Disinfection scheduling in water distribution networks considering input time-delay uncertainty

There is currently significant research interest in real-time monitoring and regulation of drinking water quality, especially after a recent EU Directive (European Union 2020), which encourages continuous monitoring of water-quality parameters in water distribution networks (WDN). According to the World Health Organization (WHO 2017), a disinfectant residual needs to be sustained throughout drinking water networks, such that it is sufficient to deactivate waterborne pathogens and, at the same time, small enough to reduce the formation of harmful disinfection byproducts (DBPs), such as trihalomethanes (THMs) and haloacetic acids (HAA) (Mouly et al. 2010).

The effective regulation of water quality in WDN is based on the use of mathematical models that facilitate the estimation of disinfectant concentration in the water at different parts of the network (Elsherif et al. 2022). This is a challenging task, since water-quality models are inaccurate and complex, especially in large-scale networks (Frankel et al. 2023). This is why heuristic algorithms (Li et al. 2021) and machine learning models (Sun et al. 2019; Li et al. 2023) have been used for the detection and source identification of contaminants or other water-quality issues in water networks. A significant challenge when regulating the concentration of a disinfectant in a WDN is the significant delay between the time that a chemical is injected at the input node and the time that the concentration is measured at the monitoring output nodes. Moreover, the time delays are time-varying and uncertain due to varying pipe flows, which depend mainly on consumer demands that are an uncontrolled and typically unknown input to the system (Eliades et al. 2023). In addition, the disinfectant reaction dynamics are usually only partially known and the reaction parameters may vary due to the use of water originating from diverse sources or due to fluctuations in temperature (Monteiro et al. 2017).

One of the first studies addressing the challenge of large and varying time delays when controlling water quality is by Polycarpou et al. (2002), where the proposed input–output (I–O) model of disinfectant concentration is used to adaptively learn periodic parameters to characterize the time delay. The input–output approach can be thought of as a model-reduction technique that overcomes the challenge of using a small number of measurements compared to the system states, to design a water-quality control algorithm (Wang et al. 2006). More recently, Wang et al. (2021) derived a state-space model of chlorine concentration evolution in WDN, which traces over time all water-quality states; i.e., the concentration at all the links and nodes of the network. To do this, they solve the hydraulic equations to obtain the steady-state flow solution for a hydraulic-step, and then create the state-space equations, which are time-varying due to the varying flows. The drawback of this approach in practice is the lack of observability for estimating the complete state vector, and the need for using model-reduction techniques to calculate a control law. A simulation-based model is used in Xie & Brdys (2015) to compute hydraulics and water quality, while nonlinear model predictive control (MPC) within a hierarchical control framework is then used to control disinfectant levels. The authors in Xie et al. (2018) use a multi-input multi-output modeling approach, which considers also the formation of DBPs.

The aforementioned approaches, however, cannot guarantee that the concentration of a chemical will remain within a specified range at the output, even though bounds on water flows uncertainty, and, by extension, the water-quality input–output time-delay uncertainty, may be known. The quantification of water-quality state uncertainty is typically achieved through Monte Carlo Simulations (MCS) of hydraulic scenarios, due to the model complexity in large-scale networks (Hart et al. 2019). However, this approach is limited by computational constraints in large systems. Thus, researchers have looked into alternative methods, such as spectral propagation (Braun et al. 2020). Recently, an algorithmic approach has been proposed by the authors that guarantee the validity of calculated water-quality bounds, as opposed to MCS (Vrachimis et al. 2021). Given these developments, a water-quality regulation approach is needed that is not reliant on MCS to maintain water-quality states within predefined value sets.

In this work, we use the input–output modelling approach to develop a disinfectant injection methodology that guarantees a bounded output disinfectant concentration, given bounds on water-flow uncertainty. The proposed methodology creates an input–output model which considers time-delay uncertainty by utilizing recent results on estimating bounds on water-quality states using the backtracking approach (Vrachimis et al. 2021) Following the model predictive scheduling framework, an optimization problem is formulated and solved to find an input schedule that keeps the disinfectant concentration within predefined bounds for a specified time horizon. Simulation results illustrate the effectiveness of this approach in an example network case study.

The following notation is followed throughout this work: matrices are denoted with capital letters, vectors with lower bold letters, and scalars with italic letters. Sets and graphs are denoted by calligraphic capital letters. Estimated parameters are denoted with a hat; i.e., denotes the estimated values of the parameter vector ⁠. Uncertain parameters are represented by a continuous interval of values defined by a lower and upper bound. Intervals are accompanied by a tilde as follows: ⁠, where is the lower bound vector and is the upper bound vector, such that: ⁠, and is the size of the vector.

