sewer overflow pollution.Result shows that the released COD mass loading during the overflow process could be 1 110 kg, which is approximately 4% lower than before the update (1 039 kg).However, the retrofit length of pipes could be decreased from 437 m to 265 m, reaching about 40% reduction.It can be seen that the coupling of simulation-optimizing model and Sobol sensitivity analysis would help decision-makers to identify the dominant LPUs and respond to the sewer overflow abatement in a cost-effective way.

Conclusions
This study presented a combination of simulationoptimizing model and Sobol algorithm to help identify the dominant LPUs within sewer system, and design retrofitting strategies to cost-effectively abate CSO pollution.The following are the major findings: Under the designed rainfall scenario, the evolution process of sewage delivery of the studied sewer system was presented.Result shows that the sewage conveyance failure successively occurred along the receiving waters.As a result, a large amount of sewage would be discharged into the nearby waterbodies.Based on the time-variable simulation, LPUs at different region were identified.
Focusing one river that was mostly within the studied catchment, this paper used Sobol sequence to design a series of retrofitting strategies for LPUs that located at the bank of the river.After embedding these strategies into sewer system model, the optimal retrofitting strategy was obtained, which could reduce the overflow COD mass loading per overflow event by 40.6%.
The effects of LPU retrofits on CSO pollution reduction were further assessed by SSI to identify the dominant LPUs.As a result, 10 dominant LPUs were screened from all LPUs and further used to update the optimal retrofitting strategy.Using the updated strategy that only associated with the dominant LPUs, the pipe retrofitting length could be decreased by 40%, while the reduction rate of COD mass loading (36.6%) was close to that before the update (40.6%).
Overall, this paper provides a method to identify LPUs that need to be preferentially retrofitted within sewer systems, and then help cost-effectively mitigate CSO pollution during wet-weathers.Future research will be directed to the identification and retrofit of LPUs at the system-wide level, for the purpose of comprehensively promote the quality of water environment of urban catchment.

Fig. 1 (
Fig. 1 (Color online) Description of the study site capacities of lifting sewage pump m , M is the number of sewage lifting pumps to be retrofitted,

Fig. 3 (
Fig. 3 (Color online) Architecture of simulation-optimizing model and the coupling with Sobol sensitivity analysis

1
1.For non-additive models in which the interactions exist, i T S is larger than i S with the sum of all i T

2 R
4(b), 4(c), and the large values indicate the model's acceptability.The calibrated parameters for the established sewer model are shown in Table

Fig. 4 (
Fig. 4 (Color online) Precipitation event (a) and modeling calibration results for (b) in-sewer flow and (c) COD concentration sanitary sewage inflow/(mgL 1 ) 167.0 10:00-12:30 (as shown in Fig. 5(a)).The modeling validation results calculated by the time-series flow at WWTP inlet (i.e., site f1) and COD concentration data (16 sewage samples) at the site c1 are shown in Figs.5(b), 5(c).The large 2 R values reflected the acceptance of the validated model, and thus it can be applied to analyze issues of concern in other similar scenarios.

Fig. 5 (
Fig. 5 (Color online) Precipitation event (a) and modeling verification results for (b) in-sewer flow and (c) in-sewer COD concentration

Fig. 8 (
Fig. 8 (Color online) The current values and allowable ranges of decision-making variables of LPUs

Fig. 9 (
Fig. 9 (Color online) Simulated overflow COD mass loading during the optimal process