Optimizing Speed Limits Towards Sustainable Urban Mobility: A Mathematical Model and Numerical Application

From a sustainable planning perspective, the traffic congestion is not a capacity issue, but a multidimensional problem, affecting environmental quality and public health. (Stop and go) Waves typical in congested flow, contribute significantly to greenhouse gas emissions and fuel wastage. Therefore, Link-Specific Speed Limits are increasingly recognized as a green traffic management strategy. However, by harmonizing traffic speeds and suppressing shockwaves variable speed limits (VSL) can reduce acceleration and deceleration cycles, thereby lowering emissions and enhancing the overall sustainability of the urban transport system.

1.1 Urban mobility and sustainability imperative

In the rapid urbanizing landscapes of the 21st century, the governance of urban mobility has transcended traditional traffic engineering to become a “central pillar” of sustainable development worldwide [1]. Furthermore, as cities densify, the conflicting demands between vehicle mobility and urban livability have intensified prompting a global “re-evaluation” of streets' function under the framework of the “United Nations Sustainable Development Goals (SDGs)”, particularly Goal 11.2, which mandates a safe and accessible transport system for all [2]. However, modernized cities currently, face a lot of challenges, such as population growth, exacerbating environmental risks, the need to address very complex “social issues”, “energy efficiency” and “security”, ageing municipal infrastructure, worsening urban traffic, and many other challenges [3]. In the same context, contemporary urban planning literature argues, that the historic prioritization of the high speed arterial corridors has severed community cohesion and prioritized mechanical movement over social interaction necessitating a paradigm shift towards “Human-Centric” planning [4]. Moreover, speed management is now recognized not merely as a safety intervention, but also as a critical lever for achieving “broader environmental” and “economic sustainability” targets, in modern metropolitan areas [5].

1.2 Evolution of speed paradigms: From design to safety

Speed limits were established using the “85th percentile rule” which is a method predicated on the assumption that the majority of drivers should behave reasonably and that limits should reflect their natural speed choice [6]. However, this engineering centric approach, has faced mounting criticism for failing to account for the kinetic energy principle of the “safe system” approach and “vision zero” strategies adopted globally [5]. Empirical evidence from recent meta-analysis demonstrates that relying on driver preference often leads to “socially inefficient” speeds, particularly in mixed- use zones where the presence of vulnerable road users (VRUS) requires a drastic reduction in kinetic impact forces [7]. The condition may be worse, when “transport demands” and “roadside activities” increase, which lead to inefficient traffic performance and increased “Side friction”, which is refers to a variable indicating the degree of interactions between the “roadside activities” and the “traffic flow” [8]. Thus, the modern consensus advocates for limits derived from biomechanical tolerance and social welfare optimization rather than free-flow behavior [9, 10].

1.3 The economic valuation of travel time vs. externalities

The determination of optimal speed is fundamentally an economic trade-off. On one hand, higher speeds reduce travel time, which is traditionally valued highly in cost-benefit analysis (CBA) as a contributor to economic productivity [11, 12]. On the other hand, speed generates negative externalities-costs imposed on third parties-that rise exponentially with velocity. Leading transportation economists, argue that conventional models often overestimate the Value of Travel Time Savings (VTTS), while underestimating the marginal costs of “crash risk” and “environmental degradation” [13, 14]. The “Socio-Economic Optimization” framework, seeks to resolve this by internalizing these conflicting variables into a unified total cost function shifting the focus from maximizing flow to minimizing total social loss [15].

1.4 The underestimated burden of noise and pavement interaction

While safety is frequently analyzed the environmental cost of traffic noise remains an under addressed externality especially in developing nations [16]. Recent epidemiological reports by the “European Environment Agency (EEA)”, have reclassified traffic noise from a minor annoyance to a major public risk linking it to cardiovascular diseases, cognitive impairment, and sleep disturbance [17]. Crucially, the noise generation mechanism is heavily dependent on the tire-pavement interaction. In regions with high pavement roughness typical of many developing infrastructures, the noise cost curve creates a binding constraint on optimal speed that is often stricter than safety constraints, a factor frequently ignored in standard international [18]. Many developing urban centers have witnessed rapid urbanization with a significant increase in vehicle ownership and population creating a lag in urban management. Furthermore, these cities often face significant traffic flow problems especially in “Central Business District (CBD) areas”, which decreases air quality and threatens healthy urban [19].

1.5 Research gap in developing contexts

Despite the theoretical maturity of these optimization models their application remains predominantly concentrated in developed economies. In the same context, developing countries, characterized by heterogeneous traffic streams, aggressive driving behaviors, varying infrastructure quality, and a lack of localized optimization tools [20]. However, current policies in such contexts typically utilize static uniform speed limits across diverse road hierarchies are ignoring the distinct functional and economical characteristics of commercial vs arterial roads [21]. This study aims to bridge this gap, by developing a site specific socio-economic optimization model, for a heterogeneous urban network providing empirical evidence to support the transition to context- sensitive “Link-Specific Speed Limits”. Furthermore, a sensitivity analysis was conducted to verify the model’s stability and examine the effect of the "value of time (VOT)" on the optimization results.

