# Promoting Liquefied Natural Gas (LNG) Bunkering for Maritime Transportation: Should Ports or Ships Be Subsidized?

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Basic Scenario with Homogeneous Ports and Ships

#### 3.1. Model Formulation

- (i)
- $\beta q<\gamma Q$. That is, the fuel cost per unit sailing distance using LNG is lower than using MDO.
- (ii)
- After retrofitting, the LNG tank capacity of a ship is $qL$. Therefore, a ship will refill a full tank of LNG whenever it visits a port with the LNG bunkering infrastructure.
- (iii)
- $N(\gamma QL-\beta qL)>c$, that is, ship owners will retrofit the ships with LNG engines when all of the ports have constructed LNG bunkering infrastructure. Note that this assumption implies Assumption i.
- (iv)
- $M(\beta -\alpha )qL>C$, that is, if all of the ships have been retrofitted, then all ports will have the motivation to construct LNG bunkering infrastructure.

**Lemma**

**1.**

**Lemma**

**2.**

#### 3.2. Mixed Subsidy Plan 1: Fully Subsidize Ports First and Then Partially Subsidize Ships

**Step 1**: The government fully subsidizes n ports to construct LNG bunkering infrastructure, $n=0,\cdots ,{N}^{min}$, i.e., provides a subsidy of C USD to each of the n port.**Step 2**: The government subsidizes ${M}^{min}$ ships by providing a subsidy of x USD to each ship’s owner, $0\le x\le c$, making sure that the ship owner has the motivation to spend $c-x$ USD in retrofitting the ship in view that n ports have already constructed the LNG bunkering infrastructure.**Step 3**: The remaining $N-n$ ports construct LNG bunkering infrastructure by themselves because they will make a profit from it and the remaining $M-{M}^{min}$ ships will be retrofitted with LNG engines by their owners to save fuel costs.

**Proof.**

#### 3.3. Mixed Subsidy Plan 2: Fully Subsidize Ships and Then Partially Subsidize Ports

**Step 1**: The government fully subsidizes m ships to be retrofitted with LNG engines, i.e., provides a subsidy of c USD to each of the m ships. Since the total number of ships M is large, we allow m to be a fractional quantity, i.e., $m\in [0,{M}^{min}]$.**Step 2**: The government provides a subsidy of y USD to each of ${N}^{min}$ ports, $0\le y\le C$, making sure that the port has the motivation to spend $C-y$ USD in constructing LNG bunkering infrastructure, in view that m ships have already been retrofitted with LNG engines.**Step 3**: The remaining $M-m$ ships will be retrofitted with LNG engines by their owners because they will thereby save fuel costs and the remaining $N-n$ ports will construct LNG bunkering infrastructure by themselves because they will make a profit from it.

**Proof.**

#### 3.4. Main Findings in the Basic Scenario

## 4. Scenario with Heterogeneous Ships

- (i)
- After retrofitting, the LNG tank capacity of ship j is ${q}_{j}L,j=1,\cdots ,M$. Therefore, a ship will refill a full tank of LNG whenever it visits a port with LNG bunkering infrastructure.
- (ii)
- $N(\gamma {Q}_{j}L-\beta {q}_{j}L)\ge N(\gamma {Q}_{M}L-\beta {q}_{M}L)>c,j=1,\cdots ,M$, that is, ship owners will retrofit the ships with LNG engines when all of the ports have constructed LNG bunkering infrastructure.
- (iii)
- $M(\beta -\alpha ){\int}_{{q}_{\mathrm{M}}}^{{q}_{0}}x\phantom{\rule{0.166667em}{0ex}}dxL=\left(\beta -\alpha \right)\frac{{q}_{0}^{2}-{q}_{\mathrm{M}}^{2}}{2}L>C,j=1,\cdots ,M$, that is, if all of the ships have been retrofitted, then all ports will have the motivation to construct LNG bunkering infrastructure.

