# Distributed Optimization of Joint Seaport-All-Electric-Ships System under Polymorphic Network

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## Abstract

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## 1. Introduction

- (1)
- Considering AESs’ as prosumer, and taking the carbon trading mechanism into account, a joint seaport-AESs power system model is established. The precise mathematical model of power dispatching is established and resolved through fully distributed algorithms.
- (2)
- This paper validates a distributed algorithm based on parameter projection that evades KKT conditions or the Lagrange multiplier operator to slash the difficulty of calculation due to a reduction in variables’ dimensions and applies it to a proposed joint seaport–AES system where AESs’ roles shift to become prosumers.

## 2. Problem Formulation

#### 2.1. Polymorphic Joint Seaport-AESs System

#### 2.2. Objective Function

- The diesel cost of DG k in MG i is approximated with a quadratic function as [17]:$${\mathrm{C}}_{DG}\left({P}_{ik}^{t}\right)={a}_{ik}P{{}_{ik}^{t}}^{2}+{b}_{ik}{P}_{ik}^{t}+{c}_{ik}$$
- The carbon-trading cost of DG k in MG i:$${\mathrm{C}}_{CT}\left({Q}_{ik}^{t}\right)=\pi \left({Q}_{{}_{ik}}^{t}-{Q}_{ik-rated}\right)$$Carbon-trading cost is regarded as an indicator of the expenditure or benefits of the system’s total carbon emissions in the carbon market. Namely, when the system’s carbon emissions quota are greater than its allocations, the excess carbon emissions need to be offset by means of purchasing additional quota; when the system carbon emission is less, the surplus can be sold to the carbon market.
- The cost of MG i for buying power from the neighbor MG j:$${\mathrm{C}}_{Buy}\left({P}_{j\to i}^{t}\right)={{\lambda}_{j}}^{t}{P}_{j\to i}^{t}$$
- The revenue of MG i for selling power to the neighbor MG j:$${\mathrm{C}}_{Sell}\left({P}_{i\to j}^{t}\right)={{\lambda}_{i}}^{t}{P}_{i\to j}^{t}$$

#### 2.3. Constraints

- Power balance constraint of the integrated seaportAESs’ multiMG system:$$\sum _{i\in {N}_{MG}}\left(\sum _{k\in N{i}_{DG}}{P}_{ik}^{t}+{P}_{iB}^{t}\right)=\sum _{i\in {N}_{MG}}{P}_{iL}^{t}-\sum _{i\in {N}_{MG}}{P}_{iR}^{t}$$For each MG (like the seaport or an AES), the power balance constraint will be decomposed as follows:$$\sum _{k\in N{i}_{DG}}{P}_{ik}^{t}+{P}_{iB}^{t}+{P}_{iR}^{t}={P}_{iL}^{t}+\sum _{j\in {N}_{i}}{P}_{i\to j}^{t}-\sum _{j\in {N}_{i}}{P}_{j\to i}^{t}$$
- Power and ramp rate limit constraints of DG k in MG i:$${P}_{ik}^{min}\le {P}_{ik}^{t}\le {P}_{ik}^{max},\forall i\in {N}_{MG},k\in N{i}_{DG},t\in T$$$$-{r}_{ik}^{max}\le {P}_{ik}^{t}-{P}_{ik}^{t-1}\le {r}_{ik}^{max},\forall i\in {N}_{MG},k\in N{i}_{DG},t\in T$$
- Energy constraints of battery energy storage system (BESS) in MG i (the definition of SOC, the definition of BESS capacity, capacity limit constraint, SOC limit constraint and power limit constraint, in order):$$SO{C}_{iB}^{t}={E}_{iB}^{t}/\phantom{{E}_{iB}^{t}{E}_{rated}}\phantom{\rule{0.0pt}{0ex}}{E}_{rated},\forall i\in {N}_{MG},t\in T$$$$SO{C}_{min}\le SO{C}_{iB}^{t}\le SO{C}_{max},\forall i\in {N}_{MG},t\in T$$$${E}_{iB}^{t+1}={E}_{iB}^{t}-{P}_{iB}^{t+1}\Delta T,\forall i\in {N}_{MG},t\in T$$$${E}_{iB}^{min}\le {E}_{iB}^{t}\le {E}_{iB}^{max},\forall i\in {N}_{MG},t\in T$$$$0\le {P}_{iB}^{t}\le {P}_{iB}^{max},\forall i\in {N}_{MG},t\in T$$
- Constraint of power transaction between grids i and j:$$0\le {P}_{i\to j}^{t}\le \sum _{k\in N{i}_{DG}}{P}_{ik}^{t},\forall i\in {N}_{MG},k\in N{i}_{DG},t\in T$$

## 3. Distributed Optimization Solution Methodology

#### 3.1. Graph Theory

#### 3.2. Parameter Projection

#### 3.3. Parameter Projection Distributed Optimization (PPDO)

## 4. Numerical Results

#### 4.1. Text System

#### 4.2. Simulation Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Position | ${\mathit{a}}_{\mathit{i}}$ | ${\mathit{b}}_{\mathit{i}}$ | ${\mathit{c}}_{\mathit{i}}$ | ${\mathit{P}}_{min}$ | ${\mathit{P}}_{max}$ |
---|---|---|---|---|---|

Seaport | 0.5 | 0 | 0 | 0 | 50/12 |

AES1 | 0.5 | 2 | 2 | 0 | 11 |

AES2 | 0.5 | 1 | 0.5 | 0 | 12 |

AES3 | 0.5 | 1 | 0.5 | 0 | 13 |

AES4 | 0.5 | 3 | 4.5 | 0 | 14 |

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**MDPI and ACS Style**

Xia, W.; Shan, Q.; Xiao, G.; Tu, Y.; Liang, Y.
Distributed Optimization of Joint Seaport-All-Electric-Ships System under Polymorphic Network. *Sustainability* **2022**, *14*, 9914.
https://doi.org/10.3390/su14169914

**AMA Style**

Xia W, Shan Q, Xiao G, Tu Y, Liang Y.
Distributed Optimization of Joint Seaport-All-Electric-Ships System under Polymorphic Network. *Sustainability*. 2022; 14(16):9914.
https://doi.org/10.3390/su14169914

**Chicago/Turabian Style**

Xia, Wenjia, Qihe Shan, Geyang Xiao, Yonggang Tu, and Yuan Liang.
2022. "Distributed Optimization of Joint Seaport-All-Electric-Ships System under Polymorphic Network" *Sustainability* 14, no. 16: 9914.
https://doi.org/10.3390/su14169914