Braess' Paradoxes in Coupled Power and Transportation Systems

TL;DR

This work develops a graph-based framework that couples transportation and power networks via EV charging to study Braess' paradox in a shared infrastructure. By formulating a generalized user equilibrium (GUE) that links route/charging choices with economic dispatch, the authors characterize how capacity expansions can paradoxically degrade system performance, even across the two subsystems. They derive necessary and sufficient conditions for various Braess' paradox types across uncongested, fully congested, and general network topologies, and propose convex-program-based methods to screen for paradoxes. The paper also proposes pricing-based mitigation strategies, including system-optimal and static charging pricing, and demonstrates the concepts through numerical studies on a coupled IEEE 9-bus and Bay Area transportation network. Overall, the work provides actionable insights for infrastructure planning under electrified mobility, highlighting cross-system externalities and mitigation pathways.

Abstract

Transportation electrification introduces strong coupling between the power and transportation systems. In this paper, we generalize the classical notion of Braess' paradox to coupled power and transportation systems, and examine how the cross-system coupling induces new types of Braess' paradoxes. To this end, we model the power and transportation networks as graphs, coupled with charging points connecting to nodes in both graphs. The power system operation is characterized by the economic dispatch optimization, while the transportation system user equilibrium models travelers' route and charging choices. By analyzing simple coupled systems, we demonstrate that capacity expansion in either transportation or power system can deteriorate the performance of both systems, and uncover the fundamental mechanisms for such new Braess' paradoxes to occur. We also provide necessary and sufficient conditions of the occurrences of Braess' paradoxes for general coupled systems, leading to managerial insights for infrastructure planners. For general networks, through characterizing the generalized user equilibrium of the coupled systems, we develop efficient algorithms to detect Braess' paradoxes and novel charging pricing policies to mitigate them.

Figures (17)

• Figure 1: 2-Route 2-Bus example. The arrow on the power line indicates the positive flow direction.
• Figure 2: Partial derivatives of social cost $\Phi_s$, $s \in \{\mathrm{T},\, \mathrm{P},\, \mathrm{C}\}$ with respect to $\alpha_1$ when the model parameters are set as $\alpha_1 = 100, \alpha_2 = 1, \bar{f} = 0.2$, and $\rho \in [0.5,10]$.
• Figure 3: 2-Route 3-Bus example. The arrows on power lines indicate the positive flow direction.
• Figure 4: GUE and social cost metrics change with $\bar{f}_3$. Model parameters are set as $\bm \alpha := [1,10]^\top, \rho := 6, \mathbf{Q} := \mathrm{diag}([2,1,1])$, and $\mathbf{\bar{f}} := [0.1,0.3,\bar{f}_3]^\top$, where $\bar{f}_3 \in [0.04,2]$.
• Figure 5: GUE and power system social cost change with $\bar{f}_3$. Model parameters are set as $\bm \alpha := [1,1]^\top, \rho := 6, \mathbf{Q} := \mathrm{diag}([0,1,1])$, and $\mathbf{\bar{f}} := [0.1,0.3,\bar{f}_3]^\top$, where $\bar{f}_3 \in [0.04,0.2]$.

Theorems & Definitions (53)

• Definition 1: Transportation GUE given LMPs
• Lemma 1: Uniform Cost
• Definition 2: GUE for the Coupled System
• Definition 3: Generalized Braess' Paradox
• Remark 1: Derivative-based BP
• Proposition 1: 2-Route 2-Bus System: Uncongested Case
• Proposition 2: 2-Route 2-Bus System: Congested Case
• Proposition 3: No Type P-T and P-P in the 2-Route 2-Bus System
• Theorem 1: BPs in 2-Route 3-Bus Coupled System
• Remark 2: Type T-C BP