Digital transition, digital twin and digital water: history, concepts and overview for the application to aqueducts

ABSTRACT

Introduction

The words digital transition, digital twin and the concept called digital water, quickly came into common use due to a profoundly modified global context following the pandemic event which accelerated changes already previously underway, including the digital transition itself.

Objectives

This document aims to give technical-scientific substance to the word digital water in a critical way, starting from the foundations of current changes and taking as a specific reference the management of aqueduct systems, without losing the generality of the discussion.

Method

To the purpose, the concepts behind the words digital transition and digital twin are discussed and revised starting from the salient points of the scientific history of the past centuries and decades, which were the foundation of these concepts. This is the only way to develop the technical-scientific meaning of digital transition and digital twin critically and usefully in the technical world of water systems, to achieve the same objectives intrinsic to the concepts of digital transition and digital twin.

1. Context

The management of the infrastructure of the integrated water system, with reference in this document to the aqueducts, is going through a moment of substantial change under the impetus of multiple concomitant events and opportunities for innovation.

In Italy, for example, in 2017 the National Authority, called ARERA, was established, with the consequent regulation of technical quality, which uses macro-indicators to evaluate the performance of the integrated water service and the related rewarding and penalising factors for water companies. The ARERA regulation, therefore, triggered the change in the management of aqueducts by introducing the need for “measurements”for example, to evaluate the linear or density indicator of water loss. For the first time in Italy, a process of “optimisation” of investments started to achieve management performance of aqueducts through their engineering and the technical rationalisation of the interventions indicated by the technical quality rules of a National Authority. Subsequently, the pandemic, which began in March 2020, was a sad and critical event that also had useful aspects:

• revealing and recalling the inalienability and centrality of scientific research in the complexity of today’s world and the very potential of machine learning when all of humanity looked at research centre for the quick develop, in few months, of vaccines that in the past took some years.

• boosting the need to face the complexity of the current world, to make it sustainable for future generations, through the concepts of ecological, energy and digital transitions, and boosting the research and the activities for recovery plans.

• anticipating and accelerating the digital transition, which was already underway, since for a long time, during the pandemic event, the information and communication technologies were the only window of sociality and relationship with the real world.

Furthermore, the policies to support the post-pandemic recoveryfor example, the National Recovery and Resilience Plan (PNRR) in Italy, led to a significant increase in investments both for research and innovation and for water infrastructures, which today must be implemented in a scenario made even more complex by water emergency, which will probably increase in the future, as they are likely triggered by climate change.

2. Digital transition

Digital transition is meant as the review of processes, using products based on digital technologies (hardrware) and strategies (software), to increase efficiency. The simpler, more accessible, and representative collection and evaluation of data relating to processes is the knowledge base to provide useful information for efficiency.

The digital transition, therefore, is not achieved only through the implementation of products and monitoring systems based on digital technologies and strategies, but when they allow achieving efficiency of the processes. In water systems, achieving efficiency implies to support decision for technical activities and human capital information and education, and water company reorganisation.

The world in the 90s has already gone through a digital transition accompanied by the affordable diffusion of the first personal computers. Therefore, digitisation has started the process of making systems more efficient for more than three decades.

The current digital transition, therefore, can be considered more rationally as a continuation of the event that began in the last century, albeit characterised by an acceleration generated by various conditions, including the large investments and profits originating from the pandemic context.

Certainly, the advances in basic electronics are the foundation of the evolution and development of digital tools and allow ever greater calculation, transmission, and data storage speeds to support the digital transition, i.e., the process efficiency through digital technologies and products.

Consequently, the development of digital technologies and products, i.e., the technological event, does not explain and is not alone the digital transition. In fact, at the basis of today’s digitalisation there are always humans who developed the theories, paradigms and concepts that generated the strategies, methods, and algorithms. The latter have not only allowed the development of digital products, together with supporting the evolution of electronics, but strategies, methods, and algorithms of the digital transition are also the basis to make efficient the processes themselves.

The scientific studies that have generated the possibility of today’s digital transition, impacting definitively on the development of both digital technologies/strategies and products, refer to the fields of mathematical logic and mathematics. They have developed throughout the last few centuries.

It is therefore useful to report some fundamental stages, which, in the past, often not recent, have given rise to the scientific and technical conditions of the digital transition.

Alan Turing, already in the thirties of the ‘900, introduces the concepts of algorithms and calculating machines that will later lead to the development of computers and today to the digital transition. In fact, he is considered the father of information technology and of the concept of machine learning which has nowadays entered everyday life with the idea of artificial intelligence. It is useful to clarify, without dwelling excessively on the subject, that machine learning or data-modelling or data-driven are more appropriate terms than the generic one of artificial intelligence. The latter is an abused slogan, for reasons distant from both scientific and technical reality, which should not be used for at least two reasons:

• it is misleading with respect to the possibility of machines, or more generally of digital tools, to make water systems more efficient using an unsupervised strategy, therefore, to implement the digital transformation without the intervention of the human reasoning, which is mandatory in the management of water systems. Indeed, the hydraulic systems are complex because of the governing physical laws, i.e., from the phenomenological point of view to understand the behaviour. Furthermore, the spatial widespread as well as the interaction with the environment and with society itself makes them very different from an industrial plant, where unsupervised machines and digital automation, under certain conditions, can allow reaching the efficiency of processes as for the digital transformation aim.

• It is not yet known what human intelligence is, both at the mechanistic and biological level of brain functioning; therefore, it is impossible to build machines that simulate something that is unknown in the intrinsic mechanisms that, in addition, generate human consciousness, for example, a concept itself that is difficult to understand.

The idea of artificial intelligence was born with McCulloch and Pitts in 1943 (McCulloch and Pitts 1943) when they published a work showing a simple system of artificial neurons able to perform basic logical functions. At least in theory, this system could learn in the same way that humans learn by using experience through the trial and error that strengthens or weakens the connections between neurons. Artificial neural networks are machine learning referencing this idea of McCulloch and Pitts. They concretely become the today well-known artificial neural networks that were already programmed in the first personal computers when Rumelhart, Hinton and Williams developed the Error Back-Propagation method (Rumelhart et al. 1986), in 1986, to train them, or rather to calibrate the weights of the “synapses” that connect the neurons simulating, in a “very simplified way”, the functioning of the human brain. Note that artificial neural networks can be seen today as a category of machine learning strategies which are based on the original paradigm with developments of the mathematical structure and learning strategies.

The same complex network theory (Newman 2010), which we use every day without being fully aware, for example with the Google Map app or in social networks, originated in the graph theory founded by Leonhard Euler, with the work “Seven Bridges of Königsberg” (Euler 1736), which later became the complex network theory during the last century with the first characterisation studies of the different topological types of networked systems (Erdӧs and Rėnyi 1959, Barabási and Albert 1999, Barthelemy 2010, Watts and Strogatz 2011).

