Drinking water utilities are aimed at continuously delivering good quality water with sufficient pressure to its customers at minimal cost [1
]. Raw water after it has gone through several treatment processes is supplied to consumers via a water distribution system (WDS). A WDS is made up of various components such as pumps, valves, pipes, storage tanks, reservoirs and several fittings. The optimal configuration settings of these components are required to ensure maximum performance at minimal cost. Hydraulic models are frequently used to study and analyse the behaviour of a WDS. The applications of these models include, but are not limited to, the investigation of various management scenarios, extension of the WDS, determination of the optimal network settings and identification of the critical areas for rehabilitation [2
]. Hence, a hydraulic model of a WDS acts as a decision support tool to improve the system’s performance and reliability. The reliability of hydraulic model predictions greatly depends on the level of calibration. Under different modelling applications, a different level of calibration is required; hence, it should be done based on the intended purpose of use of the model. Ormsbee and Lingireddy [3
] suggested that a maximum deviation of 10% between the observed and the model-simulated values can be considered as a satisfactory calibration in most planning applications, while a deviation of 5% is highly desirable for system design, operation and the application of water quality modelling.
WDS hydraulic model calibration is performed by adjusting several parameters that control the hydraulic behaviour of the network. These parameters are adjusted such that the model closely represents the behaviour of the real system [4
]. A well calibrated hydraulic model will reliably reproduce both flow and pressure values at the point of interest [5
]. Hydraulic model calibration is a considerably complex task as there are a large number of parameters involved in the calibration process. Hence, manual calibration of a WDS hydraulic model can be a time-consuming task. The calibration of a WDS hydraulic model can be automated by incorporating readily available optimisation algorithms to the hydraulic model. These algorithms are broadly classified into local and global optimisation. A local optimisation searches the minimum value of the objective function within a local region of the input space. In contrast, global optimisation searches the minimum objective function within the whole input space. A majority of the global optimisation algorithms, including the covariance matrix adaptation evolution strategy (CMAES) and the shuffled complex evolution (SCE), belong to the class of evolutionary algorithm, which is a population-based metaheuristic algorithm [6
]. The gradient-based methods such as the Gauss–Levenberg–Marquardt algorithm (GLMA) are commonly used in local optimisation [12
]. The global optimisation algorithms are proven to be very reliable, despite the limitation that they cannot effectively handle a large number of parameters [12
There have been many studies that describe the detailed calibration procedure of a WDS hydraulic model [2
]. The objective function of a WDS hydraulic model calibration is non-linear and the optimisation problem can be considered as ill-posed due to the fact that the number of measurements is much smaller than the variables [14
]. Using a genetic algorithm (GA), Do et al. [15
] proposed an approach to deal with the ill-posed problem and suggested that averaging the result of multiple simulation can be considered as a good solution. Khedr, Tolson and Ziemann [13
] compared a manual vs an automatic approach using multiple objective functions where the better performance was achieved under automatic calibration using the pareto archived dynamically dimensioned search (PA-DDS) algorithm. Similarly, Do et al. [16
] and Letting et al. [17
] used particle swarm optimisation to estimate the water demand in a WDS hydraulic model. Liong and Atiquzzaman [6
] compared the performance of several optimisation algorithms including SCE, GA and simulated annealing (SA) and concluded that SCE has the potential to optimise the WDS hydraulic model. An issue with the hydraulic simulation is that zero flows can cause a computation failure while solving the non-linear equations of the model [18
]. Currently, hydraulic models are equipped with a low flow adjustment function to prevent the situation of the solution becoming ill-conditioned or not converged.
A major challenge faced by the water utilities is to manage water losses in the network that occur via leakage. According to a survey by the International Water Services Association (IWSA), between approximately 8 and 24% of water losses are due to leakage in most developed countries, while in developing countries this could be vary between 25 and 45% [19
]. Several reasons can trigger leakage in the network including aging infrastructure, corrosion in pipes, mechanical damage due to excessive loads, material defects, assembling errors and seasonal variation in temperature [20
]. There are various leakage analysis methods and management strategies available in the literature [20
]. Water leaks in a WDS can affect the quality and quantity of water [20
]. They can increase the operating cost of the network as more energy is required to ensure the minimum operating pressure. Leakage is also an environmental, sustainability and health and safety issue [22
]. Hydraulic modelling tools allow the simulation of leakage in the network using an emitter connected to a junction. An emitter allows modelling of the flow through a nozzle or orifice that discharges water to the atmosphere. The discharge by the emitter is expressed using a power function of the nodal pressure. For each leakage node, an emitter coefficient and a global emitter exponent are used to model the leakage flow, and they are optimised to represent the leakage more accurately. However, this study focused on modelling variable demand patterns and leakage was excluded from the analysis considering the current optimisation problem size. This is because several additional parameters need to be included in the model to simulate the leakage in the network and calibrating all of them with the other model parameters would be computationally very extensive.
