Optimization of Chlorine Injection Schedule in Water Distribution Networks Using Water Age and Breadth-First Search Algorithm
1. Introduction
Previous studies have presented effective methods for maintaining residual chlorine levels and reducing mass disinfection at the water source. This study emphasizes the significance of optimizing the initial chlorine injection at the source and reducing computational difficulties in WDN water quality simulations. Computational difficulties primarily refer to the challenges posed by dealing with many variables when multiple chlorine boosters are employed, each requiring optimization to work effectively with the dynamic behavior of the WDN. This study focuses on a single variable—chlorine injection at the source—by considering water age as a factor. Research on the combined use of water age and chlorine level in water quality simulations is increasing. However, the incorporation of water age in estimating the initial chlorine at the source has not been considered despite its substantial potential for enhancing water quality assessment in WDNs.
This study focuses on methodologies for estimating the initial chlorine injection at the source as part of chlorine injection scheduling using water age. The objectives of this study are as follows: (1) to assess water age performance in estimating the initial chlorine injection and (2) to develop an optimal schedule for chlorine injection at the source to maintain chlorine levels of 0.2–1.0 mg/L at demand nodes. This approach estimates the initial chlorine injection based on water age while considering the first-order decay equation. Furthermore, the individual wall decay for each pipe is calculated using the breadth-first search (BFS) algorithm by tracing the possible water delivery pathways from the source to each demand node. Subsequently, the optimal chlorine injection schedule is determined using a genetic algorithm (GA). This study introduces a new approach for optimizing chlorine injection, and enhancing the efficiency and practicality of WDNs.
2. Materials and Methods
2.1. Chlorine Decay
where represents the chlorine concentration (mg/L) at time , is the initial chlorine concentration (mg/L), is the residence time within the network (day), and is the chlorine decay constant (day−1). The decay constant, , is determined by the sum of two components: the bulk reaction constant () and pipe wall reaction constant (), representing reactions within the bulk fluid and with the pipe wall material, respectively. Mathematically, can be expressed as
where is the mass transfer coefficient (m/day), is the Sherwood number, is the molecular diffusivity responsible for the spreading component owing to molecular motion (m2/day), and is the pipe diameter (m). The value of varies with the type of fluid flow. In a laminar flow, is calculated as follows:
where represents the Reynolds number, represents the Schmidt number (), characterizing the kinematic viscosity of water divided by the molecular diffusivity, and denotes the pipe length (m). For a turbulent flow, the expression for becomes
where denotes the wall reaction rate (m/day) and represents the pipe radius (m).
2.2. Study Area
2.3. Hydraulic and Water Quality Simulation
2.4. Water Delivery Pathway Identification
where represents the value for node , is the value of pipe obtained using Equation (6), is the flow rate at pipe , and is the number of pipes in the flow path from the source to node .
2.5. Chlorine Injection Estimation
While the variables remained the same as in Equation (1), and were redefined in this study. represents the required chlorine concentration to be injected at the source at time to maintain the target chlorine level at the demand node. In this study, the target chlorine residue at the node was set to 0.2 mg/L, whereas the maximum chlorine level was limited to 1.0 mg/L. In Equation (8), is the water age of the target node at time .
Notably, each node has distinct hourly requirements and applying all requirements at the source is unfeasible. To optimize the chlorine injection schedule at the source, the maximum required chlorine for each hour was collected from all demand nodes, resulting in 24 specific values. The minimum (i.e., 0.2 mg/L) and maximum values were then set as boundaries for the chlorine injection optimization process described in the next section.
2.6. Chlorine Injection Optimization
subject to
where represents the number of nodes in the network, denotes the residual chlorine concentration at node at time , and denotes the target chlorine concentration (0.2 mg/L). A smaller MAPE value indicates a more uniform chlorine distribution within the WDN, with zero being the ideal value. The GA was executed with a population size of 100 and run for 1000 generations with the assistance of the Pymoo library [34].
3. Results and Discussion
Chlorine injection was used to assess the water age approach. Two scenarios were presented, each with three different values, to reflect the water treatment plant operation and account for seasonal temperature changes. The chlorine injection was optimized to obtain the optimal schedule.
3.1. Required Chlorine Injection Estimation
3.2. Potential Source-Injection Chlorine Range
3.3. Chlorine Injection Optimization
3.4. Spatiotemporal Chlorine Residuals
4. Conclusions
This study established a chlorine injection optimization model for maintaining uniform residual chlorine concentrations and reducing mass injection. By incorporating water age in estimating the required chlorine injection concentration, this method offers a more efficient approach for determining the optimal chlorine injection schedule by predetermining the potential required source-injection chlorine rate. The performance of this method was assessed based on percentage errors, yielding promising results, with an average error of less than 10% observed across all demand nodes.
