# On the Rising Interdependency between the Power Grid, ICT Network, and E-Mobility: Modeling and Analysis

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## Abstract

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## 1. Introduction

- A new flow-based network model for the interdependency between the power grid, the ICT network, and e-mobility.
- A common failure propagation index, to identify critical nodes and links within single CIs and between multiple interdependent ones.
- A new hybrid case study test system that combines a power distribution grid, an ICT network, and a standard transportation test network, which can be used for further analysis and research that integrate these three CIs.

## 2. Modeling Structural Interdependencies

#### 2.1. Individual Critical Infrastructure (CI) Models

#### 2.2. The Integrated Structural Interdependency Model

## 3. Modeling Operational Interdependencies

#### 3.1. Interdependency Influence Graphs

#### 3.2. Failure Propagation Index (FPI)

#### 3.3. Microgrids and Energy Storage Systems

#### 3.4. The Integrated Operational Interdependency Model

## 4. Case Study

## 5. Results and Discussion

#### 5.1. Scenario I

#### 5.2. Scenario II

## 6. Conclusions

## Funding

## Conflicts of Interest

## Nomenclature

${\mathit{G}}_{\mathit{P}}$, ${\mathit{G}}_{\mathit{C}}$, ${\mathit{G}}_{\mathit{T}}$ | Graphs representing the power, ICT, and transportation networks, respectively |

${\mathit{V}}_{\mathit{P}}$, ${\mathit{V}}_{\mathit{C}}$, ${\mathit{V}}_{\mathit{T}}$ | Sets of power nodes, ICT nodes, and transportation nodes, respectively |

${\mathit{E}}_{\mathit{P}}$, ${\mathit{E}}_{\mathit{C}}$, ${\mathit{E}}_{\mathit{T}}$ | Sets of power grid links, ICT links, and transportation links, respectively |

${\mathit{n}}_{\mathit{P}}$, ${\mathit{n}}_{\mathit{C}}$, ${\mathit{n}}_{\mathit{T}}$ | Number of power grid nodes, ICT nodes, and transportation nodes, respectively |

${\mathit{l}}_{\mathit{P}}$, ${\mathit{l}}_{\mathit{C}}$, ${\mathit{l}}_{\mathit{T}}$ | Number of power grid links, ICT links, and transportation links, respectively |

${\mathit{A}}^{\mathit{P}}$, ${\mathit{A}}^{\mathit{C}}$, ${\mathit{A}}^{\mathit{T}}$ | Adjacency matrices of the power grid, ICT, and transportation networks, respectively |

${\Gamma}^{\mathit{P}}$, ${\Gamma}^{\mathit{C}}$, ${\Gamma}^{\mathit{T}}$ | Laplacian matrices for ${\mathit{G}}_{\mathit{P}}$, ${\mathit{G}}_{\mathit{C}}$ And ${\mathit{G}}_{\mathit{T}}$, respectively |

${\mathit{D}}^{\mathit{P}}$, ${\mathit{D}}^{\mathit{C}}$, ${\mathit{D}}^{\mathit{T}}$ | Degrees of the power grid, the ICT network, and the transportation network, respectively |

${\mathit{V}}_{\mathit{G}}^{\mathit{P}}$ | Set of generator nodes |

${\mathit{V}}_{\mathit{N}\mathit{D}\mathit{L}}^{\mathit{P}}$ | Set of non-dispatchable load nodes |

${\mathit{V}}_{\mathit{D}\mathit{L}}^{\mathit{P}}$ | Set of dispatchable load nodes |

${\mathit{V}}_{\mathit{D}\mathit{E}\mathit{R}}^{\mathit{P}}$ | Set of DER nodes |

${\mathit{V}}_{\mathit{\mu}\mathit{G}}^{\mathit{P}}$ | Set of microgrid nodes |

${\mathit{V}}_{\mathit{E}\mathit{V}\mathit{S}\mathit{E}}^{\mathit{P}}$ | Set of EVSE nodes |

${\mathit{V}}_{\mathit{R}\mathit{c}}^{\mathit{P}}$ | Set of subway substation nodes |

${\mathit{V}}_{\mathit{I}\mathit{N}}^{\mathit{P}}$ | Set of nodes connecting the power grid to infeed from a higher-voltage |

