A parking pricing scheme considering parking dynamics

Abstract

Powerful parking management approaches are vital in metropolitan areas where parking resource is limited and congestion is severe. To date, parking pricing is considered as an efficient tool for improving parking management by moderating parking demand. Hence, this paper proposes a parking charging method involving both the parking demand and supply. We aggregate the individual parking behaviors into the parking demand and consider the trade-offs between multiple parking behaviors. Then, a maximization problem is formulated to fulfill the benefits of parking demand as well as stabilizing parking supply. The results of case study indicate that the share of parking patterns has shifted positively, and an increment in the total benefit of respondents (5.00%) is obtained. Considering parking dynamics, the proposed method can serve as an effective and feasible manner to equilibrate spatial and temporal distribution of parking demand thereby alleviating parking pressure. This study also makes a methodological contribution to parking pricing policy design.

Appendix

See Tables 8 and 9

Table 8 Estimation results of the utility function

Table 9 Nomenclature

• (1) The calculation of parking facility cost

In this paper, the on-street parking facility cost is expressed in (19), which is comprised of the land price range, the road-usage cost and the corresponding taxes.

$${\sum }_{i}\left[{{C}_{i}}^{\mathrm{consistent}}+{{\int }_{0}^{T}\left(1+s\right)}^{-t}{C}_{i}(t)dt\right]={\sum }_{i}\left[{{\int }_{0}^{T}\left(1+s\right)}^{-t}{{C}_{i}}^{\mathrm{on}}(t)dt\right]={\sum }_{i}\left[{C}_{\mathrm{ran},i}^{\mathrm{on}}\sigma {N}_{i}^{\mathrm{on}}+{{\int }_{0}^{T}\left(1+s\right)}^{-t}({C}_{\mathrm{roa},i}^{\mathrm{on}}(t){N}_{i}^{\mathrm{on}}\times \text{365} + {C}_{\mathrm{tax},i}^{\mathrm{on}}(t){N}_{i}^{\mathrm{on}}\sigma )dt\right](i\hspace{0.17em}=\hspace{0.17em}3, 7)$$

(19)

The off-street parking facility cost in this paper includes the annual expenditures, social cost and other cost like cost of parking enforcement.

$${\sum }_{i}\left[{{C}_{i}}^{\mathrm{consistent}}+{{\int }_{0}^{T}\left(1+s\right)}^{-t}{C}_{i}(t)dt\right]={\sum }_{i}\left[{{\int }_{0}^{T}\left(1+s\right)}^{-t}{{C}_{i}}^{\mathrm{off}}(t)dt\right]={\sum }_{i}\left[{{\int }_{0}^{T}\left(1+s\right)}^{-t}{C}_{\mathrm{spo},i}^{\mathrm{off}}(t){N}_{i}^{\mathrm{off}}dt\right](i\hspace{0.17em}=\hspace{0.17em}1, 5) $$

(20)

where the time distribution function of each off-street parking facility cost ${C}_{\mathrm{spo},i}^{\mathrm{off}}(t)$ can be expressed as follows:

$${C}_{\mathrm{spo},i}^{\mathrm{off}}(t)=\frac{{C}_{\mathrm{exp},i}^{\mathrm{off}}(t)+{C}_{\mathrm{exp},i}^{\mathrm{off}}(t)\bullet s+{C}_{\mathrm{soc},i}^{\mathrm{off}}(t)}{{N}_{i}^{\mathrm{off}}}(i\hspace{0.17em}=\hspace{0.17em}1, 5)$$

(21)

• (2) The calculation of average parking rate per trip in the parking pattern i (R_i):

  $${R}_{1}=\frac{{r}_{1}^{\mathrm{fh}}}{{H}_{1}}+\frac{{H}_{1}-1}{{H}_{1}}{r}_{1}^{\mathrm{nfh}}$$

  (22)
  $${R}_{2}=\frac{{\delta }_{\mathrm{P}/\mathrm{OP}}}{{H}_{2}}{r}_{1}^{\mathrm{fh}}+\frac{{H}_{2}-1}{{H}_{2}}{\delta }_{\mathrm{P}/\mathrm{OP}}{r}_{1}^{\mathrm{nfh}}$$

  (23)
  $${R}_{3}=\frac{{\delta }_{\mathrm{S}/\mathrm{OS}}}{{H}_{3}}{r}_{1}^{\mathrm{fh}}+\frac{{H}_{3}-1}{{H}_{3}}{\delta }_{\mathrm{S}/\mathrm{OS}}{r}_{1}^{\mathrm{nfh}}$$