The chlorine schedule was optimized using the GA under two operational scenarios: single and four-interval injection. The results demonstrated that the chlorine injection concentration and demand pattern were inversely correlated, demonstrating that chlorine injection should be increased during low-demand periods owing to long water retention times in the pipe. The four-interval injection slightly outperformed the single-injection scenario owing to its flexibility in injection decisions. This study focused on the flexibility of the method to accommodate fluctuating demands, offering an efficient, simple, and fast method for estimating chlorine injection for water treatment operations.
Analysis of residual chlorine concentrations across the network revealed spatial variability, with elevated levels at nodes close to the source. The distribution was generally acceptable for lower values (0.1056 and 0.1872 day−1 for winter and spring/fall seasons, respectively), with MAPE values of 18 and 39% from the targeted value (0.2 mg/L), respectively. A substantial increase in the error at 158% was observed for = 0.576 day−1 (operation in summer). Nevertheless, the results demonstrated that the proposed method could effectively manage chlorine injection under fluctuating seasonal temperature conditions.
The findings of this study provide valuable insights into effectively managing chlorine levels and operations of WDNs. Using spatiotemporal uniformity facilitated by water age is a promising strategy for enhancing network performance and chlorine distribution. This research paves the way for utilizing water age in chlorine estimation, with the potential for further refinement and application for water quality management in WDNs. For future research, a significant improvement direction will involve expanding the scope to include multiple sources and storage tanks. Additionally, introducing more objectives in the optimization approach aims to enhance the comprehensiveness of the model’s results. Another essential aspect is assessing the model by comparing it with existing approaches, providing a substantial justification for the model’s benefits, and identifying areas for improvement based on other methodologies. These endeavors can offer valuable insights and pave the way for potential avenues in future research.
Author Contributions
Conceptualization, F.D.F. and D.K.; methodology, F.D.F. and M.S.M.; software, F.D.F. and M.S.M.; investigation and data analysis, F.D.F., M.S.M. and D.K.; writing—original draft, F.D.F.; writing—review and editing, M.S.M. and D.K. All authors have read and agreed to the published version of the manuscript.
Funding
This study was supported by (1) the Korea Institute of Energy Technology Evaluation and Planning (KETEP) and the Ministry of Trade, Industry, and Energy (MOTIE) of the Republic of Korea (20224000000260), and (2) the Korea Environment Industry & Technology Institute (KEITI) through the Water Management Program for Drought, funded by the Korea Ministry of Environment (MOE) (RS-2023-0023194).
Data Availability Statement
The data presented in this study are available upon request from the corresponding author. The data are not publicly available due to privacy concerns.
Conflicts of Interest
The authors declare no conflicts of interest.
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Figure 1.
Flowchart of chlorine injection schedule optimization using water age and BFS algorithm.
Figure 1.
Flowchart of chlorine injection schedule optimization using water age and BFS algorithm.
Figure 2.
Application network.
Figure 2.
Application network.
Figure 3.
Demand pattern and injection schedule interval of Scenario 2.
Figure 3.
Demand pattern and injection schedule interval of Scenario 2.
Figure 4.
Demonstration of BFS in a sample WDN.
Figure 4.
Demonstration of BFS in a sample WDN.
Figure 5.
Required chlorine injection estimation in a sample WDN: (a) required chlorine injection at hour 4; (b) required chlorine injection at hour 19.
Figure 5.
Required chlorine injection estimation in a sample WDN: (a) required chlorine injection at hour 4; (b) required chlorine injection at hour 19.
Figure 6.
Required source-injection chlorine concentration for each node at hours 4 and 19: (a) required chlorine injection at hour 4; (b) required chlorine injection at hour 19.
Figure 6.
Required source-injection chlorine concentration for each node at hours 4 and 19: (a) required chlorine injection at hour 4; (b) required chlorine injection at hour 19.
Figure 7.
Chlorine injection assessment: minimum, average, and maximum percentage of error (average value): (a) kb = 0.1056 day−1; (b) kb = 0.1872 day−1; (c) kb = 0.576 day−1.
Figure 7.
Chlorine injection assessment: minimum, average, and maximum percentage of error (average value): (a) kb = 0.1056 day−1; (b) kb = 0.1872 day−1; (c) kb = 0.576 day−1.