Sustainability and efficiency are two major aspects that need to be considered while optimising a hydraulic model. The performance of a WDS can be assessed using several key performance indicators (KPIs) including the quality of and access to the services, efficiency in operation and business management, and financial and environmental sustainability [24
]. A study by Dandy et al. [27
] suggested that optimising a WDS with consideration of the KPI variables can significantly reduce the CO2
emissions and consumption of non-renewable resources. While a water column is moving through the pipe network, it involves energy expenditure and minimum energy is sought for sustainability and efficiency aspects. A WDS subjected to low flows with high frequency can potentially reduce the energy footprint and CO2
]. Controlling the pressure throughout the network also helps to improve the efficiency of the system [26
]. The calibration objective function of a WDS hydraulic model should constitute the KPI variables to ensure sustainability and efficiency. While developing a hydraulic model, the operating rules in the model should be defined to reflect the KPIs. For instance, pump operating rules should be aligned with the minimum energy price of the day in cases where the energy price changes throughout the day to ensure cost and energy efficiency. Similarly, tank operating rules should be defined such that they correspond to a minimum water age to ensure the quality of water. The parameters representing the variable demand patterns in a WDS hydraulic model can be used with other available KPIs to formulate the objective function and optimise the model parameters.
A WDS serves a variety of customers, hence, different consumption patterns are observed in different parts of the network. Typical consumption patterns in a WDS are classified as residential, commercial and industrial [17
]. These patterns are comprised by multiplication factors that represent the hourly or sub-hourly water consumption within a 24-h cycle with respect to a baseline value. Among these patterns, the residential demand pattern is highly variable in space and time and probably one of the most dominating factors controlling the hydraulic behaviour of the network. According to Kang and Lansey [28
], water demand and pipe roughness are the most uncertain parameters as they are not directly measurable. Therefore, successful calibration mostly depends on the fine tuning of these parameters. Data analysis shows that variable demand patterns for an extended period of hydraulic simulation can be better modelled using a linear modelling approach than using the multiplication factors only. Therefore, the objectives of the study are: (i) to develop a WDS hydraulic model and to incorporate variable demand patterns using linear modelling and (ii) to access the calibration performance under different optimisation algorithms. The proposed strategy has not been investigated before, hence, it brings a new knowledge to WDS hydraulic modelling.
3. Results and Discussion
The EPANET hydraulic model for the TBK WDS was calibrated for a period of four weeks in early February–March, 2021. The flow pattern from a flowmeter during this period is presented in Figure 4
a and the estimated average daily demand multipliers of all days are presented in Figure 4
b. Both figures indicate the residential consumption mainly characterising the flow pattern in the area. There is a peak flow in the morning and in the evening, which is similar to a diurnal curve. A strong correlation was observed between the flows on different days with the correlation matrix ranged between 0.95 and 0.41. The maximum and minimum flows during this period were 13.29 L S−1
and 5.06 L S−1
, respectively, with a mean value of 7.94 L S−1
and standard deviation of 1.64 L S−1
. Several reasons, including seasonal variation of temperature, the day in the week (weekdays or weekend) and holidays can influence people’s water consumption pattern; hence, the demand multipliers on different days can also change.
Further investigation was conducted to model the relationship between the flow patterns on different days. Figure 5
shows the plot of demand multipliers on the first day against the same on all other days during the study period. In the figure, it is evident that a linear relationship exists between the demand multipliers on different days. The trend lines of these linear models have an offset in the y-axis to the origin.
Hence, using the daily multiplication factor may not be adequate to represent their relationship. Therefore, the intercept term was also considered in modelling the demand pattern relationship. Both the slope and the intercept terms were considered as calibration parameters, and their values were searched through the calibration process that closely represents their relationship. This range of parameters related to demand modelling allows additional pathways for further reduction of the objective function during calibration. However, this increases the number of parameters to optimise, hence, increasing the solution space. Running the optimisation task in the HPC environment can effectively handle this kind of large problem. Non-linear functions were not considered as they introduce more parameters than linear functions, consequently, the problem size would also increase.
