# Optimal System Frequency Response Model and UFLS Schemes for a Small Receiving-End Power System after Islanding

^{*}

## Abstract

**:**

## 1. Introduction

## 2. System Model and Formulation

#### 2.1. Typical System Frequency Response Model

_{L}is increment of load power, ΔP

_{G}is total output of governors, and H is the equivalent inertia. The Load represents the frequency dependent model of load.

#### 2.2. Complex Speed Governor-Prime Mover Model

#### 2.3. Complex Load Model

#### 2.4. Simplified System Frequency Response Model Incorporating UFLS

_{L}, which is the system load frequency damping factor. Governors and prime movers are demonstrated by a short-term first-order model approximation. Generally, the equivalent inertia He and the equivalent gain Ke of the simplified SFR model can be calculated from the individual generator and governor-prime mover as:

_{i}is the inertia constant of the i

^{th}generator with a power base S

_{i}and K

_{i}is the gain of the i

^{th}governor-prime mover with a power base S

_{i}. S is the system power base.

_{shed}is the amount of load shedding. However, D

_{L}and T are obtained with difficultly because of the variety of load dynamic behaviors and the different governors-prime movers installed in the different generators. The model and method used to extract the accurate parameters of the simplified SFR model will be proposed in the next section.

## 3. Models for Parameters’ Identification and UFLS Schemes’ Optimization

#### 3.1. A Review of Particle Swarm Optimization (PSO)

- (1)
- Evaluating the objective value of each particle;
- (2)
- Updating the local and global best objective and positions;
- (3)
- Updating the velocity and the position of each particle.

^{th}particle is updated based on the following Equation:

_{1}and c

_{2}are weighting acceleration constants, ${X}_{pbest}^{\left(k\right)}$ is the best position that a particle of the group has achieved up to now, ${X}_{gbest}^{\left(k\right)}$ is the best position that the whole group has achieved up to now, and r

_{1}and r

_{2}are random parameters ranging from [0,1] , with values produced for each particle in each iteration.

#### 3.2. Parameter Identification Strategy for Simplified SFR

^{s}is the actual output, x is the vector of unknown parameters to be identified and m is the number of recorded data sets.

^{T}are the four unknown parameters to be identified, ${f}_{i}$ and ${f}_{i}^{s}$ are the system frequency response at the COI of the full scale system output and that of the simplified SFR model output, respectively, and n is the number of sampled data.

#### 3.3. Optimal UFLS Scheme Design

- Number of load shedding stages, k
_{min}≤ k ≤ k_{max}; - Limits on frequency thresholds, f
_{th_min}≤ ${f}_{th}^{j}$ ≤ f_{th_max}; - Range of load shedding sizes for each stage, Δd
_{min}≤ Δd^{j}≤ Δd_{max}; - Range of time delay for each stage, t
_{d_min}≤ ${t}_{d}^{j}$ ≤ t_{d_max}; - Margin between two consecutive UFLS threshold frequencies, ${f}_{th}^{j}$ − ${f}_{th}^{j+1}$ ≥ f
_{ε}; - Minimum steady state frequency, f
_{ss}≥ f_{ss_min}; - Minimum transient frequency, ${f}_{ts}^{min}$ ≥ f
_{ts_min}; - Maximum transient frequency, ${f}_{ts}^{min}$ ≤ f
_{ts_max};

_{e}and C

_{ls}are the constant cost coefficient of system energy and load shed, respectively.

#### 3.4. Solving the Proposed Optimization Model Using PSO

**Step 1:**- Input PSO parameters, system data, active power disturbance set, the maxima imported active power, and the upper and lower boundaries of each variable.
**Step 2:**- Time-domain simulation for a period of 50 s is carried out on the full scale model system to determine the system frequency response at COI for the given contingency set.
**Step 3:**- Initialize the swarm comprising the four design parameters, x = [H D K T]
^{T}, of the simplified SFR model, and relay the parameter Z of the UFLS scheme, with all parameters satisfying their constraints. **Step 4:**- For each particle in the swarm, the simplified SFR model is used to determine the system frequency response for the given contingency set.
**Step 5:**- Evaluate the objective g(x) of each particle utilizing the objective function in Equation (6).
**Step 6:**- Find the best position of each particle, pbest, by comparing the evaluation value of each particle with the one in pbest of the last swarm. If the evaluation value is better, then set pbest as gbest.
**Step 7:**- Check the termination condition. If the termination condition is satisfied, go to Step 10.
**Step 8:**- The evolved existing population or swarm velocity and position are developed employing Equations (3) and (4).
**Step 9:**- Return to Step 4 to repeat the evaluation process with the evolved position.
**Step 10:**- Set the latest gbest as the parameters of the simplified SFR model with UFLS.
**Step 11:**- Compute the system frequency response using the simplified SFR model with UFLS for each particle in the swarm.
**Step 12:**- Evaluate the objective F(Z) of each particle using the objective function in Equation (7).
**Step 13:**- Select the gbest of the relay parameters of the UFLS utilizing the process in Step 6.
**Step 14:**- If the termination condition is satisfied, go to Step 17.
**Step 15:**- Update the evolved existing swarm velocity and position using Equations (3) and (4).
**Step 16:**- Return to Step 11 to repeat the evaluation process with the evolved position.
**Step 17:**- The particle that gives the latest gbest has the relay parameters of the optimal UFLS scheme.

