water Article Optimal Control Strategy of a Sewer Network Iulian Vasiliev 1,2, *, Laurentiu Luca 1 , Marian Barbu 1 , Ramon Vilanova 2 and Sergiu Caraman 1 1 Department of Automation and Electrical Engineering, Dunarea de Jos University of Galati, Domneasca No. 47, 800008 Galati, Romania; laurentiu.luca@ugal.ro (L.L.); marian.barbu@ugal.ro (M.B.); sergiu.caraman@ugal.ro (S.C.) 2 Department of Automation Systems and Advanced Control Research, Autonomous University of Barcelona, 08193 Barcelona, Spain; ramon.vilanova@uab.cat * Correspondence: iulian.vasiliev@ugal.ro Abstract: This paper proposes a series of methods to increase the efficiency of the operating of a sewer network that serves a medium-sized city with a population of 250,000 inhabitants. The sewer network serves five areas of the city and consists of seven tanks that communicate with one another and with the treatment plant through pipes. The controls are applied to the process by valves and pumps. The main objective of this paper is to determine the optimal controls to minimize two performance criteria: volume of overflow, and overflow quality index. The sewer network was modeled in the BSMSewer environment. The optimization of the operating of the sewer network was carried out in the conditions of an influent computed in relation to the number of inhabitants and to the area served, using genetic algorithms as a method of optimization. Five optimization strategies were analyzed by numerical simulation. The analysis of the five strategies was done by comparison of their results with one another, as well as in relation to the case where all of the controls were set at maximum values of 100%. The simulations showed that the third strategy produced the best results in relation to each of the two criteria.   Keywords: sewer network; wastewater; control strategy; optimization; genetic algorithm Citation: Vasiliev, I.; Luca, L.; Barbu, M.; Vilanova, R.; Caraman, S. Optimal Control Strategy of a Sewer Network. Water 2022, 14, 1062. https://doi.org/10.3390/ 1. Introduction w14071062 The development of human communities has led to an increase in the volume of polluted water that must be treated and discharged into natural receptors (rivers, lakes, Academic Editors: Harsha Ratnaweera and Scott A. Wells groundwater etc.). On average, a person uses between 79 and 307 L of water every day for drinking, cooking, washing, cleaning, and hygiene. Naturally, this water used must Received: 19 February 2022 be collected and treated before returning to the environment. Industrial activities also Accepted: 24 March 2022 produce wastewater that can endanger the environment when discharged. This has led to Published: 28 March 2022 the adoption of strict environmental protection legislation regarding the volume and the Publisher’s Note: MDPI stays neutral quality of water discharged into natural receptors. Currently, the issue of collecting and with regard to jurisdictional claims in treating polluted water is treated in an integrated way, as clearly set forth by the European published maps and institutional affil- Water Quality Directive adopted by the European Union in 2000. iations. The main issues that arise regarding wastewater are as follows: 1. The wastewater collection from various domestic or industrial activities and/or stormwater. It should be noted that in case of large rainfall events and significant amounts of stormwater, it is possible to overload water collection systems, leading Copyright: © 2022 by the authors. to discharges of the excess water into the environment—discharges that can cause Licensee MDPI, Basel, Switzerland. flooding or lead to soil and natural water pollution; This article is an open access article distributed under the terms and 2. The wastewater treatment in treatment plants, after which it can be discharged into conditions of the Creative Commons natural receptors. Attribution (CC BY) license (https:// The wastewater collection is carried out within the sewer networks, which consist of creativecommons.org/licenses/by/ pipes, tanks, pumps, etc. Several factors influence the design of a sewer network, including 4.0/). ground conditions, topography, effluent composition, and construction sequencing. These Water 2022, 14, 1062. https://doi.org/10.3390/w14071062 https://www.mdpi.com/journal/water Water 2022, 14, 1062 2 of 25 factors determine whether networks are transverse, perpendicular, parallel, zonal, or radial. Wastewater that flows through the sewer system can be transported either through gravity flow or by implementing pumping stations in situations where gravity flow is not possible. Next, the wastewater is transported to the treatment plants, where the objective is to obtain an effluent that complies with the legislation in terms of its loading with pollutants before discharge. Current wastewater collection systems are of two types [1]: (1) Combined systems that collect both wastewater from human activity and stormwater, which is then transported to the wastewater treatment plants. Such wastewater collection systems are very sensitive to the quantities of stormwater they collect, with the water transported through them significantly changing both their flow and quality during the rainfall events; (2) separate systems—in these systems only the wastewater is transported to the treatment plant, with untreated stormwater being discharged into natural receptors. Regarding the modeling of the wastewater collection systems, detailed models have been adopted that describe the phenomena that take place in the sewer pipes in depth, as well as simplified models that can be used in control algorithms [2,3]. The most complex models are the hydrodynamic ones, which are based on Saint-Venant-type equations for one-dimensional non-stationary flow. Other models used are those based on hydrological models, in which the pipes are described by first-order differential equations or those described in [4], where the sewer network is modeled as a hybrid system containing both subsystems, with continuous states as well as discrete states. The authors of [5] present a dynamic model that describes the reactions of the com- pounds in the sewer pipes. The purpose of the study was to analyze the reaction rates involved in these reactions (the growth rate of microorganisms, feeding rate, consumption rate, oxygen supply rate, etc.). In [6], an analysis of the pollution level of discharged water during pluviometric events is made. A number of variables are considered (the maximum turbidity, the total event rainfall, the previous dry weather period, etc.), with the relations between these variables being defined through a statistical model. The authors of [7] propose a simplified semi-distributed urban water quality model, EmiStatR, which brings uncertainty and sensitivity analyses of urban drainage water quality models within reach of practitioners. In [8], the authors use “deep learning” methods to predict water quality in urban sewer networks. Water quality analysis and prediction is based on the following indicators: environmental indicators (such as area and diameter); social indicators (such as popula- tion); water quantity indicators (such as drinking water supply, wastewater flow, water velocity, and liquid level), and water quality indicators that are easy to monitor (e.g., pH, temperature, and conductivity). The authors of [9] analyzed how the measurement and the life cycle of an integrated system (wastewater treatment plant (WWTP) and combined sewer system) could be influ- enced by rainfall. This study was based on correlation analyses and regression models to assess the impact of precipitation on the environmental quality. In [10], a benchmark simulation model (BSMSewer) is described in order to evaluate control strategies for the urban catchment and sewer network. It contains various modules describing wastewater generation in the catchment, along with its subsequent transport and storage in the sewer system. The authors of [11] present a library of dynamic modeling tools for estimating micropollutant fluxes within an integrated urban wastewater and stormwater system (including a drainage network, stormwater treatment units, wastewater treatment plants, sludge treatment, and the receiving water body). In [12], detailed models of sewers, treatment plants, and receiving waters were created in order to describe the performances according to the individual needs and objectives. The literature reports various control strategies. A flow control system that operates on an existing sewer system is presented in [13]. The practically implemented method is model-based control. Online radar data and radar prognosis are used to model the input flow into the sewer system. Based on these data, the model computes the optimized Water 2022, 14, 1062 3 of 25 set-points for flow control at 12 stormwater retention tanks in the sewer system. The system’s operation significantly reduces the amount of rainwater discharged into rivers and water bodies, and more rainwater is treated in the wastewater treatment plant. The authors of [14] present an automatic control strategy (equal filling degree) that can be applied to any type of sewer network, regardless of its size. This method can be a starting point for additional control (e.g., by optimization or manual control). In [15] a method of water pollution control is presented that has the drawback of the need for a large-capacity storage tank. The control method (water-quality-based double-gates