Determining the Impact of Personal Mobility Carbon Allowance Schemes in Transportation Networks

Abstract

Personal mobility carbon allowance (PMCA) schemes are designed to reduce carbon consumption from transportation networks. PMCA schemes influence the travel decision process of users and accordingly impact the system metrics including travel time and greenhouse gas (GHG) emissions. We develop a multi-user class dynamic user equilibrium model to evaluate the transportation system performance when PMCA scheme is implemented. The results using Sioux-Falls test network indicate that PMCA schemes can achieve the emissions reduction goals for transportation networks. Further, users characterized by high value of travel time are found to be less sensitive to carbon budget in the context of work trips. Results also show that PMCA scheme can lead to higher emissions for a path compared with the case without PMCA because of flow redistribution. The developed network equilibrium model allows to examine the change in system states at different carbon allocation levels and to design parameters of PMCA schemes accounting for population heterogeneity.

Appendices

Appendix A: Appendix Cell transmission model

The CTM divides each link of the network into finite number of homogeneous cells. Each cell has a length at least equal to the distance traveled in a single time interval at free flow speed. Also, the time horizon is finite and discretized into a number of intervals.

• (a) Notations and symbols:

Indices:

• w: index of origin-destination pairs, $w \in \mathbb {W}:\{1, \dots , W\}$;

• p: index of paths, $p \in \mathbb {P}:\{1,\dots , P\}$;

• i, j, k: index of cells;

• t: index of discrete time intervals,

• m: index of user group, $m \in \mathbb {M}:\{1,\dots , M\}$.

Parameters:

• ${\alpha _{1}^{m}}$: unit cost of travel time for user class m;

• ${\alpha _{2}^{m}}$: unit cost of early arrival for user class m;

• ${\alpha _{3}^{m}}$: unit cost of late arrival for user class m; where ${\alpha _{2}^{m}}<{\alpha _{1}^{m}}<{\alpha _{3}^{m}}$;

• t ^∗w: preferred arrival time for O-D pair w (this is same for all user classes); ${\beta _{T}^{m}}$: Estimated parameter for travel time cost in the generalized cost function for class m;

• ${\beta _{C}^{m}}$: Estimated parameter for carbon cost in the generalized cost function for class m;

• ${\beta _{R}^{m}}$: Estimated parameter for the effect of remaining carbon budget for class m;

• ${\beta _{H}^{m}}$: Estimated parameter for the effect of available money to spend in the carbon market for class m;

• d ^{w, m}: total demand from O-D pair w for class m;

• ς: infinitesimal flow to avoid zero denominator;

• t _e : maximum departure time (loading time);

• t _f : maximum time horizon (network clearance time);

• N ⁱ: jam density of cell i;

• Q ⁱ: flow capacity out of cell i;

• δ: ratio of backward to forward shockwave propagation.

Sets:

• C: set of cells;

• C _O : set of ordinary cells;

• C _R : set of source cells;

• C _S : set of sink cells;

• C _D : set of diverging cells;

• C _M : set of merging cells;

• E: set of links or cell-connectors;

• E _O : set of ordinary links;

• E _D : set of diverging links;

• E _M : set of merging links;

• ${\Gamma }^{-1}_{i}$: set of predecessors of cell i;

• Γ _i : set of successors of cell i;

• M: set of all user classes;

• W: set of all O-D pairs;

• P ^w: set of paths for O-D pair w;

• P: set of all the paths, P = ∪ _{w ∈ W} P ^w,

• T _e : set of all departure time intervals, $T \triangleq \{0,\cdots ,t_{e}\}$

• T _f : set of all time intervals, $T_{f} \triangleq \{0,\cdots ,t_{f}\}$.

Variables:

• $x^{i,m}_{p,t}$: cell occupancy of cell i at time t for the flow on path p ∋ i for user class m,

• $y^{i,j,m}_{p,t}$: flow from cell i to cell j at time t for the flow on path p ∋ (i, j) for user class m,

• $\bar {x}^{i}_{t}$: aggregate cell occupancy of cell i at time t, i.e., $\bar {x}^{i}_{t} =\\ \hspace *{45pt}\displaystyle {\sum\limits_{\forall p \ni i}, m \in M} x^{i, m}_{p,t}, \, \forall \, i \in C; t \in T_{f}$,

• $\bar {y}^{i, j, m}_{t}$: aggregate flow from cell i to j at time t, i.e., $\bar {y}^{i,j}_{t} =\\ \hspace *{45pt}\displaystyle {\sum\limits_{\forall p \in P}, m} y^{i,j, m}_{p,t}, \,\forall \, (i,j) \in E; t \in T_{f}$,

