Qianhai Cooperation Zone is located in the southwest part of Shenzhen, where is east to Pearl River Bay area and opposite to Hong Kong SAR as shown in Figure 1 . As a prominent platform for realizing Guangdong-Hong Kong-Macao cooperative strategy, Qianhai has developed infrastructure construction and prosperous economical level, thus drawing increasing attention in recent years. This area belongs to subtropical monsoon climate, with an annual average rainfall of 1593 mm and an average temperature of 22.2˚C. Substantial rainfall brings large amount of surface runoff and several rivers flow into bay area therein.

With frequent events of urban flood and aquatic pollution, the existed pipelines system of Qianhai can’t deal with the problem of stormwater effectively. Due to the increasing water level of outfall and rainwater flowing, local pump stations fail to drain out all rainwater in time, and pollutant loads will also surpass environmental standards under certain conditions. Hence, a more developed and effective drainage system can be essential therein.

Deep tunnel system has been regarded as the large-scale solution to flooding damage and water pollution due to severe storms ( Owusu et al., 2013 ). Besides the differences in structural characteristics and topographical properties, these systems may share similar functionality and architectural configuration. Namely, the excessive inflow that overwhelms interceptors may overflow through flow diversion and splitting structures to deep tunnels draining to storage reservoirs further downstream. Flow regulators were installed to manage flow rates to the tunnels based on water levels in the system. The deep tunnel system provides temporary storage of the combined sewage which will be pumped and delivered to nearby WTPs for treatment ( Luo et al., 2021 ).

The fitting method is based on the fitting process between the pollutant convective diffusion equation and observed data, aims to work out corresponding parameters to describe the process of pollutant degradation. The original convective diffusion equation can be demonstrated as Equation (1), where A is the area of cross-section, C is the pollutant concentration, T is the amount of mass conduction, x is the axis direction, K is the degradation coefficient of pollutant, C_s is the concentration of source, and q is the flow of source.

$\frac{\partial }{\partial t}\left(AC\right)+\frac{\partial T}{\partial x}=-AKC+C_{s}q$ (1)

Considering the unchanged condition of the observed environment, the amount of mass conduction T is neglected. If the variable AC is replaced by u, the equation can be expressed as Equation (2), which is a first-order nonhomogeneous differential equation, with the form of general solution as Equation (3):

$\frac{\partial u}{\partial t}+Ku=C_{s}q$(2)

$u=\left({\displaystyle \int C_{s}q\cdot {\text{e}}^{{\displaystyle \int K\left(t\right)\text{d}t}}\text{d}t}+u_0\right){\text{e}}^{-{\displaystyle \int K\left(t\right)\text{d}t}}$(3)

Furtherly, the equation can be demonstrated as Equation (4) and Equation (5), thus the purpose of the fitting method is to work out the best parameters to fit the observed data.

$C=\left({\displaystyle \int C_{s}q\cdot {\text{e}}^{{\displaystyle \int K\left(t\right)\text{d}t}}\text{d}t}+A_0{C}_0\right){\text{e}}^{-{\displaystyle \int K\left(t\right)\text{d}t}}/A$(4)

$C=A\cdot {\text{e}}^{B{t}^2+Ct}$(5)

Compared with the traditional method of calibrating parameters between pollutant concentration and elapsed time, this work puts forward a novel algorithm to calculate degradation coefficient K under certain moment of Equation (1) and substitute it to SWMM source code for pollutant concentration computation. This optimized algorithm consists of three processes and mechanisms: the Main process, the Outlier filtration mechanism and the Gradient descent mechanism. Among which the Outlier filtration mechanism aims to improve the calculation precision and except the influence of outlier, and the Gradient descent mechanism could achieve better calculation efficiency. By obtaining the current time t of SWMM, this algorithm can treat it as an input variable and output the pollutant degradation coefficient K, then return it to the SWMM source code and keep computing for the next moment. The diagram (as shown in Figure 2 ) and demonstration of this algorithm are as follows:

1) Main process

a) The algorithm reads the elapsed time t of SWMM and the observed data with certain time;

b) Remove outlier with the mechanism (discussed later) and calculate the upper and lower limited concentration value C₀ and C after filtering outliers;

c) Calculate degradation coefficient K with the Gradient descent mechanism (discussed later), return K to SWMM source code and output the concentration of COD, NH₃-N, TN, TP of time t;

d) Refresh time t.

2) Outlier filtration mechanism

a) Obtain the observed data of time t, and traverse the sequence of concentration C;

b) For each C_i, calculate the average concentration after removing C_i;

c) Set the allowable error range a, calculate concentration window [C', C''] with Equation (6) and Judge if C_i is in this window;

$\{\begin{array}{l}{C}^{\prime }=\overline{C_i}\times \left(1+a\right)\\ C^{″}=\overline{C_i}\times \left(1-a\right)\end{array}$(6)

where a represents the allowable range of error, [C', C''] is the allowable concentration window, C' is the upper limit of this window, C'' is the lower limit of this window and $\overline{C_i}$ is the average concentration after removing C_i;

d) If C_i is not in the allowable concentration window, remove it as an outlier and back to b), otherwise continue to calculate the next C_i of this series;

e) Judge if the dataset has changed, if not then output the average value C of the filtrated series and return this $\bar{C}$ to the Main process.