CMAES and SCE optimisation were run with different population sizes and a different number of complexes. It was found that increasing the number of complexes in the SCE optimisation considerably increased the number of model runs. The global search does not require any starting points as they perform the optimisation considering multiple start points. However, providing a set of initial values considerably decreases the number of model runs. Hence, the patterns identified through data analysis for residential demand multipliers and the typical values of commercial demand multipliers were used as the initial starting point for CMAES and SCE runs, respectively. For a local search using the GLMA algorithm, R codes were composed to generate random starting points within the upper and lower boundaries of the parameters. Several batch runs were performed with 28 local searches in each batch employing all 28 processors in a single HPC node. However, none of these GLMA-based searches seemed to proceed towards the global minima, while CMAES and SCE showed comparatively more reduction of the objective function. Figure 6
presents the observed and model-simulated time-series corresponding to the best objective function at three different locations in the TBK WDS: (a) flow at pump station 2 (PS2); (b) head at Binnies tank; (c) head at Meningie tank.
As can be seen in Figure 6
, simulated time-series closely match with the observed time-series. However, Figure 6
a shows that the simulated flow was steady while the pump was running but the actual observed flow was unsteady, indicating that the variable pump speed was poorly simulated by the model. Some studies have suggested simulating the variable pump speed by modifying the rotor speed using several rule-based controls [43
]. This was not further explored in the current study considering the optimisation problem size as several additional parameters would have been needed to be included in the model to simulate the variable pump speed. Furthermore, the simulated flow between 160 to 190 h suggests that the pump was run twice, while the observed flow data shows zero flows during this period. The tank connected to the pump (Figure 6
b) also indicates a rise of head twice between 160 to 190 h. This suggest that there could be possible SCADA outage or instrument failure during this period, hence, no data was recorded. Figure 7
presents the plot of observed vs model-simulated values at these locations. The correlation between the observed and the simulated flows at PS2 and heads at Binnies and Meningie tanks were 0.89, 0.96 and 0.88, respectively. The mean of the observed flow and heads at these locations were 57.12 L S−1
, 136.77 m and 42.53 m, respectively, while the same obtained through model simulation were 54.71 L S−1
, 136.78 m and 42.54 m, respectively. These statistics indicate that the observed data were reasonably reproduced.
The modelling of variable demand patterns and the corresponding calibration performance was compared using two different criteria: (i) daily demand multiplication factors and (ii) a linear modelling approach. GOF tests were used to assess the fit between the observed and the model simulated values. The GOF statistics for both cases are presented in Figure 8
where an improved fit is evident under second criterion. The value of NSE, R2
and RMSE for the first criterion at PS2 were 0.73, 0.77 and 27.63, respectively, while for the second criterion they were 0.77, 0.80 and 26.80, respectively. Similarly, at Binnies tank, NSE, R2
and RMSE between the observed and simulated time-series were 0.86, 0.89 and 0.26, respectively, under criterion 1, while for criterion 2, they were 0.91, 0.92 and 0.24, respectively. On the other hand, at Meningie tank, the values of NSE, R2
and RMSE for criterion 1 were 0.72, 0.76 and 0.20, respectively, while for criterion 2 they were 0.74, 0.77 and 0.19, respectively. These statistics indicate that the proposed method to model the variable demand patterns can improve the fit between the observed and simulated time-series. Although the calibration performances under both criteria during the studied period do not differ much, incorporating the linear modelling parameters into the calibration process allows modelling of the demand patterns on days with a relatively flatter peak.
Both CMAES and SCE algorithms showed different levels of optimisation performance. Compared to CMAES, it took considerably longer to complete the optimisation process by the SCE algorithm. Up to five complexes and 90,000 runs were tried but the overall achievement was poorer than CMAES. Increasing the number of complexes may improve the performance of SCE optimisation. Likewise, for SCE algorithm settings, increasing the population size was found to improve the CMAES performance. Hence, both of the CMAES and SCE algorithms’ performances were found to be subjective, and they varied with many factors, including the algorithm runtime settings. Furthermore, no leakage analysis was performed in the current study. Modelling and including the emitter parameters to the calibration can ensure water loss through leakage is reasonably accounted for, which can further improve the overall calibration performance. Table 2
presents a summary of the calibration run information using the different optimisation algorithms.