Since the beginning of the century, the complex network theory has been a field of scientific research in great progress, as well as very vast and articulated, which generates paradigms, algorithms and tools that have a daily impact in the digital transition. In the last decade complex network theory was also used to improve and support analysis, planning and management of water systems (Yazdani and Jeffrey 2012, Giustolisi et al. 2017, 2020). The water supply and distribution systems, for example, are complex networked systems; the works (Giustolisi and Ridolfi 2014, Laucelli et al. 2017) of 2014 and 2017 presented the theoretical foundations for the use of digital strategies for optimal design of district metering areas (DMAs) through the modularity index (Newman and Girvan 2004) of the complex network theory and the evolutionary optimisation.

Evolutionary computing or optimisation is another idea that was born in the last century and is nowadays a relevant component of process efficiency strategies. In 1973, Ingo Rechenberg was the pioneer of evolutionary calculation and artificial evolution (Rechenberg 1971), whose theories were taken up again in 1975 by John Holland, who develops the theory of genetic algorithms reported in the book Adaptation in Natural and Artificial Systems (Holland 1975). In 1989, David Goldberg, a student of John Holland and hydraulic engineer, wrote a book (Goldberg 1989) which became the milestone for the use of genetic algorithms.

It is not a coincidence that the genetic algorithms strategy was proposed for the first time in the field of hydraulic studies already in 1995 by Simpson, Dandy & Murphy (Simpson et al. 1994) and then in 1997 by Savic & Walters (Savic and Walters 1997) to demonstrate the possibility of optimal pipe sizing, at the minimum cost with constraints, for water distribution networks. Nowadays there are many evolutionary optimisation strategies even if they all refer to the main idea of Rechenberg and Holland.

The tools allowing the optimisation of processes with evolutionary calculation strategies, such as genetic algorithms, are essential for obtaining system efficiency. In fact, they allow cost–benefit problems (efficiency) to be solved by considering more than a single objective, contrary to most of the classical techniques. The mathematical expressions of the objectives and of the same effectiveness constraints can be written in a complex form also drawing on logical operators and not worrying about the differentiability of the expressions used. Furthermore, evolutionary strategies lend themselves better to the solution of combinatorial problems, as mentioned with multiple objective functions, exploiting the computational potential available today. Finally, these strategies return the so-called Pareto front of optimal or efficient solutions (Pareto 1906) from the cost–benefit point of view or more solutions with different cost–benefit trade-offs that become a decision support for the efficiency of any process, especially in technical systems, such as water systems. For example, Laucelli et al. (2023) report the application to the technical world, the first time at an international level, of the digital water strategy for optimal design of the DMAs and optimal planning of pipes to replace. The advanced hydraulic modelling for aqueducts integrated with evolutionary optimisation using genetic algorithms allowed to build a decision support through the concept of the Pareto front, as reported in Figure 1 for the pipe’s replacement plans.

Figure 1. Decision support for pipes replacement plans: pareto front replacement plans versus leakage reduction.

Finally, in 1992 John Koza developed the paradigm of genetic programming, i.e., he shows in a book (Koza 1992) the possibility of creating machines that programme themselves to solve problems postulated by humans. Genetic programming integrates machine learning, in a wider sense with respect to the original studies, with evolutionary optimisation in an original way. Much of what is proposed today as artificial intelligence refers to the paradigm of genetic programming.

The use of genetic programming in the hydraulic field was proposed in 2000 by Babovic & Keijzer (Babovic and Keijzer 2000). It is a specific application of Koza’s paradigm to obtain models by means of the integration of machine learning and genetic algorithms to obtain from data symbolic formulas (symbolic modelling of data) that can be evaluated as such by the expert, as opposed to what happens for artificial neural networks, which are general mathematical structures characterised by the “universal” ability to interpolate data, but, for this reason, not suitable for the interpretation of the results with respect to the physical knowledge of the expert about the modelled cause-effect phenomenon.

Later, in 2006 and 2009, Giustolisi and Savic (2006, 2009) introduced a technique called Evolutionary Polynomial Regression (EPR) which, still in the context of Koza’s paradigm, allows the symbolic modelling of data, i.e., to obtain formulas for models from the data. For example, EPR was used in 2008 for Anglian Water (Berardi et al. 2008) to obtain the pipe break prediction model in relation to asset data such as pipe age, diameter, length, number of connected properties, etc. Even in 2006 (Savic et al. 2006) for the development of formulas for models for different types of failure for sewers.

We reported so far, a brief survey of the history of scientific studies, which generated theories, paradigms and concepts for digital tools and strategies, the basis of digital transition as process efficiency. Furthermore, the application to the hydraulic field of those theories, paradigms, and concepts, adapted to technical ground, during the last decades, was briefly reported. The current world compared to the past offers greater challenges and opportunities.

Nonetheless, to be aware of the scientific story, which allowed the present digital transition, is educational for all of us to avoid slipping off the shoulders of the giants who wrote that story with results that could be extremely negative with respect to the goal of the digital transition itself.

3. Data and information

The digital transition, i.e., the opportunity for process efficiency through digitalisation, is today boosted by the increased possibility of collecting, transmitting, and evaluating much larger quantities of data than in the recent past. This collection, nevertheless, must not be driven by quantities; it must be a representative knowledge base of processes being made more efficient. Therefore, there is also a concept of efficiency, or cost-benefit, in data collection that must always be kept in mind.

Today, the risk is a recurring thought that the digital transition and the data collection are detached from the knowledge of technical processes, to be made more efficient, and from the governing physical laws of the water systems. The implicit assumption that the process efficiency is proportional to the quantity of collected data is very often dominating. Consequently, the monitoring data are not collected considering the quality in terms of representativeness of the physical phenomenon underlying the functioning of the water systems. Furthermore, it is assumed that the amount of data collected is a substitute for the knowledge of the physics of water systems and of technical processes.

The monitoring data and knowledge of the asset sometimes become a mechanistic integration to the digital tools and strategies with the naive optimism that the digital transition is consequently achieved.

On the contrary, the digital transition must be considered a progressive path where the data and digital tools (technologies and strategies), which transform them into information, must, the former be commensurate in quantity and quality and the latter in complexity, to the progressive objectives of efficiency of the processes. Note that the data and information are also strategic for the knowledge of the technical processes and functioning of water systems, which is consistent with the progressivity concept of digital transition as progressive increasing of the technical process efficiency.

Furthermore, the human capital of water companies must have education and information over time to be able to exploit data and digital tools, so there is also a progressivity of digital transition determined by the human factor.

Otherwise, the digital transition can simply generate traffic of technology (excess of measurement and calculation tools) and data (excess of collected data, which are not representative of the water system behind the process being made more efficient) aimed at justifying the expenditure rather than the perspective of the investment. The traffic of technology and data drives away from the desired efficiency challenge/opportunity through the digital transition. Furthermore, it poses problems about the future management of the costs for data collection and maintenance of digital tools that have proved to be useless. Additionally, the digital transition asks to an additional energy consumptionfor example, when collecting data, and this needs to be paid in terms of efficiency results.

The relevant issue of the digital transition relates therefore to the optimality and progressivity over time of data collection and the use of correct digital tools to generate information which can support decision-making beings enhanced compared to the past.

