Transient-Aware Multi-Objective Optimization of Water Distribution Systems for Cost and Fire Flow Reliability

Abstract

Urban water distribution systems, as integral parts of underground pipeline networks, face challenges from aging infrastructure, operational demands, and transient pressure surges that can compromise structural integrity and service reliability. This work introduces a cost-oriented multi-objective design framework that explicitly accounts for both the likelihood of fire flow failure and the risks posed by transient pressures. The approach links a probabilistic reliability model with a transient pressure evaluation module, and couples both within a non-dominated sorting genetic algorithm to generate Pareto-optimal design solutions. Design solutions are constrained to maintain transient pressures within permissible limits, ensuring enhanced pipeline safety while optimizing capital costs. Case studies show that adopting a minimum 150 mm distribution main improves fire flow capacity and reduces transient-induced failure risks. The proposed method provides a predictive, computational tool that can be integrated into digital twin environments, supporting sustainable infrastructure planning, long-term monitoring, and proactive maintenance for resilient urban water supply systems.

1. Introduction

Water distribution systems (WDS) are critical components of urban underground pipeline networks, delivering water from sources to consumers while ensuring adequate quantity, quality, and pressure. Conventional design practices typically check both extremes of system loading. Under maximum hour or maximum day demand plus a design fire flow, the network must maintain the minimum required pressure for service and fire protection. Conversely, under minimum demand conditions, pressures must remain below the maximum allowable limit to avoid structural or operational risks. Optimization techniques are then applied to select cost-effective configurations that satisfy these constraints [1,2]. The underlying principle is that using the smallest components that meet design constraints minimizes capital costs.

However, these approaches rely on implicit assumptions that can be problematic. Demand patterns and system behavior are often uncertain, yet conventional methods treat them as deterministic. Most optimization frameworks also assume steady-state or quasi-steady conditions, neglecting dynamic hydraulic transients that can undermine system reliability and safety.

A particularly critical loading condition is the needed fire flow (NFF), which introduces sudden, high-magnitude transients. Such events can cause damaging pressure surges, service interruptions, and sub-atmospheric pressures capable of drawing in contaminants [3,4,5]. Transient conditions also pose operational disruptions, including air pocket formation and intermittent flow pulsations [6]. While transient mitigation strategies exist for pumps, fire hydrants—frequent transient sources—often lack adequate protection or operational control. Recent advances [7] address this gap through automated, data-efficient transient modeling to identify hydrants with lower transient risks.

Although reliability, robustness, and resilience have been widely studied in WDS optimization [2,8,9,10], transient considerations have received comparatively less attention, often being addressed only in later design or construction phases. Critical design decisions—including pipeline routing, pipe diameter and material selection, and device selection (e.g., specific pump and valve choices)—are frequently based solely on steady-state analyses, despite the fundamental role of transients in determining system performance and lifecycle costs [11,12].

Earlier research has integrated transient analysis into pump, surge tank, pipe optimization [13,14]. More recently, advanced optimization techniques have emerged, including Genetic Algorithm-based PRV control strategies [15], multi-objective PRV placement optimization [16], and comprehensive frameworks integrating transient impacts with cost minimization [12]. Jung [17] further advanced the field by developing a worst-case design methodology that minimizes extreme transient conditions while optimizing pipe sizing. However, no study has developed a comprehensive, least-cost framework that explicitly incorporates fire flow failure probability under transient conditions.

Therefore, the central thesis of this study is that sustainable and reliable WDS design requires the explicit integration of both probabilistic fire flow reliability and transient safety considerations into the optimization process. The specific objectives are: (i) to develop a multi-objective optimization framework that simultaneously minimizes capital cost and fire flow failure probability, (ii) to constrain the design by transient pressure limits to ensure system safety, and (iii) to demonstrate the framework on representative dual and branched network case studies. By doing so, this research aims to fill a critical gap in current design practice, which often addresses transients only at later stages, and to provide a predictive tool that can be readily extended to digital twin environments for long-term planning and operation.

