# Dynamic Cooperation of Transportation and Power Distribution Networks via EV Fast Charging Stations

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## Abstract

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## 1. Introduction

#### 1.1. Background

#### 1.2. Literature Review

#### 1.3. Main Contributions

- Compared with the former literature review, it constructs a DSO model based on the Wardrop II principle. This model involves transportation and electricity operators and realized cooperation between both operators. Different from the spontaneity of vehicle drivers described in Wardrop I, Wardrop II provides an opportunity for the transportation operator to dispatch vehicles at various time intervals.
- In addition, compared with the common DTA model, the model not only considers multiple O–D pairs and multiple vehicle types, but also multiple tasks. Multi-task is a middle class under multiple O–D pairs and over vehicle types. This class determines various departing times. It is useful in the logistical department. It can assign the departing and arriving times of transportation for various goods.

## 2. Mathematical Model

#### 2.1. Model Structure

- DSO (modeling of TN)
- DOPF (modeling of PDN)
- FCS (modeling connecting two networks)

#### 2.2. Modeling of TN

_{m}and the speed of various vehicles sp

_{m}. Path q includes various links on a TN and at a FCS i (i is the bus connecting to a PDN). For GVs, the battery capacity E

_{m}is 0 and the paths q do not pass FCSs. Even though GVs and EVs have the same link that has a FCS, the paths q is different. Four classes of vehicle assignment logic are designed. The top class of vehicle assignment is O–D pair rs. The second class is various task h. For the same O–D pair rs, various tasks can be assigned to different time periods. The third class is vehicle type m. A task can be carried out by various types of vehicles. Path q is the lowest assignment class. In the same task, various vehicles are assigned to different paths and time intervals.

_{a}(t) is the number of vehicles on a link a at time t. q is the sum of all paths passing through a link a at time interval t. Its object is to minimize the total travel cost of TN. It is the sum of the travel cost of in the TN for a time period 0 to T. While a vehicle is on a link a, the vehicle has a cost c

_{a}to travel from the entrance to the exit of the link. For a more realistic explanation, the travel cost can be the cost of time, fuel, or electric energy.

_{a}and the driving speed sp

_{m}, the link travel time (neglecting congestion) τ

_{am}given by Equations (9) and (10), this group of vehicles m would exit at time t + τ

_{am}.

_{m}over the real charging power of the charging pile in the FCS.

_{a}(t), the total number of vehicles x

_{a}(t), and the total exit flow rate v

_{a}(t) on link a. These are the sum of path q, vehicle type m at task h, and O–D pair rs. An O–D pair rs includes various vehicle types m. The path is designed to refer to vehicle type.

#### 2.3. Modeling of PDN

_{g}[P

_{g}(t)] is the generating cost function versus the real power generation P

_{g}(t) of the generating unit g at time interval t. Commonly, its form includes linearity, quadratic, cube, and piecewise. Quadratic is common in transmission systems and linear is common in distribution systems.

_{ij}(t) and reactive power flow Q

_{ij}(t). If bus i is the input end and bus j is the output end of branch ij, (17) describes the input end (from bus) and (18) describes the output end (to bus). For an end, taking bus i as an example, the sum of the square of the real power flow ${P}_{ij}^{2}\left(t\right)$ and the square of the reactive power flow ${Q}_{ij}^{2}\left(t\right)$ is equal to the product of the square of the voltage magnitude ${V}_{i}^{2}\left(t\right)$ at this end and the square of the current magnitude ${I}_{ij}^{2}\left(t\right)$. ${V}_{i}^{2}\left(t\right)$ and ${I}_{ij}^{2}\left(t\right)$ are taken as independent variables in this model. The ends of power flow input end and output end are self-defined. Both ends can be defined as the real input end or the output end. If one end is defined as the input but power flow output is actually from that end, its power flow in mathematics is negative, otherwise, it is positive. There is not sign constraint in the real and reactive power flow of both ends in this model.

_{ijmax}is the complex power flow limit of branch ij.

_{j}is the total shunt capacity connecting bus j. It is half of the sum of the shunt capacity of branches connecting to the bus. The other difference is ${Q}_{j}^{g}\left(t\right)$ and ${Q}_{j}^{ch}\left(t\right)$ can be less than zero. For b

_{j}> 0 and ${V}_{j}^{2}\left(t\right)$ > 0, ${V}_{j}^{2}\left(t\right){b}_{j}$ > 0. Reactive compensation in a system decreases reactive power output in a system.