To clarify what has been reported, an example of excessive data collection due to the approximate technical-scientific knowledge of the functioning of the aqueducts is that which derives from the concept of hydraulic modelling and prediction of the behaviour of the aqueducts in “real-time” or “near real-time”, also implying the possibility to control such water systems on a time scale of a few seconds or minutes. This ambition is naïve from the scientific point of view as it neglects the fact that the information on the changes in flow rate and pressure is transmitted in the aqueducts at a celerity which has the order of magnitude of 1 km per second, since the physical substrate is the water, having an intrinsic mass. Therefore, even if assuming to control pressure every minute by means of a pressure reducing valve, the information, named unsteady flow perturbation, starts travelling from that device at the reported celerity. It would reach all the points of the aqueduct in several minutes (depending on its size) bouncing between the nodes of the network representing the domain of water system, due to the transmission and refraction of the perturbation waves, to attenuate to the equilibrium state after several minutes.

It is easy to understand that all of this would find a system changed in its main boundary conditions, i.e., the customer water requests, which are of impulsive nature at the scale of few seconds and minutes. On the other hand, the customer water demands are themselves cause of the generation of unsteady flow perturbations travelling into the aqueduct carrying the information on the variability of consumption to the water sources.

Therefore, until we can remove mass, as a paradox, from water to make it more like the electrons of electric currents, we will not be able to control aqueducts on short time scales and this will remain a fact that cannot be changed by any technology or methodology being only an abstraction confined into a computer. In other words, we will never be able to control a device on Mars, from Earth, in less than 25 minutes or talk intensely from Earth with a Martian until we find a way to transmit information faster than that of light.

To support this, it is to report that the technical-scientific research developed hydraulic modelling, as will be better explained later, in the awareness of physical phenomena. The hydraulic models for the management of aqueducts are “steady state” (stationary) and assuming technical uniform flow of the water in the pipes. This means that they capture the stationarity of the system on a time scale of several minutes, a scale whose choice is also a function of the purpose of the modelling analyses. Furthermore, this scale assumes that the unsteady flow perturbations are negligible (because averaged out to zero on the chosen scale), while integrating over time the customers’ consumption, impulsive for the single user, to have average flow rates of consumptions that are statistically stable at the chosen scale of the analysis.

Ultimately, the tools of the digital transition are “powered” by monitoring data that come from the real world, but the collection of data must consider: their representativeness with respect to the phenomena of the physical systems, whose management processes are being made more efficient; the quantity compatible with the result and, no less important, the education and information of the personnel who will manage the data and processes. Furthermore, the digital transition is also a challenge to progressive efficiency and not a “one shot” event. The lack of awareness and consideration of the above will generate an adverse effect on the digital transition which is what we have called technological and data traffic.

The data, as mentioned, relate also to the knowledge of the consistency of the asset of hydraulic systems through survey activities, which must follow the same principles as the monitoring data to be able to build an efficient and effective digital representation of water systems. The concept of digital twin is powered by data, information, system representation and digital tools (technologies and strategies). The historical and methodological aspects of digital twin are explored in the next paragraph.

4. Digital twin and digital water service

A concept alike digital twin, although implemented in physical and non-digital form, has been known since at least the 1960s. In fact, the twin concept was first used in the NASA Apollo programme as reported in (Rosen et al. 2015). NASA built two identical spacecraft: the vehicle that remained on Earth, called twin, was used as a prototype for replicating real operating conditions and simulating different scenarios on Earth, to support astronauts in decisions during critical situations. In 1991, David Gelernter, a specialist in the field of parallel computing, introduced, borrowing the philosophical principles elaborated by John Baudrillard (Jean 1981), the concept of “Mirror World” (Gelernter 1993) as a city-scale model of reality that is continuously elaborated and enhanced by in situ acquired data, allowing the user to stretch or shrink the image on a computer screen so as to obtain “one dense, lively, pulsating, teeming, moving and changing image”.

Few years later, in 2003, Michael Grieves presented a conceptual model that included real space, virtual space, virtual subspaces, and the flow of data between virtual and real spaces for industrial product lifecycle management (PLM). The same model was introduced in the first course on PLM in 2003 at the University of Michigan and was defined as a mirrored spaced model (Grieves 2006) and, subsequently, formalised with the term digital twin (Grieves 2011). The model proposed by Michael Grieves extends the twin concept of the Apollo programme by using a digital copy of the real product and defining the virtual space in which the relationships between the real and digital worlds are established.

NASA itself, in 2019, suggested adopting the concept of digital twin as a standard technology for the digital transition of the aerospace sector, defining it as an integrated multi-physics, multi-scale, probabilistic simulation of a vehicle or system using the best available physical models, sensor updates, fleet history, etc., to reflect the life of its twin in flight. The digital twin is ultra-realistic and can consider one or more important and interdependent vehicle systems, including propulsion/energy storage, avionics, life support, vehicle structure, thermal management/TPS, etc.

In the following years, with the diffusion of tools and services offered by Industry 4.0 (Kagermann et al. 2013) to various types of production and management sectors, the concept of the digital twin proposed by Michael Grieves was further expanded and generalised: the introduction of sensors for the acquisition of data in real time and the application of machine learning made it possible to improve knowledge of the dynamics occurring in the real system, thus improving decision support. Qi et al. (Qi et al. 2018), introduced the concept of digital twin services, as interface tools between the digital twin of systems and users.

In the field of water system management of aqueducts, the concept of digital twin services was independently developed and named digital water services, presented by Orazio Giustolisi in 2022 in the plenary session of the Hydroinformatics world congress (HIC2022) (Giustolisi 2022). These services are conceived following the same logic of the DTaaS (Digital Twin as a Service) paradigm, with the aim of providing support both for functional activities, such as the hydraulic analysis of water networks and the calibration of hydraulic models, and for management activities for medium and long term, such as optimal districtualization (optimal DMA design) and optimal planning of pipes substitution (planning of rehabilitation). The adoption of this strategy by the manager facilitates the involvement of the various stakeholders, including technical and professional staff, simplifying the consultation processes. This facilitation is made possible thanks to the ability to access individual services, view the results obtained and consult the relevant information. In addition, the flexibility in subscribing to specific services allows to optimise the management costs of the digital twin environment, thus eliminating the need to create and maintain a dedicated IT infrastructure for the digital twin.

The concept of digital water service, among other things, has already been applied to the technical world within the digital water strategy for the calibration of advanced hydraulic models and the optimal: design of DMA, pressure control and replacement plans (Ciliberti et al. 2021, 2023).

Mies et al. (2016) suggested the possibility of incorporating the digital twin in a single, fully automated environment for integration between various systems and services.

The above shows that even in the case of the concept of digital twin it is necessary to go more decades back in time to understand its origins which are contemporary with the first personal computers. The denomination and broad formalisation of digital twin was then introduced by Michael Grieves at the beginning of the millennium, to be further expanded and generalised recently with the concept of digital twin services and, in the field of water system management, digital water service.

The previous components integrated into the digital twin framework have been recently formalised by (International Organization for Standardization 2023), providing a blueprint for the implementation of digital twin systems across different industries. It defines the cutting-edge aspects of the various digital twin types, their life cycle processes, and the pivotal role played by the involved stakeholders.