**Lemma**

**3.**

**Lemma**

**4.**

#### 4.1. Mixed Subsidy Plan 1 with Heterogeneous Ships

**Step 1**: The government fully subsidizes n ports to construct LNG bunkering infrastructure, $n=0,\cdots ,{N}^{min}$, i.e., provides a subsidy of C USD to each of the n port.**Step 2**: The government subsidizes ship 1 to ship ${M}^{min}$ by providing a subsidy of x USD to each ship’s owner, $0\le x\le c$, making sure that the ship owner has the motivation to spend $c-x$ USD in retrofitting the ship in view that n ports have already constructed the LNG bunkering infrastructure.**Step 3**: The remaining $N-n$ ports construct LNG bunkering infrastructure by themselves because they will make a profit from it and the remaining $M-{M}^{min}$ ships will be retrofitted with LNG engines by their owners to save fuel costs.

**Proof.**

#### 4.2. Mixed Subsidy Plan 2 with Heterogeneous Ships

**Step 1**: The government fully subsidizes m ships to be retrofitted with LNG engines, i.e., provides a subsidy of c USD to each of the m ships. Since the total number of ships M is large, we allow m to be a fractional quantity, i.e., $m\in [0,{M}^{min}]$.**Step 2**: The government provides a subsidy of y USD to each of ${N}^{min}$ ports, $0\le y\le C$, making sure that the port has the motivation to spend $C-y$ USD in constructing LNG bunkering infrastructure in view that m ships have already been retrofitted with LNG engines.**Step 3**: The remaining $M-m$ ships will be retrofitted with LNG engines by their owners because they will thereby save fuel costs and the remaining $N-n$ ports will construct LNG bunkering infrastructure by themselves because they will make a profit from it.

**Proof.**

#### 4.3. Main Findings in the Scenario with Heterogeneous Ships

## 5. Optimal LNG Selling Price

#### 5.1. Optimal Value of $\beta $

**Proof.**

#### 5.2. Impact of $\alpha $ and $\gamma $ on Optimal $\beta $

#### 5.3. Main Findings in the LNG Selling Price Optimization

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

LNG | Liquefied natural gas |

IMO | International Maritime Organization |

MDO | Marine diesel oil |

MGO | Marine gas oil |

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Condition | ${\mathit{n}}^{\mathbf{*}}$ | ${\mathit{x}}^{\mathbf{*}}$ | Total Subsidy | |
---|---|---|---|---|

Case i | $c{M}^{min}<C\frac{c}{\gamma QL-\beta qL}$, i.e., $C>(\gamma QL-\beta qL){M}^{min}$ | 0 | c | $c{M}^{min}$ |

Case ii | $C\frac{c}{\gamma QL-\beta qL}\le c{M}^{min}<C{N}^{min}$ | ${N}^{min}-1$ | $C(\frac{c}{\gamma QL-\beta qL}-{N}^{min}+1)\frac{1}{{M}^{min}}$ | $C\frac{c}{\gamma QL-\beta qL}$ |

Case iii | $C{N}^{min}\le c{M}^{min}$ | ${N}^{min}$ | 0 | $C{N}^{min}$ |

Condition | ${\mathit{m}}^{\mathbf{*}}$ | ${\mathit{y}}^{\mathbf{*}}$ | Total Subsidy | |
---|---|---|---|---|

Case i | $C{N}^{min}<\frac{cC}{(\beta -\alpha )QL}$, i.e., $c>(\beta -\alpha )QL{N}^{min}$ | 0 | C | $C{N}^{min}$ |

Case ii | $c\le (\beta -\alpha )QL{N}^{min}$ | ${M}^{min}$ | 0 | $c{M}^{min}$ |

Condition | ${\mathit{n}}^{\mathbf{*}}$ | ${\mathit{x}}^{\mathbf{*}}$ | Total Subsidy | |
---|---|---|---|---|

Case i | $c{M}^{min}<C\frac{c}{\gamma {Q}_{{M}^{min}}L-\beta {q}_{{M}^{min}}L}$ | 0 | c | $c{M}^{min}$ |

