# Sustainable Development Strategies in Power Systems: Day-Ahead Stochastic Scheduling with Multi-Sources and Customer Directrix Load Demand Response

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## Abstract

**:**

## 1. Introduction

- (1)
- Compared with traditional stochastic optimization methods where the objective function is to minimize the cost of system operation in all scenarios, the proposed model aims to obtain a dispatch solution with a stable operating cost in the base case. This dispatch solution allows for the power system to operate under the forecasted base case and can safely redispatch all units in response to real-time fluctuations in RES output.
- (2)
- We introduce the dynamic characteristics of cascade hydro units and CDL-based DR in day-ahead stochastic scheduling model with RES uncertainty for the first time. The complementary characteristics of wind and solar are considered and modeled using the t-Copula function. These all aim to improve energy efficiency, ensure stable supply, and reduce energy fluctuations through multi-sources complementary and coordinated scheduling.

## 2. Customer Directrix Load

#### 2.1. CDL-Based DR Strategy

#### 2.2. Response Evaluation and Incentives

## 3. Operation Decision Model of Multi-Sources Power System with CDL-Based DR

#### 3.1. Deterministic Scheduling Model

#### 3.1.1. Objective Function

#### 3.1.2. Constraints

_{ht}is the output of hydro unit h at t time; P

_{dt}is the demand value of electric load d at t time;

**SF**is the transfer matrix;

**PL**

_{max}is the maximum capacity matrix on each transfer line; ${P}_{\mathbf{i}\mathbf{t}},{P}_{\mathbf{h}\mathbf{t}},{P}_{\mathbf{w}\mathbf{t}},{P}_{\mathbf{s}\mathbf{t}}$ are the scheduling vectors of thermal power, hydro power, wind power, and solar power, and ${K}_{\mathbf{i}},{K}_{\mathbf{h}},{K}_{\mathbf{w}},{K}_{\mathbf{s}}$ are the correlation matrix of buses with thermal power, hydro power, wind power, and solar power; ${P}_{\mathbf{d}\mathbf{t}}$ is the dispatch vector of bus load; and ${K}_{\mathbf{d}}$ is the incidence matrix of bus load.

_{i}, sd

_{i}are the startup/shutdown costs of unit i, and ${\mathrm{DR}}_{i},{\mathrm{UR}}_{i}$ are the ramp-up and ramp-down rates of unit i.

_{ht}is the reservoir discharge volume of the hydro unit h at the moment of t; V

_{ht}is the storage capacity of the hydro unit h at t time; ${Q}_{h}^{\mathrm{min}}$ is the minimal reservoir discharge volume of the hydro unit h; ${Q}_{h}^{\mathrm{max}}$ is the maximal reservoir discharge volume of the hydro unit h; ${{\displaystyle V}}_{h}^{\mathrm{min}},{{\displaystyle V}}_{h}^{\mathrm{max}}$ is the minimum/maximum storage capacity of the hydro unit h, and r

_{ht}is the natural incoming water volume of the hydro unit h at t time.

_{ht}is the water head level of the hydro unit h at time t, the value of which is related to the physical dimensions of the terraced hydroelectric power plant, h

_{0,h}, α

_{h}is the physical constant with respect to the hydro unit h, and η

_{h}is the water-to-power conversion factor.

#### 3.1.3. Piecewise Linearization of the Hydropower Conversion Function

#### 3.1.4. Abstract Formulation

#### 3.2. Stochastic Scheduling Model

## 4. Generation and Reduction of Uncertainty Scenario Sets

#### 4.1. Scenario Generation

#### 4.2. Scenario Reduction

## 5. Case Studies

#### 5.1. Deterministic Scheduling Case

#### 5.1.1. Case 1

#### 5.1.2. Case 2

#### 5.2. Stochastic Scheduling Case

#### 5.2.1. Case 3

#### 5.2.2. Case 4

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 6.**Relevant data under different RES penetration rates: (

**a**) loss of load power; (

**b**) curtailed RES power.

**Figure 7.**RES generation scenarios via Copula theory. (

**a**) Wind generation scenarios; (

**b**) solar generation scenarios.

**Figure 8.**CDL profiles and actual profiles corresponding to different values of $E$. (

**a**) Per unit value of CDL profiles; (

**b**) per unit value of actual customers’ load profiles.

**Figure 9.**RES generation scenarios under the lower forecasting errors. (

**a**) Wind generation scenarios; (

**b**) solar generation scenarios.

Unit | Lower (MW) | Upper (MW) | Min Up/Down (h) | Ramp (MW/h) | Corrective (MW) |
---|---|---|---|---|---|

G1 | 50 | 150 | 8 | 100 | 80 |

G2 | 30 | 80 | 4 | 40 | 30 |

G3 | 10 | 50 | 3 | 40 | 15 |

Unit | a (MBtu) | b (MBtu/MWh) | c (MBtu/MW2h) | Start-Up Fuel (MBtu) | Fuel Price ($/Mbtu) |
---|---|---|---|---|---|