4.1. The digital twin for aqueducts

For aqueducts, the digital twin is the digital replica of the potential and actual physical resources of the infrastructure, i.e., of the physical twin. They are as follows: the network domain (pipes and nodes), the characteristics of the asset (pipes length, diameter and material, georeferenced coordinates and elevation of nodes), the georeferenced water sources and their asset and hydraulic characteristics (springs, reservoir, tanks, etc.), the devices and their asset and hydraulic characteristics (pressure control valves, pumping systems, isolation valves, etc.), the hydraulic characteristics (pipe resistance, pumps curve, etc.), the locations of measurements (pressure, flow, level sensors and intelligent water meters), the sensor data and computed hydraulic status, the places (maps in GIS), people (georeferenced costumier of water meters), the characteristics of costumers (presence of local storage tanks, private pumping, high and number of floors of buildings, type of contract, etc.), the processes (DMA water balances, pipes replacement plans, settings of pressure control, pumping scheduling and control, valves status, etc.), etc.

The digital twin exchanges data and information with the physical twin, both synchronously (online) and asynchronously (offline).

The digital twin, evolving in the progressivity of digital transformation, increases its features of a real digital replica of the physical twin but within the limits of the aqueduct’s needs to be made efficient for several technical activities and human knowledge of the phenomenology that governs its behaviour through the laws of physics.

The digital twin of an aqueduct is composed of data and information of its resources. Digital twin characteristics are as follows: (i) the connection between the elements of the physical and corresponding virtual components; (ii) the possibility of ubiquitous access to data and information via the web, with the possibility of analyzing and optimising processes with computing resources; and (iii) the exchange of data and information between the physical and corresponding virtual components (cybernetic), using sensors and actuators.

Digital twins can integrate the internet of things, machine learning, data analytics, etc. They must also integrate hydraulic modelling, which is updated consistently with the change of the physical counterpart, and procedures for analysis and optimisation of the processes to support the technical activities.

Finally, note that data feeding the digital twin does derive also from the most diverse sources: historical data relating to past conditions, data provided by human experts with specific and relevant knowledge, etc.

4.2. The digital twin and the phenomenological twin

In the case of water systems, the digital twin is a replica of the real world that has a complex phenomenological behaviour governed by the laws of physics. The phenomenological behaviour of the water systems cannot be replicated through statistical and relational data analyses, therefore, replaced by machine learning techniques, for some main reasons:

• It is often too complex and characterised by large numbers of independent and dependent variables.

• The technical activities need of physical understanding (interpretability of the model to understand the reason of predictions and optimal solutions) to support decision-making for efficiency improvement.

• Designs and plannings at different time horizons are key activities to increase management efficiency in hydraulic systems whose technicality is relevant. The ask for physically based models, which can predict the system behaviour due to changes of the infrastructure opposite to machine learning, that encapsulates in mathematical object relationships of the system as it is.

In fact, as shown in Figure 2, there is the benefit of the interpretability of physically based models, also called white boxes, which is rewarded by the high cost in terms of general resources. Machine-learning models, also called black boxes, are low-cost models asking for input-output data of the physical system to be modelled, which is rewarded by the scarce interpretability justified by a pragmatic vision. In the middle, there is a graduality of conceptual models, also called grey boxes, with different cost–interpretability balances; the symbolic modelling of data through the reported paradigm of Koza’s genetic programming implemented by EPR technique (Giustolisi and Savic 2006), can be considered grey box.

Figure 2. Modelling systems: interpretability versus general cost.

It is therefore necessary to expand the concept of digital twin, consistently with its philosophy of extensibility, with that of phenomenological twin, i.e., a mathematical model based on the laws of physics that govern the physical system, replicating its phenomenological behaviour, see Figure 3.

Figure 3. Phenomenological twin and digital water service.

As reported by Giustolisi (Giustolisi 2022) for the case of aqueducts, phenomenological twin is physically based model based on advanced hydraulic modelling which integrates digital tools of the complex network theory, of the evolutionary optimisation, of the machine learning, etc.

Advanced hydraulic modelling is consistent with the need of advanced analysis more effective to obtain a replica of the phenomenological behaviour of the aqueduct, to extract more information from collected data to support system optimisation, i.e., the decision in “what to do” perspective opposite to “what if” of the simple trial and error analysis of the first personal computers.

The phenomenological twin is also fed by data, and it progressively develops with the knowledge that derives from the creation of the digital twin, both from the point of view of the best mathematical definition and of the phenomenological knowledge of the physics of the system.

Ultimately, the phenomenological twin, integrated with the tools of the digital transition, allows the development of the concept of digital water service (Ciliberti et al. 2021, 2023) which can easily and immediately support technical and decision-making actions in more complex environments than industrial ones, where the concept of digital twin was initially developed.

4.3. The water management system

In recent years, the concept of digital twin has spread together with water management systems (WMSs). The digital twin should be the heart of WMSs. Nonetheless, it is necessary to remember that the digital twin and WMSs are tools of the digital transition, which needs of a strategy of progressivity of the implementation and development. It is mandatory to accompany such strategy with the education and information of human capital and with the efficient reorganisation of companies (management transition).

Furthermore, as reported in the paragraph Data and Information, data collection must be subject to cost–benefit assessment as for the progressivity concept of the digital transition and must be representative of the physical behaviour of water systems to be made more efficient. The representativeness of data is not a trivial objective to reach being the water systems based on complex nonlinear physical laws mandatory requiring the phenomenological twin to be exploited.

There is a connection between the phenomenological knowledge of water systems and the progressivity of the digital transition. The former supports the representative collection of data, and the latter might allow increasing the phenomenological knowledge and the phenomenological twin.

Current WMSs are complex digital tools capable of organising the collection and management of data and the development of the concept of digital twin in a sophisticated way from an information technology (IT) point of view. Nonetheless, the passionate tension towards IT and the virtual world creates a naive optimism with respect to the result of the digital transition, imagining being able to perform without the need of the phenomenological knowledge of the water systems, therefore, of the phenomenological twin and, also, of the education and information of human capital of technicians.

Unfortunately.,the development of current WMSs is essentially IT, neglecting the centrality of the phenomenological twin or digital water service concept. Hydraulic of water systems is replaced by a simple hydraulic modelling, mainly developed with concepts of the last century, neglecting the limited ability to replicate physical phenomena, impairing the effort, also economic, of the digital twin implementation and development. The basic idea is the assumption of the possibility of managing water systems directly through data analysis, with machine learning and/or classical statistics techniques, calling it data-driven management. That idea lacks both scientific and technical foundation.

For example, one of the WMS aims is to manage real water loss of aqueducts through DMA water balance strategy. The balance becomes more accurate with the measurement of the daily volumes of water costumier consumption through smart meters and returns the daily water losses, as better explained later in the document. Water loss depends on the water consumption level, which determines the pressure status of each DMA, as well as on operating conditions and status of hydraulic devices, increasingly used in aqueducts for pressure control. Therefore, the variation in the level of water loss cannot be univocally attributed to a system anomaly, for example through machine learning methods, but the phenomenological twin is necessary to understand if the variation of water loss is caused by the surrounding conditions of the DMA or anomalies such the real increase of leakage outflows.

