# Coordinated Design of Type-2 Fuzzy Lead–Lag-Structured SSSCs and PSSs for Power System Stability Improvement

^{1}

^{2}

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^{*}

## Abstract

**:**

## 1. Introduction

- i.
- A new controller structure, known as type-2 fuzzy lead–lag (T2FLL), is proposed in this paper for PSS- and SSSC-based controllers for power system stability improvement.
- ii.
- The design task is taken as an optimization problem, and controller parameters are optimized by a recently proposed hADE-PS method.
- iii.
- Technique-wise, the hDE-PS method is compared with GA, PSO, and DE methods, and controller-wise, T2FLL is compared with type-1 fuzzy lead–lag (T1FLL) and widely used lead–lag (LL) controllers.
- iv.
- Various disturbance scenarios and changed loading/fault locations are simulated for both single-machine infinite-bus (SMIB) and MMPS, and it is seen that improved damping is attained with T2FLL related to T1FLL and LL controllers for all scenarios.

## 2. Systems under Investigation

#### 2.1. The SMIB System

_{S}and V

_{T}, respectively; while V

_{1}and V

_{2}are the bus voltages, the SSSC converter output voltage and the DC voltage source are represented by V

_{cnv}and V

_{DC}, respectively; the line current is I; the real powers in the transmission lines and one line are each represented by P

_{L}and P

_{L1}, respectively. The generator is provided with a turbine and governor, an excitation system, and a PSS. The excitation system contains a voltage regulator and exciter [37]. The system parameters are specified in Appendix A.

#### 2.2. Kundur’s Test System (Four-Machine Two-Area)

## 3. The Proposed Approach

#### 3.1. Type-2 Fuzzy Logic Overview

_{Var}

_{Var}is expressed as:

#### 3.2. Structure of T2FLL Controller

_{q}.

#### 3.3. Optimization Problem

_{L}and Δω

_{I}are the speed deviations of MMPS related to local and inter-area modes, respectively, and t

_{sim}is the simulation time.

## 4. Overview of Hybrid Adaptive DE and PS Technique

#### 4.1. Mutation

#### 4.2. Crossover

#### 4.3. Selection

^{i}and Cr

^{i}, mx (maximum), mn (minimum), and md (median) are engaged in a generation as indicated in Equations. (21) and (22).

^{i}and Cr

^{i}values are set which affects the mutation, cross-over, and selection process for the newly generator vector ${y}_{}^{j}{|}_{i+1}$. Due to the easy generation of adaptive values for F and Cr, it can maintain the time complexity as compared to conventional DE.

## 5. Outcomes

#### 5.1. SMIB System

^{−3}) are obtained with the hADE-PS technique compared to GA (44.3216 × 10

^{−3}), PSO (40.9781 × 10

^{−3}) and DE (37.7718 × 10

^{−3}) techniques. So with the identical system and controller structure, the percentage reduction in ITAE value with hADE-PS compared to GA, PSO, and DE techniques are 23.56%, 17.53%, and 10.53%, respectively. This demonstrates the dominance of hADE-PS over GA, PSO, and DE techniques. The speed deviation response is revealed in Figure 6, from which it is evident that the best system response is attained with hADE-PS related to GA, PSO, and DE.

^{−3}with T1FLL, and the least ITAE of 28.6961 × 10

^{−3}is attained with the T2FLL controller. So there is a reduction of 10.13% and 15.08% in J value with T2FLL related to the T1FLL and LL controller, respectively.

#### 5.1.1. Scenario 1: Large Disturbance Condition

_{0}= 48.5°). A 3-phase fault of 5 cycles is applied at the midpoint of one transmission line. The fault is removed by opening the line for 5 cycles. The responses with the hADE-PS-tuned LL, T1FLL, and T2FLL controllers are revealed in Figure 7a–e. For comparison, the responses without any control are also shown in Figure 7a–e. Figure 7a–e displays the speed deviation response ($\Delta \omega $) in p.u., power angle response (δ) in degrees, tie-line power in the line (P

_{L}) in MW SSSC output voltage (V

_{q}) in p.u., and PSS output (V

_{s}) in p.u. under above severe disturbance.

