# Vulnerability Analysis to Maximize the Resilience of Power Systems Considering Demand Response and Distributed Generation

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## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

#### 1.2. Literature Review

#### 1.3. Contributions and Paper Organization

- It complements previous works reported in the specialized literature regarding the solution of the EGIP by considering AC modeling of the problem as well as simultaneous attacks on lines and generators.
- New metrics are proposed for the assessment of power system resiliency under deliberate attacks.
- Enhancement of grid resiliency is proposed by introducing the effect of DERs as a reaction strategy of the system operator.

## 2. Outline of the EGIP

#### 2.1. Normal Operative Conditions

- The resources that the network operator allocates for the protection of lines and generators in the system are known to the disruptive agent.
- The disruptive agent is aware of the bilateral agreements for voluntary load shedding.
- The attacks performed on the system are 100% effective.
- The network operator considers DR to be a mechanism for immediate mitigation of the attack plan and reduction of network operating costs.

#### 2.2. Scenarios for the Case Study

- Scenario 1: There is no agreement for voluntary load shedding. Under this condition, the most severe attack plan of the vulnerability analysis is executed without taking mitigation actions by the network operator.
- Scenario 2: There is a bilateral agreement between the network operator and some system loads to voluntarily disconnect a percentage of the total load. From this condition, the disruptive agent executes the most severe attack of the vulnerability analysis. In the post-attack stage, the network operator takes no action to decrease load shedding.
- Scenario 3: There is no agreement for voluntary load shedding. Under this condition, the most severe attack plan of the vulnerability analysis is executed. In the post-attack stage, the network operator optimizes the location and sizing of distributed generators to reduce load shedding.
- Scenario 4: There is a bilateral agreement between the network operator and some system loads to voluntarily disconnect a percentage of the total load. From the condition, the disruptive agent executes the most severe attack of the vulnerability analysis. In the post-attack stage, the network operator optimizes the location of distributed generators and reallocates demand response to reduce load shedding.

## 3. Mathematical Modeling and Solution Approach

#### 3.1. Vulnerability Analysis

#### 3.1.1. Genetic Algorithm

#### 3.1.2. Upper-Level Optimization Problem

_{g}is the cost of the power delivered by generator g; P

_{g}is the power delivered by g; C

_{RDn}is the cost of the dispatchable load at node n; P

_{RDn}is the demand response at the node n; P

_{Dm}is the load shedding at node m; and C

_{Dm}is the cost of load shedding at node m. The disruptive agent strategy is modeled through an interdiction vector for lines and generators. In Equation (2) and Equation (3), the interdiction vector for lines and generators, respectively, is defined as a vector of binary variables, where 1 represents the elements under attack. In this case, δ

_{L}(l) and δ

_{G}(g) are the interdiction vectors for the set of lines and generators, respectively. Constraint Equation (4) describes the destructive resources of the attacking agent, where M

_{l}is the cost of attacking a line, while M

_{g}is the cost of attacking a generator. L is the set of lines; G is the set of generators; M represents the total resources of the attacker; N is the set of buses; and NRD is the set of buses with the demand response.

#### 3.1.3. Lower-Level Optimization Problem

#### 3.2. Allocation of Costs in Loads

#### 3.3. Resilience of an Electrical Power System

#### 3.4. Strategies for Maximizing Network Resilience Following a Disruptive Event

## 4. Tests and Results

#### 4.1. Results with a 5-Bus Power System

#### 4.1.1. Normal Operation and Disruptive Event

^{5}, where load shedding represents 93.5% of the total cost and only 20% of the load is served.