2. Materials and Methods

2.1. Analytical Assessment of Fire Flow Failure Probability

Opening a fire hydrant during a fire creates a sudden, but somewhat uncertain, demand on the network. The precise estimation of NFF is difficult because fire incidents arise unpredictably across locations, and each location may demand different levels of fire protection. In North America, NFF is estimated with Insurance Services Office (ISO), Iowa State University (ISU), and Fire Underwriters Survey (FUS) methods [18,19]. However, the actual NFF required to fight any particular fire remains uncertain, since the conditions in any given fire are variable.

The uncertainty in NFF at network node j (Q_fj) is represented by a probability density function (PDF) in Figure 1a. In this model, ${\mu }_{Q_{fj}}$ is the mean of NFF at node j and ${\sigma }_{Q_{fj}}$ is the standard deviation of NFF at node j, representing a measure of uncertainty in NFF. WDS are also required to supply fire flow above a minimum required pressure for fire protection (usually 14 m H₂O or 20 psi). Meeting the minimum pressure requirement is governed by factors such as pump capacity, the placement and dimensions of storage facilities, and the diameters of both trunk mains and local distribution pipes. The critical flow, Q_cj, in Figure 1a is the fire flow at node j that corresponds to the minimum required pressure head, H_{min j} at node j (Figure 1b). If the fire flow exceeds the critical flow (Q_fj > Q_cj), the pressure head is below the minimum required pressure head, H_{min j}, due to the inverse relationship between fire flow and available pressure head at node j. In this study, fire flow failure probability corresponds to the chance that the nodal pressure head H_j drops below its minimum allowable level H_{min j} during a fire event at node j. The value is calculated through the cumulative distribution function (CDF) of H_j evaluated at H_{min j}, as shown in Figure 1c.

$$F_{H_j}(H_{\min j})=P[H_j\le H_{\min j}]$$

(1)

An explicit measure of the failure probability in WDS is challenging because of the complexity and non-linearity of WDS. In addition, no widely accepted measure methodology has been introduced in WDS [8,9,10]. In this paper, an analytical probabilistic approach of Jung et al. [20] is applied to calculate the fire flow failure probability in a branched distribution network. The analytical approach is carried out quickly and precisely in the branch pipe without simplifying assumptions and approximations when compared to other computationally expensive methodologies. The available pressure head H_j during a fire at node j connecting its upstream pipe i with NFF Q_fj in addition to MDD conditions in the branched system is calculated in an analytical manner such that:

$$\begin{array}{c}{H}_j=H_s+h_p-{\displaystyle \sum _{l=1}^i{h}_{Ll}}\\ =H_s+A-B/n^2(Q_{fj}+{\displaystyle \sum _{k=1}^{N_n}{Q}_k}{)}^2-{\displaystyle \sum _{l=1}^i{K}_l{(Q_{fj}+q_l)}^2}\end{array}$$

(2)

where H_s = water source (e.g., reservoir, clear well) level; h_p = pressure head provided by n parallel pumps of identical type; A and B denote the coefficients describing the quadratic pump performance curve expressed as hp = A − B(Q/n)²; h_Ll = head loss in pipe l; N_n = number of nodes in the network; Q_k = MDD at node k; $K_l=f_l{L}_l/2g{D}_l{A}_l^2$ representing the hydraulic resistance of pipe l; f_l, L_l, D_l and A_l are Darcy-Weisbach friction factor, pipe length, the pipe diameter and the pipe cross-sectional area of pipe l, respectively; q_l = flow at pipe l. Combining Equation (1) with (2) and then solving for the NFF Q_fj with the quadratic formula yields:

$$F_{H_j}(H_{\min j})=1-F_{Q_{fj}}\left({\displaystyle \frac{-b+\sqrt{b^2-a(H_{\min j}-c)}}{a}}\right)$$

(3)

where $a=B/n^2+{\displaystyle \sum _{l=1}^i{K}_l}$, $b=B/n^2{\displaystyle \sum _{k=1}^{N_n}{Q}_k}+{\displaystyle \sum _{l=1}^i{K}_l{q}_l}$ and $c=H_s+A-B/n^2{\left({\displaystyle \sum _{k=1}^{N_n}{Q}_k}\right)}^2-{\displaystyle \sum _{l=1}^i{K}_l{\left(q_l\right)}^2}$.