control strategy for combined sewer overflows pollution control) uses the chemical oxygen demand (COD) as a control criterion. The control method gives good results even in rainfall conditions. In [16], neural networks were used to predict the discharges in the sewer systems. Different neural networks were investigated using different sets of input and output data, resulting in the neural network with 50 neurons giving the best performance. In [17], a three-level hierarchical control structure for municipal wastewater systems is proposed; the first level contains sensors and actuators connected to the PLCs, the second level consists of advanced process control systems for wastewater treatment plants and sewer systems, and the third is dedicated to the administrative level. The authors of [18] present another three-level control structure: The first level is the adaptation level, in which the prediction of the network input flows based on the precipitation measurements takes place. The second level is the optimization level that calculates the trajectories for the state and control variables for the entire sewer network, and the third control level is the distributed control level, where the controls calculated by level 2 represent set-points for the decentralized control of each tank. The issue of controlling the wastewater collection networks has been addressed in many cases by using different optimization algorithms to avoid discharges, efficient use of the treatment plant, and efficient use of the storage tank network [3,18,19]. In [20], a quick pluvial flood warning system using rainfall thresholds is proposed. A tabu search algorithm is implemented with initial boundary conditions based on hydrological analysis in order to optimize the rainfall thresholds. The choice of the optimization method depends on the system’s characteristics, the available data, and the objectives of the control strategy. Thus, in [21], linear programming was used to control the storage capacity in a large wastewater collection system. Dynamic programming has been used to design a minimum cost sewer network [22] and to control the San Francisco sewer system [23]. The theory of squared linear control has been used in the multivariable linear control of wastewater collection systems [24,25]. Last, but not least, artificial intelligence, the Internet of things, and genetic algorithms have been used [19,26,27] for the design of wastewater collection systems and the control of their discharges into natural receptors, as well as multi-agent systems in optimization problems (e.g., negotiation mechanisms between various polluting entities for their access to the wastewater treatment plant) [28]. In this paper, the authors propose five optimization strategies based on genetic algo- rithms of a sewer network for a medium-sized city in Romania with 250,000 inhabitants. The network was modeled in the BSMSewer environment. The rest of this paper is structured as follows: Section 2 presents the structure and the characteristics of the sewer network, the influent provided by the five areas of the city, the performance criteria, and a brief description of the optimization method; Section 3 shows the simulation results obtained with the five optimization strategies, and the last two sections are dedicated to the discussions and conclusions. 2. Materials and Methods 2.1. Structure and Characteristics of the Sewer Network As stated above, a sewer network serving a medium-sized city with a population of 250,000 inhabitants was considered as a case study. The structure of this network is presented in detail in [29]. It contains 7 tanks, as shown in Figure 1, noted as TKi , i = 1...7. 2. Materials and Methods 2.1. Structure and Characteristics of the Sewer Network As stated above, a sewer network serving a medium‐sized city with a population of Water 2022, 14, 1062 4 of 25 250,000 inhabitants was considered as a case study. The structure of this network is pre‐ sented in detail in [29]. It contains 7 tanks, as shown in Figure 1, noted as TKi, 𝑖 1. . .7. Figure 1. Figure 1. The Thestructure structureofofthe thesewer network. sewer network. It was was considered considered that thatthe the77tanks tankscollect collectwastewater wastewaterand andstormwater stormwater from from 5 collec‐ 5 collection tion areas. It should also be mentioned that the first 4 tanks collect domestic wastewater areas. It should also be mentioned that the first 4 tanks collect domestic wastewater and and stormwater, stormwater, while while the the 5th 5th tanktank collects collects industrial industrial wastewater.InInFigure wastewater. Figure1,1,the thefollowing fol‐ lowing notations notations are used:are used: • ddii,, 𝑖i = 1. . .5, is 1...5, is the the inflow inflowof tank𝑖i from oftank from the thecorresponding correspondingcollection collectionarea; area; • u i, 𝑖 1. . .7, is the outflow of ui , i = 1...7, is the outflow of tank i;tank 𝑖; • qqover,i, ,𝑖 i =1.1...7, over,i . .7, represents representsthe theoverflow overflowofof each tank. each tank. Each collection area is characterized by its surface and the number of inhabitants Each collection area is characterized by its surface and the number of inhabitants living on that surface, as stated in Table 1. living on that surface, as stated in Table 1. Table 1. The characteristics of the collection areas. Table 1. The characteristics of the collection areas. Collection Area Collection Area Surface The Population Served Number (ha) Collection Area Surface (Number The Population Served of Inhabitants) Collection Area Number (ha) (Number of Inhabitants) 1 2500 75,000 2 1 5000 2500 50,000 75,000 3 2 6000 5000 40,000 50,000 4 3 7500 6000 60,000 40,000 5 3500 25,000 4 7500 60,000 Tanks 1–45 and tank 7 communicate via free 3500fall of the polluted water, using 25,000a valve at the outlet to control the outflow, while the other 2 tanks (tanks 5 and 6) use pumps to drainTanks the wastewater. 1–4 and tank 7 communicate via free fall of the polluted water, using a valve at the outletvolume The of each to control tank, 𝑉 while the outflow, , is computed the otherconsidering 2 tanks (tanksa value of6)0.05 5 and usempumps 3 for each to drain inhabitant of the wastewater.the collection area served by tank 𝑖 if it directly serves a collecting area or, conversely, 0.05 m3 for each of the average number of inhabitants served by the tanks The volume of each tank, Vi , is computed considering a value of 0.05 m3 for each whose outflow is part of the influent of tank 𝑖. For the tank that serves the area contain‐ inhabitant of the collection area served by tank i if it directly serves a collecting area or, ing the industrial zone, an additional capacity was added with the same value as the av‐ conversely, 0.05 m3 for each of the average number of inhabitants served by the tanks erage industrial wastewater production per day—𝑄 2500 m /day. whose outflow is part of the influent of tank i. For the tank that serves the area containing the industrial zone, an additional capacity was added with the same value as the average industrial wastewater production per day—Qind = 2500 m3 /day. The maximum flow at the output of tank i, Umax,i , is computed as follows: • For the tanks that use pumps to drain the wastewater, a constant value is considered representing the maximum flow of the pump; • For the gravitational tanks, the maximum flow is considered to be 3 times higher than the average wastewater production of the population served by tank i, or by Water 2022, 14, 1062 5 of 25 the tanks whose outflow is part of tank i’s inflow—the average domestic water Q pe = 0.15 m3 /(day·inhabitant). The volumes and the maximum output flow of the 7 tanks are given in Table 2. Table 2. The characteristics of the tanks of the sewer network. The Tank Volume Maximum Flow at the Tank Output Tank Number (m3 ) (m3 /day) 1 3750 33,750 2 2500 22,500 3 2000 18,000 4 3000 15,000 5 3750 18,750 6 2083 25,000 7 2500 112,500 The following assumptions were considered for the transport component of each collection area [10]: 1. A 15 min delay until the wastewater travels from one person to the tank corresponding to the collection area; 2. Twenty-five percent of TSS can accumulate in the piping, but no more than 10 kg/ha. The accumulated particulates will be flushed away when the flow increases over 8 times compared to the average domestic flow of the corresponding collection area (e.g., during pluviometric events). 