• $\tilde {x}^{i,j}_{t}$: aggregate cell occupancy at diverging cell i at time t proceeding to cell j,

• $r_{p,t}^{m}$: departure rate at time t for the flow using path p for user class m,

• T T _{p, t} : travel time for the flow using path p at time t,

• $\tau _{p,t,t^{\prime }}$: auxiliary variable for maximum travel time estimation,

• $\hat {\tau }_{p,t,t^{\prime }}$: auxiliary variable for maximum travel time estimation: $\hat {\tau }_{p,t,t^{\prime }} = 1 - \tau _{p,t,t^{\prime }}$,

• ${\theta }_{p,t,t^{\prime }}$: auxiliary variable for maximum travel time estimation,

• ${\nu }_{p,t,t^{\prime }}$: auxiliary variable for average travel time estimation.

An ordered pair of cells (i, j) represents a link and an ordered collection of links or an ordered collection of cells represents a path (similar to (Ukkusuri et al. 2012)). Also, a cell i ∈ p implies that path p must contain cell i and a link (i, j) ∈ p implies that path p must go through link (i, j).

• (b) Initialization: At the beginning of the simulation (t = 0), the cell occupancies are set to zero for all paths and for all user classes in the network.

  $$\begin{array}{@{}rcl@{}} &&x^{i, m}_{p,0} = 0, \quad \quad \forall i \in C, p \in P, m \in M; \end{array} $$

  (20)
  $$\begin{array}{@{}rcl@{}} &&y^{i,j,m}_{p,0} = 0, \quad \quad \forall (i,j) \in E, p \in P, m \in M. \end{array} $$

  (21)

Source cells (C _R ):

During network loading vehicles get into the source cells according to the demand pattern $r_{p,t}^{m}$. Based on the capacity (in and outflow rate), vehicles move to the next cell. For each path p ∈ P containing source cells i ∈ C _R , the occupancy updates can be expressed as:

$$\begin{array}{@{}rcl@{}} x^{i, m}_{p,t} &=& r^{m}_{p,t-1} + x^{i, m}_{p,t-1} - y^{i, j, m}_{p, t-1}, \quad \forall j \in {\Gamma}_{i}, t = \{1,\dots, t_{e} + 1 \}, m \in M, \end{array} $$

(22)

$$\begin{array}{@{}rcl@{}} x^{i, m}_{p,t} &=& x^{i, m}_{p, t-1} - y^{i, j, m}_{p,t-1}, \quad \forall j \in {\Gamma}_{i}, t=\{t_{e}+2,\dots, t_{f}\}, m \in M. \end{array} $$

(23)

Now, each O-D pair has unique demand values for each user class m ∈ M of the network. The cumulative departure rate should be equal to the total demand for a OD pair.

$$\begin{array}{@{}rcl@{}} \sum\limits_{p \in P^{w}}\sum\limits_{t=0}^{t_{e}} r_{p,t}^{m} &=& {d_{w}^{m}}, \quad \forall w \in W, m \in M; \end{array} $$

(24)

$$\begin{array}{@{}rcl@{}} \sum\limits_{m \in M}{d_{w}^{m}} &=& d_{w}, \quad \forall w \in W. \end{array} $$

(25)

Ordinary cells (C _O ): An Ordinary cell, i ∈ C _O has one incoming link and one outgoing link. The following equation updates the cell occupancy of an ordinary cell for a user class m ∈ M:

$$ x^{i, m}_{p,t} = x^{i, m}_{p,t-1} + y^{k, i, m}_{p,t-1} - y^{i,j, m}_{p,t-1}, \quad \forall p \ni i, k \in {\Gamma}^{-1}_{i}, j \in {\Gamma}_{i}, t=\{1,\dots, t_{f}\}. $$

(26)

Also, if i not part of path p, then $x^{i, m}_{p,t} = 0$. Diverging-merging cells (C _D ): The occupancy of any diverging and merging cell, i ∈ C _D ∪C _M for a user class, $m \in \mathbb {M}$ is updated updated as follows:

$$ x^{i, m}_{p,t} = x^{i, m}_{p,t-1} + \sum\limits_{k \in {\Gamma}^{-1}_{i}} y^{k,i, m}_{p,t-1} - \sum\limits_{j \in {\Gamma}_{i}}y^{i,j, m} _{p,t-1}, \quad \forall p \ni k,i,j; k \in {\Gamma}^{-1}_{i}; j \in {\Gamma}_{i}; t=\{1,\dots, t_{f}\}. $$

(27)

Sink cells (C _S ): A sink cell i ∈ C _S has limited in-flow capacity Q ^s, however the storage capacity is unlimited (N ^s→∞). The cell occupancy $x^{i, m}_{p,t}$ equals with the cumulative arrivals of path p in the period from 0 to t for user class $m \in \mathbb {M}$.