#### 5.1.2. Scenario 2: Small Disturbance Condition

#### 5.1.3. Scenario 3: Changed Loading Condition and Fault Location

_{0}= 38.2°) condition. Under this contingency, Δω variation is revealed in Figure 9. It is noticed that oscillations are damped quickly with T2FLL in comparison with the T1FLL and LL controllers and without the controllers.

#### 5.2. Extension to Multi-Machine System (MMPS)

#### 5.2.1. Scenario 1: Three Phase Fault Disturbance Condition

#### 5.2.2. Scenario 2: Line Outage Disturbance Condition

#### 5.2.3. Scenario 3: Small Disturbance Condition

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

SMIB Parameters |

Generator: Nominal power (S_{B}),voltage (V_{B}) and frequency (f): 2100 MVA, 13.8 kV, 60 HzParameters: Stator resistance (R _{S}): 2.8544 × 10^{−3}, Reactances: X_{d} = 1.305, X_{q} = 0.474, X’_{d} = 0.296, X’_{q} = 0.243, X’’_{d} = 0.252, X’’_{q} = 0.18, T_{d} = 1.01s, T’_{d} = 0.053s, T’’_{qo} = 0.1s.Inertia constant (H) and pole pairs (p): 3.7 s and 32 Excitation System:Gains (K _{A}) and time constant (T_{A}) of regulator: 200 and 0.001 sGains (K _{e}) and time constant (T_{e}) of exciter: 1 and 0 sGains (K _{f}) and time constant (T_{f}) of damping filter: 0.001 and 0.1 sLow-pass filter time constant (T _{LP}): = 0.02 s, Transient gain reduction (T _{b}, T_{c}): 0Regulator output limits (E _{fmax}/E_{fmin}) and gain (K_{p}): 7/0 and 0Hydraulic Turbine and Governor:Gains (K _{a}) and time constant (T_{a}) of Servo-motor: 3.33 and 0.07 sLimits of Gate opening (G _{max}/G_{min} and V_{gmax}/V_{gmin}): 0.97518/0.01 and 0.01/−0.1 pu/sPermanent droop (R _{p}): = 0.05, Hydraulic turbine: $\beta $ = 0, ${T}_{w}$ = 2.67 sPID regulator (K_{p}, Ki, Kd, T_{d}): 1.163, 0.105, 0, 0.01 sTransformer: Nominal power (S _{B}) = 2100 MVA, Winding connection: D_{1}/Y_{g}, Primary and secondary voltage (V_{1}/V_{2}): 13.8/500 kV, Resistance (R): 0.002 p.u., Inductances (L_{1}/L_{2}): 0/0.12, Magnetization resistance (R_{m}) and reactance (L_{m}): 500 Ω Transmission line:Line length and no. of phases: 300 km, 3-Ph, Resistance per unit length (R _{1}/R_{0}): 0.02546/0.3864 Ω/ km, Inductance per unit length (L_{1}/L_{0}): 0.9337 × 10^{−3} /4.1264 × 10^{−3} H/km, Capacitance per unit length (C_{1}/C_{0}): 12.74 × 10^{−9}/7.751 × 10^{−9} F/ kmLoad at Bus2:250 MW (500 kV, 60 Hz, Y-grounded) |