2.2. The Influent of the Sewer Network The influent of the sewer network comes from the 5 collection areas previously men- tioned. It is defined on a time horizon of 28 days. Domestic wastewater and stormwater are collected from the first 4 areas, while industrial wastewater is collected from the 5th area, with loads specific to a medium-sized brewery industry [30]. Daily and yearly variations in the influent were also considered. Thus, all three components of the collected wastewater (domestic, industrial, and stormwater) can be modeled by time series based on previous information and/or on meteorological information. Figure 2 presents the influent generated for the 1st collection area, while Figure 3 shows the influent generated for the 5th collection area (the industrial part of the city). Table 3 gives the average loads and the average flow values of the entire time horizon of 28 days (39,968 values for each tank), expressed in kg/m3 . Table 4 shows the same average loads for all of the collection areas, expressed as concentrations. Table 3. Influent medium loads and flow values. Area CODsol CODpart NH+4 PO34− Flow Number (kg/day) (kg/day) (kg/day) (kg/day) (m3 /day) 1 720.71 4607.6 220.84 57.05 7786.2 2 487.97 3104.7 143.80 38.26 5283.0 3 421.18 2693.0 121.91 30.95 4400.5 4 589.17 3789.9 182.30 44.36 6369.4 5 2742.60 2051.3 132.67 61.22 4771.9 water are collected from the first 4 areas, while industrial wastewater is collected from the 5th area, with loads specific to a medium‐sized brewery industry [30]. Daily and yearly variations in the influent were also considered. Thus, all three components of the collected wastewater (domestic, industrial, and stormwater) can be modeled by time se‐ ries based on previous information and/or on meteorological information. Figure 2 pre‐ Water 2022, 14, 1062 6 of 25 sents the influent generated for the 1st collection area, while Figure 3 shows the influent generated for the 5th collection area (the industrial part of the city). Water 2022, 14, x FOR PEER REVIEW 6 of 27 (a) (b) (c) (d) (e) Figure 2. The influent corresponding to the 1st collection area: (a) soluble COD; (b) particulate Figure 2. The influent corresponding to the 1st collection area: (a) soluble COD; (b) particulate COD; COD; (c) ammonia; (d) phosphate; (e) flow of the 1st collection area. (c) ammonia; (d) phosphate; (e) flow of the 1st collection area. Table 4. Influent medium loads as concentrations. CODsol CODpart NH+4 PO34− Area Number (g/m3 ) (g/m3 ) (g/m3 ) (g/m3 ) 1 92.57 591.78 28.67 7.33 2 92.37 587.68 27.22 7.25 3 93.67 611.97 27.71 7.04 4 92.50 595.03 28.62 6.97 5 (a) 574.74 429.87 (b)27.81 12.83 2.3. Performance Criteria of a Sewer Network The efficiency of the operation of a sewerage network is given by 10 performance criteria implemented for BSMSewer [10], which are presented in Table 5. (c) (d) (e) Water 2022, 14, 1062 7 of 25 Figure 2. The influent corresponding to the 1st collection area: (a) soluble COD; (b) particulate COD; (c) ammonia; (d) phosphate; (e) flow of the 1st collection area. (a) (b) Water 2022, 14, x FOR PEER REVIEW 7 of 27 (c) (d) (e) Figure 3. The Figure influent 3. The corresponding influent to the corresponding to 5th collection the 5th area:area: collection (a) soluble COD; (a) soluble (b) particulate COD; (b) particulate COD; COD; (c) ammonia; (d) phosphate; (e) flow of the 5th collection area. (c) ammonia; (d) phosphate; (e) flow of the 5th collection area. Table 3 gives the average loads and the average flow values of the entire time hori‐ zon2.4. of 28Genetic Algorithm days (39,968 values for each tank), expressed in kg/m3. Table 4 shows the same The genetic average loads for all ofalgorithm is a metaheuristic the collection algorithm areas, expressed commonly used to find solutions as concentrations. to the optimization problems, inspired by the biological natural selection process. It uses Table 3. Influent medium a population loads and flowconsisting of chromosomes, values. of genes, that is a genetic representation of the solutions to the optimization CODsol problem. It 𝐍𝐇 CODpart uses biologically 𝐏𝐎𝟑𝟒 inspired Flow operators such as 𝟒 Area Number mutation, and selection to evolve from crossover, the initial population to a population (kg/day) (kg/day) (kg/day) (kg/day) (m3/day) containing 1 solutions 720.71 that minimize 4607.6 a fitness 220.84 function associated 57.05with a chromosome. 7786.2 During each generation, the algorithm uses the selection operator to select a part of the existing 2 487.97 3104.7 143.80 38.26 5283.0 population to breed a new generation. A new generation solution is obtained by applying 3 421.18 2693.0 121.91 30.95 4400.5 the crossover function to its pair of parents, and by using the mutation operator to randomly 4 589.17 3789.9 182.30 44.36 6369.4 change the values of some genes [31]. 5 2742.60 2051.3 132.67 61.22 4771.9 Table 4. Influent medium loads as concentrations. CODsol CODpart 𝐍𝐇𝟒 𝐏𝐎𝟑𝟒 Area Number (g/m3) (g/m3) (g/m3) (g/m3) 1 92.57 591.78 28.67 7.33 2 92.37 587.68 27.22 7.25 3 93.67 611.97 27.71 7.04 4 92.50 595.03 28.62 6.97 5 574.74 429.87 27.81 12.83 2.3. Performance Criteria of a Sewer Network Water 2022, 14, 1062 8 of 25 Table 5. Performance criteria. Criteria Unit Description Novf 1/year Frequency of overflow Tovf Days/year Duration of overflow Vovf m3 /year Volume of overflow OQI kg of polluting units/day Overflow quality index Cmax,TSS gCOD/m3 Maximum concentration of TSS/hour Cmax,TKN gN/m3 Maximum concentration of TKN/hour Cmax,PO4 gP/m3 Maximum concentration of PO4 /hour Texc,TSS Days/year Annual duration of exceedances for TSS Texc,TKN Days/year Annual duration of exceedances for TKN Texc,PO4 Days/year Annual duration of exceedances for PO4 The genetic algorithm used to minimize the volume of overflow and the overflow quality index for a sewer network has the following characteristics: • The chromosome number and structure of genes is chosen according to each control strategy approach, as can be seen in the following sections; • The population size is chosen to consist of 20 individuals; • The initial population is randomly generated with a uniform distribution; • The crossover function that generates the offspring chromosome (C) from two parent chromosomes (P1 and P2) generates a binary vector (S) with random values that have the same length as the chromosome. Considering this randomly generated binary vector, if S(k) = 0, then the gene k of the offspring chromosome inherits the value of gene k of the first parent, or if S(k) = 1, the gene k of the offspring chromosome inherits the gene k value of the second parent (Figure 4). • The parent chromosomes are randomly chosen from the population, using for each candidate a probability directly proportional to its fitness function; • To each gene of the parents, he mutation function adds a random value chosen from a Gaussian distribution with zero mean and standard deviation calculated for each gene as follows:   k σk = σk−1 × 1 − (1) Gen where # k—the current generation number; # Gen—the maximum number of generations; # σk —standard deviation of the Gaussian distribution at the generation k (at the first generation, σ1 is the size of the gene range); • The genetic algorithm stops when the maximum number of generations is reached (100 generations), or when the best fitness does not improve in the last 30 generations. Remark: For each of the control strategies, the genetic algorithm was run twice, with two different fitness functions. The first time, it looked for the optimal controls to minimize the volume of overflow (Vovf ), while the second time it minimized the overflow quality index (OQI).  The initial population is randomly generated with a uniform distribution;  The crossover function that generates the offspring chromosome (C) from two par‐ ent chromosomes (P1 and P2) generates a binary vector (S) with random values that have the same length as the chromosome. Considering this randomly generated bi‐ nary vector, if S(k) = 0, then the gene k of the offspring chromosome inherits the Water 2022, 14, 1062 9 of 25 value of gene k of the first parent, or if S(k) = 1, the gene k of the offspring chromo‐ some inherits the gene k value of the second parent (Figure 4). Water 2022, 14, x FOR PEER REVIEW 9 of 27  The genetic algorithm stops when the maximum number of generations is reached (100 generations), or when the best fitness does not improve in the last 30 genera‐ tions. Remark: For each of the control strategies, the genetic algorithm was run twice, with two different fitness functions. The first time, it looked for the optimal controls to mini‐ mize the volume of overflow (V ), while the second time it minimized the overflow quality index (OQI). Figure Figure4.4.Genetic Geneticalgorithm: algorithm:crossover operator. crossover operator. 3. Simulation Results 3. Simulation The parent Results Theoptimization chromosomes are randomly of the operation chosen of the sewer from the network population, is based on theusing for each prediction of candidate the influent a probability The optimization directly proportional of the operation (as shown in Section 2.2) and onofthe to its thecontrol fitness function; sewercomputation network is using basedgenetic on thealgo‐ prediction of the rithms To(Figure each gene influent of shown 5). (as the parents, he mutation in Section function 2.2) and on theadds a random control value chosen computation from using genetic a Gaussian distribution algorithms (Figure 5). with zero mean and standard deviation calculated for each gene as follows: 𝑘 𝜎 𝜎 1 (1) 𝐺𝑒𝑛 where o k—the current generation number; o Gen—the maximum number of generations; o 𝜎 —standard deviation of the Gaussian distribution at the generation k (at the Figure5. 5. Figure Optimal Optimal control first generation, control of 𝜎the of the issewer sewer the network. size network.of the gene range); Five Five strategies strategies forfor optimizing optimizing the the operating operating ofsewer of the the sewer network network were approached. were approached. Theanalysis The analysis ofof thethe sewer sewer network network efficiency efficiency waswas focused focused mainly mainly onthe on 2 of 2 of the performance performance criteria—V criteria— andOQI—of V ovfand OQI—of thethe 10 mentioned 10 mentioned in Section in Section 2.3. 2.3. 