  (24)
  $${R}_{4}=\frac{{\delta }_{\mathrm{S}/\mathrm{OS}}{\delta }_{\mathrm{P}/\mathrm{OP}}}{{H}_{4}}{r}_{1}^{\mathrm{fh}}+\frac{{H}_{4}-1}{{H}_{4}}{\delta }_{\mathrm{S}/\mathrm{OS}}{\delta }_{\mathrm{P}/\mathrm{OP}}{r}_{1}^{\mathrm{nfh}}$$

  (25)
  $${R}_{5}=\frac{{\delta }_{\mathrm{C}/\mathrm{NC}}}{{H}_{5}}{r}_{1}^{\mathrm{fh}}+\frac{{H}_{5}-1}{{H}_{5}}{\delta }_{\mathrm{C}/\mathrm{NC}}{r}_{1}^{\mathrm{nfh}}$$

  (26)
  $${R}_{6}=\frac{{\delta }_{\mathrm{C}/\mathrm{NC}}{\delta }_{\mathrm{P}/\mathrm{OP}}}{{H}_{6}}{r}_{1}^{\mathrm{fh}}+\frac{{H}_{6}-1}{{H}_{6}}{\delta }_{\mathrm{C}/\mathrm{NC}}{\delta }_{\mathrm{P}/\mathrm{OP}}{r}_{1}^{\mathrm{nfh}}$$

  (27)
  $${R}_{7}=\frac{{\delta }_{\mathrm{C}/\mathrm{NC}}{\delta }_{\mathrm{S}/\mathrm{OS}}}{{H}_{7}}{r}_{1}^{\mathrm{fh}}+\frac{{H}_{7}-1}{{H}_{7}}{\delta }_{\mathrm{C}/\mathrm{NC}}{\delta }_{\mathrm{S}/\mathrm{OS}}{r}_{1}^{\mathrm{nfh}}$$

  (28)
  $$ R_{8} = \frac{{\delta_{{{\text{C}}/{\text{NC}}}} \delta_{{{\text{S}}/{\text{OS}}}} \delta_{{{\text{P}}/{\text{OP}}}} }}{{H_{8} }}r_{1}^{{{\text{fh}}}} + \frac{{H_{8} - 1}}{{H_{8} }}\delta_{{{\text{C}}/{\text{NC}}}} \delta_{{{\text{S}}/{\text{OS}}}} \delta_{{{\text{P}}/{\text{OP}}}} r_{1}^{{{\text{nfh}}}} $$

  (29)

(3) The calculation of (17)

$$ \begin{gathered} \frac{\partial L}{{\partial r_{1}^{{{\text{fh}}}} }} = \frac{1}{{e^{{V_{2} }} + \cdots + e^{{V_{8} }} }}\left( \begin{gathered} \frac{{\delta_{{\text{P/OP}}} }}{{H_{2} }}\gamma_{2,15} e^{{V_{2} }} + \frac{{\delta_{{\text{S/OS}}} }}{{H_{3} }}\gamma_{3,15} e^{{V_{3} }} + \frac{{\delta_{{\text{S/OS}}} \delta_{{\text{P/OP}}} }}{{H_{4} }}\gamma_{4,15} e^{{V_{4} }} + \frac{{\delta_{{\text{C/NC}}} }}{{H_{5} }}\gamma_{5,15} e^{{V_{5} }} \hfill \\ + \frac{{\delta_{{\text{C/NC}}} \delta_{{\text{P/OP}}} }}{{H_{6} }}\gamma_{6,15} e^{{V_{6} }} + \frac{{\delta_{{\text{C/NC}}} \delta_{{\text{S/OS}}} }}{{H_{7} }}\gamma_{7,15} e^{{V_{7} }} + \frac{{\delta_{{\text{C/NC}}} \delta_{{\text{S/OS}}} \delta_{{\text{P/OP}}} }}{{H_{8} }}\gamma_{8,15} e^{{V_{8} }} \hfill \\ \end{gathered} \right) \hfill \\ \mathop \int \limits_{0}^{T} (1 + s)^{ - t} Q_{1} \left( t \right)dt - \frac{{\lambda_{1} }}{{H_{1} }}\mathop \int \limits_{0}^{T} (1 + s)^{ - t} Q_{1} \left( t \right)dt + \mathop \sum \limits_{i = 2}^{8} \left[ {\left( {\gamma_{i,15} - \lambda_{1} } \right)} \right. \times \left. {\frac{{\partial R_{i} }}{{\partial r_{1}^{{{\text{fh}}}} }}\mathop \int \limits_{0}^{T} (1 + s)^{ - t} Q_{i} \left( t \right)dt} \right] + \lambda_{2} - \lambda_{3} + \lambda_{6} = 0 \hfill \\ \end{gathered} $$