It is important to note that the NFF is modeled as a random variable to account for its inherent uncertainty. For analytical tractability, we assume: (1) no more than one fire takes place simultaneously within the network, and (2) the fire event coincides with MDD. Since MDD is inherently uncertain, a full reliability assessment would require a joint probability distribution of both domestic and fire demands. However, due to the significant magnitude difference between fire flow and MDD, the uncertainty in fire flow remains the dominant factor. This simplification, thus, remains justified while providing meaningful insights into system performance under critical loading conditions.

2.2. Optimization Methodology

The multi-objective transient pipeline optimization under fire flow conditions is developed next. The first objective, described by Equation (4) below, minimizes capital investment in pipes and pumps by optimizing both pipe sizes and pump quantities as decision parameters. The second objective described by Equation (5) seeks to minimize the fire flow probability of failure on a fire flow event. The probabilities of fire flow failure at all fire flow nodes are calculated with Equation (3) and then averaged with the assumption that fire events are uniformly distributed and mutually exclusive (i.e., only single fire occurs at any time). The average fire flow failure probability is ranged from 0 to 1. A probability value of zero indicates that fire flow failure does not occur, whereas larger values denote a greater chance of such failure.

$$\mathrm{Minimize}{C}_{capital}\left(D,n\right)={\sum }_{i=1}^{N_p}u\left(D_i\right)L_i+n\cdot c_p$$

(4)

$$\mathrm{Minimize}{\displaystyle \frac{1}{N_n}}{\sum }_{j=1}^{N_n}{F}_{H_j}\left(H_{\min j}\right)$$

(5)

Subject to:

$$D_i\in \left\{\mathrm{D}\right\},i=1,\dots ,N_p$$

(6)

D_i ≥ D_i+1, i = 1, …, N_p − 1

(7)

$${\displaystyle \sum Q_{in}-{\displaystyle \sum Q_{out}=Q_j+}}{Q}_{fj},j=1,\dots ,N_n$$

(8)

$$H_j\ge H_{\min j}^{\#},j=1,\dots ,N_n$$

(9)

$$V_i\le V_i^{\max },i=1,\dots ,N_p$$

(10)

$$H_{\min }^{*}\le H_j\le H_{\max }^{*},j=1,\dots ,N_n$$

(11)

$${\displaystyle \frac{1}{g{A}_i}}{\displaystyle \frac{\partial Q}{\partial t}}+{\displaystyle \frac{\partial H}{\partial x}}+{\displaystyle \frac{f_{i}Q\left|Q\right|}{2g{D}_i{A}_i^2}}=0$$

(12)

$${\displaystyle \frac{\partial H}{\partial t}}+{\displaystyle \frac{a_i^2}{g{A}_i}}{\displaystyle \frac{\partial Q}{\partial x}}=0$$

(13)

where D_i = pipe diameters chosen from the standard set of commercially available sizes; u(D_i) = unit cost of pipe i with diameter D_i; L_i = length of pipe i; N_p = number of pipes; c_p = cost of single pump; n = number of pumps; $H_{\min j}^{\#}$ = minimum required pressure under MHD condition; V_i = velocity at pipe i; $V_i^{\mathrm{m}\mathrm{a}\mathrm{x}}$ = maximum allowable velocity at pipe i; $H_{\mathrm{m}\mathrm{a}\mathrm{x}}^{*}$= maximum allowable pressure during transient; $H_{\mathrm{m}\mathrm{i}\mathrm{n}}^{*}$= minimum allowable pressure during transient; x = distance along the centerline of conduit; t = time; a_i = celerity of the shock wave of pipe i. Equation (7) is to ensure the engineering practicality that upstream pipe size, D_i, is not smaller than its downstream pipe size, D_i+1. Equation (8) states the mass balance requirement at every node j. Equation (9) ensures that the pressure head H at each node j remains at or above the prescribed minimum threshold $H_{\min j}^{\#}$ under the steady state MHD condition. Equation (10) limits the fluid velocity (e.g., to 3 m/s) in order to avoid scouring of the pipe wall. Equation (11) is a transient constraint during the fire flow event with maximum permissible heads (e.g., pipe ratings or negative pressures for health concerns). Equations (12) and (13) describe the governing momentum and continuity relations for unsteady flow in pressurized pipelines. Note that the hydraulic constraints in Equations (8)–(10) are under steady state condition to be solved analytically, which is consistent with the EPANET hydraulic formulation but simplified here due to the network structure. For transient conditions, the continuity and momentum equations [Equations (12) and (13)] were solved using the Method of Characteristics (MOC), a widely accepted approach for simulating unsteady flows in pressurized pipeline systems [21].