3.1.Operation 3.1. Operation of the of the Sewer Sewer Network Network withwith Constant Constant Controls Controls at Maximum at Maximum ValuesValues (100%)(100%) Thiscase This casewas wasconsidered considered inin [29]. [29]. NoNo control control system system waswas taken taken into into account; account; therefore, there‐ all valves fore, at the all valves tanktank at the outputs were outputs wereconsidered to to considered bebefully fullyopened, opened,while whileallallofofthe thepumps were running pumps at 100% were running as long at 100% as the as long wastewater as the wastewater level levelininthe thetank tank was over0.5 was over 0.5m.m. The simulation The simulationresults areare results presented presentedininTable Table6.6. Simulation Table6.6.Simulation Table results results (no(no control control is taken is taken intointo account). account). Tank Number Tank Number 𝐕𝐨𝐯𝐟Vovf (m3(m 3 /year) /year) 𝐎𝐐𝐈 OQI 11 0 0 0 0 2 22,729 192 2 22,729 192 3 50,202 743 43 50,202 55,134 1365 743 54 055,134 0 1365 65 241,6000 3830 0 76 0 241,600 0 3830 Global 369,665 6131 7 0 0 Analyzing Global 369,665 the sewer network structure from Figure 1 and the results from6131 Table 6, it can be observed that minimizing the two performance criteria of the sewer network can be done only in relation Analyzing withnetwork the sewer the output controlsfrom structure of tanks Figure3, 4, and 5. 1 and theSetting resultsthe output from Table 6, it control of the other tanks to a value smaller than 100% does not help in decreasing can be observed that minimizing the two performance criteria of the sewer network any of can the beperformance done only incriteria. The relation outflows with of tanks the output 1, 2, andof6 tanks controls represent 3, 4,the andinflows for tank 5. Setting the7output (Figure 1) and, because tank 7 does not overflow when no control is used, setting the control of the other tanks to a value smaller than 100% does not help in decreasing any of output control of these tanks to a value smaller than 100% does not help in decreasing the performance criteria. The outflows of tanks 1, 2, and 6 represent the inflows for tank any of the performance criteria. The only effect of decreasing the controls of tanks 1, 2, 7 (Figure 1) and, because tank 7 does not overflow when no control is used, setting the and 6 is to increase the overflows of those tanks. The outflow of tank 7 is connected di‐ Water 2022, Water 14,14, 2022, 1062 x FOR PEER REVIEW 10 of 25 10 of 27 Water 2022, 14, x FOR PEER REVIEW 10 of 27 rectly to output the wastewater control treatment of these tanks plant input, to a value so than smaller keeping 100%thisdoes tank’s notoutput help incontrol set to any decreasing 100% is the best solution in relation with the sewer network performance. of the performance criteria. The only effect of decreasing the controls of tanks 1, 2, and 6 is to increase rectly the overflows to the wastewater of those treatment tanks. plant The input, outflow this so keeping of tank tank’s7 is connected output controldirectly set to to the 3.2. Control Strategy wastewater treatment#1—Constant plant Controls input, so keeping this tank’s output control set to 100% is the 100% is the best solution in relation with the sewer network performance. In this strategy, best solution the with in relation controls the at the outputs sewer networkofperformance. tanks 3, 4, and 5 are maintained con‐ 3.2.stant at certain Control Strategyvalues that are Controls #1—Constant determined by the genetic algorithm in order to minimize 3.2. the Control fitness Strategy The function. #1—Constant chromosome Controls containsofthree genes In this strategy, the controls at the outputs tanks 3, 4, (Figure 6): maintained con‐ and 5 are stant atIn this strategy, certain the are values that controls at the outputs determined of tanks by the genetic 3, 4, andin5order algorithm are maintained to minimizeconstant theatfitness certain values The function. thatchromosome are determined by the contains genetic three genes algorithm (Figure 6): in order to minimize the fitness function. The chromosome contains three genes (Figure 6): Figure 6. Chromosome structure in Strategy #1. Figure 6. Chromosome structure in Strategy #1. where 𝑢 ∈ 0, 1 is the percentage control of the valve/pump at the outlet of tank i. Figure 6. Chromosome structure in Strategy #1. where u ∈ [0, 1] is the percentage control of the valve/pump at the outlet of tank i. i 3.2.1. Volume of Overflow Minimization (Strategy #1) where 𝑢 ∈ 0, 1 is the percentage control of the valve/pump at the outlet of tank i. Characteristics: 3.2.1. Volume of Overflow function: V ; the fitness Minimization algorithm (Strategy #1) stopped after 69 generations. In Figure 3.2.1. Volume 7, the of Overflow best and the average Minimization fitness of each of the generations can be ob‐ Characteristics: fitness function:(Strategy Vovf ; the#1) algorithm stopped after 69 generations. served. Characteristics: fitness function: V ; the algorithm stopped after 69 generations. In Figure 7, the best and the average fitness of each of the generations can be observed. In Figure 7, the best and the average fitness of each of the generations can be ob‐ served. Figure 7. Figure 7. Best Best and and average averagefitness fitnessevolution evolutioninin control strategy control #1 when strategy minimizing #1 when V V minimizing . ovf . The following optimal controls were obtained: Figure 7.The Bestfollowing optimal and average controlsinwere fitness evolution obtained: control strategy #1 when minimizing V .  𝑢∗∗ 0.7 𝑢∗∗ 0.8743 𝑢∗ ∗ 0.6543  (2) u3 = 0.7 The following optimal controls 4 = 0.8743 u5 = 0.6543 wereuobtained: (2) The simulation results are presented in Table 7. 𝑢∗ are The simulation results 𝑢∗ 0.8743 0.7presented 𝑢∗ 7. 0.6543 in Table (2) The simulation results are presented in Table 7. Water 2022, 14, 1062 11 of 25 Water 2022, 14, x FOR PEER REVIEW 11 of 27 Table 7. Simulation results when the volume of overflow is minimized (Strategy #1). Table 7. Simulation Tank Number * overflow results when the volumeVof 3 is minimized (Strategy #1).OQI ovf (m /year) ∗ 1 Tank Number 𝐕𝐨𝐯𝐟 0 (m3/year) 𝐎𝐐𝐈 0 12 0 22,750 0 192 23 22,750 63,727 192 886 3 63,727 886 4 61,099 1376 4 61,099 1376 5 12,924 451 5 12,924 451 66 130,855 130,855 22232223 77 0 0 0 0 Global Global 291,355 291,355 51285128 * *Optimal Optimalvalue valueofof V. . Vovf 3.2.2. Overflow 3.2.2. Overflow Quality QualityIndex IndexMinimization Minimization(Strategy #1)#1) (Strategy Characteristics: fitness function: OQI; the algorithm stopped after 91 generations. Characteristics: fitness function: OQI; the algorithm stopped after 91 generations. Figure 8 presents the best and the average fitness evolution during the generations. Figure 8 presents the best and the average fitness evolution during the generations. Figure 8. Figure 8. Best Best and and average averagefitness fitnessevolution evolutioninin control strategy control #1 when strategy minimizing #1 when OQI.OQI. minimizing The values The values of of the theoptimal optimalcontrols controlsare asas are follows: follows:  𝑢∗∗ 0.7062 𝑢∗∗ 0.6872 𝑢∗ ∗ 0.6742  (3) u3 = 0.7062 u4 = 0.6872 u5 = 0.6742 (3) The simulation results are presented in Table 8. The simulation results are presented in Table 8. Table 8. Simulation results when the overflow quality index is minimized (strategy #1). 3.3. Control Strategy #2 Tank Number 𝐕 (m3/year) 𝐎𝐐𝐈 ∗ The outputs of tanks 3, 4, and 5 𝐨𝐯𝐟 are considered to vary in time, depending on the 1 0 0 liquid level in the tank, as follows: if the liquid level is below a certain value hi,LI M , the 2 22,740 valve/pump control will be ui , while if the level is above hi,LI M , then193 the control will be 3 63,189 considered 1 (100%). This way, when the liquid level is over a certain 852limit, the output control is kept4 to 100%, so the overflow 106,189 of the current tank will be1789 as low as possible. 5 In order to prevent the quick change of 11,876 the control when the tank level 422is around h i,LI M , 6 considered, as presented a hysteresis was 103,051 in Figure 9. 1811 Water 2022, 14, 1062 12 of 25 Water 2022, 14, x FOR PEER REVIEW 12 of 27 Water 2022, 14, x FOR PEER REVIEW 12 of 27 Table 8. Simulation results when the overflow quality index is minimized (strategy #1). 7 0 0 7 0 0 Global Tank Number Vovf (m3 /year) 307,044 5066 OQI * Global * Optimal value of OQI. 307,044 5066 1 * Optimal value of OQI. 0 0 3.3. Control Strategy2 #2 22,740 193 3.3. Control Strategy #2 The outputs3of tanks 3, 4, and 5 are considered 63,189 to vary in time, depending 852 on the Thelevel liquid outputs in theof tank, tanksas3, follows: 4, and 5ifare the considered liquid leveltoisvary in atime, below depending certain value ℎ on the , the 4 106,189 1789, liquid level in the tank, as follows: if the liquid level is below a certain valve/pump control will be 𝑢 , while if the level is above ℎ , , then the control value ℎ , , will thebe valve/pump 5 will be 𝑢 , while if the level 11,876 ℎ , a, then 422 be considered control 1 (100%). This way, when the liquidislevel aboveis over thelimit, certain controlthewill output considered 1 (100%). 6 This way, when the liquid level 103,051 is over a certain limit, control is kept to 100%, so the overflow of the current tank will be as low as possible. the outputIn 1811 control is kept to 100%, so the overflow of the current tank will be as low as possible. In order to prevent 7the quick change of the control0 when the tank level is around0 ℎ , , a order to prevent considered,change hysteresis wasGlobalthe quick of the in as presented control Figure 9. the tank level is around5066 when ℎ, ,a hysteresis was considered, as presented in Figure 307,044 9. * Optimal value of OQI. Figure 9. Strategy #2 controls depending on the liquid level in the tanks. Figure 9. Strategy Figure #2 controls 9. Strategy depending #2 controls on theon depending liquid the level liquidinlevel the tanks. in the tanks. For each of the three tanks, ℎ , and 𝑢 , 𝑖 3 … 5, are determined by the optimi‐ Foralgorithm zation eacheach For of the of three inthe order tanks, three ℎ , hi,LI tanks, to minimize and the 𝑢 , 𝑖 uifunction. and M fitness ,3 i…=5,3are determined . . The . 