_{ij}is the resistance of branch ij. x

_{ij}is the reactance of branch ij. The power loss is to add the power flow of both sides i and j together. In a specific branch and a specific time interval, the real or reactive power flow is certain from one side to the other side. If one side’s real or reactive power flow is positive (negative), the other side must be negative (positive) and the absolute real or reactive power flow in the positive side must be larger than the negative side for the positive side is power input and the negative side is power output. The branch losses some power from the power input. Because of the opposite of both sides, the real or reactive power loss is the sum of both sides.

#### 2.4. Modeling of FCS

#### 2.5. Multi-Objective Optimization

## 3. Results

#### 3.1. Parameter Settings

_{a}[x

_{a}(t)] is the time cost. Table 3 provides the parameters of this TN case.

_{a}of link e and e′ are 240 km. The other links are all 120 km. This means that vehicles passing e and e′ at the speed of 120 km/h need 2 h. Passing the other 8 links needs 1 h. FCS ① is located at the exit of link a and the entrance of link a′. The distance from ① to the exit of link a is 0 and to the exit of link a′ is 120 km. FCS ② is located at the middle of link e and e′. The distance from ② to the exits of link e and e′ is 120 km each.

#### 3.2. Results and Analysis

#### 3.2.1. Optimal Solution versus Weight

#### 3.2.2. Optimal Solution While TN Prior

#### 3.2.3. Optimal Solution While PDN Prior

#### 3.2.4. Changing Charging Power of FCS ②

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature and Abbreviations

Notations in the Transportation Network Model | |

Sets and Indices | |

a | Link |

A_{l} | Set of links whose tail node is l (l ≠ r ∪ l ≠ s) |

A_{r} | Set of links whose tail node is origin node r |

A_{s} | Set of links whose tail node is destination node s |

B_{l} | Set of links whose head node is node l (l ≠ r ∪ l ≠ s) |

B_{r} | Set of links whose head node is origin node r |

B_{s} | Set of links whose head node is destination node s |

c_{a}[x_{a}(t)] | Travel cost on link a |

h | Task |

l | Node excluding origin r and destination s |

m | Vehicle type |

q | Path |

r | Origin node |

s | Destination node |

t | Time interval index |

${t}_{am}^{ch}$ | Charging time interval index of vehicle m at the charging station on link a |

Parameters | |

d_{a} | Distance of link a |

E_{m} | Battery capacity of electric vehicle m |

${F}_{mh}^{rs}$ | Total number of vehicle m in task h with origin r and destination s |

${F}_{rmh}^{rs}$ | Total number of vehicle m in task h departing from origin r towards destination s |

${F}_{smh}^{rs}$ | Total number of vehicle m in task h arriving at destination s from origin r |

sp_{m} | Speed of vehicle m |

τ_{am} | Link a travel time of vehicle m without congestion |

T | Final time |

w(DSO) | Weight of dynamic system optimal objective function |

Variables | |

${f}_{rmh}^{rs}\left(t\right)$ | Instantaneous departing flows of vehicle m number in task h departing from origin r towards destination s at time t |

${f}_{smh}^{rs}\left(t\right)$ | Instantaneous arriving flows of vehicle m number in task h arriving at destination s from origin r at time t |

u_{a}(t) | Total inflow rate on link a over path q at time t |

${u}_{amhq}^{rs}\left(t\right)$ | Inflow rate on link a over path q which belongs to vehicle m number in task h from origin r and destination s at time t |

x_{a}(t) | Total number of vehicles travelling on link a at time t |

${x}_{amhq}^{rs}\left(t\right)$ | Number of vehicles on link a over path q which belong to vehicle m number in task h with origin r and destination s at time t |

v_{a}(t) | Total exit flow rate from link a at time t |

${v}_{amhq}^{rs}\left(t\right)$ | Exit flow rate from link a over path q which belongs to vehicle m number in task h with origin r and destination s at time t |

