Hierarchical Pricing Game for Balancing the Charging of Ride-Hailing Electric Fleets

Abstract

Due to the ever-increasing popularity of ride-hailing services and the indisputable shift towards alternative fuel vehicles, the intersection of the ride-hailing market and smart electric mobility provides an opportunity to trade different services to achieve societal optimum. In this work, we present a hierarchical, game-based, control mechanism for balancing the simultaneous charging of multiple ride-hailing fleets. The mechanism takes into account sometimes conflicting interests of the ride-hailing drivers, the ride-hailing company management, and the external agents such as power-providing companies or city governments that will play a significant role in charging management in the future. The upper-level control considers charging price incentives and models the interactions between the external agents and ride-hailing companies as a Reverse Stackelberg game with a single leader and multiple followers. The lower-level control motivates the revenue-maximizing drivers to follow the company operator's requests through surge pricing and models the interactions as a single leader, multiple followers Stackelberg game. We provide a pricing mechanism that ensures the existence of a unique Nash equilibrium of the upper-level game that minimizes the external agent's objective at the same time. We provide theoretical and experimental robustness analysis of the upper-level control with respect to parameters whose values depend on sensitive information that might not be entirely accessible to the external agent. For the lower-level algorithm, we combine the Nash equilibrium of the upper-level game with a quadratic mixed integer optimization problem to find the optimal surge prices. Finally, we illustrate the performance of the control mechanism in a case study based on real taxi data from the city of Shenzhen in China.

Figures (6)

• Figure 1: Schematic sketch of the problem setting with companies $\mathcal{C}=\left\{\mathcal{C}_1,\mathcal{C}_2\right\}$ and charging stations $\mathcal{M}=\left\{\mathcal{M}_1, \mathcal{M}_2, \mathcal{M}_3\right\}$. The central body, e.g., the government or the power company, wants to balance the vehicle load on different charging stations by properly setting the pricing policies $p_i(x^1,x^2):\mathcal{X} \rightarrow \mathds{R}^3$ for $i\in\left\{1,2\right\}$. Under the provided pricing policies, each ride-hailing company wants to minimize its own operational cost by steering its set of vehicles $\mathcal{V}_i$ to different charging stations. The blocks $K_F^i$ calculate the optimal splits $x^{i*}\in\mathcal{X}_i\subseteq\mathcal{P}_{\mathcal{M}}$ of the ride-hailing fleets in a decentralized manner with little information exchange and based on the parameters $A_i$, $B_i$, $c_i$, $D_i$, $f_i$. The blocks $K_M^i$ calculate the surge price vectors $\rho^v_i\in\mathds{R}_+^{\left|\mathcal{M}\right|}$ for every vehicle $v\in\mathcal{V}_i$.
• Figure 2: Map of Shenzhen - The network topology used in the case study consists of 1858 intersections connected by 2013 road segments divided in 4 regions around charging stations $\mathcal{M}=\left\{\mathcal{M}_1,\mathcal{M}_2,\mathcal{M}_3,\mathcal{M}_4\right\}$ according to the Voronoi partitioning of the city. Color of the nodes within each region indicates the total number of ride-hailing requests whose origin is in that region. From the left part of the figure, it is clear that the highest number of ride-hailing requests occurs in the region around charging station $\mathcal{M}_1$ and the lowest number of requests occurs in the region around $\mathcal{M}_4$. The right part of the figure shows the macroscopic fundamental diagram (MFD) for the city of Shenzhen according to doi:10.3141/2422-01.
• Figure 3: Evolution of the total number of vehicles at each charging station when no action is taken, i.e., $p=p_{\text{base}}=\left[3.0,3.0,3.0,3.0\right]^T$. The dashed line represents the desired value of the vehicle accumulation whereas the solid line represents the attained value.
• Figure 4: Performance of the reverse Stackelberg game pricing mechanism. The upper subplot shows the convergence of the government's loss function whereas the lower subplot shows the evolution of the total number of vehicles at each charging station. Here, $\sigma_j\left(x\right)$ corresponds to the total number of vehicles at charging station $\mathcal{M}_j$, dash line corresponds to desired and solid line to attained value of the total number of vehicles.
• Figure 5: Robustness plot for different levels of perturbation magnitude $\alpha$. For every $\alpha$, we sample $w_{k}$ 100 times and report the mean value of the government's loss in the Nash equilibrium when the system optimal pricing policies, $\overline{p}_1=\left[2.75, 1.625, 2.208, 1.0\right]$ and $\overline{p}_2=\left[4.03, 2.8, 3.49, 2.24\right]$ are applied.

Theorems & Definitions (24)

• Definition 1
• Definition 2: $\varepsilon$-Nash equilibrium
• Definition 3: Reverse Stackelberg game
• Definition 4: Stackelberg game
• Theorem 1
• proof
• Proposition 1
• proof
• Definition 5: System Optimal Pricing Policies
• Theorem 2