The phenomenological twin itself, thanks to the ability to replicate the phenomenological behaviour of real water loss, can be used to optimise the pressure controls, the design of DMAs, to plan pipe replacement, to pumping scheduling and control, etc.

The phenomenological twin can be adjusted over time by means of the monitoring data and information of the asset changes collected in the digital twin.

Ultimately, the phenomenological twin needs of advancing hydraulic modelling over time, which is consistent with the need to replicate the phenomenological behaviour, which allows the digital twin to be a more effective replica of the physical twin for the digital transition aim. The advanced hydraulic analysis of the phenomenological twin enhances the so-called “what if” strategy and the phenomenological twin, incorporating the concept of digital water service (Ciliberti et al. 2021, 2023, Giustolisi 2022), supports technical and decision-making activities in the complex environments, using the so-called “what to do” strategy.

The phenomenological twin, therefore, is a relevant component of the extension of the digital twin concept to water systems to avoid obscuring the management decision and boosting the education and information of human capital during the digital transition.

5. The digital transition and the water companies

Water companies are key players of the digital transition because they are responsible for implementing the plans and managing the integrated water service.

In the past, the economic, social, and environmental impact of water companies was not fully perceived socially and little politically. The increase of the awareness of people about the importance and strategic role of planning and management of the integrated water service for socio-economic and environmental sustainability changed the perception of the strategic role of the water companies. For example, in several European countries the huge investment with respect to past requires a prompt action in spending considering the deadline of 2026 decided by the European Commission.

The purpose of the investments is to boost the management of the integrated water service towards socio-economic and environmental sustainability for the next generations, targeting the concepts of ecological and energy transition, through the opportunities offered by the digital transition which must be associated with the management transition, i.e., a new internal organisation to water companies to support the changes of the managements and the new perspectives of the digitalisation.

If digital transition means making technical processes more efficient, these processes need to be made more rational over time. Doing so, the processes can increase scalability, replicability and integrability features over time, considering that flexibility and adaptability, especially for strategic planning and decisions, are also relevant issues being inherent the uncertainty in the management of water systems (Pellegrino et al. 2018). This is the true path to obtain efficiency together with its effectiveness.

The management of water infrastructures, then, must be supported by real industrial plans for operational (short term), tactical (medium term), and strategic (long term) activities to address the economic-financial and technical-scientific complexity, the only way to achieve sustainability.

The digital transition represents an opportunity to support the changes if, rather than being a mere updating of technologies from classic to digital, it is placed at the service of system engineering, and not vice versa, as can happen because of the emphasis of the historic moment.

The water sector is to grasp the request for “change management”. To this end, education and information aimed especially at top management level, which today has the duty to encourage the “change management” and accomplish the changes through huge investments, must have a central role, to be able to interpret the future and not suffer it; otherwise, change will be pursued without making it an opportunity and with poor overall results.

6. Aqueduct systems: consistency, functioning and modelling

Aqueduct systems are hydraulic infrastructures made up of pipes in which water flows for drinking purposes, confined and generally under pressure, from water sources to civil and industrial users. Modern aqueducts are the result of the technological evolution of the first industrial revolution, which allowed the construction of pipes, generally cylindrical, of different materials capable of withstanding the stress induced by internal fluids at higher pressure than the external ones. This has made it possible to overcome the technical problems of the aqueducts of past civilisations with reference to the flexibility of the routes and to the transportable unit volumes of water.

6.1. Consistency

The aqueducts are made up of water transmission and distribution systems: the former is generally made up of long pipes of greater diameter without connections to costumers, the latter are generally composed of shorter pipes of smaller diameter with a significant number of connections to costumers.

The water transmission systems have the function of transporting the water from the water sources closer to the consumption centres. The transmission pipelines are network systems, that from a topological point of view are not, generally, particularly complex. A real water transmission system might not exist when the water sources are, for example, on-site aquifers.

The water distribution systems have the function to bring water to the individual costumers of consumption centres, which, as we will discuss later, will be monitored with so-called smart meters in the next future due to digital transition. The distribution pipelines are network systems whose complexity, from a topological point of view, generally grows with the size of the consumption centres. It consists of a small percentage of larger diameter pipes, which are the skeleton of the water distribution network. The smaller diameter pipes feed the delivery points, through meters to costumers.

The water transmission systems are already subject to remote control through monitoring systems and remotely controlled hydraulic devices given the spatial extension of these infrastructures, because of the technological developments of the digital transition in the 90s.

The water distribution systems will have to become the object of remote control because of the developments of today’s digital transition through their engineering, i.e., the implementation of the DMAs, and remote control of their water balances, of the hydraulic devices. The monitoring system of pressures, flows, tank levels and consumptions (smart meters) will be the database to be transformed into information to support and streamline of decisions and activities water companies.

6.2. Functioning of the water distribution systems

Water distribution systems, as mentioned, are piping networks that reach individual costumier. There are always water transmission pipelines, hydraulically upstream, composed of pipes with greater diameter and lower density of costumier connections, which carries larger volumes of water to the whole system. The other part of the water distribution network, hydraulically downstream, is composed of pipes with smaller diameters and generally higher density of costumer connections. Those pipes gradually have less importance in the transport of volumes of water, up to the peripheral ones, that feed a few costumers.

The single water request is impulsive and stochastic from the scale of a few seconds to that of a few minutes. Every time the user changes the water request by opening/closing a tap, the water request information is a change in the state of the system which is transmitted to the water sources by crossing the aqueduct network from downstream to upstream. From the hydraulic point of view, it is a flow rate and pressure variations, travelling at the celerity of the order of 1 km per second depending on fluid substrate and pipe stiffness, which are the waves of unsteady flow.

Therefore, considering the intrinsic stochastic nature of the water requests of the individual costumers, innumerable waves are generated every second in the system which, bouncing between the nodes in the aqueduct network and to the costumers themselves, due to their transmission and refraction, by attenuating and reaching a state of equilibrium after several minutes. Note that, the state of equilibrium is never reached due to the continuous overlapping of water demands.

Furthermore, even today’s control of water distribution systems, through control devices such as, for example, pressure reducing valves, has the same physical-hydraulic mechanism of information transmission above reported. Therefore, we can visualise the hydraulic system as continuously travelled by waves of unsteady flow whose amplitude is one of the technical issues to be addressed to contain the effects of the fatigue, which lead to its deterioration and, therefore, to excessive increase of real water loss over time.

The same technical-scientific research developed hydraulic modelling, as will be better explained later, in the awareness of physical phenomena. Therefore, the hydraulic models for the management of aqueducts are “steady state” (stationary) and with the simplification of uniform technical motion of the water flows in the pipes. That is, they capture the “technical” stationarity of the water distribution system on the time scale of several minutes, a scale whose choice is also a function of the purpose of the modelling analyses, assuming a condition of negligibility of the perturbations of unsteady (i.e., assuming they average to the null value on the scale of minutes).

Under these conditions, water costumer consumption is integrated over time on the scale of the “technical” stationarity of the system, of several minutes, becoming increasingly suitable, statistically speaking, to be treated in their average values since they belong to continuous probability distributions.