Figure 8.
Potential source-injection chlorine ranges for two injection scenarios: (a) single injection range; (b) 4-interval injection range.
Figure 8.
Potential source-injection chlorine ranges for two injection scenarios: (a) single injection range; (b) 4-interval injection range.
Figure 9.
Comparison of chlorine injection schedules at the source for two scenarios: (a) kb = 0.1056 day−1; (b) kb = 0.1872 day−1; (c) kb = 0.576 day−1.
Figure 9.
Comparison of chlorine injection schedules at the source for two scenarios: (a) kb = 0.1056 day−1; (b) kb = 0.1872 day−1; (c) kb = 0.576 day−1.
Figure 10.
System-wide temporal variation in residual chlorine concentration: (a) Scenario 1—kb = 0.1056 day−1; (b) Scenario 2—kb = 0.1056 day−1; (c) Scenario 1—kb = 0.1872 day−1; (d) Scenario 2—kb = 0.1872 day−1; (e) Scenario 1—kb = 0.576 day−1; (f) Scenario 2—kb = 0.576 day−1.
Figure 10.
System-wide temporal variation in residual chlorine concentration: (a) Scenario 1—kb = 0.1056 day−1; (b) Scenario 2—kb = 0.1056 day−1; (c) Scenario 1—kb = 0.1872 day−1; (d) Scenario 2—kb = 0.1872 day−1; (e) Scenario 1—kb = 0.576 day−1; (f) Scenario 2—kb = 0.576 day−1.
Figure 11.
Average nodal chlorine concentration for kb = 0.1056 day−1: (a) Scenario 1; (b) Scenario 2.
Figure 11.
Average nodal chlorine concentration for kb = 0.1056 day−1: (a) Scenario 1; (b) Scenario 2.
Figure 12.
Average nodal chlorine concentration for kb = 0.1872 day−1: (a) Scenario 1; (b) Scenario 2.
Figure 12.
Average nodal chlorine concentration for kb = 0.1872 day−1: (a) Scenario 1; (b) Scenario 2.
Figure 13.
Average nodal chlorine concentration for kb = 0.576 day−1: (a) Scenario 1; (b) Scenario 2.
Figure 13.
Average nodal chlorine concentration for kb = 0.576 day−1: (a) Scenario 1; (b) Scenario 2.
Table 1.
Chlorine parameter data.
Table 1.
Chlorine parameter data.
Temp (°C) | (day−1) | (m2/s) | (m2/s) |
---|---|---|---|
4.5 | 0.1056 | 1.55 × 10−6 | 6.74 × 10−10 |
18.0 | 0.1872 | 1.06 × 10−6 | 1.14 × 10−9 |
25.0 | 0.576 | 9.03 × 10−7 | 1.38 × 10−9 |
Table 2.
Chlorine injection scenarios.
Table 2.
Chlorine injection scenarios.
Scenario | Operation Method | Number of Decision Variable | Interval Time Step (hour) |
---|---|---|---|
1 | Single Injection | 1 | 24 |
2 | 4-Interval Injection | 4 | 8–6–4–6 |
Table 3.
Statistics of optimal results for two scenarios.
Table 3.
Statistics of optimal results for two scenarios.
Statistics | kb = 0.1056 | kb = 0.1872 | kb = 0.576 | |||
---|---|---|---|---|---|---|
Sce.1 | Sce.2 | Sce.1 | Sce.2 | Sce.1 | Sce.2 | |
Mean Concentration (mg/L) | 0.243 | 0.237 | 0.279 | 0.285 | 0.521 | 0.516 |
Standard Deviation Concentration (mg/L) * | 0.044 | 0.038 | 0.086 | 0.081 | 0.332 | 0.328 |
Minimum Concentration (mg/L) | 0.206 | 0.200 | 0.204 | 0.200 | 0.200 | 0.200 |
Maximum Concentration (mg/L) | 0.260 | 0.270 | 0.320 | 0.330 | 0.730 | 0.740 |
Dosage Input (kg/day) [A] | 92.7 | 90.5 | 114.1 | 111.9 | 260.3 | 258.1 |
Dosage Output (kg/day) [B] | 87.0 | 84.8 | 102.0 | 99.9 | 188.4 | 186.7 |
Dosage Decay (kg/day) [A-B-C] | 8.0 | 7.9 | 14.8 | 14.6 | 77.2 | 76.6 |
Dosage Storage Change (kg/day) [C] | −2.26 | −2.21 | −2.68 | −2.63 | −5.20 | −5.15 |
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