Notations in the Power Distribution Network Model | |

Sets and Indices | |

fb | From bus |

g | Generating unit |

i | Bus i |

j | Bus j |

k | Bus k |

tb | To bus |

Parameter | |

b_{ij} | Susceptance at branch from bus i to bus j |

b_{j} | Sum of susceptance caused by shunt capacitors at bus j |

fb | From bus |

P_{gmin} | Minimum real power output of unit g |

P_{gmax} | Maximum real power output of unit g |

${P}_{j}^{d}$ | Real load at bus j |

P_{ramp} | Ramp of real power output per time frame |

Q_{gmin} | Minimum reactive power output of unit g |

Q_{gmax} | Maximum reactive power output of unit g |

${Q}_{j}^{d}$ | Reactive load at bus j |

Q_{ramp} | Ramp of reactive power output per time interval |

r_{ij} | Resistance of branch from bus i to bus j |

S_{ijmax} | Complex power flow limit of branch ij. |

${\mathrm{T}}_{am}^{ch}$ | Charging time of vehicle m in charging station on link a |

tb | To bus |

${V}_{imin}^{2}$ | Minimum value of square of voltage magnitude at bus i |

${V}_{imax}^{2}$ | Maximum value of square of voltage magnitude at bus i |

w(DOPF) | Weight of dynamic optimal power flow objective function |

x_{ij} | Reactance of branch from bus i to bus j |

z_{ij} | Impedance of branch from bus i to bus j |

Variables | |

c_{g}[P_{g}(t)] | Cost of real power output of unit g at time t |

${\dot{I}}_{ij}\left(t\right)$ | Complex current in branch from bus i to bus j at time t |

${\dot{I}}_{ij}^{*}\left(t\right)$ | Conjugate complex current in branch from bus i to bus j at time t |

${I}_{ij}^{2}\left(t\right)$ | Square of current magnitude in branch from bus i to bus j at time t |

P_{g}(t) | Real power output of unit g at time t |

P_{ij}(t) | Real power flow of branch from bus i to bus j at time t |

P_{jk}(t) | Real power flow of branch from bus j to bus k at time t |

${P}_{j}^{g}\left(t\right)$ | Real power output of unit g at bus j at time t |

Q_{g}(t) | Reactive power output of unit g at time t |

Q_{ij}(t) | Reactive power flow of branch from bus i to bus j at time t |

Q_{jk}(t) | Reactive power flow of branch from bus j to bus k at time t |

${Q}_{j}^{g}\left(t\right)$ | Reactive power generation of unit g at bus j at time t |

${\tilde{S}}_{ij}\left(t\right)$ | Complex power flow of branch from bus i to bus j at time t |

${\dot{V}}_{i}\left(t\right)$ | Complex voltage at bus i at time t |

${\dot{V}}_{j}\left(t\right)$ | Complex voltage at bus j at time t |

${V}_{i}^{2}\left(t\right)$ | Square of voltage magnitude at bus i at time t |

${V}_{j}^{2}\left(t\right)$ | Square of voltage magnitude at bus j at time t |

Notations in the Fast Charging Station Model | |

Parameter | |

${d}_{a}^{\gamma}$ | Distance from the charging station on link a to the link’s exit |

${p}_{i}^{ch}$ | Real power of charging pile in charging station on link a |

${\tau}_{am}^{\gamma}$ | Link travel time of vehicle m from the charging station on link a to the link’s exit without congestion |

${x}_{amax}^{ch}$ | Maximum number of vehicles charging in charging station on link a |

Variables | |

${P}_{i}^{ch}\left(t\right)$ | Real power demand of charging station connecting to bus i at time t |

${Q}_{j}^{ch}\left(t\right)$ | Reactive power demand of charging station connecting to bus j at time t |

${x}_{a}^{ch}\left(t\right)$ | Number of vehicles staying in charging station on link a |