${\mathit{G}}_{\mathit{P}}^{\mathit{M}}$ | Graph representing a community microgrid |

${\mathit{V}}_{\mathit{P}}^{\mathit{M}},{\mathit{E}}_{\mathit{P}}^{\mathit{M}}$ | Nodes and links of the community microgrid, respectively |

${\mathit{V}}_{\mathbb{Z}}^{\mathit{C}}$ | Measurement nodes |

${\mathit{V}}_{\mathbb{R}}^{\mathit{C}}$ | Router nodes |

${\mathit{V}}_{\u2102}^{\mathit{C}}$ | Control center nodes |

${\mathit{V}}_{\mathfrak{j}}^{\mathit{T}}$ | Road junctions |

${\mathit{V}}_{\mathfrak{s}}^{\mathit{T}}$ | Passenger stations |

${\mathit{E}}_{\mathfrak{j}}^{\mathit{T}}$ | Road sections between adjacent |

${\mathit{E}}_{\mathfrak{s}}^{\mathit{T}}$ | Train railway sections between adjacent stations |

${\U0001d4d6}_{\mathit{S}}$ | The integrated graph |

${\mathbb{V}}_{\mathit{S}},{\mathbb{E}}_{\mathit{S}}$ | Nodes and links of ${\U0001d4d6}_{\mathit{S}}$, respectively |

${\U0001d4d8}_{\mathit{S}}^{\mathit{P}\mathit{C}}$ | Links between power nodes and their corresponding ICT nodes |

${\U0001d4d8}_{\mathit{S}}^{\mathit{P}\mathit{T}}$ | Links between power nodes and their corresponding transportation nodes |

${\mathit{G}}_{\mathit{P}}^{\mathit{I}}$ | Graph to model the operational interdependencies of the power grid |

${\mathit{V}}_{\mathit{P}}^{\mathit{l}}$ | Set of nodes that represent links of ${\mathit{G}}_{\mathit{P}}$ |

${\mathit{V}}_{\mathit{P}}^{\u2132}$ | Set of fictitious nodes that are added to ${\mathit{G}}_{\mathit{P}}^{\mathit{I}}$ To model alternative sources of power, if any |

${\mathit{E}}_{\mathit{P}}^{\mathit{l}}{}_{\mathit{i}\mathit{j}}$ | Links that signify the probability of node ${V}_{P}^{l}{}_{i}$ To fail following a disturbance at node ${V}_{P}^{l}{}_{j}$ |

FPI | Failure propagation index |

${\mathit{E}}_{\mathit{L}\mathit{P}}^{\mathit{P}}$ | Set of links from ${V}_{P}^{l}$ Nodes to ${V}_{P}$ Nodes |

${\mathit{P}}_{\mathit{i}}$ | The net power injection at node $\mathit{i}$ |

${\mathit{F}}_{\mathit{i}\mathit{k}}$ | The power flow through the link connecting nodes $\mathit{i}$ and $\mathit{k}$ |

${\mathit{C}}_{\mathit{i}\mathit{k}}$ | The power flow capacity of the link connecting nodes $\mathit{i}$ and $\mathit{k}$ |

$\mathit{B}$ | Imaginary component of the bus admittance matrix |

${\mathit{E}}_{\mathit{P}\mathit{L}}^{\mathit{P}}$ | Set of links from ${V}_{P}$ Nodes to ${V}_{P}^{l}$ Nodes |

$\Delta \mathit{F}$ | A vector combining the power variation in the various lines for changes in the power injection vector |

${\mathit{E}}_{\mathit{L}\mathit{L}}^{\mathit{P}}$ | Set of edges from ${\mathit{V}}_{\mathit{P}}^{\mathit{l}}$ nodes to other ${\mathit{V}}_{\mathit{P}}^{\mathit{l}}$ ones |

$\mathit{\varsigma}$ | Line outage distribution factor $\mathit{\varsigma}$ |

$\stackrel{\u23de}{\mathit{F}}$ | Array combining all elements of $\mathit{F}$ as a vector |

${\mathit{E}}_{\mathit{P}\mathit{\mu}}^{\mathit{P}}$ | Set of links microgrid nodes and their corresponding fictitious nodes |

${\mathit{P}}_{\mathit{\mu}\mathit{G}}^{\U0001d4dc}$ | Maximum power capacity of local microgrid resources |