Figure 2 depicts a flowchart of the multi-objective transient pipeline optimization approach. The procedure begins with the initialization of a population of candidate solutions, where each individual represents a set of pipe diameters and pump numbers as decision variables. Each individual is then evaluated for the two objectives—capital cost [Equation (4)] and average fire flow failure probability [Equation (5)]—while pipe size availability and practicality [Equations (6) and (7)], hydraulic feasibility under steady-state [Equations (8)–(10)] and transient constraints [Equations (11)–(13)] are also checked. The NSGA [22] improves the population iteratively by applying four key operators: (i) selection, where individuals are chosen based on a binary tournament using both Pareto rank and crowding distance; (ii) crossover, which recombines parent solutions to produce offspring with mixed pipe and pump configurations; (iii) mutation, introducing small random changes to maintain diversity; and (iv) elitist sorting, where the combined parent and offspring populations are re-ranked by non-domination, and the best individuals are retained for the next generation. Within each Pareto front, crowding distance is calculated to preserve diversity and avoid premature convergence. This evolutionary cycle continues until successive generations converge toward a Pareto-optimal frontier, representing a set of trade-off solutions that balance system cost and reliability while satisfying all constraints.

3. Results

3.1. Case 1: Dual Pipeline System

The multi-objective optimization approach was tested on a dual pipeline system illustrated in Figure 3. The test network consists of a reservoir with a fixed water level of 10 m, a set of parallel pumps, six pipes, and seven nodes located at zero elevation. For a single pump, the nominal discharge Q and head rise hp are 20 L/s and 100 m, respectively, leading to the characteristic curve for n pumps in parallel: hp = 133 − 0.083(Q/n)². Each pipe is 1000 m long with a Darcy–Weisbach roughness factor of f = 0.02. Each of the nodes N2–N7 (Figure 3) serves 100 detached dwellings, with an assumed four residents per household and an individual water use of 400 L/day. The corresponding average daily demand per node is 1.9 L/s. Using a peaking factor of 1.7, the maximum day demand becomes 3.2 L/s per node. The mean NFF to control a fire in a single-family residential unit is 32 L/s with a duration of 2 h [19]. The NFF is uncertain and assumed to be normally distributed (without loss of generality) with a standard deviation of 8 L/s (coefficient of variation of 25%). It was assumed that the reservoir storage volume is adequately sized for the 2 h required fire flow duration. For fire flow conditions, each node must maintain a minimum pressure of 14 m. The maximum hour demand (MHD) is modeled using a peaking factor of 3.4, and under this loading, the minimum nodal pressure requirement remains 14 m. Transient conditions are triggered by opening hydrant with the mean of NFF under the steady state MDD conditions. All six fire flow events at Nodes N2 to N7 are considered separately to maintain transient pressure heads within the maximum permissible heads $H_{max}^{*}$ and $H_{min}^{*}$ in Equation (11) which are assumed as 200 m and 0 m, respectively, for the whole system. To access the severity of transients, three different hydrant opening times—5, 10 and 20 s—were used in this analysis. In industrial practices, typical hydrant opening time is near 10 s so faster transient at 5 s and slower transient at 20 s are included. The case of excluding the transient constraint in Equation (11), denoted as no transient, is also included to compare with the three different severities of transients.

NSGA population size was set to 50 and the number of generations to 200. These values were determined through preliminary convergence tests, which showed no further improvement of the Pareto front beyond 200 generations with population sizes greater than 50. The mutation probability (0.025) and uniform crossover probability (0.9) follow commonly used ranges in evolutionary optimization studies [22] and were confirmed to provide stable and diverse solutions in this application. The chromosome length of 24 corresponds to the encoding of six decision variables (pipe diameters) and one pump variable, ensuring sufficient resolution of the search space.