$$ x^{i, m}_{p,t} = x^{i, m}_{p,t-1} + y^{k,i, m}_{p,t-1}, \quad \forall i \in C_{S}; p \ni i; k \in {\Gamma}^{-1}_{i}; t=\{1,\dots, t_{f}\}. $$

(28)

Ordinary links (E _O ): At the aggregate flow level, we have:

$$ \bar{y}^{i,j}_{t} = \min\left( \bar{x}^{i}_{t}, Q^{i}, Q^{j}, \delta(N^{j}-\bar{x}^{j}_{t}) \right) \ \ \ \forall (i,j) \in E_{O}; t=\{1,\cdots, t_{f}\} $$

(29)

At the disaggregate level, we use the proportion of path-based cell occupancy $x^{i,m}_{p,t}$ and aggregate cell occupancy $\bar {x}^{i}_{t}$ to determine the path flow $y^{i,j,m}_{p,t}$

$$ y^{i,j,m}_{p,t} = \min\left( \bar{x}^{i}_{t}, Q^{i}, Q^{j}, \delta(N^{j}-\bar{x}^{j}_{t})\right) \times \frac{x^{i,m}_{p,t}} {\bar{x}^{i}_{t} + \sigma} \ \ \ \ \ \forall (i,j) \in E_{O}; p \ni i; j \in {\Gamma}_{i}; t=\{1,\cdots, t_{f}\} $$

(30)

Here ς > 0 is an infinitesimal number used to make sure that the denominator is different from 0.

Diverging links (E _D ): The updates are similar to Ukkusuri et al. (2012) and aggregate occupancies are computed as follows:

$$ \tilde{x}^{i,j}_{t} = \displaystyle{ \sum\limits_{\forall p \ni (i,j), m \in M} x^{i,m}_{p,t}} \ \ \ \ \ \forall \, i \in C_{D}; j \in {\Gamma}_{i}; t=\{1,\cdots, t_{f}\} $$

(31)

$$ \bar{x}^{i}_{t} = \sum\limits_{j \in {\Gamma}_{i}} \tilde{x}^{i,j}_{t} \ \ \ \ \ \forall i \in C_{D}; t \in \{1,\cdots,t_{f}\}, $$

(32)

For the diverging flows: For all i ∈ C _D ;j ∈ Γ _i ;t = {1,⋯ ,t _f },

$$ \bar{y}^{i,j}_{t} = \min(\tilde{x}^{i,j}_{t}, Q^{j}, \delta(N^{j}-\bar{x}^{j}_{t})) \times \min \left( 1, \frac{Q^{i}}{{\sum}_{j^{\prime} \in {\Gamma}_{i}} \left( \min(\tilde{x}^{i,j^{\prime}}_{t}, Q^{j^{\prime}}, \delta(N^{j^{\prime}}-\bar{x}^{j^{\prime}}_{t}))\right)+\mu}\right) $$

(33)

(Ukkusuri et al. 2012) use the proportional rule to obtain the flow for each path from each O-D pair:

$$ y^{i,j, m}_{p,t} = \bar{y}^{i,j}_{t} \times \frac{x^{i,m}_{p,t}} {\tilde{x}^{i,j}_{t}+\sigma} \ \ \ \ \ \forall \, i \in C_{D}; p \ni i; j \in {\Gamma}_{i}; t \in \{1,\cdots,t_{f}\} $$

(34)

Merging links (E _D ): The updating for merging links are computed by the following equations:

For all $i \in C_{M}; k \in {\Gamma }^{-1}_{i}; t=\{1,\cdots , t_{f}\}$,

$$ \bar{y}^{k,i}_{t} = \min(Q^{k},\bar{x}^{k}_{t}) \times \min\left( 1,\frac{\min{\left( Q^{i}, \delta(N^{i} - \bar{x}^{i}_{t})\right)}}{{\sum}_{k^{\prime} \in {\Gamma}^{-1}_{i}} \left( \min(Q^{k^{\prime}},\bar{x}^{k^{\prime}}_{t})\right)+\sigma}\right). $$

(35)

The flow for each path of each O-D pair can be computed as:

$$ y^{k,i,m}_{p,t} = \bar{y}^{k,i}_{t} \times \frac{x^{k,m}_{p,t}} {\bar{x}^{k}_{t}+\sigma} \ \ \ \ \ \forall \, i \in C_{M}; p \ni i; k \in {\Gamma}^{-1}_{i}; t \in \{1,\cdots,t_{f}\} $$

(36)

Appendix B: Travel time estimation using CTM

Ramadurai (2009) and (Han et al. 2011) provide details on computing average travel time within the path-based CTM model. For the completeness of our discussion we mention the equations from Ukkusuri et al. (2012).

