# Cold Load Pickup Model Adequacy for Power System Restoration Studies

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## Abstract

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## 1. Introduction

- To provide parameter fits of CLPU models for additional disturbance events beyond those 10 published in [8], now totalling 31 events;
- To provide parameter fits based on actual measurements of a real load composition to the delayed exponential decay model of CLPU according to [13], where it was analytically derived for space heating only;
- To determine the degree to which the load profile, caused by CLPU, can be adequately represented using simplified models and to quantify the impact of these simplifications on simulation results with respect to frequency nadir and active power sharing in the use case demonstrated in [12] with a grid structure similar to [14,15].

## 2. Cold Load Pickup Models

#### 2.1. Exponential Decay Models

#### 2.2. Other Models

#### 2.3. Interaction of CLPU with Time Variable Demand

## 3. Measurement Data

- Outage duration at least 5 min;
- Reconnection after the outage occurred with (nearly) the same switching status as before the disturbance;
- Measurements are available for at least 30 min, requiring an unchanged switching status after reconnection;
- The time resolution of the measurements is at least 1 measurement per minute.

## 4. Parameter Determination

#### 4.1. Under-Determined Parameter Fit

#### 4.2. Distribution of Parameters

#### 4.3. Correlation of Outage Duration and Parameters

## 5. Simulation Model for Restoration Study

- Operation of a small island system energized by a blackstart capable gas turbine power plant (blackstart unit, BSU) with a directly coupled synchronous generator and significant renewable generation behaving according to current German grid connection standards for inverter coupled units;
- Active power sharing among generators based on the $P\left(f\right)$-characteristics of the generators;
- Reconnection of unsupplied grid areas subject to CLPU.

#### 5.1. System Model

#### 5.2. Gas Turbine Power Plant Model

#### 5.3. Renewable Generation Model

#### 5.4. Load Model

## 6. Simulation Case Study

#### 6.1. Base Case without Any CLPU

#### 6.2. Comparison Case with CLPU

#### 6.3. Sensitivity Study: Variation of CLPU Modelling and Parameters

- Application of recorded current time series;
- Exponential decay (ED) model;
- Delayed exponential decay (DED) model;
- Active power step to the expected initial load;
- Active power step directly to the expected steady-state load value.

- The frequency nadir ${f}_{min}$ should safely remain above the relevant threshold, which is (in the European case) 49 Hz if under frequency load shedding is to be avoided and 47.5 Hz if only the disconnection of generators and subsequent network collapse is of concern.
- The steady-state frequency indicates how much CLPU influences the steady state of the power system. In the case without CLPU, it is very close to 50.2 Hz.
- The steady-state power ${P}_{end}$ of the synchronous generator is relevant for stable operation of the gas turbine power plant. Without CLPU, at the chosen initial operating point of the plant, it is always above 14.3 MW. Uncertainty of CLPU behaviour creates an additional need to operate the plant at a higher initial power and thus further away from critical minimum load.

## 7. Discussion and Outlook

#### Regulatory Approaches to Reduce CLPU

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AVR | Automatik Voltage Regulator |

BSU | Blackstart Unit |

CLPU | Cold Load Pickup |

ED | Exponential Decay |

DED | Delayed Exponential Decay |

OHL | Overhead Line |

PLL | Phase-Locked Loop |

PV | Photovoltaic |

RMS | Root Mean Square |

RMSE | Root Mean Square Error |

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**Figure 1.**Exponential decay of CLPU with and without delay, $a=1$ and $\tau =1$, ${t}_{d}=0$ or ${t}_{d}=1$, respectively.

**Figure 2.**Combination of time-dependent “normal load” and CLPU with $a=1$ and $\tau =1\phantom{\rule{0.166667em}{0ex}}\mathrm{h}$.

**Figure 5.**Example of CLPU curve without delay and with two different values of ${t}_{d}$ that fit equally well to measurements.

**Figure 6.**Distribution of fitted model parameters: Red crosses are outliers beyond 1.5 time the interquartile range. The y-axis is compressed for values beyond 3.

**Figure 12.**Base case simulation of load connection subject to CLPU (${P}_{0}=5\phantom{\rule{0.166667em}{0ex}}\mathrm{MW}$, $a=2.5$, $\tau =5\phantom{\rule{0.166667em}{0ex}}\mathrm{min}$ and ${t}_{d}=3\phantom{\rule{0.166667em}{0ex}}\mathrm{min}$).

**Figure 13.**Distribution of fitted model parameters: Red crosses are outliers beyond 1.5 time the interquartile range. Distribution of frequency nadir and deviation of frequency nadir and steady-state power in recorded time series and respective deviation using ED or DED CLPU models or approximating load steps to the steady-state value of 5 MW or the maximal value $(a+1)\xb75$ MW, respectively.

Parameter | Mean | Median | Standard Deviation |
---|---|---|---|

a in p.u. (ED model) | 0.63 | 0.62 | 0.39 |

$\tau $ in hours (ED model) | 776 | 0.16 | 3182 |

a in p.u. (DED model) | 0.59 | 0.59 | 0.39 |

$\tau $ in hours (DED model) | 521 | 0.15 | 2804 |

${t}_{d}$ in hours (DED model) | 0.05 | 0.00 | 0.09 |

Parameter | Correlation with Outage Duration in DE Model | Correlation with Outage Duration in DED Model |
---|---|---|

a in p.u. | −0.05 | −0.06 |

$\tau $ in hours | −0.17 | −0.21 |

${t}_{d}$ in hours | N/A | 0.15 |

**Table 3.**Simulation results and mean deviation depending on CLPU model: RMSE between simulation result with respective model and simulation results using recorded current time series.

CLPU Model | ${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$ in Hz | ${\mathit{f}}_{\mathit{e}\mathit{n}\mathit{d}}$ in Hz | ${\mathit{P}}_{\mathit{e}\mathit{n}\mathit{d}}$ in MW |
---|---|---|---|

Mean result | 49.72 | 50.24 | 11.62 |

Standard deviation of result | 0.176 | 0.023 | 1.73 |

RMSE ED model | 0.180 | 0.025 | 1.78 |

RMSE DED model | 0.183 | 0.023 | 1.69 |

RMSE step to maximal load $(5\xb7(a+1)$ MW) | 0.180 | 0.043 | 3.21 |

RMSE step to steady-state load (5 MW) | 0.304 | 0.043 | 3.22 |

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**MDPI and ACS Style**

Hachmann, C.; Becker, H.; Braun, M.
Cold Load Pickup Model Adequacy for Power System Restoration Studies. *Energies* **2022**, *15*, 7675.
https://doi.org/10.3390/en15207675

**AMA Style**

Hachmann C, Becker H, Braun M.
Cold Load Pickup Model Adequacy for Power System Restoration Studies. *Energies*. 2022; 15(20):7675.
https://doi.org/10.3390/en15207675

**Chicago/Turabian Style**

Hachmann, Christian, Holger Becker, and Martin Braun.
2022. "Cold Load Pickup Model Adequacy for Power System Restoration Studies" *Energies* 15, no. 20: 7675.
https://doi.org/10.3390/en15207675