Available pipe sizes were restricted to 16 standard diameters with unit cost data provided in [20]. Each pump was assigned a fixed cost of $5000. Consequently, with six pipes and up to 16 pump alternatives, the total solution space spans 16⁷ (approximately 2.68 × 10⁸) possible configurations. At the start of each optimization run, NSGA creates an initial population of candidate designs, where the decision variables are the diameters of the pipes and the number of pumps. For every candidate, the algorithm evaluates pipe and pump costs, the mean fire flow failure probability, and any constraint violations, and uses these results to evolve the next generation of solutions.

The solutions of four different optimization run with transient severities—three different hydrant opening times of 5 s, 10 s and 20 s and no transient—after 200 generations are shown in Figure 4. On the x-axis is the total pipe and pump cost defined in Equation (4), while the y-axis represents the average fire flow failure probability given by Equation (5).

Table 1. Representative Pareto-optimal solutions for Case 1.

Type	Solution	Pipe Diameter (mm)						Number of Pumps	Cost (105 $)	Average Fire Flow Failure Probability
		P1	P2	P3	P4	P5	P6
No transient	A1	150	150	150	150	100	100	2	3.24	0.724
	A2	150	150	150	150	100	100	4	3.34	0.504
	A3	200	150	150	150	100	100	3	3.49	0.372
	A4	200	150	150	150	150	100	3	3.69	0.254
	A5	200	150	150	150	150	150	5	3.99	0.114
	A6	200	200	200	200	150	150	4	4.54	0.037
	A7	250	200	200	200	200	200	5	5.18	0
Transient (20 s)	B1	200	150	150	150	150	150	2	3.84	0.377
	B2	200	150	150	150	150	150	4	3.94	0.135
	B3	200	200	200	200	150	150	4	4.54	0.037
	B4	250	200	200	200	200	200	5	5.18	0.000
Transient (10 s)	C1	250	200	200	200	150	150	2	4.63	0.123
	C2	250	200	200	200	150	150	3	4.68	0.040
	C3	250	200	200	200	200	200	5	5.18	0
Transient (5 s)	D1	250	200	200	200	200	200	2	5.03	0.023
	D2	250	200	200	200	200	200	5	5.18	0

shows Pareto-optimal pipe diameters, number of pumps, total cost (pipe and pump) and fire flow failure probability of representative of four different optimization runs. Under the conditions of no transient, the optimal total cost of $324,000 in Table 1 (Solution A1) can satisfy the MHD constraint but its resulting fire flow failure probability is high as 0.724, suggesting most nodes cannot maintain NFF conditions. When the total cost is increased from $324,000 to $518,000 (Solution A1 to A7), NFF conditions, as well as the MHD constraint, are satisfied. The trade-off results indicate the fire flow in the dual pipeline system is more dominant and critical loading condition than the MHD conditions. In addition, the low fire flow failure probability solutions are consistent with the common engineering standard of adopting a 150 mm minimum diameter for distribution mains to guarantee fire flow capacity.

Under the conditions of three different transient severities, the rapider hydrant opening (the severer transient) restrains the Pareto-optimal front range with the higher total cost: $324,000 (Solution A1) to $384,000 (Solution B1), $463,000 (Solution C1) and $503,000 (Solution D1). This requires larger pipes, for example, from 100 mm (Solution A1) to 150 mm (Solutions B1 and C1) and 200 mm (Solution D1) at Pipes P5 and P6. This reduces fire flow failure probability accordingly from 0.724 to 0.377, 0.123 and 0.023, respectively.

The results confirm that a minimum 150 mm diameter watermain is necessary to both mitigate transient responses and meet the NFF requirements. All solutions that consider transient constraints (Solutions B1 to D2) consistently result in a minimum pipe diameter of 150 mm.

3.2. Case 2: Branched Network

To further test the method, the optimization framework was implemented on a larger branched network (Figure 5), representative of an actual water supply system with 28 pipes, 29 junctions, parallel pumping units, and one source. This network was derived from the looped EPANET benchmark case [23] through strategic skeletonization and conversion to a branched configuration while preserving hydraulic equivalence with the original model. Preliminary analyses confirmed that the skeletonization had a negligible impact on transient behavior, thereby validating the analytical fire flow failure probability assessment.

For optimization efficiency, the 28 pipes were grouped into 8 clusters (

Table 2. Pipe and node data for branched network Case 2.