To understand which data acquisition scale is useful for the representativeness, with respect to stationary assumption, and parsimony, with respect to the technical result, of the measurement data collection, it must be said that:

• the pressure measurement values must be sampled and averaged on a rather smaller scale than that of the model analyses since they are more subject to the variations due to the transit of unsteady flow waves. Acquisition of high-frequency pressure values is useful for monitoring the operation of hydraulic devices.

• The flow rates can be measured at much larger scales (consistent with the type of analysis of the model) since they are less variable, referring to mass quantities, and representative of the volumes transmitted at the scale of analysis. Note that the transmission pipelines are characterised by flow rates depending on a larger spatial integration of the water costumier consumption. Consequently, flow rate measurements are characterised by greater statistical “stability” of the average values; on the contrary, the peripheral pipes are subjected to the operation of a few costumers and are characterised by a lower statistical “stability” of the flow rate average values.

• For the smart meters the same reasoning can be reported considering that acquisitions of data at the minute level become statistically insignificant and a matter of research only.

• Tank-level measurements generally follow the same indications of flow measurements as they are also phenomenologically linked to displacements of water volumes.

6.3. Advance hydraulic modelling to support the digital transition

In the early twentieth century the construction of the aqueducts was functional to increase human health and well-being, but also to support the economic development of industrial activities and provide fire protection. Therefore, at the basis of the modelling studies there was the need to develop hydraulic verification criteria for aqueduct design, i.e., pipes sizing in relation to the ability to satisfy the water requests (statistical requests) of the various types of water costumers (civil, commercial, and industrial) and, especially in Anglo-Saxon technical culture, the need for fire protection. Consequently, the aim of hydraulic analysis was the calculation of the operating pressures at the nodes of the water network for fixed pipes resistance and water demands. Ultimately, the verification was an assessment of the pressures at the nodes with respect to the minimum pressures for a correct service to the costumier and the minimum residual flows and pressures for a correct hydraulic performance of the hydrants for fire protection. With the advent of automatic calculation, in 1988, Todini and Pilati (Todini and Pilati 1988) invented the global gradient algorithm, which a few years later became the “hydraulic engine” of EPANET, developed by the US agency EPA Current software packages are generally based on that “hydraulic engine” or similar strategies. Therefore, all the classic hydraulic simulators have been developed for hydraulic verification, and not for management purposes; they assume fixed demands at the nodes of the aqueduct network and are called demand-driven, i.e., the pressures calculated at the nodes are “driven by the demands fixed a priori at the nodes”.

Progressive urban development produced increasingly large, complex, and outdated aqueduct networks, implying management needs, today the focus of the concept of aqueduct engineering, with respect to real water loss, reliability, water quality, energy optimisation, rehabilitation, etc., more and more. Todini (2003) introduced the need for a hydraulic modelling to evaluate the “actual” water demands that can be supplied to the costumers in conditions of pressure lower than the minimum pressures for a correct service. The corresponding hydraulic modelling is called pressure-driven, i.e., the water demands calculated at nodes are “driven by the pressures and not by the priori fixed demands”.

Later, Giustolisi et al. (2008) extended Todini’s concept to real water loss and developed the representation of the relative component in the hydraulic modelling depending on pressures at each single pipe. In this way, the accuracy of the hydraulic modelling accounts for the information of water loss at pipes level, since it is indispensable in various management activities such as the designs of DMAs and pressure control, as reported in Figure 4, and planning pipe replacement (rehabilitation).

Figure 4. Water loss volume of five days of the calibrated model (left) and after designing DMAs and pressure control (right).

Therefore, the classic demand-driven hydraulic simulators (e.g., EPANET) are not suitable tools for hydraulic simulation for management purposes even if the technical-scientific inertia sometimes does not consider that fact, although in practice the modeller need of innovation in hydraulic modelling, especially during this era of digital transition asking for digital twins which should be a replica, also phenomenological speaking (phenomenology twin), of the physical twin.

Advanced hydraulic modelling concept (Ciliberti et al. 2021, 2023, Giustolisi 2022) is the answer for the digital twin request to be an accurate replica of the physical twin implementing the phenomenological twin. Advancing hydraulic modelling of water systems is today mandatory for enhancing the data collection of the digital transition, to be transformed into useful information for efficiency replacing over time heuristics with rationality in the management activities better supporting decision-making.

Figure 5 reports and example of useful advancement to enhance the digital twins for aqueducts to be a better replica of the physical twin; advanced hydraulic modelling allows to avoid the concentration in the model nodes of the costumier water demands. This fact allows maintaining their individuality, which is consistent with GIS representation, being very effective for any management action always aimed at the increasing service efficiency with respect to the same costumers. This is relevant, as will be reported later, for integrating smart metering data of costumers into modelling analysis, such as the DMA balances, consistently with the inspiring principles of the digital twin concept.

Figure 5. Georeferenced costumers in the hydraulic model.

Finally, from a technical point of view, the “smart” management of the water systems asks for a true valorization of relevant data for the representativeness of the phenomena, i.e., in the true concept of smart metering.

The decision must be rationalised and supported by advanced hydraulic models capable to enhance the value extraction from the smart metering, by making that data become management information. The “what if” strategy, based on the hydraulic analyses only, is enhanced by the concept of advanced hydraulic modelling. Whilst the innovation path must be completed with the “what to do” strategy, integrating theories, paradigms, and concepts of the digital transition, to achieve a more and more efficient support to technical activities.

6.4. Smart metering of water consumption: water balance of DMAs, rehabilitation planning and new service for costumers

The aqueducts distribute volumes of water at the request of the customers, therefore, their functioning is conditioned by the life cycle of human beings which is daily, since the planetary revolution around the Earth’s Sun has such cadence in the night-day succession. The daily life cycle of human beings is quite stable, albeit with some differences between holidays, weekdays, summer, winter, etc. Figure 6 shows the variability of the daily inflow of a consumption centre. Note that Figure 6 reports the inflow of a centre, which is not characterised by a huge variability.

Figure 6. Variability of the daily water inflow of a consumption centre.

For this reason, it is correct to analyse the aqueducts taking the daily operating cycle as a basis and considering the operational differences and the boundary conditions that determine their functioning.

The previous paragraph is useful for understanding the fact that the most relevant boundary condition is the water consumption which varies stochastically, although driven by social and individual behaviour, both in the total volume and in the spatial and temporal distribution within the operating cycle.

This clarifies that, maybe, the most important technological innovation of digital transformation is the smart metering of water consumption capable of collecting the real water costumer consumption data on a daily, if not hourly, time scale, making them quickly available in digital twins, although the prompt availability of such data depends on the implemented data transmission technology.

It is necessary to remember that the consumption data available up to the recent past was on an annual scale, in some rare cases monthly, and presented significant problems in its use on a daily scale. This fact caused the need of studying demand patterns a priori fixed for different type of costumers remaining unmodified in the hydraulic modelling as well as the base demands, while demand patters and base demands varies over days, as demonstrated by the varying daily inflow of Figure 6.