3) Gradient descent mechanism

a) After obtaining the upper and lower limited concentration value C₀ and C after filtering outliers, set the initial K, variant range b, and define gradient expression f(K) with the above concentration;

b) Calculate K' and K'' after variation with Equation (7), and obtain gradient after variation f(K'), f(K'')

$\{\begin{array}{l}{K}^{\prime }=K\times \left(1+b\right)\\ K^{″}=K\times \left(1-b\right)\end{array}$(7)

where K' is the upper limit of the variation value, K'' is the lower limit of the variation value.

c) Judge if min(f(K'), f(K'')) has reached the allowed error range, if not then set K matches min(f(K'), f(K'')), otherwise refresh K and back to b).

The rainfall series of study area is based on the rainfall intensity formula of Shenzhen, which is demonstrated in Equation (8). By obtaining such rainfall model, the corresponding rainfall series is input into SWMM for simulation, and results are output for validating the accuracy of this model.

$i=\frac{7.34\times \left(1+0.524\lg P\right)}{{\left(t+5.707\right)}^{0.551}}$(8)

where i is the rainfall intensity, mm/min; P is the rainfall recurrence period, a; t is the rainfall time, min.

As discussed in 2.2.2, the allowable error range a can be regarded as a key parameter for validating the efficiency of this algorithm: if this range is set too vast then such mechanism won’t filter the outlier effectively, while if the range gets too narrow it may delete some ordinary value. Therefore, various parameter a was set to simulate and generate results.

Meanwhile, simulation results of fitting method can also be incorporated to compare the simulated value and observed value, thus to validate the accuracy of this study. For furtherly carrying error analysis, the indicators of RMSE, MSE and MAE are selected as indicators to present the efficiency (Equation (9)-Equation (11)).

$\text{RMSE}=\sqrt{\frac{{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(Y_i^{obs}-Y_i^{sim}\right)}^2}}{n}}$(9)

$\text{MRE}=\frac{{\displaystyle \underset{i=1}{\overset{n}{\sum }}\left|\frac{Y_i^{obs}-Y_i^{sim}}{Y_i^{obs}}\right|}}{n}$(10)

$\text{MAE}=\frac{\sqrt{{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(Y_i^{obs}-Y_i^{sim}\right)}^2}}}{n}$(11)

where $Y_i^{obs}$ represents the observed data, $Y_i^{sim}$ represents the value of prediction and $\overline{Y_i^{obs}}$ is the average value of the observed data.

During the water quality experiments, COD, NH₃-N, TN and TP are chosen as pollutant indicators of the observed data. Accordingly, the aforementioned indicators are simulated and output in SWMM by substituting different allowable error range a into the source code. Experiment data of laboratory is regarded as observed value and modelling data of SWMM is treated as simulated value. All the error analysis results are presented in Figure 3 .

As shown in Figure 3 , the varying trend of simulation error tends to increase before decreasing with a increases, and reaches its peak when a equals 0.3, which matches with the facts mentioned before. Furtherly, nearly all pollutant indicators have the least error value when a equals 0.5, and commonly behave better than the fitting method. Although the amount of data is limited, such algorithm has still obtained ideal results as for simulation accuracy compared with the fitting method, thus it’s expected to work better with substantial observed data.

The optimized algorithm utilizes an iterative method to work out the best pollutant degradation coefficient K, while the fitting method directly calculates such a parameter, thus the algorithm may increase the computation consumption accordingly. For solving this problem, the gradient descent mechanism is put forward to decrease iterative times. To present the effect of optimization, the average iterative amounts and average iterative time are selected as representative indicators to demonstrate, the comparing results between the gradient descent mechanism and the ordinary iterative method are shown in Figure 4 .

As shown in Figure 4 , the gradient descent mechanism has obvious optimized effects compared with the common iterative method. In terms of iterative amounts, the gradient descent mechanism has about 2500 times of iteration for four pollutant indicators, while that of the common iterative method is about 33,000, thus the gradient descent mechanism nearly reduces 92.42% of the amounts. As for the iterative time, the gradient descent mechanism takes nearly 0.004s to converge while that of the common iterative method approaches 0.22 s, which denotes such an optimized algorithm improves 55 times of the computation efficiency. Similarly, the amount of data has constrained the function of this algorithm as discussed in the Section 3.2.1, while it still shows ideal performance as for the comparative results, thus it may harvest more in the latter research.