Kundur’s 4-machine 2-area system |

Generator: Nominal powers: 900 MVA each, Nominal voltages: 20 kV each, frequency (f): 60 HzParameters: Stator resistance (R _{S}): 2.8544 × 10^{−3}, Reactances: X_{d} = 1.8, X_{q} = 1.7, X’_{d} = 0.3, X’_{q} = 0.55, X’’_{d} = 0.25, X’’_{q} = 0.25, T’_{do} = 8 s, T’’_{do} = 0.03 s, T’_{qo} = 0.4 s, T’’_{qo} = 0.05 s, Stator resistance (R_{S}): 0.0025 ΩExcitation Systems: Each same as SMIB systemSteam Turbine and Governor:Regulator Gain (K _{p}) = 1, Permanent drooop (R_{p}): 0.05, Dead zone (D_{z}): 0, Speed relay and servo-motor time constants (T_{sr}/T_{sm}): 0.001/0.15 s, Limits of gate opening (G_{max}/G_{min} and V_{gmax}/V_{gmin}): 4.496/0 and 0.01/−0.1 pu/s, time constants of steam turbine (T_{1}, T_{2}, T_{3}, T_{4}): 0, 10, 3.3, 0.5 s, Turbine torque fractions (F_{1}, F_{2}, F_{3}, F_{4}): 0, 0.36, 0.36, 0.28Transformers: Nominal powers = 900 MVA each, Winding connection: D _{1}/Y_{g}, Primary and secondary voltage (V_{1}/V_{2}): 20/230 kV, Resistance (R): 1 × 10^{−6}, Inductances (L_{1}/L_{2}): 0/0.15, Magnetization resistance (R_{m}) and reactance (L_{m}): 500 ΩTransmission lines:Distributed parameter line (110 km line sections) and PI section line (10 km and 25 km line sections) Line length and n0. of phases: 300 km, 3-Ph, Resistance per unit length (R _{1}/R_{0}): 0.0529/1.61 Ω/km, Inductance per unit length (L_{1}/L_{0}): 0.0014/0.0061 H/km, Capacitance per unit length (C_{1}/C_{0}): 8.7749 × 10^{−9}/5.2489 × 10^{−9} F/kmLoads:Area-1: Active power: 967 MW, Inductive reactive power: 100 MVAR, Capacitive reactive power: 387 MVARArea-2: Active power: 1767 MW, Inductive reactive power: 100 MVAR, Capacitive reactive power: 537 MVAR |

SSSC Data |

Converter rating: = 100 MVA, Nominal voltage: 500 kV, Frequency (f): 60 Hz, Maximum rate of change in reference voltage = 3 pu/s, Converter impedances (R/L): 0.00533/0.16, DC link voltage: 40 kV, DC link equivalent capacitance: 375 × 10^{−6} F, Injected Voltage regulator gains (K_{P} /K_{I}): 0.00375/0.1875, DC Voltage regulator gains (K_{P}/K_{I} ): 0.1 × 10^{−3}/20 × 10^{−3}, Limits of injected voltage: ± 0.2 |

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**Figure 7.**System response of SMIB for Scenario 1: (

**a**) Δω, (

**b**) Δδ, (

**c**) P

_{L,}(

**d**) SSSC output, (

**e**) PSS output.

**Figure 12.**System response of MMPS system for Scenario 1: (

**a**) local mode; (

**b**) inter-area mode. (

**a**) local mode of oscillation of MMPS for Scenario 1. (

**b**) Inter-area mode of oscillation of MMPS for Scenario 1.

**Figure 13.**System response of MMPS system for Scenario 2: (

**a**) local mode; (

**b**) inter-area mode. (

**a**) local mode of oscillation of MMPS system for Scenario 2. (

**b**) Inter-area mode of oscillation of MMPS system for Scenario 2.

**Figure 14.**System response of MMPS system for Scenario 3: (

**a**) local mode; (

**b**) inter-area mode. (

**a**) Local mode of oscillation of MMPS system for Scenario 3. (

**b**) Inter-area mode of oscillation of MMPS system for Scenario 3.

ė | EXN | LN | ZER | LP | EXP | |
---|---|---|---|---|---|---|

e | ||||||

EXN | EXN | EXN | LN | LN | ZER | |

LN | EXN | LN | EXN | ZER | LP | |

ZER | LN | LN | ZER | LP | LP | |

LP | LN | ZER | LP | LP | EXP | |

EXP | ZER | LP | LP | EXP | EXP |

Technique | Controller | K_{i} | T_{1i} | T_{2i} | T_{3i} | T_{4i} | J Value × 10^{−3} |
---|---|---|---|---|---|---|---|

