# A PSO-Based Approach for Optimal Allocation and Sizing of Resistive-Type SFCLs to Enhance the Transient Stability of Power Systems

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## Abstract

**:**

## 1. Introduction

- (1)
- The optimization problem is solved for two cases, considering three and five RSFCLs due to a trade-off between technical and economic issues. Results are compared based on a detailed analysis.
- (2)
- Two main criteria are considered for the assessment of transient stability enhancement including the critical fault clearing time (CCT) as well as the generators’ maximum rotor angle deviations. Comprehensive simulations and investigations are performed for both criteria.
- (3)
- The three-phase short-circuit fault location is carefully selected regarding the maximum normal condition power flow of the selected transmission line, which will result in a larger power swing in the event of its short-circuit fault occurrence, leading to more serious transient instability issues.

## 2. Resistive Superconducting Fault Current Limiters

_{Shunt}, in parallel with a superconductor resistance of R

_{SC}[11,28]. R

_{Shunt}and R

_{SC}are zero in normal steady-state condition. Once the quenching is occurred, they will have non-zero time-varying values [20]. Total resistance of the RSFCL during a fault depends on the series-connected number of units.

_{0}is the quenching start time, and t

_{1}and t

_{2}are the first and second recovery times, respectively. T

_{SC}is typically considered to be 1 m.s. [11].

## 3. Problem Formulation

#### 3.1. Objective Function

#### 3.2. Decision Variables

#### 3.3. Constraints

## 4. Problem Solution

#### 4.1. Particle Swarm Optimization

_{1}and r

_{2}are random values in [0, 1]. C

_{1}and C

_{2}are typically considered 2.05 (C

_{1}+ C

_{2}= 2.1), and the constriction factor C is determined as follows [43]:

#### 4.2. The Proposed PSO-Based Optimization Algorithm

## 5. Numerical Studies

## 6. Simulation Results and Discussion

#### 6.1. CCT Enhancement

#### 6.2. Improvement of Rotor Angle Deviations

#### 6.2.1. Scenario 1: Optimization of Three Candidate RSFCLs

#### 6.2.2. Optimization of 5 Candidate RSFCLs

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

- Abbreviations

FCL | Fault current limiter |

SFCL | Superconducting fault current limiter |

RSFCL | Resistive-type superconducting fault current limiter |

CCT | Critical fault clearing time |

PSO | Particle swarm optimization |

FCT | Fault clearing time |

p.u. | Per unit |

- B.
- Indices and sets

N_{G} | Set of generation buses |

N_{B} | Set of power system buses |

N_{L} | Set of power system loads |

N_{CR} | Set of candidate RSFCLs |

$C{L}_{RSFCL}$ | Set of locations for candidate RSFCLs |

${S}_{CR}$ | Set of size decision variables (continuous variables) |

${L}_{CR}$ | Set of location decision variables (integers) |

N_{nsg} | Set of non-slack generators |

j | Index of non-slack generators |

X | Set of decision variables, $X=\left[{S}_{CR},{L}_{CR}\right]$ |

i | Index of buses |

- C.
- Parameters and variables

δ | Rotor angle of generator in electrical radians |

P_{m} | Mechanical power input of generator in p.u. |

P_{e} | Electrical power output of generator in p.u. |

H | Inertia constant of generator in MW-s/MVA |

ω_{0} | Nominal speed of generator in electrical radian/s |

OF | Objective function |

δ_{j}(t) | Rotor angle magnitude of the j^{th} generator at time t with reference to the slack generator |

n | Number of generators in the power system |

G_{SL} | Slack bus generator (reference machine) |

CR | Number of candidate RSFCLs to be optimized |

${P}_{Gi},{Q}_{Gi}$ | Generated active power (MW) and reactive power (MVAR) at bus i |

${P}_{Li},{Q}_{Li}$ | Active (MW) and reactive (MVAR) loads at bus i |

${P}_{Gi}^{\mathrm{min}},{Q}_{Gi}^{\mathrm{min}}$ | Minimum active power (MW) and reactive power (MVAR) of generator i |

${P}_{Gi}^{\mathrm{max}},{Q}_{Gi}^{\mathrm{max}}$ | Maximum active power (MW) and reactive power (MVAR) of generator i |

$\left|{V}_{i}\right|,{\theta}_{i}$ | Voltage magnitude (p.u.) and angle (degree) of bus i |

$\left|Vj\right|,{\theta}_{j}$ | Voltage magnitude (p.u.) and angle (degree) of bus j |

$\theta {}_{ref}$ | Voltage angle (degree) of the reference bus |

$\left|{Y}_{ij}\right|,{\phi}_{ij}$ | Admittance Amplitude (p.u.) and angle (degree) of line between buses i and j |

${S}_{RSFCL,k}$ | Value of candidate RSFCLs (p.u.) |

${S}_{RSFCL,k}^{\mathrm{max}}$ | Maximum value of candidate RSFCL (p.u.) |

${L}_{RSFCL,k}$ | Location of the candidate RSFCLs |

$\psi $ | Inertia weight in the PSO |

r_{1} | Cognitive factor |

r_{2} | Social factor |

C | Constriction factor |

C_{1}, C_{2} | Acceleration constants |

${\overrightarrow{v}}_{i}(t)$ | Current velocity of the i^{th} particle |

${\overrightarrow{v}}_{i}(t+1)$ | Next velocity of the i^{th} particle |

${\overrightarrow{x}}_{i}(t)$ | Current position of the i^{th} particle |

${\overrightarrow{x}}_{i}(t+1)$ | Next position of the i^{th} particle |

R_{m} | The maximum resistance that the RSFCL can inject into the power system (p.u.) |

T_{Sc} | The time of transition from the superconducting state to the normal state in the RSFCL |

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**Figure 3.**Possible applications of SFCLs in the power system [32].