On the other hand, it is necessary to understand that the absolute novelty of smart meters at costumers entails both management costs and the collection of data from several users which, already for a small consumption centre, amounts to thousands. Consequently, the collection of data should be parsimonious to the aim of the cost–benefit result. For example, if the decision is for a daily collection, each smart meter is characterised by 365 annual values, while for hourly collection each smart meter is characterised by 8,760 annual values. Such decision, then, impact on data transfer and useful life of the smart meter battery, but also on general data management.

Also, not least, it is necessary to remember that the real world, the physical twin, is more complex and uncertain than the digital twin; therefore, the decision on the collection of consumption data should account for their loss for the most disparate reasons.

7. Digital transformation and innovation of the management of aqueducts: few examples

7.1. Smart metering of customers

The meters reading and transmitting the costumier data to the digital twin from the physical twin aims at: (1) improving the system management; and (2) implementing additional services to customers.

Due to tradition and not complete understanding of their usefulness in the engineering of aqueducts, point (2) is most recognised as useful both for the improvement of billing, with a rapid economic advantage for the water company, and for the detection of consumption anomalies, with elaborations of data, e.g., via machine learning. This last circumstance, perhaps, made them become “intelligent” in the common jargon.

The possibility of truth deserving the characterisation of intelligent is much greater exploring the improvements of the aqueduct management through highly representative data of its hydraulic functioning. Two relevant examples are following reported: (1) the calibration of advanced hydraulic models; and (2) the analysis of water balance of DMAs.

7.1.1. Model calibration and smart meters

The calibration of hydraulic models aimed at management needs ask for effectiveness to capture the water flow field of the aqueduct varying over day being associated to spatial and temporal water costumer consumption variability and consequent real water loss variability. In the past, design of aqueducts and fire protection ask for a calibration mainly aimed at evaluating the hydraulic resistances of the pipes, i.e., the hydraulic capacity of the system.

In fact, the management of the aqueducts asks both for global and spatial analyses, through the DMAs, of the real water loss and of water costumer consumption, which are the two main components of the daily water inflow, e.g., those reported in Figure 6.

Figure 7 shows the modelling separation of two components of the daily inflow: the real water loss and water costumer consumption. The former is deterministic following physical laws the latter is stochastic depending on users.

Figure 7. Main components of the daily inflow: real water loss and water costumer consumption.

The separation of such components, also at DMA level, is already possible through advanced hydraulic modelling using annual water consumption data and identifying the actual daily consumption pattern by optimisation, and this is already an innovation consistent with the novel tools of the digital transition (Ciliberti et al. 2021, 2023, Giustolisi 2022). Nonetheless, smart meters will be a further support for increasingly accurateness and quickness of the model calibration, also considering a greater of DMAs in the future. This fact is consistent with the concept of progressivity of the digital transformation. In this line, the daily volume will certainly be sufficiently effective, optionally integrated with further verification data, to obtain a great benefit compared to the past.

7.1.2. DMA water balance and smart metering

The monitoring of pressures and flow rates of the aqueducts is necessary to “observe” the varying behaviour of the hydraulic system during an operating cycle and the changes, over operative cycles, due to the boundary conditions variability such water consumption, status of hydraulic devices, tank levels, asset deterioration status, etc.

Therefore, it is a relevant issue to select positions of measurement sensors considering both the representativeness and parsimony concepts of the digital twin implementation and the need technical observability. The implementation of the DMAs divides the network structure of aqueducts into districts to better observe, at a zonal level, the hydraulic behaviour, thus improving the rationality of the activities by a better technical understanding. The understanding for the management of the aqueducts, in fact, increases through the monitoring of the water balances.

The water balance of DMAs has three volume components: (1) volume of water inflow and outflow; (2) volume of water loss and (3) volume of water consumption due to costumers.

The volume entering and leaving the DMAs (1) can be evaluated through flow measurements and it is a matter of optimal design of DMAs in terms of the minimum number of flow meters and closed gates in the boundary, for a given number of devices. As previously mentioned, a high frequency of data collection is not necessary since this will be integrated on the scale of the water balance (volumes) analysis, which can be daily or hourly.

The difference between the inflow and outflow water volumes (1) subtracted from the water consumption volumes (3) makes it possible to evaluate the water losses (real and apparent) (2).

As today happens, the consumption data are on a larger scale (e.g., annual) than the DMA balance analysis (e.g., daily, or hourly). Therefore, it is necessary to reconstruct daily level of consumption and demand patterns, as previously reported for calibration.

The accuracy of daily-level reconstruction of consumption and demand patterns is lower decreasing the time scale of acquisition of the consumer data; therefore, smart metering will increase the model predicted water balance at DMA and the accuracy to predict anomalies by means of flow and pressure measurements.

7.1.3. Requirements of water balance for system management

The basic scale useful for the water balance of DMAs is daily and indicates the minimum requirement for water consumer data collection, i.e., smart meters.

The daily scale allows comparing model forecast water balance with that calculated by daily measurement data. The analysis of the comparison is an anomaly detection method and can support the analysis of the water loss level and increase, of improper consumption, of overflows from tanks, etc., thus implementing the proactive management vision of the digital transition.

Note that the advanced hydraulic modelling, the phenomenological twin within the digital twin, allows the robust and consistent prediction of the physical behaviour of the system underlying the comparison.

The daily consumer volume data are the minimum requirement for the calibration of an advanced hydraulic model that allows a better support to investments, as will be clarified in the next paragraph.

8. Perspective: AMSI for the effective support of investments in the era of digital transition

The present section aims at introducing the novel AMSI (Asset Management Support Indicator), which inherits the scalability and physically based properties of the linear or density indicator of water loss, in italy for example named M1, allowing to drive investments and technical activities at any system scale.

M1, in m³/km/day, is the indicator of the volume of water loss divided by the length of the portion of the aqueduct under analysis, e.g., DMA or consumption centre or entire hydraulic system. It is possible to transform M1 in m³/m/sec, i.e., the outflow Q_Leak in m³/sec for unit length of the system under analysis, considering that 1 km = 1,000 m and in 1 day = 86,400 sec. In addition, it is possible to write the model of the leakage flow, Q_Leak, for the unit length by means of the Germanoupulos formulation (1985) (Germanopoulos 1985), without impairing the generality of the formulation strategy,

(1) $\begin{array}{rl}& \mathrm{M}1=86,400,000\cdot Q_{\mathit{Leak}}=8.6\cdot {10}^7\cdot \mathit{\beta }\cdot P^{\mathit{\alpha }\approx 1}\Rightarrow 1.157\cdot {10}^{-8}\\ & \mathrm{M}1=\mathit{\beta }\cdot P^{\mathit{\alpha }\approx 1}\end{array}$(1)

Equation (1)(1) $\begin{array}{rl}& \mathrm{M}1=86,400,000\cdot Q_{\mathit{Leak}}=8.6\cdot {10}^7\cdot \mathit{\beta }\cdot P^{\mathit{\alpha }\approx 1}\Rightarrow 1.157\cdot {10}^{-8}\\ & \mathrm{M}1=\mathit{\beta }\cdot P^{\mathit{\alpha }\approx 1}\end{array}$(1) makes explicit the dependence of M1 on the deterioration parameter of the asset, β, and on the mean pressure, P, of the system under analysis, i.e., on the physically based model, although empirical, of leakage outflow for unit length, Q_Leak. α is an exponent that can be approximated to unit value as reported in recent studies (Ciliberti et al. 2021, 2023, Giustolisi 2022) about the Fixed And Variable Area Discharge (FAVAD) formulation of leakage model.