5, are by the determined chromosomal optimi‐ by the optimiza- structure is zation tionalgorithm algorithm in order in presented in Figure 10. to order minimize to the minimize fitness the function. fitness The function. chromosomal The structure chromosomal is structure is presented in Figure presented 10. 10. in Figure Figure 10. Chromosomal structure in strategy #2. Figure 10. Chromosomal structure in strategy #2. Figure 10. Volume 3.3.1. Chromosomal structure in of Overflow strategy #2. (Strategy #2) Minimization 3.3.1. Volume of Overflow Minimization Characteristics: (Strategy #2) 3.3.1. Volume of Overflowfitness function:(Strategy Minimization Vovf ; the #2)algorithm stopped after 93 generations. Characteristics: fitness function: V ; the algorithm In Figure 11 the best and the average fitness of each stopped of theafter 93 generations. generations can be observed. Characteristics: In The Figure 11 fitness the best function: and the V ; thefitness average algorithm of stopped each of after the 93 generations. generations can be ob‐ following optimal solutions were obtained: In Figure 11 the best and the average fitness of each of the generations can be ob‐ served. served.  ∗ ∗ ∗  h3,LI M = 4.9192 h4,LI M = 0.2053 h5,LI M = 4.9896 (4) u3∗ = 0.7716 u4∗ = 0.7983 u5∗ = 0.6096 The simulation results are presented in Table 9. Water Water 2022, 2022, 14,14, x FOR PEER REVIEW 1062 13 of 27 13 of 25 Figure 11. Figure 11. Best Best and andaverage averagefitness fitnessevolution evolutionin in control strategy control #2 when strategy minimizing #2 when V .Vovf . minimizing TableThe followingresults 9. Simulation optimal solutions when were of the volume obtained: overflow is minimized (strategy #2). ℎ∗ , 4.9192 ℎ∗ , * 0.2053 ℎ∗ , 4.9896 Tank Number Vovf (m3 /year) OQI (4) 𝑢∗ 0.7716 𝑢 ∗ 0.7983 𝑢∗ 0.6096 1 0 0 The simulation results are presented in Table 9. 2 22,730 192 Table 9. Simulation 3 results when the volume of overflow 57,059 is minimized (strategy #2). 784 ∗ 4 Tank Number 𝐕𝐨𝐯𝐟 (m55,156 3/year) 𝐎𝐐𝐈1365 15 0 15,512 0 508 26 22,730 141,430 192 2307 3 57,059 784 7 0 0 4 55,156 1365 Global 291,887 5157 5 15,512 508 * Optimal value of Vovf . 6 141,430 2307 7 0 0 3.3.2. Overflow Quality Index Minimization (Strategy #2) Global 291,887 5157 Characteristics: * Optimal value of V . fitness function: OQI; the algorithm stopped after 100 generations. Figure 12 shows the best and the average fitness of each of the generations. 3.3.2.The following Overflow optimal Quality Indexcontrols were obtained: Minimization (Strategy #2) Characteristics:  fitness function: OQI; ∗ the= algorithm stopped after 100 generations. h∗ M = 4.8716 h4,LI ∗  4.7138 h5,LI = 3.3904 Mthe Figure 12 shows3,LI the ∗ best and the ∗ M fitness average of each of ∗ generations. (5) u3 = 0.6206 u4 = 0.7312 u5 = 0.9158 3.4. Control Strategy #3 The controls for the outputs of tanks 3, 4, and 5 are considered to vary in time, depending on the liquid level in the tank, as follows: if the liquid level in the tank is below a certain level hi,LI M , the valve/pump control will be ui,B , while if the level is above hi,LI M , then the control will be considered ui,A . No inequality constraints for ui,A and ui,B were considered. In this way, when the liquid level is approaching the limit, the new control value can be higher than the old one, in order to make the overflow of the current tank smaller, or it can be lower in order to help the tank whose inflow is the outflow of the current tank to overflow less. To prevent the control quickly changing when the tank level is around hi,LI M , a hysteresis was considered around this level, as presented in Figure 13. Water Water 2022, 2022, 14,14, x FOR PEER REVIEW 1062 14 of 27 14 of 25 Figure12. Figure 12. Best Best and and average averagefitness fitnessevolution evolutioninin control strategy control #2 when strategy minimizing #2 when OQI.OQI. minimizing The following optimal controls were obtained: Table 10. Simulation results when the overflow quality index is minimized (strategy #2). ℎ∗ , 4.8716 ℎ∗ , 4.7138 ℎ∗ , 3.3904 (5) Tank Number 𝑢∗ ∗ (m /year) 𝑢∗ 3 OQI * 0.6206 V𝑢ovf 0.7312 0.9158 1 The simulation 0 results are presented in Table 10. 0 2 22,728 192 Table 10. Simulation results when the overflow quality index is minimized (strategy #2). 3 66,739 878 ∗ Tank Number 4 𝐕𝐨𝐯𝐟 (m 3/year) 76,156 𝐎𝐐𝐈1488 1 0 0 5 535 15 2 22,728 192 6 156,137 2607 3 66,739 878 74 0 76,156 1488 0 Water 2022, 14, x FOR PEER REVIEW 5 Global 535 322,295 155180 15 of 27 * Optimal value of6OQI. 156,137 2607 7 0 0 Global 322,295 5180 * Optimal value of OQI. 3.4. Control Strategy #3 The controls for the outputs of tanks 3, 4, and 5 are considered to vary in time, de‐ pending on the liquid level in the tank, as follows: if the liquid level in the tank is below a certain level ℎ , , the valve/pump control will be 𝑢 , , while if the level is above ℎ , , then the control will be considered 𝑢 , . No inequality constraints for 𝑢 , and 𝑢 , were considered. In this way, when the liquid level is approaching the limit, the new control value can be higher than the old one, in order to make the overflow of the cur‐ rent tank smaller, or it can be lower in order to help the tank whose inflow is the outflow of the current tank to overflow less. To prevent the control quickly changing when the tank level is around ℎ , , a hysteresis was considered around this level, as presented in Figure 13. Figure 13.13. Figure Strategy #3 controls Strategy depending #3 controls on the depending onliquid level in the liquid the in level tanks. the tanks. For each of the three tanks, ℎ , , 𝑢 , , and 𝑢 , , 𝑖 3 … 5 are determined by the optimization algorithm to minimize the fitness function. In this case, the chromosomal structure is presented in Figure 14. Water 2022, 14, 1062 15 of 25 Figure 13. Strategy #3 controls depending on the liquid level in the tanks. For each of the three tanks, ℎ , , 𝑢 , , and 𝑢 , , 𝑖 3 … 5 are determined by the For each of the three tanks, hi,LIfitness optimization algorithm to minimize the M , ui,Afunction. , and ui,BIn = 3case, , ithis . . . 5the arechromosomal determined by the optimization algorithm to minimize structure is presented in Figure 14. the fitness function. In this case, the chromosomal structure is presented in Figure 14. Figure 14. Strategy #3 chromosomal structure. Figure 14. Strategy #3 chromosomal structure. Water 2022, 14, x FOR PEER REVIEW 3.4.1. Volume of Overflow Minimization (Strategy #3) 16 of 27 3.4.1. Volume of Overflow Minimization (Strategy #3) Characteristics: fitness function: Vovf ; the algorithm stopped after 100 generations. Characteristics: fitness function: V ; the algorithm stopped after 100 generations. Figure 15 shows the best and the average fitness of each of the generations. Figure 15 shows the best and the average fitness of each of the generations. Figure 15. Figure 15. Best Best and andaverage averagefitness fitnessevolution evolutionin in control strategy control #3 when strategy minimizing #3 when V .V . minimizing ovf The following The followingoptimal optimalsolutions solutionswere wereobtained: obtained: ∗ ℎ , 3.507 ℎ∗ , 4.3294 ℎ , ∗ 2.0954 ∗ h∗ ∗ ∗ ∗ M = 3.507𝑢 , h4,LI 𝑢∗ , h4,LI M = 4.3294 M = 2.0954   𝑢 , 3,LI0.6997 0.6910 0.51 (6) 𝑢∗ u∗ 0.8423 ∗ = 0.6997𝑢 , ∗ = 0.6910 ∗ u4,A 0.8927 𝑢 , ∗ = 0.51 u5,A 0.9836 (6) , 3,A  ∗ = 0.8423 u3,B ∗ u4,B = 0.8927 ∗ u5,B = 0.9836 The simulation results are presented in Table 11. The simulation results are presented in Table 11. Table 11. Simulation results when the volume of overflow is minimized (strategy #3). ∗ 3.4.2. Overflow 𝐕𝐨𝐯𝐟 Quality Index Minimization Tank Number (m3/year] 𝐎𝐐𝐈 1 Characteristics: fitness function: OQI; 0the algorithm stopped after 0100 generations. Figure 16 2shows the best and the average 22,759 fitness of each of the generations. 193 The following 3 optimal solutions were obtained: 90,614 1197 4  85,118 1819 ∗ h3,LI ∗ ∗ M = 3.3937 h4,LI M = 0.8911 h5,LI M = 3.9285727  5 21,207  u∗ = 0.5510 ∗ u4,A = 0.8598 ∗ = 0.6190  u5,A (7) 6 3,A 24,667 318 u ∗ ∗ = 0.8414 ∗ = 0.9961 7 3,B = 0.9157 u4,B 0 u5,B 0 Global 244,365 4254 The simulation results are presented in Table 12. * Optimal value of V . 3.4.2. Overflow Quality Index Minimization Characteristics: fitness function: OQI; the algorithm stopped after 100 generations. Figure 16 shows the best and the average fitness of each of the generations. Water 2022, 14, 1062 16 of 25 Table 11. Simulation results when the volume of overflow is minimized (strategy #3). Tank Number V*ovf (m3 /year) OQI 1 0 0 2 22,759 193 3 90,614 1197 4 85,118 1819 5 21,207 727 6 24,667 318 7 0 0 Water 2022, 14, x FOR PEER REVIEW 17 of 27 Global 244,365 4254 * Optimal value of Vovf . Figure 16. Figure 16. Best Best and andaverage averagefitness fitnessevolution evolutioninin control strategy control #3 when strategy minimizing #3 when OQI.OQI. minimizing Table The following optimal 12. Simulation solutions results when were obtained: the overflow quality index is minimized (strategy #3). ℎ∗ , 3.3937 ℎ∗ , 0.8911 3 ℎ∗ , 3.9285 Tank Number ∗ ∗ Vovf (m /year) ∗ OQI * 𝑢 , 0.5510 𝑢 , 0.8598 𝑢 , 0.6190 (7) 1 𝑢∗ , 0.9157 𝑢∗ , 0.8414 0 𝑢∗ , 0.9961 0 2 The simulation results are presented in22,772 Table 12. 