(30)

$$ \begin{gathered} \frac{\partial L}{{\partial r_{1}^{{{\text{nfh}}}} }} = \frac{1}{{e^{{V_{2} }} + \cdots + e^{{V_{8} }} }}\left( \begin{gathered} \frac{{H_{2} - 1}}{{H_{2} }}\delta_{{\text{P/OP}}} \gamma_{2,15} e^{{V_{2} }} + \frac{{H_{3} - 1}}{{H_{3} }}\delta_{{\text{S/OS}}} \gamma_{3,15} e^{{V_{3} }} + \frac{{H_{4} - 1}}{{H_{4} }}\delta_{{\text{S/OS}}} \delta_{{\text{P/OP}}} \gamma_{4,15} e^{{V_{4} }} + \hfill \\ \frac{{H_{5} - 1}}{{H_{5} }}\delta_{{\text{C/NC}}} \gamma_{5,15} e^{{V_{5} }} + \frac{{H_{6} - 1}}{{H_{6} }}\delta_{{\text{C/NC}}} \delta_{{\text{P/OP}}} \gamma_{6,15} e^{{V_{6} }} + \frac{{H_{7} - 1}}{{H_{7} }}\delta_{{\text{C/NC}}} \delta_{{\text{S/OS}}} \gamma_{7,15} e^{{V_{7} }} + \hfill \\ \frac{{H_{8} - 1}}{{H_{8} }}\delta_{{\text{C/NC}}} \delta_{{\text{S/OS}}} \delta_{{\text{P/OP}}} \gamma_{8,15} e^{{V_{8} }} \hfill \\ \end{gathered} \right) \hfill \\ \mathop \int \limits_{0}^{T} (1 + s)^{ - t} Q_{1} \left( t \right)dt - \frac{{H_{1} - 1}}{{H_{1} }}\lambda_{1} \times \mathop \int \limits_{0}^{T} (1 + s)^{ - t} Q_{1} \left( t \right)dt + \mathop \sum \limits_{i = 2}^{8} \left[ {\left( {\gamma_{i,15} - \lambda_{1} } \right)\frac{{\partial R_{i} }}{{\partial r_{1}^{{{\text{nfh}}}} }}\mathop \int \limits_{0}^{T} (1 + s)^{ - t} Q_{i} \left( t \right)dt} \right] \hfill \\ + \lambda_{4} - \lambda_{5} - \lambda_{6} = 0 + \lambda_{4} - \lambda_{5} - \lambda_{6} = 0 \hfill \\ \end{gathered} $$

(31)

$$ \begin{gathered} \frac{\partial L}{{\partial \delta_{{\text{C/NC}}} }} = \left( {\gamma_{5,15} - \lambda_{1} } \right)\left( {\frac{1}{{H_{5} }}r_{1}^{fh} + \frac{{H_{5} - 1}}{{H_{5} }}r_{1}^{nfh} } \right)\mathop \smallint \limits_{0}^{T} (1 + s)^{ - t} Q_{5} \left( t \right)dt + \left( {\gamma_{6,15} - \lambda_{1} } \right)\left( {\frac{{\delta_{{\text{P/OP}}} }}{{H_{6} }}r_{{1}}^{{{\text{fh}}}} + \frac{{H_{6} - 1}}{{H_{6} }}\delta_{{\text{P/OP}}} r_{{1}}^{{{\text{nfh}}}} } \right) \hfill \\ \mathop \int \limits_{0}^{T} (1 + s)^{ - t} Q_{6} \left( t \right)dt + \left( {\gamma_{7,15} - \lambda_{1} } \right)\left( {\frac{{\delta_{S/OS} }}{{H_{7} }}r_{1}^{fh} + \frac{{H_{7} - 1}}{{H_{7} }}\delta_{{\text{S/OS}}} r_{{1}}^{{{\text{nfh}}}} } \right)\mathop \int \limits_{0}^{T} (1 + s)^{ - t} Q_{7} \left( t \right)dt \hfill \\ + \left( {\gamma_{8 - 15} - \lambda_{1} } \right)\left( {\frac{{\delta_{{\text{S/OS}}} \delta_{{\text{P/OP}}} }}{{H_{8} }}r_{1}^{fh} + \frac{{H_{8} - 1}}{{H_{8} }}\delta_{{\text{S/OS}}} \delta_{{\text{P/OP}}} r_{{1}}^{{{\text{nfh}}}} } \right) \times \mathop \int \limits_{0}^{T} (1 + s)^{ - t} Q_{8} \left( t \right)dt + \lambda_{7} - \lambda_{8} = 0 \hfill \\ \end{gathered} $$