Consequently, it is possible to write:

(2) $8.6\cdot {10}^7\cdot \mathit{\beta }=\mathrm{AMSI}\left(\mathit{asset}\right)=\frac{\mathrm{M}1}{{\mathit{P}}^{\mathit{\alpha }\approx 1}}$(2)

where AMSI is a scaled, with respect to deterioration factor of the asset, β, having the same dependencies. It depends on the number and size of leakage outflows in the mains and connections to the properties and depends on asset variables, like age, pipes material and diameter, number of connections to properties, system length, etc. Its increase of rate over time depends on pressure, effects of fatigue (e.g., pressure variation due to unsteady flow or pressure control; traffic; etc.), environmental factors, etc. as reported in the vast technical-scientific literature.

For sake of completeness, AMSI can be also formulated using FAVAD formulation of the leakage model as follows:

(3) $8.6\cdot {10}^7\cdot \mathit{\beta }=\mathrm{AMSI}\left(\mathit{asset}\right)=\frac{\mathrm{M}1}{\left(1+\mathit{LN}\right)\sqrt{P}}$(3)

where LN is the leakage number of Van Zyl and Cassa (2014), which better accounts for the system material and average pressure being FAVAD more physically based model than the Germanoupulos’ one (Germanopoulos 1985).

AMSI represents the daily volume of leakages for km of the system under analysis and for its unit pressure, and correctly characterises its deterioration allowing to support the decision about the activities of asset management in any portion of the entire hydraulic system as reported in the following example.

AMSI is a relevant indicator because is the ratio of the density of water loss, M1, and mean pressure of the system under analysis. It benefits of the fact that, being the pressure usually give in meters of water column, the numerator and denominator in Equations (1)(1) $\begin{array}{rl}& \mathrm{M}1=86,400,000\cdot Q_{\mathit{Leak}}=8.6\cdot {10}^7\cdot \mathit{\beta }\cdot P^{\mathit{\alpha }\approx 1}\Rightarrow 1.157\cdot {10}^{-8}\\ & \mathrm{M}1=\mathit{\beta }\cdot P^{\mathit{\alpha }\approx 1}\end{array}$(1) and (2(2) $8.6\cdot {10}^7\cdot \mathit{\beta }=\mathrm{AMSI}\left(\mathit{asset}\right)=\frac{\mathrm{M}1}{{\mathit{P}}^{\mathit{\alpha }\approx 1}}$(2) ) has the same order of magnitude.

For example, if we have three consumption centres having a rather high M1 = 50 m³/km/day and we assume that p = 25, 100 and 50 m, respectively, AMSI = 2, 0.5 and 1.

Therefore, Equation (2)(2) $8.6\cdot {10}^7\cdot \mathit{\beta }=\mathrm{AMSI}\left(\mathit{asset}\right)=\frac{\mathrm{M}1}{{\mathit{P}}^{\mathit{\alpha }\approx 1}}$(2) provides insights though the interpretable M1, offering guidelines for asset management in the following manner: (1) in the first town, water loss depends on system deterioration. Consequently, it needs prioritised plans for pipes replacement and implementing active leakage detection actions; (2) in the second town, water loss depends on high system pressure level. As a result, it requires pressure control actions, and (3) in the third town, water loss depend on both system pressure and system deterioration. This requests a combination of the approaches previously exposed to address the issue effectively.

Therefore, the structure and scalability of M1 facilitates a robust approach for steering investments for asset management of drinking water infrastructure, as today required worldwide. Achieved through the simple Equations (1)(1) $\begin{array}{rl}& \mathrm{M}1=86,400,000\cdot Q_{\mathit{Leak}}=8.6\cdot {10}^7\cdot \mathit{\beta }\cdot P^{\mathit{\alpha }\approx 1}\Rightarrow 1.157\cdot {10}^{-8}\\ & \mathrm{M}1=\mathit{\beta }\cdot P^{\mathit{\alpha }\approx 1}\end{array}$(1) and (2(2) $8.6\cdot {10}^7\cdot \mathit{\beta }=\mathrm{AMSI}\left(\mathit{asset}\right)=\frac{\mathrm{M}1}{{\mathit{P}}^{\mathit{\alpha }\approx 1}}$(2) ), this method inherits the effectiveness of M1 and the scalability to focus on different portions of the entire hydraulic system, including DMAs or consumption centres. Furthermore, the novelty of AMSI, with respect to their indicators, resides in the capability of supporting decision about the efficiency of the type of investment. In fact, it allows ex post and ex ante evaluations of the efficiency of investments about pipe replacement and active leakage detection versus pressure control, while M1 alone is much influenced by the efficiency of pressure control. AMSI also allows comparing different drinking water systems and understanding the type of investments for each water utility and where to allocate them inside each one as useful for water agencies.

The dependency of AMSI on the asset variables can be modelled with machine learning through the data stored in the digital twin and it is supported by the advanced hydraulic modelling through the concept of phenomenological twin inside the most general of digital twin, consistently with the digital transformation aim (Ciliberti et al. 2023).

9. Final remarks and perspectives

The concepts of digital transition and digital twin are the foundations of the digital water strategy to increase the efficiency of water systems management using digital technologies and products. This paper revisited the concepts of digital transition and digital twin and reported the highlights of the scientific history of the past centuries and decades that underpinned these concepts. In this way, the technical-scientific meaning of the digital transition and the digital twin were critically discussed with the specific application to aqueduct systems.

It has been demonstrated that, since digital transformation is linked to the objective of increasing the efficiency of the processes with the collection of data representative of the physical systems, it is intrinsically progressive with the greater constraint of information and training for human capital, especially for the top management of water companies, that is, the most influential decision makers.

Furthermore, the need for efficiency in data collection with respect to the representativeness of the functioning and behaviour of the water systems was also discussed as this collection is not the digital transition.

Regarding the digital twin concept, it was recalled that it is an evolving and expanding concept generated for the first time in the industry. The perspective in water systems is to advance knowledge and modelling within the new concept of phenomenological twin using the concept of progressiveness of digital transformation and to expand the concept of digital twin as a service, digital water services, to increase the support for decision makers and technicians through the growth of digital transformation information.

Acknowledgments

I wish to thank Prof. Philippe Gourbesville, Prof. Gabriele Freni, Prof. Luigi Berardi, Prof. Daniele Biagio Laucelli, Dr. Andres David Ariza, Dr. Francesco Gino Ciliberti, Dr. Laura Vanessa Enriques, Dr. Alessia Giustolisi, Dr. Francesca Mastromarino, Dr. Gianfredi Mazzolani and Dr. Antonietta Simone for the revision of the document and the translations. Their suggestions and comments were very useful.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The author confirms that the data supporting the findings of this study are available within the article.