Consider a WDN whose topology is modeled by a directed graph denoted as ⁠. Let be the set of all nodes, where is the total number of nodes. These represent junctions of pipes, consumer water demand locations, reservoirs, and tanks. Moreover, let represent the tank nodes, where is the total number of tanks. Finally, let be the set of links, where is the total number of links. These represent network pipes, water pumps, and pipe valves.

The hydraulic state associated with a node at each time instant is the hydraulic head, denoted by ⁠. The hydraulic state associated with a link is the water flow, denoted by ⁠. In general, the complete set of hydraulic states can be expressed as ⁠. Each node is associated with a water consumer demand at the node location, denoted by ⁠. Water demands drive the hydraulic dynamics of a WDN and in this work are considered as an unknown and uncontrolled input to the system. It is noted that some demands may be zero if the node is not associated with any consumers. Background leakages that may exist in the network are modeled as part of the unknown water demands. Pumps and valves are the main hydraulic actuators in a WDN and are modeled at the network links. Their control settings are indicated using the hydraulic input vector ⁠.

Specific hydraulic states are measured at regular time intervals using hydraulic sensors installed in the network. The interval is in the range of minutes, and hydraulic analysis is typically performed in discrete time. The measured states may be flow, pressure, or tank level and are indicated by the hydraulic output vector ⁠, where indicates the discrete hydraulic time-step associated with time intervals ⁠.

Water quality refers to the concentration of various chemical substances and biological species in drinking water that may affect human health. In general, the concentration of these substances in the water should be within predefined bounds, defined by regulating authorities. Disinfectants are the only controllable substance in the water. The existence of a sufficient concentration level of disinfectant in the water indicates that any infectious substances have reacted with the disinfectant and have been neutralized. In this work, water-quality state refers to the concentration of a disinfectant in the water, specifically at the nodes of the network. The water-quality state vector at each time instant is denoted by ⁠.

Water quality is regulated using chemical dosing pumps, designed to inject specific amounts of disinfectant into the water at a set of network nodes ⁠, where is the total number of controlled inputs. Water quality is monitored using sensors (e.g., chlorine sensors) installed at nodes ⁠, where is the total number of quality sensors. Control and measurement signals are transmitted at discrete time-steps ⁠. For simplicity, in this work, we assume that is equal to the hydraulic step, i.e., ⁠; to ensure the stability of the solver, the water-quality time step must be sufficiently small compared to the water transport times in the network pipes (Rossman 2000). The water-quality control input vector is denoted by ⁠, while the measurement vector of quality states is denoted by ⁠.

A hydraulic model is a set of equations that describe the hydraulic dynamics. These equations are formulated considering the conservation of energy and mass laws (Vrachimis et al. 2019). The relationship between the network topology and the physical parameters denoted by the vector (pipe roughness, length, diameter, node elevations, tank dimensions) are also described by the model. Hydraulic parameters are typically time-invariant for the time scales considered in this work. The hydraulic model used in this work is indicated by ⁠.

The installed hydraulic sensors measuring flow and pressure in the network typically do not satisfy hydraulic-state observability conditions (Díaz et al. 2017). This is because water demands are typically unknown and only an approximate pattern of consumption at nodes is available a priori. The approximate consumption patterns are used as pseudo-measurements to achieve hydraulic-state observability. Due to the high uncertainty of pseudo-measurements, an interval of possible demand values for each node is used at each time step instead, indicated by ⁠. Moreover, the network’s physical parameters are not exactly known. An estimate of these parameters is used instead, indicated by ⁠. The uncertainty of these parameters is considered bounded in this work, thus the range of values they may take is indicated by the interval vector ⁠, with ⁠.

A water-quality model describes the dynamics of species transport and reaction in the water. Transport refers to the movement of substances in water due to the water flow in pipes. Substances can be tracked throughout the network when water flows are known for a number of time-steps, referred to as the memory of the system ⁠. This is defined as the maximum number of discrete time-steps needed for any input node to affect any output and can be computed given the minimum flow rate of pipes during the system operation.

Many substances, such as various disinfectants, react with other substances in the water and materials attached to pipe walls. Thus, reaction dynamics need to be taken into account when estimating their concentration in the water. Reaction rates are typically unknown due to the multiple species existing in the water and the different characteristics of the water inserted in the network. We assume the availability of bounds on the uncertain estimates of disinfectant reaction rates in the water and with pipe walls. The range of disinfectant reaction rates is indicated by the interval vector ⁠. Finally, the water-quality model is indicated by ⁠.

An optimization formulation is proposed to calculate the control input for a prediction horizon using the uncertain IR matrix ⁠. The objective is to find an input that minimizes the deviation of expected output disinfectant concentration from a reference concentration ⁠. For simplicity, we assume that the past inputs of the system be zero, thus the reference concentration ⁠; note that this assumption is not limiting because any initial disinfectant concentration decays after a period, specific to each water network, thus only the controlled injection of disinfectants affects the water-quality state. Moreover, the rapid changes of disinfectant injection should be minimized. The objectives should be achieved under the following constraints: (a) The output should not violate the defined lower bound for all possible scenarios given uncertainty bounds on water demands, and (b) The control input should reside within a predefined interval given by ⁠.