193 3 156,472 1982 Table 12. Simulation results when the overflow quality index is minimized (strategy #3). 4 69,088 1526 ∗ Tank Number 5 𝐕 𝐨𝐯𝐟 (m 3/year) 9111 𝐎𝐐𝐈 299 1 0 0 6 17,609 280 2 22,772 193 7 0 0 3 156,472 1982 Global 4 275,052 69,088 15264280 * Optimal value of5OQI. 9111 299 6 17,609 280 3.5. Control Strategy 7 #4 0 0 The controls Global for the outputs of tanks 275,052 3, 4, and 5 are considered 4280to vary in time, depending on the * Optimal value liquid level in the tank, as follows: if the liquid level in the tank is below of OQI. a certain level hi,LOW , the valve/pump control will be ui,B ; if the level is above hi,H IGH , then 3.5. Control Strategy #4 The controls for the outputs of tanks 3, 4, and 5 are considered to vary in time, de‐ pending on the liquid level in the tank, as follows: if the liquid level in the tank is below a certain level ℎ , , the valve/pump control will be 𝑢 , ; if the level is above ℎ , , then the control will be 𝑢 , and if the level is between ℎ and ℎ , the control Water 2022, 14, 1062 17 of 25 Water2022, Water 2022,14, 14,x xFOR FORPEER PEERREVIEW REVIEW 1818ofof2727 the control will be ui,A , and if the level is between hi,LOW and hi,H IGH , the control will be scaled accordingly between ui,B and ui,A (Figure 17). Figure Figure Figure17. 17. Strategy Strategy 17. #4#4control Strategy control #4 depending depending control onthe on depending the liquid onliquid levelin level the liquid inthe thein level tank. tank. the tank. For For each each For eachofofthe ofthe three three the tanks, tanks, three tanks,ℎℎ, ,hi,LOW , ,ℎℎ, ,, hi,H, ,IGH 𝑢𝑢, , ,, ,u and i,A 𝑢, , , 𝑖,u𝑖 3,3…i…5 and, 𝑢and i,B =5are 3are determined . . determined .5 are determined by by the the optimization optimization algorithm algorithm toto minimize minimize the the fitness fitness by the optimization algorithm to minimize the fitness function. The two limits, function. function. The The two two limits, limits, ℎℎ, , hi,LOW and and ℎℎ, , , , must must always always respect respect thethe constraint constraint ℎ ℎ, , ℎℎ, , and hi,H IGH , must always respect the constraint hi,LOW < hi,H IGH . This requires changing . . This This requires requires changing changing somecharacteristics some characteristicsofofthe thegenetic geneticalgorithm algorithm(e.g.,(e.g.,initial initialpopulation, population,crossover crossoverfunction, function, some characteristics of the genetic algorithm (e.g., initial population, crossover function, mutationfunction) mutation function)totomake makesuresurethat thatall allsolutions solutionsare arevalid. valid.To Toavoid avoidthis, this,ititmust must be mutation function) to make sure that all solutions are valid. To avoid this, itbemust be consideredthat considered thatℎℎ, , 𝑚𝑖𝑛 andℎℎ, , 𝑚𝑖𝑛ℎℎ, ,, ℎ, ℎ, , and 𝑚𝑎𝑥ℎℎ, ,, ℎ, ℎ, , , ,where 𝑚𝑎𝑥 whereℎℎ, , and andℎℎ, , considered that hi,LOW = min(hi,1 , hi,2 ) and hi,H IGH = max(hi,1 , hi,2 ), where hi,1 and hi,2 areprovided are providedby bythethegenetic geneticalgorithm. algorithm.Therefore, Therefore,the thechromosome chromosomecontains contains12 12genes, genes,asas are provided by the genetic algorithm. Therefore, the chromosome contains 12 genes, as presentedininFigure presented Figure18. 18. presented in Figure 18. Figure 18. Strategy #4 chromosomal structure. Figure18. Figure 18.Strategy Strategy#4#4chromosomal chromosomalstructure. structure. 3.5.1. Volume of Overflow Minimization 3.5.1.Volume 3.5.1. VolumeofofOverflow OverflowMinimization Minimization Characteristics: fitness function: Vovf ; the algorithm stopped after 100 generations. Characteristics: Characteristics: fitness fitnessthefunction:VV ; ;the function: thealgorithm algorithmstopped stoppedafter after100 100 generations. Figure 19 presents best and the average fitness of each of thegenerations. generations. Figure Figure 19 presents the best and the average fitness of each of the generations. The 19 presentsoptimal following the best solution and the average fitness of each of the generations. is obtained: ∗ ∗ ∗   h3,LOW = 1.5247 h4,LOW = 2.6187 h5,LOW = 0.0654  ∗ h3,H IGH = 4.9168 ∗ h4,H IGH = 2.8814 ∗ h5,H IGH = 1.4234  (8)    ∗ = 0.6678 u3,A ∗ = 0.6564 u4,A ∗ = 0.6403 u5,A    ∗ = 0.9829 u3,B ∗ = 0.9976 u4,B ∗ = 0.8426 u5,B The simulation results are presented in Table 13. 3.5.2. Overflow Quality Index Minimization Characteristics: fitness function: OQI; the algorithm stopped after 100 generations. Figure 20 shows the best and the average fitness of each of the generations. Water Water 2022, 2022, 14,14, x FOR PEER REVIEW 1062 19 of 27 18 of 25 Figure 19. Figure 19. Best Bestand andaverage averagefitness fitnessevolution evolutionin in control strategy control #4 when strategy minimizing #4 when V .V . minimizing ovf TableThe following optimal 13. Simulation solution results when is obtained: the volume of overflow is minimized (strategy #4). ℎ∗ , 1.5247 ℎ∗ , 2.6187 ℎ∗ 0.0654 ⎡ ∗ Tank Number ∗ V*ovf (m3 /year)∗ , ⎤ OQI ⎢ℎ , 4.9168 ℎ , 2.8814 ℎ , 1.4234⎥ (8) ⎢1 𝑢∗ , 0.6678 𝑢∗ , 0 0.6564 𝑢∗ , 0.6403 ⎥ 0 ⎢ ∗ ⎥ ⎣2 𝑢 , 0.9829 𝑢 , ∗ 0.9976 22,743 𝑢∗ , 0.8426 ⎦ 193 3 The simulation results are presented in87,101 Table 13. 1168 4 86,162 1882 Table 13. Simulation results when the volume of overflow is minimized (strategy #4). 5 26,715 883 ∗ Tank Number 6 𝐕𝐨𝐯𝐟 (m 3/year) 40,314 𝐎𝐐𝐈503 1 0 0 7 0 0 Water 2022, 14, x FOR PEER REVIEW 2 22,743 193 20 of 27 Global 263,034 4629 3 87,101 1168 * Optimal value of Vovf . 4 86,162 1882 5 26,715 883 6 40,314 503 7 0 0 Global 263,034 4629 * Optimal value of V . 3.5.2. Overflow Quality Index Minimization Characteristics: fitness function: OQI; the algorithm stopped after 100 generations. Figure 20 shows the best and the average fitness of each of the generations. Figure 20. Figure 20. Best Bestand andaverage averagefitness evolution fitness in control evolution strategy in control #4 when strategy #4minimizing OQI. when minimizing OQI. The following optimal solutions were obtained: ℎ∗ , 2.265 ℎ∗ , 0.3715 ℎ∗ , 1.5886 ⎡ ∗ ⎤ ⎢ℎ , 3.672 ℎ∗ , 3.6782 ℎ∗ , 2.53 ⎥ ⎢ 𝑢∗ (9) 0.622 𝑢∗ 0.8374 𝑢∗ 0.6492 ⎥ Water 2022, 14, 1062 19 of 25 The following optimal solutions were obtained: ∗ ∗ ∗   h3,LOW = 2.265 h4,LOW = 0.3715 h5,LOW = 1.5886  h∗ ∗ ∗  3,H IGH = 3.672 h4,H IGH = 3.6782 h5,H IGH = 2.53  (9)  ∗ = 0.622 ∗ = 0.8374 ∗ = 0.6492  u3,A u4,A u5,A    ∗ = 0.8871 u3,B ∗ = 0.9323 u4,B ∗ = 0.9937 u5,B The simulation results are presented in Table 14. Table 14. Simulation results when the overflow quality index is minimized (strategy #4). Tank Number Vovf (m3 /year) OQI * 1 0 0 2 22,759 193 3 126,250 1609 4 69,097 1547 5 15,677 539 6 18,166 258 7 0 0 Global 251,949 4173 * Optimal value of OQI. 3.6. Control Strategy #5 This strategy is similar to strategy #3 (Figure 13), but this time, it is considered that hi,LI M is equal to 3.5 m. This way, the number of genes of the chromosome is smaller, decreasing the number of dimensions of the solution space. This should improve the con- vergence of the algorithm and decrease the probability of getting stuck at a local minimum. Water 2022, 14, x FOR PEER REVIEW For each of the three tanks, ui,A and ui,B , i = 3 . . . 5, are determined by the21optimization of 27 algorithm to minimize the fitness function. In this case, the structure of the chromosome is as presented in Figure 21. Figure 21. Strategy #5 chromosomal structure. Figure 21. Strategy #5 chromosomal structure. 3.6.1. Volume of Overflow Minimization Volume of Overflow Minimization Characteristics: fitness function: Vovf ; the algorithm stopped after 100 generations. Characteristics: fitnessthe Figure 22 presents function: best andV the ; the algorithm average stopped fitness after of each of 100 the generations. generations. Figure 22 presents The following the bestsolutions optimal and the average fitness of each of the generations. were obtained:  ∗ = 0.6085 u3,A ∗ = 0.8033 u4,A ∗ = 0.5609 u5,A  ∗ = 0.9462 ∗ = 0.9008 ∗ = 0.7024 (10) u3,B u4,B u5,B The simulation results are presented in Table 15. Figure 21. Strategy #5 chromosomal structure. Volume of Overflow Minimization Water 2022, 14, 1062 Characteristics: fitness function: V ; the algorithm stopped after 100 generations. 20 of 25 Figure 22 presents the best and the average fitness of each of the generations. Figure 22. Figure 22. Best Best and andaverage averagefitness fitnessevolution evolutioninin control strategy control #5 when strategy minimizing #5 when V .V . minimizing ovf Table The following optimal 15. Simulation solutions results when were obtained: the volume of overflow is minimized (strategy #5). 𝑢∗ , 0.6085 𝑢∗ , 0.8033 𝑢∗ , 0.5609 ∗ V* (m3 /year) (10) 𝑢∗ Tank Number , 0.9462 𝑢 , ovf 0.9008 𝑢∗ , 0.7024 OQI 1 The simulation 0 results are presented in Table 15. 0 2 22,759 193 Table 15. Simulation results when the volume of overflow is minimized (strategy #5). 3 113,086 1489 ∗ Tank Number 4 𝐕𝐨𝐯𝐟 (m 3/year) 72,589 𝐎𝐐𝐈1610 1 0 0 5 27,726 924 2 22,759 193 6 10,983 157 3 113,086 1489 74 0 72,589 1610 0 5 Global 27,726 247,143 9244373 * Optimal value of6Vovf . 