#### 4.1.2. Disruptive Event with DR

#### 4.1.3. DG Allocation to Increase Resiliency without DR

#### 4.1.4. Location and Sizing of DG along with DR to Improve Resiliency

#### 4.1.5. Quantification of Resilience in Terms of Operating Cost and Percentage of the Total Load

#### 4.2. Results with the IEEE RTS-24 Bus Power System

## 5. Discussion of Results

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Acronyms and Abbreviations

AC | Alternating Current |

DC | Direct Current |

DERs | Distributed Energy Resources |

DG | Distributed Generation |

DR | Demand Response |

EPS | Electric Power Systems |

GA | Genetic Algorithm |

PSO | Particle Swarm Optimization |

## Nomenclature

Indices and sets | |

L | Set of lines |

G | Set of generators |

$M$ | Total resources of the attacker |

$N$ | Set of buses |

$NRD$ | Set of buses with demand response |

${\Psi}_{G}{}^{n}$ | Set of generators connected to node n |

${\Psi}_{D}{}^{n}$ | Set of demands connected to node n |

${\Psi}_{L}{}^{n}$ | Set of lines connected to node n |

Parameters and constants | |

${C}_{g}$ | Cost of the power delivered by generator g |

${C}_{RD}{}_{n}$ | Cost of the dispatchable load at node n |

${C}_{D}{}_{m}$ | Cost of load shedding at node m |

${M}_{l}$ | Cost of attacking a line |

${M}_{g}$ | Cost of attacking a generator |

$c1,$c2 | Uncertainty costs |

${C}_{gd}$ | Cost of demand |

${P}_{g}d$ | Power demanded |

Variables | |

${P}_{g}$ | Power delivered |

${P}_{RD}{}_{n}$ | Power demand response at node n |

${P}_{D}{}_{m}$ | Power Load shedding at node m |

${Q}_{g}$ | Reactive power delivered by generator g |

${\delta}_{L}\left(l\right)$ | Interdiction vector for the set of lines |

${\delta}_{G}\left(g\right)$ | Interdiction vectors for the set of generators |

${P}_{d}$ | Active power demand |

${Q}_{d}$ | Reactive power demand |

${S}_{l}{}^{Br}$ | Apparent power flow in line $\mathrm{l}$ |

Wn | Power scheduled for generator n |

x1, x2 | Power demanded |

m1, m2 | Cost operation results and minimum and maximum power at the loads |

μ1 | Represents the ability of the system to adequately manage the optimal power flow to meet the demand |

μ2 | Shows that the power grid has mechanisms to minimize mandatory load shedding as the worst case scenario for the network operator and the loads |

μ | Quantify respectively a fully resilient and a zero resilient network |

${V}_{n}$ | Voltage magnitude at bus n |

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$\mathit{\mu}$ | Resiliency Degree |
---|---|

$\mu =0$ | None |

$0<\mu \le 0.25$ | Deficient |

$0.25<\mu \le 0.5$ | Poor |

$0.5<\mu \le 0.75$ | Regular |

$0.75<\mu <1$ | Good |

$\mu =1$ | Excellent |

Bus | Type | P Load (MW) | Q Load (MVAR) | Voltage (p. u) |
---|---|---|---|---|

1 | PV | 0 | 0 | 1.07 |

2 | PQ | 300 | 98.61 | 1.08 |

3 | PV | 300 | 98.61 | 1.09 |

4 | Slack | 400 | 131.47 | 1.06 |

5 | PV | 0 | 0 | 1.06 |

Generator | Pg (MW) | Qg (MVAR) | Cost (USD/MWh) |
---|---|---|---|

G1 | 40 | 30 | 14 |

G2 | 170 | 127.5 | 15 |

G3 | 324.5 | 390 | 30 |

G4 | 0 | −10.8 | 40 |

G5 | 470.69 | −165 | 10 |

Resource | Costs of Attacking Elements, DR, and Load Shedding |
---|---|

Cost of attacking a line (USD/line) | 50 |

Cost of attacking a generator (USD/generator) | 100 |

Total resources of the attacker (USD) | 300 |

Load shedding costs buses 2, 3, 4 (USD/MWh) | 100, 100, 400 |

DR at buses 2, 3, 4 (%) | 0, 50, 25 |

Cost of DR at buses 2, 3, 4 (USD/MWh) | 0, 50, 50 |

Attack | Attacked Lines | Attacked Generators | Operation Cost × 10^{5} (USD) | % of Load Served |
---|---|---|---|---|

1 | L1, L2, L5, L6 | G4 | 1.8365 | 52 |

2 | L2, L3, L5, L6 | G4 | 1.7485 | 60 |

3 | L2, L4, L5, L6 | G4 | 1.7203 | 60 |

4 | L2, L5, L6 | G4 | 1.7013 | 60 |

5 | L1, L2, L6 | G3 | 1.4800 | 20 |

6 | L1, L2, L3, L6 | G3 | 1.4800 | 20 |

7 | L1, L2, L4, L6 | G3 | 1.4800 | 20 |

8 | L1, L2, L5, L6 | G3 | 1.4800 | 20 |

9 | L1, L2 | G3, G4 | 1.3287 | 22.34 |

10 | L2, L3, L4, L6 | G3 | 1.3023 | 40.88 |

Bus | Generation (MW) | Generation Cost (USD) | Served Load (MW) | Served Load (%) | Cost of Load Shedding (USD) |
---|---|---|---|---|---|

1 | 0 | 0 | - | - | - |

2 | - | - | 220 | 73.33 | 8000 |

3 | 520 | 15,600 | 300 | 100 | 0 |

4 | 0 | 0 | 0 | 0 | 160,000 |

5 | 0 | 0 | - | - | - |

Attacked Elements | Operation Cost (USD) | Served Load (MW) |
---|---|---|

L1, L2, L5, L6, G4 | 14,464 | 700 |

Node | Power Supplied (MW) | DR (MW) | Load Shedding (MW) | Cost (USD) | Served Load (%) |
---|---|---|---|---|---|