The first constraint is the key to addressing the uncertain time-delay issue. Essentially it represents the physics of the system using the I–O model, i.e., the impulse response matrix. However, using the uncertain IR matrix in the formulation, the physics of multiple models are considered simultaneously. The use of path-IR matrices in the formulation is essential, in order to distinguish between models that represent a different reality, and models that represent different paths in the same reality.

We compare the proposed approach with a scheduling algorithm that uses the nominal model of the network and does not consider flow uncertainty (Munavalli & Kumar 2003). The implementation of the ‘nominal’ scheduling algorithm in this work considers a single-input single-output system. Essentially, it uses the formulation of (13), with the first constraint substituted by an IR model, as in (7), that is derived using the nominal hydraulic model of the network.

The results of Figure 4 illustrate the benefits of considering time-delay uncertainty when calculating chlorine injection inputs. When the nominal model is used (Figure 4(a)), we observe the following deficiencies in a number of uncertainty scenarios compared to when the proposed scheduling approach is used (Figure 4(b)):

• ⁠: (a) there are lower bound violations during abrupt changes; (b) the proposed approach predicts the worst-case delay and avoids violations.

• ⁠: (a) there are lower bound violations because of the effect of varying time-delay on chlorine decay; (b) there are small violations when using the proposed approach due to the proposed approach not considering modeling and discretization uncertainty ⁠.

• ⁠: (a) There is an overshoot of chlorine concentration in some scenarios due to the effect of path 2 starting earlier in some scenarios; (b) The same effect is observed here since there is no violation of the lower bound.

• ⁠: (a) There is a significant drop in chlorine concentration and lower bound violation due to flow reversal occurring at link 6 in some scenarios; (b) In contrast, the proposed approach anticipates the flow reversal and keeps the chlorine concentration above the threshold in all considered scenarios.

The scalability of the proposed approach is demonstrated using the ‘CY-DMA’ network (illustrated in Figure 5), a DMA of a real network on Cyprus (Vrachimis et al. 2020), with pipes, junctions, and 1 reservoir which servers as network water source. The disinfectant input of this network is administered at the inlet node ⁠, while the sensor is positioned at the far end of the network, at node ⁠. Node demands follow a diurnal pattern. The network has multiple flow paths from input to output, with some links changing direction during simulation. The network is simulated for the duration of h, with a time-step corresponding to min.

In this work, we have demonstrated a new disinfection scheduling methodology, which considers the time-delay uncertainty when regulating the disinfectant concentration at specific locations of a water distribution network. The results indicate that this approach is able to avoid violations of predefined bounds on disinfectant concentration and regulates the output to follow closely the reference signal, even when the modeling uncertainty on water demands is up to ⁠. The use of a modified version of the uncertainty bounding backtracking algorithm proposed in Vrachimis et al. (2021) ensures the consideration of all different flow scenarios when calculating the scheduling input, as opposed to using a Monte-Carlo simulation approach that only samples a subset of potential scenarios.

While this work addresses the challenge associated with time-delay uncertainty when regulating disinfectant concentration, it is important to acknowledge certain limitations. Firstly, the methodology focuses on single-input single-output water-quality systems, thus limiting its applicability to networks with multiple disinfectant booster stations and multiple sensors. Additionally, the approach assumes networks without water tanks or those where disinfectant concentrations in tanks are known. Another limitation is the exclusion of uncertainty in decay rates, as our model primarily accounts for time-delay uncertainty arising from fluctuating water flows. Finally, as indicated in the case studies, the proposed scheduling algorithm may exhibit a degree of conservatism in disinfectant concentration regulation, depending on the given uncertainty of water flows.

Future work will develop this methodology to consider multiple inputs and outputs in the network and study its performance and computational efficiency when applied to large-scale water networks. Moreover, a natural extension of this work is the consideration of disinfectant measurements in a feedback loop, to improve the water-quality model and provide better estimations, thus also improving the disinfectant regulation. Finally, the effect of modeling and discretization uncertainty on the performance of the proposed methodology will be analyzed.

This work was funded by the European Research Council (ERC) under the ERC Synergy Grant ‘Water-Futures’ (Grant agreement No. 951424) and by the European Union Horizon 2020 programme Grant Agreement No. 739551 (KIOS CoE), and the Government of the Republic of Cyprus through the Deputy Ministry of Research, Innovation and Digital Policy.

The models, code and data generated during this study are available in https://doi.org/10.5281/zenodo.10226047.

The authors declare there is no conflict.