(32)

$$ \begin{gathered} \frac{\partial L}{{\partial \delta_{{\text{S/OS}}} }} = \left( {\gamma_{3,15} - \lambda_{1} } \right)\left( {\frac{1}{{H_{3} }}r_{{1}}^{{{\text{fh}}}} + \frac{{H_{3} - 1}}{{H_{3} }}r_{1}^{{{\text{nfh}}}} } \right)\mathop \int \limits_{0}^{T} (1 + s)^{ - t} Q_{3} \left( t \right)dt + \left( {\gamma_{4,15} - \lambda_{1} } \right)(\frac{{\delta_{{\text{P/OP}}} }}{{H_{4} }}r_{1}^{{{\text{fh}}}} + \hfill \\ \frac{{H_{4} - 1}}{{H_{4} }}\delta_{{\text{P/OP}}} r_{1}^{{{\text{nfh}}}} )\mathop \smallint \limits_{0}^{T} (1 + s)^{ - t} Q_{4} \left( t \right)dt + \left( {\gamma_{7,15} - \lambda_{1} } \right)\left( {\frac{{\delta_{{\text{C/NC}}} }}{{H_{7} }}r_{1}^{{{\text{fh}}}} + \frac{{H_{7} - 1}}{{H_{7} }}\delta_{{\text{C/NC}}} r_{1}^{{{\text{nfh}}}} } \right)\mathop \int \limits_{0}^{T} (1 + s)^{ - t} Q_{7} \left( t \right)dt + \hfill \\ \left( {\gamma_{8,15} - \lambda_{1} } \right) \times \left( {\frac{{\delta_{{\text{C/NC}}} \delta_{{\text{P/OP}}} }}{{H_{8} }}r_{1}^{fh} + \frac{{H_{8} - 1}}{{H_{8} }}\delta_{{\text{C/NC}}} \delta_{{\text{P/OP}}} r_{1}^{{{\text{nfh}}}} } \right) \times \mathop \int \limits_{0}^{T} (1 + s)^{ - t} Q_{8} \left( t \right)dt + \lambda_{9} - \lambda_{10} = 0 \hfill \\ \end{gathered} $$

(33)

$$ \begin{gathered} \frac{\partial L}{{\partial \delta_{{\text{P/OP}}} }} = \left( {\gamma_{2,15} - \lambda_{1} } \right)\left( {\frac{1}{{H_{2} }}r_{1}^{{{\text{fh}}}} + \frac{{H_{2} - 1}}{{H_{2} }}r_{1}^{{{\text{nfh}}}} } \right)\mathop \int \limits_{0}^{T} (1 + s)^{ - t} Q_{2} \left( t \right)dt + \left( {\gamma_{4,15} - \lambda_{1} } \right)(\frac{{\delta_{{\text{S/OS}}} }}{{H_{4} }}r_{{1}}^{{{\text{fh}}}} + \hfill \\ \frac{{H_{4} - 1}}{{H_{4} }}\delta_{{\text{S/OS}}} r_{1}^{{{\text{nfh}}}} )\mathop \int \limits_{0}^{T} (1 + s)^{ - t} Q_{4} \left( t \right)dt + \left( {\gamma_{6,15} - \lambda_{1} } \right)\left( {\frac{{\delta_{{\text{C/NC}}} }}{{H_{6} }}r_{1}^{{{\text{fh}}}} + \frac{{H_{6} - 1}}{{H_{6} }}\delta_{{\text{C/NC}}} r_{1}^{{{\text{nfh}}}} } \right)\mathop \int \limits_{0}^{T} (1 + s)^{ - t} Q_{6} \left( t \right)dt \hfill \\ \left( {\gamma_{8,15} - \lambda_{1} } \right)\left( {\frac{{\delta_{{\text{C/NC}}} \delta_{{\text{S/OS}}} }}{{H_{8} }}r_{1}^{{{\text{fh}}}} + \frac{{H_{8} - 1}}{{H_{8} }}\delta_{{\text{C/NC}}} \delta_{{\text{S/OS}}} r_{1}^{{{\text{nfh}}}} } \right) \times \mathop \int \limits_{0}^{T} (1 + s)^{ - t} Q_{8} \left( t \right)dt + \lambda_{11} - \lambda_{12} = 0 \hfill \\ \end{gathered} $$

