Urban traffic networks and signal optimisation

The setting

Urban traffic is awkward to model. First-order models (LWR-type) miss important phenomena like stop-and-go waves and capacity drops at bottlenecks. Microscopic agent-based simulations capture detail but scale poorly and are hard to optimise analytically. Second-order continuum models like Payne–Whitham occupy an interesting middle ground — but adapting them to networked urban roads with traffic lights introduces real mathematical subtleties at intersections.

Data

The model is calibrated and applied to the Stara Ochota district of Warsaw, Poland — a central urban area with 21 signalised intersections. Road network topology and traffic-light locations were extracted from OpenStreetMap; driver turning fractions at each junction were estimated using the Traffic Simulation Framework. Fundamental diagram parameters (free-flow speed, critical density, speed-decay exponent) were calibrated against 5-minute traffic counter measurements from Ljubljana, Slovenia, provided by the Faculty of Civil and Geodetic Engineering at the University of Ljubljana. Model initial conditions — segment-level free-flow and average speeds — were sourced from TomTom's Flow Segment Data API.

Approach

We extend the Payne–Whitham framework to urban networks where junctions are governed by traffic lights with controllable phase. Traffic dynamics on each road segment follow a second-order PDE system, while signals impose time-varying boundary conditions that couple adjacent segments. We formulate the resulting optimisation problem — choosing signal phases to minimise total travel time or congestion — and study its structure.

Results

The PW network model captures phenomena that first-order LWR models cannot represent: stop-and-go wave formation at bottlenecks and queue build-up upstream of red lights. Both are consequences of the second-order pressure term and emerge naturally from the calibrated model without hand-tuning.

Applied to the Stara Ochota district of Warsaw (21 signalised intersections), the optimised signal timing achieves a 34% improvement in average traffic speed and a 24% reduction in queue length compared to fixed-cycle baseline timing, under the calibrated demand scenario. The key mathematical contribution enabling this is a tractable gradient structure for the PW network PDE with signal-coupled boundary conditions, which makes the optimisation amenable to standard gradient-based methods rather than requiring exhaustive search.

What this means

Second-order continuum models are underused in signal optimisation. The mathematical difficulty of coupling a PDE system to discrete signal dynamics has kept the literature on simpler first-order models or agent-based simulations. This work shows that the coupling can be formulated tractably, and that the resulting optimiser produces meaningful improvements on a realistic urban district.

The pipeline uses only publicly available data sources — OpenStreetMap topology, TomTom flow data, and traffic counter measurements — making it straightforward to adapt to other cities.

Publications