10,983 157 7 0 0 3.6.2. Overflow Global Quality Index Minimization 247,143 4373 * Optimal value of V .3.6.2. Characteristics: fitnessOverflow Quality function: OQI; Index the Minimization. algorithm stopped after 100 generations. Figure 23 shows the best and the average fitness of each of the generations. The following optimal solutions were obtained:  ∗ = 0.6626 u3,A ∗ = 0.4811 u4,A ∗ = 0.9417 u5,A  ∗ = 0.9060 ∗ = 0.9955 ∗ = 0.9013 (11) u3,B u4,B u5,B The simulation results are presented in Table 16. Water 2022, 14, x FOR PEER REVIEW 22 of 27 Water 2022, 14, 1062 Characteristics: fitness function: OQI; the algorithm stopped after 100 generations. 21 of 25 Figure 23 shows the best and the average fitness of each of the generations. Figure 23. Figure 23. Best Bestand andaverage averagefitness fitnessevolution evolutionin in control strategy control #5 when strategy minimizing #5 when OQI.OQI. minimizing TableThe following optimal 16. Simulation solutions results when were obtained: the overflow quality index is minimized (strategy #5). 𝑢∗ , 0.6626 𝑢∗ , 0.4811 𝑢∗ , 0.9417 Tank Number 𝑢∗ 0.9060 𝑢 , 3 ∗ Vovf (m /year) 0.9955 𝑢∗ , 0.9013 OQI ∗ (11) , 1 The simulation 0 16. results are presented in Table 0 2 22,741 193 Table 16. Simulation results when the overflow quality index is minimized (strategy #5). 3 96,103 1275 ∗ Tank Number 4 𝐕𝐨𝐯𝐟 (m 3/year) 105,689 𝐎𝐐𝐈2218 1 0 0 5 2844 89 2 22,741 193 6 45,448 574 3 96,103 1275 47 0 105,689 2218 0 Global 5 272,826 2844 89 4349 * Optimal value of6OQI. 45,448 574 7 0 0 4. Discussion Global 272,826 4349 The study * Optimal value ofcarried OQI. out in this paper presents a series of results regarding the optimiza- tion of the volume of overflow and the overflow quality index—results obtained on the 4. Discussion model of a sewer network corresponding to a city in eastern Romania, with a population of 250,000Theinhabitants. study carried The method out in this of genetic paper algorithms presents a serieswas used for of results optimization. regarding Simple the opti‐ controls mizationthat required of the volumethe computation of overflow of 3–12 and the parameters overflow quality were generated. index—results obtained on Figureof24a presents the model the evolutions sewer network of theto corresponding fitness a cityfunctions in easternover generations Romania, for all five with a popu‐ strategies in the inhabitants. lation of 250,000 two cases considered The methodin ofthe analysis: genetic (a) when algorithms minimizing was used Vovf , and for optimiza‐ (b) when tion. minimizing Simple OQI. controls that required the computation of 3–12 parameters were generated. Figure 24 presents the evolutions of the fitness functions over generations for all five strategies in the two cases considered in the analysis: (a) when minimizing V , and (b) when minimizing OQI. Water 2022, 14, x FOR PEER REVIEW 23 of 27 Water 2022, 14, 1062 22 of 25 (a) (b) Figure24. Figure 24.Fitness Fitnessfunction function evolutions evolutions for for all all strategies: strategies:(a) (a)when minimizingVVovf whenminimizing ; (b) when ; (b) mini‐ when minimiz- mizing OQI. ing OQI. AAsynthesis synthesisofofthe thesimulation simulationresults results obtained obtained with with the the five five optimal optimal control control strate‐ strategies, gies, in relation to the minimization of both the volume of overflow and the overflow in relation to the minimization of both the volume of overflow and the overflow quality quality index, is given in Table 17. index, is given in Table 17. Table 17. Simulation results for solutions of the five strategies with both fitness functions. Table 17. Simulation results for solutions of the five strategies with both fitness functions. ∗ 𝐕𝐨𝐯𝐟 Percentage of Percentage of V*ovf (m3/year) Percentage 𝐎𝐐𝐈 ∗ of “No Control” Percentage “No Control” of OQI * No control (m3 /year) 369,655 “No Control” 6131 “No Control” Strategy No control#1 (min V 369,655 ) 291,355 −21.18% 5128 6131 −16.36% Strategy Strategy #1 (#1 min VovfOQI) (min ) 291,355 307,044 −16.94% −21.18% 5066 5128 −17.37% −16.36% Strategy #2 (min V ) 291,877 −21.04% 5157 −15.89% Strategy #1 (min OQI ) 307,044 −16.94% 5066 −17.37% Strategy #2 (min OQI) 322,295 −12.81% 5180 −15.51% Strategy #2 (min Vovf ) 291,877 −21.04% 5157 −15.89% Strategy #3 (min V ) 244,365 −33.89% 4245 −30.61% Strategy Strategy min #2 (#3 OQIOQI) (min ) 322,295 275,052 −12.81% −25.59% 5180 4280 −15.51% −30.19% Strategy #3 (#4 Strategy min VovfV) 244,365 (min ) 263,034 −33.89% −28.84% 4245 4629 −30.61% −24.50% Strategy #4 (min OQI) Strategy #3 (min OQI ) 275,052 251,949 −31.84% −25.59% 4173 4280 −31.94% −30.19% Strategy #5 (min V ) Strategy #4 (min Vovf ) 263,034 247,143 −33.14% −28.84% 4373 4629 −28.67% −24.50% Strategy #5 (min OQI) 272,826 −26.19% 4349 −29.06% Strategy #4 (min OQI ) 251,949 −31.84% 4173 −31.94% * Optimal values of V and OQI. Strategy #5 (min Vovf ) 247,143 −33.14% 4373 −28.67% The results are analyzed Strategy #5 (min OQI ) 272,826 by comparison −26.19% with the case when 4349 the controls have −29.06% maximum values (the valves * Optimal values of Vovf and OQI. are fully opened and the pumps provide maximum flows). In the third and fifth columns (Percentage of “No control”), the differences can be found between the values The results of the performance are analyzed criteriawith by comparison computed when the case whenthe the optimization strate‐max- controls have gies are imum applied values (theand the are valves aforementioned fully openedsituation. and the pumpsFor example, provide −21.18% maximum(column #3) In flows). means the thirdthat andstrategy #1 ensures fifth columns a 21.18% lower (Percentage of “No volume of overflow control”), than in the the differences cancase of be found “No control”, between which the values of means that strategy the performance #1 is computed criteria more efficient when inthe relation to this perfor‐ optimization strategies mance are criterion. applied and the aforementioned situation. For example, −21.18% (column #3) means The results from Table that strategy #1 ensures 17 show a 21.18% lowerthat in all of volume of overflow the optimization than in thestrategies case ofapplied in “No control”, the conditions of an influent that contains the three components of wastewater which means that strategy #1 is more efficient in relation to this performance criterion. (domestic, industrial, and stormwater), The results from Table 17 significant show that improvements in all of the in the performance optimization criteria strategies were in applied observed. Of the five optimization strategies, strategy #3 ensures the most the conditions of an influent that contains the three components of wastewater (domestic, efficient oper‐ ation in relation to the volume of overflow, while strategy #4 ensures the best efficiency industrial, and stormwater), significant improvements in the performance criteria were observed. Of the five optimization strategies, strategy #3 ensures the most efficient opera- tion in relation to the volume of overflow, while strategy #4 ensures the best efficiency in relation to the overflow quality index, as highlighted in Table 17 in yellow. As the values of the overflow quality index are quite close in strategies #3 and #4 (−30.19% compared to Water 2022, 14, 1062 23 of 25 −31.94%), it can be stated that strategy #3 can be considered the best. In fact, the evolutions of the fitness functions (Figure 24a—orange curve and Figure 24b—magenta curve) show the same thing—they converge to the minimum values. Statistical parameters for the strategy that gave the best results (strategy #3, when minimizing Vovf ) were computed. The genetic algorithm was run 18 times, and the results shown in Table 18 were obtained. Table 18. 18 results of strategy #3 when minimizing Vovf . V*ovf Percentage of Percentage of Run Number OQI (m3 /year) “No Control” “No Control” 1 244,365 −33.89% 4245 −30.61% 2 261,818 −29.17% 4628 −24.51% 3 291,646 −21.10% 5003 −18.39% 4 245,079 −33.70% 4322 −29.50% 5 279,821 −24.30% 4932 −19.55% 6 269,879 −26.99% 4824 −21.31% 7 277,738 −24.86% 4858 −20.76% 8 245,616 −33.55% 4265 −30.43% 9 256,327 −30.65% 4476 −26.99% 10 241,795 −34.58% 4339 −29.22% 11 254,391 −31.18% 4554 −25.72% 12 265,862 −28.07% 4621 −24.62% 13 265,846 −28.08% 4756 −22.28% 14 253,098 −31.53% 4375 −28.93% 15 258,943 −29.95% 4429 −27.76% 16 254,574 −31.13% 4465 −27.17% 17 251,778 −31.88% 4405 −28.15% 18 248,581 −32.75% 4375 −28.64% * Optimal value of Vovf . Table 19 shows the values of the statistical parameters in relation to the two criteria (Vovf and OQI): Table 19. Statistical parameters. Maximum Minimum Mean Standard Value Value Value Deviation ∗ (m3 /year) 291,646 241,795 259,286.5 Vovf 13,710.8951 (−21.10%) (−34.58%) (−29.85%) 5003 4245 4548.4444 OQI 237.0131 (−18.39%) (−30.61%) (−25.81%) * Optimal value of Vovf . Low values of the standard deviation were obtained in relation to both Vovf and OQI, showing that the values of the two performance criteria are quite well grouped around the average values. This proves that the algorithm is robust and reliable. The optimization strategies approached in this paper assume the following: optimal controls are computed and applied in the process for the time horizon considered; during this period, a new set of controls is determined, which will be applied in the next time horizon, and so on. Compared to the present paper, in [32], the use of fuzzy-logic-based Water 2022, 14, 1062 24 of 25 structures for the control of storage tank capacities in order to reduce the volume of overflow is proposed. Although the results obtained by the authors are good, this method does not guarantee optimality. Additionally, [18] deals with a similar problem following the control of a combined sewer network by two methods: nonlinear optimal control and multivariable feedback control. The case study consisted of a sewer network of lower capacity than the one used in this paper. In the case of the first method, the authors’ attention was focused on two objectives: (1) increasing the efficiency of the sewer network operating only in terms of the volume of overflow—as opposed to the present paper, which also considered OQI—and (2) increasing the computational efficiency of the optimization algorithm. A qualitative analysis of the results obtained in the present paper and in [18] shows that they are similar, taking into account the differences in structure and scale between the two sewer networks. 