Municipalities

Each municipality (gemeente) is represented as a single junction point with positive demand, abstracting the entire secondary and tertiary distribution network within that jurisdiction. This node serves as the sole supply point for all water within the municipality’s boundaries, with characteristics such as population, land area, and housing stock consolidated at this level.

The system presents two main challenges. First, municipal parameters evolve over time as cities grow and change. Second, the network topology itself is dynamic: municipalities can merge or be absorbed by larger neighbors, causing the number of nodes to vary throughout the planning horizon.

Excursus on the Modelling Approach

To model this administrative restructuring, municipalities can only open or close on January 1st of each year. When a municipality closes, its delivery point disappears from the network.

Absorption by existing municipalities: When a municipality is absorbed by a larger neighbor that already exists, all attributes of the closing municipality (population, land area, housing stock, etc.) transfer to the destination municipality. Any pipe that previously connected these two entities becomes hidden, as it formally becomes part of the destination municipality’s internal distribution network.

Clustering into new municipalities: When multiple municipalities close and cluster together to form a new entity, all their delivery points disappear and a new supply point emerges at the location of the newly formed municipality. The new municipality inherits all pipeline connections and attributes from the closing municipalities. This modelling approach mirrors real-world dynamics in densely populated countries like the Netherlands. When a new municipality forms through clustering, typically a new city center is established while former city centers become secondary neighborhoods. These moments of urban reorganization present natural opportunities for water utilities to lay new connections and redesign substantial portions of the distribution system.

Municipalities have many attributes that influence the other modules of the system. The full list can be seen in tbl. 1, while the actual values for these variables can be inspected within the data files, which are mapped in Appendix A.

Table 1: Municipalities’ properties review.
Property Type Scope Unit
Name Static Municipality
Identifier Static Municipality
Latitude Static Municipality degrees
Longitude Static Municipality degrees
Elevation Static Municipality m
Province Static Municipality
Begin date Static Municipality date
End date Static [Optional] Municipality date
End reason and destination Static [Optional] Municipality
Population Dynamic Exogenous Municipality inhabitants
Surface land Dynamic Exogenous Municipality \(km^2\)
Surface water (inland) Dynamic Exogenous Municipality \(km^2\)
Surface water (open water) Dynamic Exogenous Municipality \(km^2\)
Number of houses Dynamic Exogenous Municipality units
Number of businesses Dynamic Exogenous Municipality units
Average Disposable Income Dynamic Exogenous Municipality \(k\text{€}\)
Average age distribution network Dynamic Endogenous Municipality years

The total municipality water demand comprises two volumetric quantities:

While these quantities represent the physical components of demand, they are not directly observable by participants. Instead, participants observe the total water demand divided into two components: consumption (\(Q\); delivered outflow) and undelivered demand (\(U\)).

For each municipality \(m\) at time \(t\), these quantities maintain the following relationships:

\[ \begin{aligned} D_m(t) &= D^\text{BIL}_m(t) + D^\text{NRW}_m(t) \\ &= Q_m(t) + U_m(t) \end{aligned} \qquad{(1)}\]

The decomposition between delivered and undelivered demand is extracted from an EPANET simulation of the network run in pressure-driven analysis (PDA) mode with a minimum pressure threshold of 30 m. Whenever there is undelivered demand, we assume that this reduces the billable component first, i.e., \(Q^\text{BIL}_m(t) = D^\text{BIL}_m(t) - U_m(t)\).

Water Demand Model

The methodology developed to generate water consumption time series builds on historical data from the Dutch association of water companies (Vewin, 2025), which provide nationwide trends in total drinking water production, sectoral water use, and non-revenue water over the period 2000–2024. Specifically, water-consumption time series generation is structured into three phases.

Phase I. The first phase estimates the annual water volume supplied to each municipality using information on households and businesses (CBS, 2025), complemented by projected data where required. These annual volumes are calibrated to match national totals reported in official statistics (Vewin, 2025) and then randomized around the calibrated value to introduce variability among municipalities.

Phase II. In the second phase, representative hourly consumption profiles are assigned to each municipality using a library of year-long, normalized profiles derived from district-metered areas and pre-processed to remove leakage effects. In greater detail, for each municipality, two residential profiles are selected from the library according to municipality population class, while a single non-residential profile is drawn from a dedicated set.

Phase III. The third phase produces the final hourly time series by applying a Fourier series-based approach which combines seasonal modulation, climate-related adjustments (accounting for the maximum yearly temperature), and random perturbations to capture temporal variability. The two residential profiles associated with each municipality are aggregated through weighted combinations, and both residential and non-residential profiles are scaled to match the previously estimated yearly volumes.