(34)

$$\frac{\partial L}{\partial {\lambda }_{1}}={\sum }_{{\mu }_{\mathrm{off}}=1}^{2}\left[{\int }_{0}^{T}(1+s{)}^{-t}{C}_{\mathrm{spo},{\mu }_{\mathrm{off}}}^{\mathrm{off}}(t){N}_{{\mu }_{\mathrm{off}}}^{\mathrm{off}}dt\right]+{\sum }_{{\mu }_{\mathrm{on}}=1}^{2}{C}_{\mathrm{ran},{\mu }_{\mathrm{on}}}^{\mathrm{on}}\sigma {N}_{{\mu }_{\mathrm{on}}}^{\mathrm{on}}+{\sum }_{{\mu }_{\mathrm{on}}=1}^{2}\left[{\int }_{0}^{T}(1+s{)}^{-t}\times \right.\left.({C}_{\mathrm{roa},{\mu }_{\mathrm{on}}}^{\mathrm{on}}(t){N}_{{\mu }_{\mathrm{on}}}^{\mathrm{on}}\times 365+{C}_{\mathrm{tax},{\mu }_{\mathrm{on}}}^{\mathrm{on}}(t)\sigma {N}_{{\mu }_{\mathrm{on}}}^{\mathrm{on}})dt\right]-{\sum }_{i=1}^{n}{\int }_{0}^{T}(1+s{)}^{-t}{R}_{i}{Q}_{i}(t)dt+{\psi }_{1}^{2}=0$$

(35)

$$\frac{\partial L}{\partial {\lambda }_{2}}={r}_{1}^{\mathrm{fh}}-{y}_{1}^{^{\prime}}+{\psi }_{2}^{2}=0$$

(36)

$$\frac{\partial L}{\partial {\lambda }_{3}}={y}_{1}-{r}_{1}^{\mathrm{fh}}+{\psi }_{3}^{2}=0$$

(37)

$$\frac{\partial L}{\partial {\lambda }_{4}}={r}_{1}^{\mathrm{nfh}}-{z}_{1}^{^{\prime}}+{\psi }_{4}^{2}=0$$

(38)

$$\frac{\partial L}{\partial {\lambda }_{5}}={z}_{1}-{r}_{1}^{\mathrm{nfh}}+{\psi }_{5}^{2}=0$$

(39)

$$\frac{\partial L}{\partial {\lambda }_{6}}={r}_{1}^{\mathrm{fh}}-{r}_{1}^{\mathrm{nfh}}+{\psi }_{6}^{2}=0$$

(40)

$$\frac{\partial L}{\partial {\lambda }_{7}}={\delta }_{\mathrm{C}/\mathrm{NC}}-{x}^{^{\prime}}+{\psi }_{7}^{2}=0$$

(41)

$$\frac{\partial L}{\partial {\lambda }_{8}}=x-{\delta }_{\mathrm{C}/\mathrm{NC}}+{\psi }_{8}^{2}=0$$

(42)

$$\frac{\partial L}{\partial {\lambda }_{9}}={\delta }_{\mathrm{S}/\mathrm{OS}}-{u}^{1}+{\psi }_{9}^{2}=0$$

(43)

$$\frac{\partial L}{\partial {\lambda }_{10}}=u-{\delta }_{\mathrm{S}/\mathrm{OS}}+{\psi }_{10}^{2}=0$$

(44)

$$\frac{\partial L}{\partial {\lambda }_{11}}={\delta }_{\mathrm{P}/\mathrm{OP}}-{w}^{^{\prime}}+{\psi }_{11}^{2}=0$$

(45)

$$\frac{\partial L}{\partial {\lambda }_{12}}=w-{\delta }_{\mathrm{P}/\mathrm{OP}}+{\psi }_{12}^{2}=0$$

(46)

$$\frac{\partial L}{\partial {\psi }_{i}}=2{\lambda }_{i}{\psi }_{i}=0$$

(47)