2 | 300 | 0 | 0 | 0 | 100 |

3 | 220 | 80 | 0 | 4000 | 100 |

4 | 0 | 100 | 300 | 125,000 | 25 |

Type | Pmax (MW) | Cost (USD/MW) |
---|---|---|

1 | 100 | 45 |

2 | 300 | 45 |

Location DG-Type | Power Generated (MW) | Operation Cost (USD) | Served Load (MW) | Load Supplied (%) |
---|---|---|---|---|

4(1), 4(2) | 100, 300 | 41,648 | 920 | 92 |

Bus | Active Power (MW) | Load Shedding (MW) | Cost (USD) | Load Supplied (%) |
---|---|---|---|---|

2 | 300 | 0 | 0 | 100 |

3 | 220 | 80 | 8000 | 73.33 |

4 | 400 | 0 | 0 | 100 |

Location DG-Type | DR Location | DR | Operation Cost (USD) | Load Supplied (MW) |
---|---|---|---|---|

4-(1), 4-(2) | 3 | 80 | 37,645 | 1000 |

Served Load (MW) | Operation Cost (USD) | Load Shedding Cost (USD) | |
---|---|---|---|

Scenario 1 | 520 | 183,650 | 168,000 |

Scenario 2 | 700 | 144,645 | 120,000 |

Scenario 3 | 920 | 41,648 | 8000 |

Scenario 4 | 1000 | 37,645 | 0 |

$\mathit{\mu}1$ | $\mathit{\mu}2$ | $\mathit{\mu}$ | Resilience | |
---|---|---|---|---|

Scenario 1 | 0.52 | 0.0852 | 0.3026 | Poor |

Scenario 2 | 0.70 | 0.1703 | 0.4351 | Poor |

Scenario 3 | 0.92 | 0.8079 | 0.8639 | Good |

Scenario 4 | 1 | 1 | 1 | Excellent |

Attacked Lines | Attacked Generators | DG (Bus-Type) | DR (Bus) | |
---|---|---|---|---|

Scenario 1 | 1, 7, 10, 15, 17, 18, 19, 25, 26, 28, 36, 37 | G21, G22 | - | - |

Scenario 2 | 1, 2, 3, 8, 10, 11, 18, 20, 21, 23, 27, 29, 36 | G23 | - | 6, 8, 9, 11, 14, 20 |

Scenario 3 | 1, 7, 10, 15, 17, 18, 19, 25, 26, 28, 36, 37 | G21, G22 | 6-1, 9-1, 12-1, 14-2, 19-2, 24-2 | - |

Scenario 4 | 1, 2, 3, 8, 10, 11, 18, 20, 21, 23, 27, 29, 36 | G23 | 6-1, 9-1, 12-1, 14-2, 19-2, 24-2 | 6, 8, 9, 11, 14, 20 |

Load Served (MW) | Load Served (%) | Load Shedding Cost (USD) | |
---|---|---|---|

Scenario 1 | 1094.5 | 38.40 | 336,464 |

Scenario 2 | 1527.2 | 53.58 | 218,736 |

Scenario 3 | 1318.0 | 46.24 | 255,394 |

Scenario 4 | 1750.7 | 61.42 | 184,442 |

$\mathit{\mu}$ | $\mathit{\mu}2$ | $\mathit{\mu}$ | Resiliency | |
---|---|---|---|---|

Scenario 1 | 0.3840 | 0.052211 | 0.21812 | Deficient |

Scenario 2 | 0.5359 | 0.131355 | 0.33360 | Poor |

Scenario 3 | 0.4625 | 0.078019 | 0.27023 | Poor |

Scenario 4 | 0.6142 | 0.147741 | 0.38095 | Poor |

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**MDPI and ACS Style**

Mosquera Palacios, D.J.; Trujillo, E.R.; López-Lezama, J.M.
Vulnerability Analysis to Maximize the Resilience of Power Systems Considering Demand Response and Distributed Generation. *Electronics* **2021**, *10*, 1498.
https://doi.org/10.3390/electronics10121498

**AMA Style**

Mosquera Palacios DJ, Trujillo ER, López-Lezama JM.
Vulnerability Analysis to Maximize the Resilience of Power Systems Considering Demand Response and Distributed Generation. *Electronics*. 2021; 10(12):1498.
https://doi.org/10.3390/electronics10121498

**Chicago/Turabian Style**

Mosquera Palacios, Darin Jairo, Edwin Rivas Trujillo, and Jesús María López-Lezama.
2021. "Vulnerability Analysis to Maximize the Resilience of Power Systems Considering Demand Response and Distributed Generation" *Electronics* 10, no. 12: 1498.
https://doi.org/10.3390/electronics10121498