GA | SSSC | 74.0740 | 0.4286 | 0.0511 | 0.4523 | 1.1098 | 44.3216 |

PSS | 19.8294 | 0.1415 | 0.0011 | 0.0630 | 1.2736 | ||

PSO | SSSC | 38.5588 | 1.0653 | 0.3218 | 0.6751 | 0.7496 | 40.9781 |

PSS | 24.2563 | 1.5546 | 1.4223 | 0.0134 | 0.4930 | ||

DE | SSSC | 50.1218 | 0.8175 | 1.1379 | 1.4961 | 0.4326 | 37.7718 |

PSS | 16.0860 | 0.1553 | 0.7317 | 1.7974 | 1.7974 | ||

hADE-PS | SSSC | 71.5265 | 1.2006 | 1.3051 | 0.0011 | 0.0012 | 33.7926 |

PSS | 18.2475 | 0.0011 | 0.5204 | 0.0011 | 0.4981 |

Controller | KSF_{1} | KSF_{2} | K_{i} | T_{1i} | T_{2i} | T_{3i} | T_{4i} | J Value × 10^{−3} |
---|---|---|---|---|---|---|---|---|

T2FLL controller | ||||||||

SSSC | 0.0105 | 0.1824 | 79.5279 | 1.9974 | 1.5774 | 1.0292 | 0.4604 | 28.6961 |

PSS | 0.0104 | 0.7241 | 49.9350 | 0.0622 | 1.6995 | 0.0012 | 1.9974 | |

T1FLL controller | ||||||||

SSSC | 1.9093 | 1.4684 | 14.2391 | 0.0154 | 0.4538 | 1.9974 | 0.9935 | 31.9342 |

PSS | 0.3209 | 1.2285 | 30.5106 | 0.2421 | 1.7827 | 0.4172 | 1.9974 |

Scenario/Controller | ISE (×10 ^{−6}) | ITAE (×10 ^{−2}) | ITSE (×10 ^{−3}) | IAE (×10 ^{−6}) | Overshoots in Δω (×10 ^{−3}) | Undershoots in Δω (×10 ^{−3}) | |
---|---|---|---|---|---|---|---|

Scenario 1 | NC | 256.6942 | 244.0563 | 43.0972 | 1493.7607 | 7.786 | −8.6124 |

LL | 7.7379 | 3.3792 | 2.3878 | 10.1409 | 5.0881 | −4.4641 | |

T1FLL | 6.8863 | 3.1934 | 2.0871 | 8.8434 | 4.998 | −4.3516 | |

T2FLL | 5.8701 | 2.86961 | 2.0413 | 7.4721 | 5.0363 | −3.3831 | |

Scenario 2 | NC | 9.3642 | 42.5216 | 8.1817 | 45.9841 | 1.7148 | 1.7312 |

LL | 0.1951 | 0.4901 | 0.3586 | 0.2514 | 0.8321 | −0.7049 | |

T1FLL | 0.1668 | 0.4862 | 0.3398 | 0.2163 | 0.7148 | −0.5189 | |

T2FLL | 0.1162 | 0.4618 | 0.3142 | 0.1528 | 0.6732 | −0.3972 | |

Scenario 3 | NC | 19.6749 | 30.5829 | 9.0338 | 48.6916 | 4.6224 | −4.3954 |

LL | 2.6978 | 1.825 | 1.3121 | 3.3519 | 4.5324 | −2.8929 | |

T1FLL | 2.4873 | 1.8011 | 1.2967 | 3.0572 | 4.5324 | −2.2581 | |

T2FLL | 2.2042 | 1.8169 | 1.2816 | 2.73604 | 4.5324 | −1.8272 |

Controller | K_{i} | T_{1i} | T_{2i} | T_{3i} | T_{4i} |
---|---|---|---|---|---|