$\U0001d4d0$ | Time-varying availability factor |

${\U0001d4d6}_{\U0001d4d8}$ | The operational interdependency influence graph |

${\mathbb{V}}_{\U0001d4d8},{\mathbb{E}}_{\U0001d4d8}$ | Nodes and links of ${\U0001d4d6}_{\U0001d4d8}$ |

${\mathit{G}}_{\mathit{C}}^{\mathit{I}}$ | Graph to model the operational interdependencies of the ICT network |

${\mathit{E}}_{\mathit{C}}^{\mathit{I}}$ | Links that determine the influence of ICT nodes on each other |

${\Re}_{\mathit{i}}^{\mathit{C}}$ | Set of possible shortest paths between node $\mathit{i}$ and the control center, excluding those that are be available due to failure of node $\mathit{j}$ |

$\U0001d4fc$ | State vector of the power grid at a given time |

$\mathit{\phi}$ | Vector of transformer phase shifts |

$\U0001d4e5$ | Voltage magnitude at every bus |

$\U0001d4e1$ | Vector of transformer off-nominal voltage ratios |

$\U0001d4f6$ | Set of power system measurements |

$\mathit{\epsilon}$ | Measurement errors |

$\Lambda $ | Square matrix whose off-diagonal elements are zeros, and diagonal elements equal the standard deviation of the measurement errors |

$\mathit{v}$ | Iteration index |

${\U0001d4d8}_{\U0001d4d8}^{\mathit{C}\u27f6\mathit{P}}$ | Link from a power grid node $\mathit{i}$ to an ICT node $\mathit{j}$ |

${\mathfrak{l}}_{\mathit{i}}^{\mathit{C}}$ | Number of ICT nodes connected to power node $i$ |

${\U0001d4ee}_{\mathit{i}}^{\mathit{C}}$ | Estimated error between the actual measurement of $\mathit{i}$, and its estimated value |

${\U0001d4d8}_{\U0001d4d8}^{\mathit{P}\u27f6\mathit{C}}$ | Interdependency link from an ICT node $\mathit{i}$ to a power grid node $\mathit{j}$ |

${\mathfrak{l}}_{\mathit{i}}^{\mathit{P}}$ | Number of power nodes connected to ICT node $\mathit{i}$ |

$\stackrel{\u02d8}{\mathit{i}}$ | Fictitious power node |

${\mathit{E}}_{\mathit{C}}^{\u2132}$ | Total energy capacity |

$\U0001d4d2$ | Total energy capacity |

$\Psi $ | Available energy |

${\mathit{G}}_{\mathit{T}}^{\mathit{I}}$ | Graph representing the transportation network |

${\mathit{V}}_{\mathit{T}}^{\mathit{l}}$ | Set of nodes representing links ${E}_{T}$ of ${\mathit{G}}_{\mathit{T}}$ |

${\mathit{E}}_{\mathit{T}\mathit{L}}^{\mathit{T}}$ | Set of links between a ${\mathit{V}}_{\mathit{T}}$ node and a ${\mathit{V}}_{\mathit{T}}^{\mathit{l}}$ one |

${\mathit{E}}_{\mathit{L}\mathit{L}}^{\mathit{T}}$ | Set of links between a ${\mathit{V}}_{\mathit{T}}^{\mathit{l}}$ Node and another ${\mathit{V}}_{\mathit{T}}^{\mathit{l}}$ One |

${\mathit{V}}_{\mathit{T}}^{\mathfrak{j}}$, ${\mathit{V}}_{\mathit{T}}^{\mathfrak{s}}$ | Road and subway nodes, respectively |

${\mathit{E}}_{\mathit{L}\mathit{L}}^{\mathfrak{j}\mathfrak{s}}$, ${\mathit{E}}_{\mathit{L}\mathit{L}}^{\mathfrak{s}\mathfrak{j}}$ | Links between ${V}_{T}^{\mathfrak{j}}$ And ${V}_{T}^{\mathfrak{s}}$, and links between ${V}_{T}^{\mathfrak{s}}$ And ${V}_{T}^{\mathfrak{j}}$, respectively |

