Fundamentals of Transportation/Analogs

Transportation networks have analogs with network processes in other systems, such as water networks, structures, and electrical networks. Some of the relationships are outlined below. Concept Transportation (traffic) Water (pipe flow / hydraulics) Structures (linear elastic / statics) Electrical (DC circuits) Node law(conservation / equilibrium) Σqin − Σqout = s(node source/sink s) ΣQin − ΣQout = S(node withdrawal/injection S) ΣFx = 0,ΣFy = 0,ΣM = 0 ΣI = 0(Kirchhoff’s Current Law, KCL) Element / link law(constitutive) Kinematics: q = k·vLink cost (often nonlinear): t = t(q) Head/pressure loss: Δp = Rh·Qn(linear case: Q = Ch·Δp) Hooke (axial): F = k·δ Ohm: V = R·IAlso: I = G·V, G = 1/R Loop / compatibility law Wardrop user equilibrium (OD pair):fr > 0 ⇒ cr = c*fr = 0 ⇒ cr ≥ c* Loop head balance:ΣΔHℓ = 0 Compatibility (parallel members):δ1 = δ2 = … ΣV = 0 around any closed loop(Kirchhoff’s Voltage Law, KVL) “Through” variable(adds at node) Flow q Discharge Q Member force F Current I “Across” variable(equal in parallel) Travel time / cost t or c Pressure p or head H Displacement δ Voltage V Resistance / conductance analog Increasing cost t(q) (generally nonlinear)Marginal: dt/dq Hydraulic resistance RhConductance Ch = 1/Rh Stiffness kCompliance 1/k Resistance RConductance G = 1/R Notes Hydrostatics (no-flow special case): p = p0 + ρgh (gauge form: p = ρgh). Symbols Transportation q = flow (vehicles/time) k = density (vehicles/length) v = speed (length/time) t, c = travel time / generalized cost fr = route flow on route r cr = cost on route r c* = minimum cost for the OD pair (under user equilibrium) Water (hydraulics) Q = discharge (volume/time) p = pressure H = hydraulic head ρ = fluid density = m/V g = gravitational acceleration h = elevation Rh = hydraulic resistance Ch = hydraulic conductance = 1/Rh n = exponent in loss law (model-dependent) Structures F = force M = moment δ = displacement/deflection k = stiffness Electrical V = voltage I = current R = resistance G = conductance = 1/R