SSSC | 92.8460 | 0.0885 | 0.0308 | 4.5494 | 7.4982 |

PSS1 | 46.8908 | 0.0430 | 0.0182 | 2.7794 | 8.8044 |

PSS2 | 48.0950 | 0.0114 | 0.0155 | 3.9735 | 3.5212 |

PSS3 | 45.8967 | 0.0712 | 0.0128 | 3.7749 | 3.2167 |

PSS4 | 27.6544 | 0.0974 | 0.0319 | 3.8884 | 3.2087 |

Controller | KSF_{1} | KSF_{2} | K_{i} | T_{1i} | T_{2i} | T_{3i} | T_{4i} |
---|---|---|---|---|---|---|---|

T2FLL controller | |||||||

SSSC | 0.0102 | 0.0148 | 41.0552 | 0.0990 | 0.0064 | 1.5070 | 2.9408 |

PSS1 | 0.7591 | 0.0126 | 24.6432 | 0.0193 | 0.0066 | 2.4321 | 2.3374 |

PSS2 | 1.9797 | 0.0468 | 16.5071 | 0.0394 | 0.0104 | 1.9728 | 1.5423 |

PSS3 | 0.0765 | 0.0284 | 37.0997 | 0.0992 | 0.0059 | 2.4474 | 3.0812 |

PSS4 | 1.2245 | 0.2568 | 11.0542 | 0.0155 | 0.0051 | 1.7857 | 0.9991 |

T1FLL controller | |||||||

SSSC | 0.2323 | 0.5164 | 2.7141 | 0.0103 | 0.0101 | 4.9493 | 2.0462 |

PSS1 | 1.0206 | 0.0871 | 11.7451 | 0.0194 | 0.0066 | 3.2127 | 3.8817 |

PSS2 | 1.5428 | 0.4353 | 40.6019 | 0.0415 | 0.0383 | 2.3425 | 3.8154 |

PSS3 | 0.9495 | 0.0100 | 49.4935 | 0.0121 | 0.0060 | 4.2398 | 4.4935 |

PSS4 | 1.4140 | 0.0492 | 35.8082 | 0.0115 | 0.0239 | 2.4098 | 3.6268 |

Scenario/Controller | Scenario 1 (×10 ^{−3}) | Scenario 2 (×10 ^{−3}) | Scenario 3 (×10 ^{−3}) |
---|---|---|---|

NC | 20,806.0798 | 21,201.1951 | 21,040.6094 |

LL | 14.5224 | 12.8586 | 17.4648 |

T1FLL | 12.6361 | 11.8245 | 13.8912 |

T2FLL | 9.9952 | 10.6961 | 10.5976 |

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## Share and Cite

**MDPI and ACS Style**

Khampariya, P.; Panda, S.; Alharbi, H.; Abdelaziz, A.Y.; Ghoneim, S.S.M.
Coordinated Design of Type-2 Fuzzy Lead–Lag-Structured SSSCs and PSSs for Power System Stability Improvement. *Sustainability* **2022**, *14*, 6656.
https://doi.org/10.3390/su14116656

**AMA Style**

Khampariya P, Panda S, Alharbi H, Abdelaziz AY, Ghoneim SSM.
Coordinated Design of Type-2 Fuzzy Lead–Lag-Structured SSSCs and PSSs for Power System Stability Improvement. *Sustainability*. 2022; 14(11):6656.
https://doi.org/10.3390/su14116656

**Chicago/Turabian Style**

Khampariya, Prabodh, Sidhartha Panda, Hisham Alharbi, Almoataz Y. Abdelaziz, and Sherif S. M. Ghoneim.
2022. "Coordinated Design of Type-2 Fuzzy Lead–Lag-Structured SSSCs and PSSs for Power System Stability Improvement" *Sustainability* 14, no. 11: 6656.
https://doi.org/10.3390/su14116656