## 4. Simulations and Results

_{1}, and c

_{2}are set to 0.8, 2, and 2, respectively.

#### 4.1. Identification Parameters of Simplified SFR after Islanding for Small Receiving-End System

#### 4.2. UFLS Design Using the Identified Simplified SFR Model

_{e}= 1.0 pu/pu, C

_{ls}= 0.1 pu/pu [9]; k

_{min}= 1, k

_{max}= 6; f

_{th_min}= 48.0 Hz, f

_{th_max}= 49.5 Hz; f

_{ε}= 0.1 Hz; Δd

_{max}= 0.3 pu, Δd

_{min}= 0.01 pu; t

_{d_min}= 0.1 s, t

_{d_max}= 0.2 s; f

_{ss_min}= 49.8 Hz, f

_{ts_min}= 48.8 Hz, f

_{ts_max}= 50.2 Hz.

#### 4.3. Time Domain Verification of the Optimal UFLS Schemes

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**General steam turbine speed governing model of IEEE (The parameters are defined in the literature [27]).

**Figure 3.**Typical model of the Hydro Turbine-Governor system (The parameters are defined in the literature [27]).

**Figure 4.**Classical gas turbine governor and prime mover (The parameters are defined in the literature [27]).

**Figure 11.**Comparison between the full scale model and simplified SFR model response: (

**a**) Peak demand; (

**b**) Valley demand.

**Figure 12.**System frequency responses with optimized UFLS schemes under different conditions: (

**a**) Peak demand; (

**b**) Valley demand.

Power Plant | MVA | Inertia Constant | Governor Model in PSS/E |
---|---|---|---|

A | 2 × 150 | 14.1 | HYGOV |

B | 1 × 330 | 4.59 | GAST2A |

C | 2 × 600 | 2.98 | IEEEG1 |

D | 2 × 600 | 2.39 | TGOV5 |

Best | Worst | Mean | Standard Deviation | Average Run Time (s) |
---|---|---|---|---|

0.1046 | 0.2118 | 0.1307 | 0.0334 | 1.27 |

Technology | He | D_{L} | Ke | T |
---|---|---|---|---|

SE and LS | 2.5501 | 0.0570 | 21.07 | 11.34 |

PSO | 2.3872 | 0.0632 | 23.08 | 13.56 |

GA | 2.5011 | 0.0485 | 22.01 | 12.36 |

Number of Load Shedding Stages | Best | Worst | Mean | Standard Deviation | Average Run Time (s) |
---|---|---|---|---|---|

2 | 1.864 | 2.061 | 1.962 | 0.047 | 2.27 |

3 | 1.285 | 1.468 | 1.307 | 0.032 | 6.63 |

4 | 1.517 | 1.736 | 1.594 | 0.054 | 10.78 |

5 | 1.717 | 2.014 | 1.885 | 0.068 | 15.32 |

6 | 2.134 | 2.387 | 2.205 | 0.073 | 20.25 |

Stage | Number of Load Shedding Stages | |||||
---|---|---|---|---|---|---|

3 | 4 | |||||

f_{th} | Δd | t_{d} | f_{th} | Δd | t_{d} | |

(Hz) | (s) | (Hz) | (s) | |||

1 | 49.50 | 7.52% | 0.1 | 49.5 | 6.62% | 0.13 |

2 | 49.31 | 9.70% | 0.1 | 49.3 | 7.11% | 0.15 |

3 | 49.11 | 6.25% | 0.1 | 49.1 | 3.88% | 0.10 |

4 | − | − | − | 48.9 | 5.89% | 0.14 |

5 | − | − | − | − | − | − |

6 | − | − | − | − | − | − |

Obj | 1.285 | 1.517 |

Condition | Peak Demand | Valley Demand | ||||
---|---|---|---|---|---|---|

Scheme | A | B | C | A | B | C |

Amount of Load Shed (MW) | 781.2 | 809.7 | 829.8 | 642.3 | 628.6 | 607.9 |

Steady Frequency (Hz) | 49.88 | 79.89 | 49.90 | 49.96 | 49.91 | 49.88 |

Minimum Frequency (Hz) | 48.92 | 48.85 | 48.81 | 49.09 | 49.10 | 49.02 |

© 2017 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 (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Yang, D.; Liu, S.; Cai, G.
Optimal System Frequency Response Model and UFLS Schemes for a Small Receiving-End Power System after Islanding. *Appl. Sci.* **2017**, *7*, 468.
https://doi.org/10.3390/app7050468

**AMA Style**

Yang D, Liu S, Cai G.
Optimal System Frequency Response Model and UFLS Schemes for a Small Receiving-End Power System after Islanding. *Applied Sciences*. 2017; 7(5):468.
https://doi.org/10.3390/app7050468

**Chicago/Turabian Style**

Yang, Deyou, Shiyu Liu, and Guowei Cai.
2017. "Optimal System Frequency Response Model and UFLS Schemes for a Small Receiving-End Power System after Islanding" *Applied Sciences* 7, no. 5: 468.
https://doi.org/10.3390/app7050468