Therefore, the total billable demand of municipality \(m\) at time \(t\) (within year \(y\)) is defined as:

\[ D^\text{BIL}_m(t) = D^\text{R1}_m(t, T_y) \cdot w_m + D^\text{R2}_m(t, T_y) \cdot (1-w_m) + D^\text{C}_m(t, T_y) \qquad{(2)}\]

where \(D^\text{R1}_m(t, T_y)\) and \(D^\text{R2}_m(t, T_y)\) represent the two residential demands, \(w_m \in [0,1]\) is the unitary weight to combine them, \(D^\text{C}_m(t, T_y)\) is the (commercial) non-residential demand, and \(T_y\) is the maximum temperature recorded in year \(y\).

Table 2: Water demand model’s properties review.
Property Type Scope Unit
Population Dynamic Exogenous Municipality inhabitants
Number of houses Dynamic Exogenous Municipality units
Number of businesses Dynamic Exogenous Municipality units
Daily per household demand Dynamic Exogenous Municipality mÂł/house/hour
Daily per business demand Dynamic Exogenous Municipality mÂł/business/hour
Max yearly temperature Dynamic Exogenous National °C

Non-Revenue Water Model

Non-revenue water (NRW) is an uncertain quantity modeled through the average age of pipe infrastructure in each municipality’s inner distribution network (IDN). Based on this average age, municipalities are assigned to one of five NRW classes as reported in tbl. 3. Each class is associated with a distinct probability distribution of NRW demands, from which daily samples are drawn to generate the volumetric NRW demand factor (\(m^3/km/day\)). Notably, older infrastructure suffers from more leaks and therefore exhibits higher NRW demand factors.

The distribution of NRW demands varies by class and is illustrated in fig. 1. The actual values of the distributions’ parameters can be inspected within the data files, which are mapped in Appendix A.

The total length of pipes in a municipality is linked to its population size through the following linear relationship:

\[ L^\text{IDN}_{m}(y) = 57.7*10^{-4} \cdot \text{inhabitants}_{m}(y) \qquad{(3)}\]

where \(m\) is the municipality index, \(y\) is the year, and \(L^\text{IDN}_{m}(y)\) is the total length of pipes (km) in municipality \(m\) at year \(y\).

The actual municipality NRW demand is also capped at twice the billable daily demand to prevent unrealistic leakage levels. Therefore, total NRW demand for municipality \(m\) at day \(d\) is calculated as:

\[ D^\text{NRW}_{m}(d) = \min\left(f^\text{NRW}_{\text{class}(m)} \cdot L^\text{IDN}_{m}(y), \, 2 \cdot \bar D^\text{BIL}_{m}(d)\right) \qquad{(4)}\]

where \(f^\text{NRW}_{\text{class}(m)}\)​ is the sampled NRW demand factor (\(m^3/km/day\)) for the municipality’s class, and \(\bar D^\text{BIL}_{m}(d)\)​ is the average daily water demand of municipality \(m\) at day \(d\). The daily leak \(D^\text{NRW}_{m}(d)\) is equally spread across the day (i.e., \(D^\text{NRW}_{m}(t)=D^\text{NRW}_{m}(d)/24\)).

Non-revenue water demand factor per class
Figure 1: Non-revenue water demand factor per class
Table 3: Non-revenue water classification by infrastructure age.
Inner distribution network - average age [years] NRW Class Probability distribution
0 - 25 A Inverted Exponential
25 - 43 B Uniform
43 - 54 C Uniform
54 - 60 D Uniform
> 60 E Exponential
Table 4: Non-revenue water model’s properties review.
Property Type Scope Unit
Inner distribution network - length to population ratio Static National \(km \cdot (10^4 \text{ inhabitants})^{-1}\)
Inner distribution network - length Dynamic endogenous Municipality \(km\)
Inner distribution network - average age Dynamic endogenous Municipality \(years\)
NRW intervention - unit cost Dynamic endogenous NRWClass, Municipality Size Class, National \(\text{€}/year/km\)
NRW intervention - effectiveness factor Static [Uncertain] NRWClass, Municipality Size Class, National
Intervention budget Option \(\text{€}/year\)
Intervention policy Option
CBS. (2025). Homepage. https://www.cbs.nl
Vewin. (2025). Homepage. In Vewin. https://www.vewin.nl/
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