Pipe Number	Pipe Group	Length, m	Node Number	Elevation, m	Demand, L/s
P1	PG1	731.5	N1	15.2	0.0
P2	PG1	243.8	N2	30.5	1.4
P3	PG1	365.7	N3	30.5	1.0
P4	PG1	822.9	N4	38.1	0.3
P5	PG2	121.9	N5	48.8	0.3
P6	PG2	213.4	N6	54.9	0.9
P7	PG2	579.1	N7	56.4	2.2
P8	PG2	182.9	N8	64.0	1.0
P9	PG2	121.9	N9	64.0	0.1
P10	PG3	91.4	N10	61.0	0.1
P11	PG3	76.2	N11	57.9	0.1
P12	PG3	182.9	N12	57.9	0.7
P13	PG3	91.4	N13	70.1	0.5
P14	PG3	182.9	N14	70.1	0.4
P15	PG3	121.9	N15	57.9	1.1
P16	PG3	121.9	N16	39.6	0.5
P17	PG4	213.4	N17	33.5	0.4
P18	PG4	91.4	N18	33.5	0.0
P19	PG5	365.7	N19	33.5	0.1
P20	PG5	304.8	N20	33.5	0.6
P21	PG6	457.2	N21	39.6	0.3
P22	PG6	426.7	N22	45.7	2.5
P23	PG7	335.3	N23	45.7	2.7
P24	PG7	396.2	N24	51.8	1.7
P25	PG7	304.8	N25	61.0	1.1
P26	PG7	121.9	N26	54.9	0.1
P27	PG8	152.4	N27	57.9	0.1
P28	PG8	304.8	N28	33.5	0.0
			N29	39.6	0.2

) based on spatial and hydraulic similarities, with each group assigned a uniform diameter during optimization—reducing decision variables to 8 pipe groups plus pump quantities. The NSGA parameters were configured with a population size of 100 and 200 generations, while mirroring Case 1’s setup for the other system and optimization parameters. When the population size was doubled from 100 to 200, the Pareto front solutions remained virtually identical. Similarly, increasing the number of generations from 200 to 300 did not produce additional non-dominated solutions. Transient events were simulated by hydrant openings at the mean NFF value across three valve operating times (5, 10, and 20 s). To balance computational effort and scenario coverage, eight critical fire flow locations were selected (Nodes 5, 9, 14, 19, 21, 23, 27, and 29), representing spatially distributed demand points.

The solutions of four different optimization run with transient severities are shown in Figure 6.

Table 3. Representative Pareto-optimal solutions for Case 2.

Type	Solution	Pipe Diameter (mm)								Number of Pumps	Cost (105 $)	Average Fire Flow Failure Probability
		PG1	PG2	PG3	PG4	PG5	PG6	PG7	PG8
No transient	A1	250	150	150	50	50	100	100	50	3	4.59	0.576
	A2	250	150	150	100	50	100	100	50	4	4.69	0.491
	A3	250	200	150	50	50	100	100	50	3	4.83	0.398
	A4	250	200	150	100	50	100	100	50	4	4.94	0.331
	A5	250	200	150	100	50	100	150	100	4	5.26	0.183
	A6	250	200	150	150	100	100	150	100	4	5.44	0.117
	A7	250	200	150	150	150	150	150	100	4	5.75	0.033
	A8	250	250	200	150	150	200	200	150	6	6.76	0
Transient (20 s)	B1	250	250	150	150	150	150	150	150	2	5.98	0.253
	B2	250	250	150	150	150	150	150	150	3	6.03	0.014
	B3	250	250	200	150	150	200	200	150	6	6.76	0
Transient (10 s)	C1	250	250	150	150	150	150	200	150	4	6.31	0.002
	C2	250	250	200	150	150	200	200	150	6	6.76	0
Transient (5 s)	D1	250	250	150	150	150	150	200	150	4	6.31	0.002
	D2	250	250	200	150	150	200	200	150	6	6.76	0

indicates the optimal pipe diameters, number of pumps, total pipe and pump cost and average fire flow failure probability for representative Pareto-optimal solutions. Similar to Case 1, reducing total cost in Case 2 increases the average fire-flow failure probability. For example, the optimization run of no transient shows that the total cost increases from $459,000 (Solution A1) to $676,000 (Solution A8) and the average fire flow failure probability decreases from 0.576 to 0. Optimization allocates large pipe diameter (250 mm) near the water source to meet MHD conditions and then increases pumps (3 to 6 pumps) and pipe capacity in the downstream to reduce the fire flow failure probability. As in Case 1, a minimum 150 mm diameter watermain is required to fully satisfy the NFF conditions.