**Figure 4.**The δ oscillation curve of a generator at time interval of (0 − T) and its corresponding area.

**Figure 8.**Convergence curve of PSO for the proposed objective function. (

**a**) Scenario 1 (3 RSFCLs), (

**b**) Scenario 2 (5 RSFCLs).

**Figure 10.**CCT of the system before optimization (for the considered short-circuit fault). (

**a**) FCT = 80 m.s. (stable), (

**b**) FCT = 81 m.s. (unstable).

**Figure 11.**CCT in Scenario 1 (3 RSFCLs) for the considered short-circuit fault. (

**a**) FCT = 91 m.s. (stable), (

**b**) FCT = 92 m.s. (unstable).

**Figure 12.**CCT in Scenario 2 (5 RSFCLs) for the considered short-circuit fault. (

**a**) FCT = 226 m.s. (stable), (

**b**) FCT = 227 m.s. (unstable).

**Figure 13.**Rotor angle oscillation of generators in Scenario 1 with reference to the reference machine “with” and “without” optimized RSFCLs. (

**a**) Generator G1, (

**b**) Generator G3, (

**c**) Generator G4, (

**d**) Generator G5, (

**e**) Generator G6, (

**f**) Generator G7, (

**g**) Generator G8, (

**h**) Generator G9, (

**i**) Generator G10.

**Figure 14.**Rotor angle oscillation of generators in Scenario 2 with reference to the reference machine “with” and “without” optimized RSFCLs. (

**a**) Generator G1, (

**b**) Generator G3, (

**c**) Generator G4, (

**d**) Generator G5, (

**e**) Generator G6, (

**f**) Generator G7, (

**g**) Generator G8, (

**h**) Generator G9, (

**i**) Generator G10.

Item | Number of Iterations: | Swarm Size: | PSO Setting Parameters: | Bounds of Variables | |||
---|---|---|---|---|---|---|---|

C_{1} | C_{2} | C | Location | Size | |||

Value | 1000 | 40 | 2.05 | 2.05 | 0.85 | (1–46) | (0–0.025) |

Item | Value |
---|---|

Fault type | Symmetrical three-phase |

Fault location | Line (21–22), close to bus 22 |

Fault clearing time: | 300 (m.s.) |

Scenario | RSFCL Location (bus i–bus i’) | RSFCL Size (p.u.) | Objective Function Value |
---|---|---|---|

Scenario 1: (3 RSFCLs) | (35–22) | 0.013885 | 1939.500 |

(36–23) | 0.021661 | ||

(38–29) | 0.006964 | ||

Scenario 2: (5 RSFCLs) | (35–22) | 0.014138 | 1793.600 |

(36–23) | 0.023494 | ||

(20–19) | 0.024524 | ||

(12–11) | 0.021002 | ||

(39–9) | 0.005074 |

Scenario | Without Employing RSFCLs | Scenario 1 (3 RSFCLs) | Scenario 2 (5 RSFCLs) |
---|---|---|---|

CCT (m.s.) | 80 | 91 | 226 |

**Table 5.**Comparison of generators’ maximum rotor angle deviations in both scenarios for the considered short-circuit fault.

Maximum Rotor Angle Deviation (Degrees) | ||||
---|---|---|---|---|

Scenario 1 (3 RSFCLs) | Scenario 2 (5 RSFCLs) | |||

Generator No. | Without RSFCLs | With RSFCLs | Without RSFCLs | With RSFCLs |

G1 | 27.4 | 24.4 | 42.5 | 42.1 |

G3 | 7.4 | 4.3 | 7.5 | 6.9 |

G4 | 25.9 | 13.5 | 33.7 | 15.1 |

G5 | 22.1 | 11.4 | 31.6 | 23.5 |

G6 | Unstable | 80 | Unstable | 97.1 |

G7 | Unstable | 70.6 | Unstable | 80.4 |

G8 | 14.6 | 9.1 | 14.5 | 13.3 |

G9 | 22.1 | 14.3 | 25.6 | 23.9 |

G10 | 12.8 | 4.7 | 14.4 | 13 |

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**MDPI and ACS Style**

Khatibi, M.; Jalilzadeh, S.; Hussain, A.; Haider, W.
A PSO-Based Approach for Optimal Allocation and Sizing of Resistive-Type SFCLs to Enhance the Transient Stability of Power Systems. *Electronics* **2022**, *11*, 3980.
https://doi.org/10.3390/electronics11233980

**AMA Style**

Khatibi M, Jalilzadeh S, Hussain A, Haider W.
A PSO-Based Approach for Optimal Allocation and Sizing of Resistive-Type SFCLs to Enhance the Transient Stability of Power Systems. *Electronics*. 2022; 11(23):3980.
https://doi.org/10.3390/electronics11233980

**Chicago/Turabian Style**

Khatibi, Masoud, Saeid Jalilzadeh, Arif Hussain, and Waseem Haider.
2022. "A PSO-Based Approach for Optimal Allocation and Sizing of Resistive-Type SFCLs to Enhance the Transient Stability of Power Systems" *Electronics* 11, no. 23: 3980.
https://doi.org/10.3390/electronics11233980