3.1 General optimization framework

The primary objective of this study is to formulate a Socially Optimal Speed (SOS) determination framework, applicable to heterogeneous urban traffic environments. Unlike traditional, traffic engineering approaches that define speed limits based on the “85th percentile speed” or design speed, this study adopts a “Socio-Economic” Optimization approach. Furthermore, the core hypothesis of this study, is that the optimal speed corresponds to the equilibrium point, where the marginal benefit of travel time savings equals the marginal social cost of safety risks and environmental externalities.

The total social cost function Z(ν) represents the summation of three distinct cost components normalized per kilometer of travel [22]. The optimization problem is defined as in Eq. (1):

Minimize Z(v) = C _Time (v) + C _Noise (v) + C _Safety(v)                    (1)

where:

V: The operating speed (km/h)

Z(v): Total generalized social cost ($/km)

C _Time: User cost related to travel time ($/km)

C _Noise: External cost of noise pollution ($/km)

C S_afety: External cost of accident risk ($/km)

The optimization seeks to find the speed v*. The first derivative of the total cost function equals zero (dz/dv = 0).

3.2 User cost modeling (time valuation)

Travel time savings represent the primary economic utility of increasing speed. However, the cost of time is modeled as an inverse function of speed. Furthermore, to ensure the framework reflects the economic reality of developing cities “VOT” was not applied uniformly. Instead, a function- based calibration was performed assigning higher “VOT” values to arterial highways (reflecting business and long distance trips) and lower to local streets [11]. The cost function is expressed as in Eq. (2) [23]:

$\mathrm{C}_{\text {Time }}=\operatorname{VOT}_{\text {class }} \times \frac{1}{V X}$                    (2)

(VOT_class) is the value of time calibration for the specific road functional class, such as commercial, collector, or arterial.

The term (1/v), represents the time required to travel one kilometer (hrs/km). This differentiation acknowledges that the urgency and economic value of trips vary significantly across the urban network [24].

3.3 Environmental cost modeling (empirical noise approach)

Instead of relying on theoretical noise prediction models such as “FHWA” and “TNM”, which often fail to account for local conditions such as, high pavement roughness and older vehicle fleets. This study developed site specific empirical noise cost functions.

In the same context, data collected from the representative numerical application Speed vs. Noise dB, were analyzed using non-linear polynomial regression in OriginPro to drive an empirical relationship for each road topology. However, the economic valuation follows the “Hedonic Pricing Method”, monetizing the annoyance level above a health threshold as in Eq. (3) [18]:

C_Noise = λ _Noise x max (0, L_eq (v)-L_Threshold)                 (3)

where:

L_eq (v) is the empirical polynomial function representing the noise level at speed v.

L_Threshold is the annoyance threshold set at 55 dB(A) following WHO guidelines.

λ _Noise is the marginal monetary cost per decibel increase.

The baseline value for λ _Noise was set at 0.05 USD/dB(A) for exposure levels exceeding the 55 dB (A) threshold; this threshold explicitly conforms to the World Health Organization (WHO), Environmental Noise Guidelines (2018). In the same context, recognizing that noise impact heavily depends on spatial context and population exposure, a functional weighting approach was applied, consistent with the methodology outlined in the European Commission’s Handbook on the External Costs of Transport. Furthermore, the “baseline cost” was adapted, to the functional classification of each link:

Link I (Arterial Road) is assigned a weight of (0.2) yielding (0.01) USD/dB(A). The better geometric design and separation from residential structures, significantly reduce direct human exposure and allow the model to prioritize mobility.

Link II (Collector Road) is assigned a weight of (0.8), yielding (0.04) USD/dB(A), this accounts for localized traffic routing through residential neighborhoods necessitating higher noise protection for residents.

Link III (Commercial Road) retains the maximum baseline penalty of (0.05) USD/dB (a) weight of (1.0) due to high daytime pedestrian activity, dense retail presence, and maximum human exposure to noise externalities.

Finally, a sensitivity analysis, was conducted on these baseline values, + 20% and +50%, to ensure the optimization model’s robustness as detailed in the results section.

3.4 Safety cost modeling (calibrated power model)

To internalize the cost of crash risks the study adopts the “Power Model”, which empirically relates speed to safety outcomes, based on kinetic energy principles. Moreover, a risk factor (α) was calibrated, according to the road’s functional classification as shown in Eq. (4) [25].

The safety cost function is defined as:

$\mathrm{C}_{\text {safety }}=\alpha_{\text {class }} \times\left(\frac{V}{V r e f}\right)^4$                 (4)

where:

Vref: a reference speed, typically the design speed or 60 km/hr

α class is a calibrated risk coefficient:

high α: assigned to commercial streets, to account for high pedestrian density and conflict points.

Low α is assigned to arterial highways, to reflect safer geometric design and segregated traffic.

This calibration, ensures that the model penalizes speed heavily in pedestrian zones, while allowing higher speeds on protected arterials.