A mild transient (hydrant opening in 20 s) increases initial total cost from $459,000 (Solution A1) to $598,000 (Solution B1) but with a reduced fire flow failure probability from 0.576 to 0.253. As the transient becomes more severe with the rapider hydrant openings (10 s and 5 s), the initial total costs increase to $631,000 (Solutions C1 and D1) while the corresponding fire flow failure probability becomes near zero (0.002). These results suggest that the rapid hydrant opening is the key factor influencing the design of water distribution networks. The optimization results also indicate that a minimum 150 mm distribution main is necessary not only to ensure adequate fire flow protection but also to effectively mitigate fire flow-induced transients.

4. Discussion

Two case studies highlight the trade-offs captured by the proposed framework. In Case 1, incorporating transient constraints increased the minimum feasible cost from $324,000 (no transient) to $503,000 (5 s opening), while reducing the average fire-flow failure probability from 0.724 to 0.023 (Table 1). Similarly, in Case 2, the least-cost design without transients was $459,000 with a failure probability of 0.576, whereas under severe transients (5–10 s opening), the cost increased to approximately $631,000 while reducing failure probability to nearly zero (Table 3). These results demonstrate that more stringent transient conditions require larger pipe sizes and additional pumps, thereby raising capital costs but significantly improving reliability. Across both cases, the optimization consistently converged to a minimum 150 mm pipe diameter as necessary to satisfy NFF requirements and mitigate transients. This provides a clear design benchmark for practitioners when balancing cost, reliability, and transient safety.

The proposed optimization framework was developed under specific assumptions and resource configurations to maintain clarity and computational tractability. The analysis focused on ensuring adequate flow and pressure during firefighting operations, without explicitly addressing water quality or routine operational resilience—factors that often involve trade-offs when optimized alongside fire protection [24]. Although detailed water quality modeling was outside the scope of this study, we computed average water age values in the analyzed variants as a proxy indicator. For example, in Case 1, the water age for average day demand across Pareto-optimal solutions ranged from approximately 7 to 17 h. Designs with smaller pipe diameters reduce water age by improving turnover. These observations highlight that including water age alongside fire flow and transient criteria could provide an integrated view of both hydraulic reliability and water quality in future optimization studies.

Transient assessment in this study was limited to fire hydrant operations; however, the methodology may be adapted to additional critical situations, including pump trips, fast valve shut-offs, or concurrent events. Several design variables, including detailed system topography, pipe material and wall thickness variations, and the placement or selection of surge protection devices (e.g., air valves, surge tanks), were intentionally excluded to isolate the effects of key hydraulic parameters.

A fully comprehensive transient optimization would require integrating these additional factors, supported by robust engineering judgment to balance model complexity, computational feasibility, and the practicality of cost-effective surge mitigation. Future work could expand the framework to incorporate these parameters, enabling broader safety assessment and enhancing the long-term sustainability and resilience of urban underground water infrastructure.

5. Conclusions

This study presented a transient-aware multi-objective optimization framework that integrates fire flow failure probability analysis with hydraulic transient assessment. By embedding an analytical probabilistic model within NSGA, the framework identifies Pareto-optimal solutions that minimize capital cost and fire flow failure probability while satisfying hydraulic and transient constraints. Application to two case studies confirmed the practicality of the approach: a minimum 150 mm distribution main consistently emerged as necessary for both fire flow reliability and transient control, and rapid hydrant opening significantly increased design costs by intensifying transient severity. These findings indicate that incorporating transient analysis at the design stage enhances safety and supports sustainable, cost-effective planning of water distribution systems.

While this work focused on fire flow conditions and selected design variables, the methodology is extensible. Future research could integrate pipe material deterioration, water quality dynamics, and operational resilience metrics, as well as additional transient scenarios such as pump failures or valve operations. Broader consideration of surge protection devices and lifecycle cost analysis would enable more comprehensive safety assessments and proactive maintenance strategies for aging urban pipeline networks, further advancing sustainable infrastructure management.