Abbreviation | |

AC | Alternating current |

DOPF | Dynamic optimal power flow |

DSO | Dynamic system optimal |

DTA | Dynamic traffic assignment |

DUO | Dynamic user optimal |

EV | Electric vehicle |

FCC | Flow conservation constraints |

FCS | Fast charging station |

FPC | Flow propagation constraints |

GV | Gasoline vehicle |

LFC | Load frequency control |

LP | Linear programming |

MILP | Mix-integer linear programming |

MIQCP | Mix-integer quadratically constrained programming |

O-D | Origin–destination |

OPF | Optimal power flow |

PDN | Power distribution network |

PEV | Plug-in electric vehicle |

PSO | Particle swarm optimization |

SFIFO | Strong first-in-first-out |

QCP | Quadratically constrained programming |

SO | User optimal |

SOC | State of charge |

TA | Traffic assignment |

TAP | Traffic assignment problem |

TN | Transportation network |

UE | User equilibrium |

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**Figure 3.**(

**a**) Flow conservation constraints (FCC) at origin r of vehicle m in task h; (

**b**) FCC at destination s of vehicle m at task h.

**Figure 9.**(

**a**) Instantaneous flow rate of task 1 for time period 0 to 10 under TN prior; (

**b**) instantaneous flow rate of task 2 for time period 4 to 14 under TN prior. (G represents GV, E represents EV, 14 and 41 are O–D pairs, r represents departing flow, and s represents arriving flow).

**Figure 11.**(

**a**) Instantaneous flow of task 1 for time period 0 to 10 under PDN prior; (

**b**) instantaneous flow of task 2 for time period 4 to 14 under PDN prior.

Wardrop I | Wardrop II | |
---|---|---|

TA (static) | UE | SO |

DTA (dynamic) | DUO | DSO |

${\mathit{f}}_{1\mathit{h}\mathit{m}}^{12}\left(0\right)$ | ${\mathit{f}}_{1\mathit{h}\mathit{m}}^{12}\left(1\right)$ | ${\mathit{f}}_{1\mathit{h}\mathit{m}}^{12}\left(2\right)$ | ${\mathit{f}}_{1\mathit{h}\mathit{m}}^{12}\left(3\right)$ | ${\mathit{f}}_{1\mathit{h}\mathit{m}}^{12}\left(4\right)$ | ${\mathit{f}}_{1\mathit{h}\mathit{m}}^{12}\left(5\right)$ | ${\mathit{f}}_{1\mathit{h}\mathit{m}}^{12}\left(6\right)$ | ${\mathit{f}}_{1\mathit{h}\mathit{m}}^{12}\left(7\right)$ | |
---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | ||||||

${v}_{ahmq}^{rs}\left(0\right)$ | ${v}_{ahmq}^{rs}\left(1\right)$ | ${v}_{ahmq}^{rs}\left(2\right)$ | ${v}_{ahmq}^{rs}\left(3\right)$ | ${v}_{ahmq}^{rs}\left(4\right)$ | ${v}_{ahmq}^{rs}\left(5\right)$ | ${v}_{ahmq}^{rs}\left(6\right)$ | ${v}_{ahmq}^{rs}\left(7\right)$ | |

${x}_{i}^{ch}\left(0\right)$ = 0 | 0 | 0 | 0 | |||||

${x}_{i}^{ch}\left(1\right)$ = 0 | 0 | 0 | 0 | |||||

${x}_{i}^{ch}\left(2\right)$ = 1 | 0 | 0 | 1 | |||||

${x}_{i}^{ch}\left(3\right)$ = 3 | 0 | 1 | 2 | |||||

${x}_{i}^{ch}\left(4\right)$ = 6 | 1 | 2 | 3 | |||||

${x}_{i}^{ch}\left(5\right)$ = 5 | 2 | 3 | ||||||

${x}_{i}^{ch}\left(6\right)$ = 3 | 3 | |||||||

${x}_{i}^{ch}\left(7\right)$ = 0 | ||||||||

${f}_{2hm}^{12}\left(0\right)$ | ${f}_{2hm}^{12}\left(1\right)$ | ${f}_{2hm}^{12}\left(2\right)$ | ${f}_{2hm}^{12}\left(3\right)$ | ${f}_{2hm}^{12}\left(4\right)$ | ${f}_{2hm}^{12}\left(5\right)$ | ${f}_{2hm}^{12}\left(6\right)$ | ${f}_{2hm}^{12}\left(7\right)$ | |