$$\begin{array}{@{}rcl@{}} \nu_{p,t,t^{\prime}} &=& \max \left( 0, \, \displaystyle{\sum\limits_{h = 0}^{t}} r_{p,h} - x_{p,t^{\prime}}^{s} \right), \\ && \forall p \in P; s \in p \cap C_{S}; t = \{0, \cdots, t_{e}\}; t^{\prime} =\{ t,\cdots,t_{f}\} \end{array} $$

(37)

$$\begin{array}{@{}rcl@{}} TT_{p,0} &=& \frac{\displaystyle{\sum\limits_{h = 0}^{T_{f}- 1}} (\nu_{p,0,h} - \nu_{p,0,h+1})h}{\displaystyle{r_{p,0}+\mu}}, \forall p \in P \end{array} $$

(38)

$$\begin{array}{@{}rcl@{}} TT_{p,t} &=& \frac{\displaystyle{\sum\limits_{h = t}^{T_{f} - 1}} (\nu_{p,t,h} - \nu_{p,t,h+1} + \nu_{p,t-1,h+1} - \nu_{p,t-1,h})(h-t)} {\displaystyle {r_{p,t} + \mu}},\\ && \forall p \in P; s \in p \cap C_{S}; t = 0, \cdots, T . \end{array} $$

(39)

Users from all classes experience the same travel time. Therefore, the average travel time computation is not specific for a class.

$$\begin{array}{@{}rcl@{}} r_{p,h} &=& \sum\limits_{m \in M} r_{p,t}^{m}, \end{array} $$

(40)

$$\begin{array}{@{}rcl@{}} x_{p, t^{\prime}}^{s} &=& \sum\limits_{m \in M}x_{p,t}^{s,m}. \end{array} $$

(41)

Now, the max operator can be replaced by the following complementarity conditions:

$$\begin{array}{@{}rcl@{}} && 0 \, \leq \, \nu_{p,t,t^{\prime}} \, \perp \, \nu_{p,t,t^{\prime}} - \left( \displaystyle{\sum\limits_{h = 0}^{t}} r_{p,h} - x_{p,t^{\prime}}^{s}\right) \geq 0 \\ && \forall p \in P; t = \{0, \cdots, t_{e}\}; t^{\prime} = \{t,\cdots,t_{f}\}. \end{array} $$

(42)

Algorithm step (Zhan and Ukkusuri 2014)

• Step 0: Set counter k = 0. Initialize a feasible departure rate.

• Step 1: Run the simulation C T M(γ ^k) and compute T C(γ ^k), C C(γ ^k), G C(γ ^k).

• Step 2: Decompose $\gamma ^{k}=({\gamma ^{k}_{1}}, {\gamma ^{k}_{2}}, ..., {\gamma ^{k}_{m}})^{T}$.

• Step 3: Find the λ ^k(γ ^k) that ensures most number of user classes and OD pairs satisfy: $||T_{i}({\gamma ^{k}_{i}})-T_{i}(\gamma ^{k-1}_{i})||\leq \alpha ||{\gamma ^{k}_{i}}- \gamma ^{k-1}_{i})||$, in which $T_{i}({\gamma ^{k}_{i}})=Pr_{{\Omega }_{i}}[{\gamma ^{k}_{i}}-\lambda ^{k} (\gamma ^{k}) {GC}_{i}(\gamma ^{k})]$

• Step 4: Update departure rate ${\gamma ^{k}_{i}}$ using mapping T _i only for those user classes and OD pairs that satisfy the condition in Step 3. For others that not satisfied, set ${\gamma ^{k}_{j}}= \gamma ^{k-1}_{j}$.

• Step 5: If none of the user classes and OD pairs are found satisfy condition in Step 3, set ${\gamma ^{k}_{j}}= \gamma ^{k-1}_{j}, \lambda ^{k}(\gamma ^{k})=\lambda ^{k-1}(\gamma ^{k-1})/2$.

• Step 6: If ||z ^∗−γ ^k|| ≤ 𝜖, terminate the algorithm, γ ^∗ = z ^∗. Otherwise γ ^k+1 = γ ^k, Set k = k + 1, go to step 1.

Appendix C: DUE Results

1.1 D.1. DUE-1 with base case: Sioux-Falls

Fig. 17

PMCA-DUE-1 results base case: no carbon cap

1.2 D.2. DUE-2 with 2 %reduction: Sioux-Falls

Fig. 18

PMCA-DUE-2 results: 2 % reduction from base case

1.3 D.3. DUE-3 with 2 % reduction: Sioux-Falls

Fig. 19

PMCA-DUE-3 results: 2 % reduction from base case