$\Delta {\mathfrak{E}}_{\mathit{i}\mathit{j}}^{\mathfrak{j}}$ | Change in the flow due to a problem in the ${\mathit{E}}_{\mathfrak{j}}^{\mathit{T}}\left(\mathit{i},\mathit{j}\right)$ link |

$\left|{\mathfrak{E}}_{\mathit{i}\mathit{j}}^{\mathfrak{j}}-{\mathit{C}}_{\mathit{i}\mathit{j}}^{\mathfrak{j}}\right|$ | Remaining capacity of link ${\mathit{E}}_{\mathfrak{s}}^{\mathit{T}}\left(\mathit{i},\mathit{j}\right)$ |

${\mathit{\sigma}}_{\mathit{i}\mathit{j}}^{\mathfrak{j}\mathfrak{s}}$, ${\mathit{\sigma}}_{\mathit{i}\mathit{j}}^{\mathfrak{s}\mathfrak{j}}$ | Probability of travelers changing their mode of travel from road to subway, and subway to road, respectively |

${\U0001d4d8}_{\U0001d4d8}^{\mathit{T}\to \mathit{P}}$ | Links that model dependence of the power grid on the transportation network |

${\mathit{\sigma}}_{\mathit{i}\mathit{j}}^{\mathfrak{j}\mathfrak{j}}\Delta {\mathfrak{E}}_{ij}^{\mathfrak{j}},{\mathit{\sigma}}_{\mathit{i}\mathit{j}}^{\mathfrak{s}\mathfrak{s}}\Delta {\mathfrak{E}}_{\mathit{i}\mathit{j}}^{\mathfrak{s}}$ | Modified demand encountered by node junctions and passenger stations located nearest to node $\mathit{j}$, respectively |

${\U0001d4d3}_{\mathit{i}\mathit{j}}^{\mathfrak{j}}$, ${\U0001d4d3}_{\mathit{i}\mathit{j}}^{\mathfrak{s}}$ | Sets representing the power demand per unit increase in the traffic and subway demand, respectively |

${\U0001d4d8}_{\U0001d4d8}^{\mathit{P}\to \mathit{T}}$ | Links that model the dependence of the transportation network on the power grid |

${\U0001d4d8}_{\U0001d4d8\mathfrak{j}}^{\mathit{P}\to \mathit{T}}$ | ${\mathcal{I}}_{\mathcal{I}}^{P\to T}$ Dependency related to EVSEs |

${\U0001d4d8}_{\U0001d4d8\mathfrak{s}}^{\mathit{P}\to \mathit{T}}$ | ${\U0001d4d8}_{\U0001d4d8}^{\mathit{P}\to \mathit{T}}$ Dependency related to subway substations |

${\mathit{\varkappa}}_{\mathit{j}\mathit{i}}$ | Probability that rectifier substation $\mathit{j}$ can support a failed substation $\mathit{i}$ |

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**Figure 1.**The integrated structural interdependency graph ${\U0001d4d6}_{S}=\left({\mathbb{V}}_{S},{\mathbb{E}}_{S}\right)$.

**Figure 2.**The operational interdependency influence graph ${\U0001d4d6}_{\mathcal{I}}=\left({\mathbb{V}}_{\mathcal{I}},{\mathbb{E}}_{\mathcal{I}}\right)$.

**Figure 3.**Average yield for Scenarios I and II. In Scenario 1, the yield following the first contingency drops from 1 to about 0.84, and then drops again to about 0.27 before the cascade stops. In Scenario II, since it involves failures within multiple CIs, the yield drops more sharply to zero.

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## Share and Cite

**MDPI and ACS Style**

Mohamed, A.A.A.
On the Rising Interdependency between the Power Grid, ICT Network, and E-Mobility: Modeling and Analysis. *Energies* **2019**, *12*, 1874.
https://doi.org/10.3390/en12101874

**AMA Style**

Mohamed AAA.
On the Rising Interdependency between the Power Grid, ICT Network, and E-Mobility: Modeling and Analysis. *Energies*. 2019; 12(10):1874.
https://doi.org/10.3390/en12101874

**Chicago/Turabian Style**

Mohamed, Ahmed Ali A.
2019. "On the Rising Interdependency between the Power Grid, ICT Network, and E-Mobility: Modeling and Analysis" *Energies* 12, no. 10: 1874.
https://doi.org/10.3390/en12101874