Case ii | $C\frac{c}{\gamma {Q}_{{M}^{min}}L-\beta {q}_{{M}^{min}}L}\le c{M}^{min}<C{N}^{min}$ | ${N}^{min}-1$ | $C(\frac{c}{\gamma {Q}_{{M}^{min}}L-\beta {q}_{{M}^{min}}L}-{N}^{min}+1)\frac{1}{{M}^{min}}$ | $C\frac{c}{\gamma {Q}_{{M}^{min}}L-\beta {q}_{{M}^{min}}L}$ |

Case iii | $C{N}^{min}\le c{M}^{min}$ | ${N}^{min}$ | 0 | $C{N}^{min}$ |

Condition | ${\mathit{m}}^{\mathbf{*}}$ | ${\mathit{y}}^{\mathbf{*}}$ | Total Subsidy | |
---|---|---|---|---|

Case i | $C{N}^{min}<\frac{cC}{(\beta -\alpha )\left({Q}_{0}^{2}-{Q}_{{M}^{\mathrm{min}}}^{2}\right)L}$, i.e., $c>(\beta -\alpha )\left({Q}_{0}^{2}-{Q}_{{M}^{\mathrm{min}}}^{2}\right)L{N}^{min}$ | 0 | C | $C{N}^{min}$ |

Case ii | $c\le (\beta -\alpha )\left({Q}_{0}^{2}-{Q}_{{M}^{\mathrm{min}}}^{2}\right)L{N}^{min}$ | ${M}^{min}$ | 0 | $c{M}^{min}$ |

Condition | ${\mathit{\beta}}^{\mathbf{*}}$ | Total Subsidy | |
---|---|---|---|

Case a | $\frac{\gamma Q}{2q}+\frac{\alpha}{2}\ge \alpha +\frac{C}{MqL}$ | $\frac{\gamma Q}{2q}+\frac{\alpha}{2}$ | $\frac{2Cc}{\gamma QL-\alpha qL}$ |

Case b | $\frac{\gamma Q}{2q}+\frac{\alpha}{2}<\alpha +\frac{C}{MqL}$ | ${\left(\alpha +\frac{C}{MqL}\right)}^{+}$ | ${\left(\frac{cCM}{M\gamma QL-\alpha qLM-C}\right)}^{+}$ |

Derivative over $\mathit{\alpha}$ | Derivative over $\mathit{\gamma}$ | |||
---|---|---|---|---|

${\mathit{\beta}}^{\mathbf{*}}$ | Total Subsidy | ${\mathit{\beta}}^{\mathbf{*}}$ | Total Subsidy | |

Case a | $\frac{1}{2}$ | $\frac{2cCqL}{{\left(\gamma QL-\alpha qL\right)}^{2}}$ | $\frac{Q}{2q}$ | $\frac{-2CcQL}{{\left(\gamma QL-\alpha qL\right)}^{2}}$ |

Case b | 1 | $\frac{cCqL{M}^{2}}{{\left(M\gamma Ql-\alpha qLM-C\right)}^{2}}$ | 0 | $\frac{-cC{M}^{2}QL}{{\left(M\gamma QL-\alpha qLM-C\right)}^{2}}$ |

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**MDPI and ACS Style**

Qi, J.; Wang, H.; Zheng, J.
Promoting Liquefied Natural Gas (LNG) Bunkering for Maritime Transportation: Should Ports or Ships Be Subsidized? *Sustainability* **2022**, *14*, 6647.
https://doi.org/10.3390/su14116647

**AMA Style**

Qi J, Wang H, Zheng J.
Promoting Liquefied Natural Gas (LNG) Bunkering for Maritime Transportation: Should Ports or Ships Be Subsidized? *Sustainability*. 2022; 14(11):6647.
https://doi.org/10.3390/su14116647

**Chicago/Turabian Style**

Qi, Jingwen, Hans Wang, and Jianfeng Zheng.
2022. "Promoting Liquefied Natural Gas (LNG) Bunkering for Maritime Transportation: Should Ports or Ships Be Subsidized?" *Sustainability* 14, no. 11: 6647.
https://doi.org/10.3390/su14116647