1 | 2 | 3 |

a | a′ | b | b′ | c | c′ | d | d′ | e | e′ | |
---|---|---|---|---|---|---|---|---|---|---|

d_{a} (km) | 120 | 120 | 120 | 120 | 120 | 120 | 120 | 120 | 240 | 240 |

${d}_{a}^{\gamma}$ (km) | 0 | 120 | 120 | 120 |

m | Type | E_{m} (kWh) | sp_{m} (km/h) |
---|---|---|---|

1 | GV | 0 | 120 |

2 | EV | 60 | 120 |

Station | Bus | ${\mathit{p}}_{\mathit{i}}^{\mathit{c}\mathit{h}}$ (kW) | Link 1 | ${\mathit{x}}_{1\mathit{m}\mathit{a}\mathit{x}}^{\mathit{c}\mathit{h}}$ | ${\mathit{d}}_{1}^{\mathit{\gamma}}$ (km) | Link 2 | ${\mathit{x}}_{2\mathit{m}\mathit{a}\mathit{x}}^{\mathit{c}\mathit{h}}$ | ${\mathit{d}}_{2}^{\mathit{\gamma}}$ (km) |
---|---|---|---|---|---|---|---|---|

① | 7 | 30 | a | 15 | 0 | a′ | 15 | 120 |

② | 2 | 30 | e | 15 | 120 | e′ | 15 | 120 |

rs | h(t) | $\mathbf{m}\left({\mathit{F}}_{\mathit{h}\mathit{m}}^{\mathit{r}\mathit{s}}\right)$ | q | a | a′ | b | b′ | c | c′ | d | d′ | e | e′ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

14 | 1 (0–10) | 1(40) | 1 | 1 | 1 | ||||||||

2 | 1 | 1 | 1 | ||||||||||

3 | 1 | 1 | 1 | ||||||||||

4 | 1 | 1 | |||||||||||

2(20) | 1 | 2 | 1 | ||||||||||

2 | 2 | 1 | 1 | ||||||||||

3 | 1 | 1 | 2 | ||||||||||

4 | 1 | 2 | |||||||||||

2 (4–14) | 1(40) | 1 | 1 | 1 | |||||||||

2 | 1 | 1 | 1 | ||||||||||

3 | 1 | 1 | 1 | ||||||||||

4 | 1 | 1 | |||||||||||

2(20) | 1 | 2 | 1 | ||||||||||

2 | 2 | 1 | 1 | ||||||||||

3 | 1 | 1 | 2 | ||||||||||

4 | 1 | 2 | |||||||||||

41 | 1 (0–10) | 1(40) | 1 | 1 | 1 | ||||||||

2 | 1 | 1 | 1 | ||||||||||

3 | 1 | 1 | 1 | ||||||||||

4 | 1 | 1 | |||||||||||

2(20) | 1 | 2 | 1 | ||||||||||

2 | 1 | 1 | 2 | ||||||||||

3 | 2 | 1 | 1 | ||||||||||

4 | 1 | 2 | |||||||||||

2 (4–14) | 1(40) | 1 | 1 | 1 | |||||||||

2 | 1 | 1 | 1 | ||||||||||

3 | 1 | 1 | 1 | ||||||||||

4 | 1 | 1 | |||||||||||

2(20) | 1 | 2 | 1 | ||||||||||

2 | 1 | 1 | 2 | ||||||||||

3 | 2 | 1 | 1 | ||||||||||

4 | 1 | 2 |

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© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chen, Z.; Han, B.; Xue, F.; Lu, S.; Jiang, L.
Dynamic Cooperation of Transportation and Power Distribution Networks via EV Fast Charging Stations. *World Electr. Veh. J.* **2023**, *14*, 38.
https://doi.org/10.3390/wevj14020038

**AMA Style**

Chen Z, Han B, Xue F, Lu S, Jiang L.
Dynamic Cooperation of Transportation and Power Distribution Networks via EV Fast Charging Stations. *World Electric Vehicle Journal*. 2023; 14(2):38.
https://doi.org/10.3390/wevj14020038

**Chicago/Turabian Style**

Chen, Zihao, Bing Han, Fei Xue, Shaofeng Lu, and Lin Jiang.
2023. "Dynamic Cooperation of Transportation and Power Distribution Networks via EV Fast Charging Stations" *World Electric Vehicle Journal* 14, no. 2: 38.
https://doi.org/10.3390/wevj14020038