5. Conclusions The present paper shows that sewer networks are very well suited to the use of optimization algorithms in order to increase their operational performance. The main idea of this paper was to determine the optimal set of controls to ensure minimal values of two of the performance criteria proposed for the BSMSewer simulation environment (volume of overflow, and overflow quality index). Five optimization strategies based on genetic algorithms were proposed and tested on a varied influent containing three components: domestic wastewater, stormwater, and industrial wastewater. In essence, our method has the advantage of simplicity in application. The accuracy of the results depends directly on the influent prediction over the chosen time horizon. In principle, the prediction horizon can be reduced so that the values of the influent have a higher degree of confidence. Through an analysis of meteorological observations, an acceptable compromise can be reached regarding the choice of the prediction horizon between the degree of confidence of the meteorological observations and the computing time. Furthermore, this paper shows that the results are better as the number of control parameters (chromosome structure) is increased (strategies #3 and #4 have 9 and 12 param- eters, respectively, compared to strategy #1 (3 parameters), strategy #2 (6 parameters), and strategy #5 (6 parameters)). The price paid is the significant increase in the running time in the case of strategies #3 and #4, with 12–16 h optimization sequences, using a computer with the following characteristics: Intel Core i5-10500T CPU @ 2.30 GHz with 8 GB of RAM. In subsequent research, the authors aim to approach other control (optimization) strate- gies for sewer networks: optimization of sewer networks using aggregate criteria (e.g., the two criteria considered in the paper aggregated in a single criterion), real-time control of sewer networks, and the use and testing of other optimization algorithms to solve the opti- mization problem, including the wastewater treatment plant in the optimization procedure. Author Contributions: Conceptualization, I.V., S.C. and R.V.; software, I.V. and L.L.; methodology, M.B.; writing—review and editing, I.V. and S.C.; validation, I.V. and M.B. All authors have read and agreed to the published version of the manuscript. Funding: This research was supported by the project "DINAMIC" financed by the Romanian Ministry of Research and Innovation, Contract no. 12PFE/2021. The research was also supported by the Spanish Government through the MICINN project PID2019-105434RB-C33. Data Availability Statement: Not applicable. Conflicts of Interest: The authors declare no conflict of interest. References 1. Barbu, M.D. Contribut, ii Privind Conducerea Automată a Sistemelor de Colectare s, i Tratare a Apelor Uzate. Ph.D. Thesis, Dunărea de Jos University, Galat, i, Romania, 2014. 2. Marinaki, M. A non-linear optimal control approach to central sewer network flow control. Int. J. Control 1999, 72, 418–429. [CrossRef] 3. Schütze, M.R.; Butler, D.; Beck, M.B. Modelling, Simulation and Control of Urban Wastewater Systems; Springer: London, UK, 2002. Water 2022, 14, 1062 25 of 25 4. Ocampo-Martinez, C. Model Predictive Control of Wastewater Systems; Springer: London, UK, 2010. 5. Pai, T.-Y.; Lo, H.-M.; Wan, T.-J.; Wang, Y.-H.; Cheng, Y.-H.; Tsai, M.-H.; Tang, H.; Sun, Y.-X.; Chen, W.-C.; Lin, Y.-P. A Sewer Dynamic Model for Simulating Reaction Rates of Different Compounds in Urban Sewer Pipe. Water 2021, 13, 1580. [CrossRef] 6. García, J.T.; Espín-Leal, P.; Vigueras-Rodriguez, A.; Castillo, L.G.; Carrillo, J.M.; Martínez-Solano, P.D.; Nevado-Santos, S. Urban Runoff Characteristics in Combined Sewer Overflows (CSOs): Analysis of Storm Events in Southeastern Spain. Water 2017, 9, 303. [CrossRef] 7. Torres-Matallana, J.A.; Leopold, U.; Klepiszewski, K.; Heuvelink, G.B.M. EmiStatR: A Simplified and Scalable Urban Water Quality Model for Simulation of Combined Sewer Overflows. Water 2018, 10, 782. [CrossRef] 8. Jiang, Y.; Li, C.; Sun, L.; Guo, D.; Zhang, Y.; Wang, W. A deep learning algorithm for multi-source data fusion to predict water quality of urban sewer networks. J. Clean. Prod. 2021, 318, 128533. [CrossRef] 9. Li, Y.; Hou, X.; Zhang, W.; Xiong, W.; Wang, L.; Zhang, S.; Wang, P.; Wang, C. Integration of life cycle assessment and statistical analysis to understand the influence of rainfall on WWTPs with combined sewer systems. J. Clean. Prod. 2018, 172, 2521–2530. [CrossRef] 10. Saagi, R.; Flores-Alsina, X.; Fu, G.; Butler, D.; Gernaey, K.V.; Jeppsson, U. Catchment & sewer network simulation model to benchmark control strategies within urban wastewater systems. Environ. Model. Softw. 2016, 78, 16–30. [CrossRef] 11. Vezzaro, L.; Benedetti, L.; Gevaert, V.; De Keyser, W.; Verdonck, F.; De Baets, B.; Nopens, I.; Cloutier, F.; Vanrolleghem, P.A.; Mikkelsen, P.S. A model library for dynamic transport and fate of micropollutants in integrated urban wastewater and stormwater systems. Environ. Model. Softw. 2014, 53, 98–111. [CrossRef] 12. Rauch, W.; Bertrand-Krajewski, J.-L.; Krebs, P.; Mark, O.; Schilling, W.; Schütze, M.; Vanrolleghem, P. Deterministic modelling of integrated urban drainage systems. Water Sci. Technol. 2002, 45, 81–94. [CrossRef] 13. Brepols, C.; Dahmen, H.; Lange, M.; Sohr, A.; Kiesewski, R.; Rohlfing, R. Model-Based Sewer Network Control-Practical Experiences; Springer: Cham, Switzerland, 2018; pp. 20–24. 14. Kroll, S.; Weemaes, M.; Van Impe, J.; Willems, P. A Methodology for the Design of RTC Strategies for Combined Sewer Networks. Water 2018, 10, 1675. [CrossRef] 15. Wei, Z.; Shangguan, H.; Zhan, J.; Lin, R.; Huang, X.; Lu, L.; Li, H.; Du, B.; Fan, G. Water Quality-Based Double-Gates Control Strategy for Combined Sewer Overflows Pollution Control. Water 2021, 13, 529. [CrossRef] 16. Van Nguyen, L.; Tornyeviadzi, H.M.; Bui, D.T.; Seidu, R. Predicting Discharges in Sewer Pipes Using an Integrated Long Short-Term Memory and Entropy A-TOPSIS Modeling Framework. Water 2022, 14, 300. [CrossRef] 17. Harremoës, P.; Hvitved-Jacobsen, T.; Lynggaard-Jensen, A.; Nielsen, B. Municipal Wastewater Systems, Integrated Approach to Design, Monitoring and Control. Water Sci. Technol. 1994, 29, 419–426. [CrossRef] 18. Marinaki, M.; Papageorgiou, M. Optimal Real-Time Control of Sewer Networks; Springer: London, UK, 2005. 19. Minzu, V.; Serbencu, A. Systematic Procedure for Optimal Controller Implementation Using Metaheuristic Algorithms. Intell. Autom. Soft Comput. 2020, 26, 663–677. [CrossRef] 20. Liao, H.-Y.; Pan, T.-Y.; Chang, H.-K.; Hsieh, C.-T.; Lai, J.-S.; Tan, Y.-C.; Su, M.-D. Using Tabu Search Adjusted with Urban Sewer Flood Simulation to Improve Pluvial Flood Warning via Rainfall Thresholds. Water 2019, 11, 348. [CrossRef] 21. Bradford, B.H. Optimal Storage Control in a Combined Sewer System. J. Water Resour. Plan. Manag. Div. 1977, 103, 1–15. [CrossRef] 22. Walters, G.A. The Design of the Optimal Layout for a Sewer Network. Eng. Optim. 1985, 9, 37–50. [CrossRef] 23. Labadie, J.W.; Chen, Y.H.; Morrow, D.M. Optimal Control of Unsteady Combined Sewer Flow. J. Water Resour. Plan. Manag. Div. 1980, 106, 205–223. [CrossRef] 24. Marinaki, M.; Papageorgiou, M.; Messmer, A. Multivariable Regulator Approach to Sewer Network Flow Control. J. Environ. Eng. 1999, 125, 267–276. [CrossRef] 25. Marinaki, M.; Papageorgiu, M. Linear-Quadratic Regulators Applied to Sewer Network Flow Control. In Proceedings of the 2003 European Control Conference (ECC), Cambridge, UK, 1–4 September 2003. 26. Zhang, D. Sewer System Control using Artificial Intelligence, Hydraulic Model, and Internet of Things. Ph.D. Thesis, Norwegian University of Life Sciences, As, Norway, 2019. 27. Liang, L.Y.; Thompson, R.G.; Young, D.M. Optimising the design of sewer networks using genetic algorithms and tabu search. Eng. Constr. Arch. Manag. 2004, 11, 101–112. [CrossRef] 28. Murillo, J.; Busquets, D.; Dalmau, J.; López, B.; Muñoz, V.; Rodríguez-Roda, I. Improving urban wastewater management through an auction-based management of discharges. Environ. Model. Softw. 2011, 26, 689–696. [CrossRef] 29. Vasiliev, I.; Luca, L.; Barbu, M.; Vilanova, R.; Caraman, S. Sewer Network Model of a City with a Medium-Sized Population. In Proceedings of the 23rd International Carpathian Control Conference, Sinaia, Romania, 29 May–1 June 2022. 30. Jern, H.W. Industrial Wastewater Treatment; Imperial College Press: London, UK, 2006; p. 17. [CrossRef] 31. Mitchell, M. An Introduction to Genetic Algorithms; MIT Press: Cambridge, MA, USA, 1998. 32. Barbu, M.; Vilanova, R.; Santín, I. Fuzzy control applied on a benchmark simulation model for sewer networks. In Proceedings of the 2016 20th International Conference on System Theory, Control and Computing (ICSTCC), Sinaia, Romania, 13–15 October 2016; pp. 180–185. [CrossRef]