G1 | 0.0044 | 13.29 | 39 | 100 | 2.5 |

G2 | 0.0459 | 15.47 | 74.33 | 40 | 2.5 |

G3 | 0.0080 | 14.50 | 42 | 40 | 2.5 |

Line | From | To | X (p.u.) | Flow Limite (MW) |
---|---|---|---|---|

L1 | 1 | 2 | 0.0370 | 200 |

L2 | 1 | 4 | 0.0160 | 200 |

L3 | 2 | 3 | 0.1015 | 175 |

L4 | 2 | 4 | 0.1170 | 175 |

L5 | 3 | 6 | 0.0355 | 175 |

L6 | 4 | 5 | 0.0370 | 200 |

L7 | 5 | 6 | 0.1270 | 200 |

Unit | H1 | H2 |
---|---|---|

Efficiency | 6.197 | 6.465 |

H_{0} | 0.82679 | 0.58434 |

$\alpha $ | 4.2 × 10−4 | 1.15 × 10^{−3} |

Max discharge (m^{3}) | 2 × 10^{5} | 2 × 10^{5} |

Min discharge (m^{3}) | 0 | 0 |

Max volume (m^{3}) | 2.4 × 10^{6} | 3.0 × 10^{6} |

Min volume (m^{3}) | 1.0 × 10^{6} | 1.2 × 10^{6} |

Ramp (MW/h) | 60 | 60 |

Min on/off time (h) | 1 | 1 |

Lower bund (MW) | 7 | 7 |

Upper (MW) | 115 | 120 |

Nature inflow (m^{3}) | 1.5 × 10^{5} | 5 × 10^{4} |

RES Penetration | Total Costs ($) | Loss of Load Penalty ($) | Curtailed RES Output Costs ($) | Thermal Power Generation Costs ($) |
---|---|---|---|---|

0 | 8,133,874.52 | 7,894,789.48 | 0 | 239,085.03 |

0.1 | 4,385,292.05 | 4,102,886.12 | 49,335.65 | 233,070.28 |

0.2 | 3,894,427.85 | 3,528,935.00 | 134,761.06 | 230,731.79 |

0.3 | 3,737,378.09 | 3,300,696.17 | 216,548.96 | 220,132.96 |

0.4 | 3,687,348.71 | 3,164,161.34 | 308,954.21 | 214,233.16 |

0.5 | 3,777,943.05 | 3,164,161.34 | 404,495.06 | 209,286.65 |

0.6 | 3,867,915.24 | 3,164,161.34 | 499,074.22 | 204,679.68 |

RES Penetration | Total Costs ($) | Loss of Load Penalty ($) | DR Costs ($) | Curtailed RES Output Costs ($) | Thermal Power Generation Costs ($) |
---|---|---|---|---|---|

0.1 | 228,093.07 | 0 | 16,515.37 | 0 | 211,577.70 |

0.3 | 286,240.79 | 0 | 27,775.37 | 86,842.88 | 171,622.54 |

0.6 | 529,491.71 | 0 | 28,715.36 | 347,585.02 | 153,191.33 |

$\mathbf{Scenario}\mathsf{\xi}$ | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

${\rho}^{\xi}$ | 0.205 | 0.237 | 0.179 | 0.225 | 0.154 |

$\mathit{E}$ | Total Costs ($) | Loss of Load Penalty ($) | DR Costs ($) | Curtailed RES Output Costs ($) | Thermal Power Generation Costs ($) |
---|---|---|---|---|---|

0.75 | 786,499.26 | 347,849.23 | 9979.60 | 189,504.68 | 239,165.75 |

0.80 | 637,888.21 | 204,554.24 | 10,966.03 | 187,600.48 | 234,767.46 |

0.85 | 551,383.35 | 134,914.78 | 12,007.54 | 175,994.67 | 228,466.36 |

0.90 | 450,929.18 | 59,744.99 | 13,179.74 | 163,780.85 | 214,223.60 |

0.95 | 372,115.47 | 0 | 14,655.71 | 152,900.71 | 204,559.05 |

0.99 | 326,324.58 | 0 | 16,559.33 | 132,819.44 | 176,945.81 |

Total Costs ($) | Loss of Load Penalty ($) | DR Costs ($) | Curtailed RES Output Costs ($) | Thermal Power Generation Costs ($) |
---|---|---|---|---|

418,091.17 | 49,232.03 | 13,179.74 | 163,324.90 | 192,354.50 |

Cases | 1 | 2 | 3 | 4 |
---|---|---|---|---|

Time (s) | 4.09 | 4.78 | 458.59 | 145.41 |

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**MDPI and ACS Style**

Liu, J.; Huang, S.; Shuai, Q.; Gu, T.; Zhang, H.
Sustainable Development Strategies in Power Systems: Day-Ahead Stochastic Scheduling with Multi-Sources and Customer Directrix Load Demand Response. *Sustainability* **2024**, *16*, 2589.
https://doi.org/10.3390/su16062589

**AMA Style**

Liu J, Huang S, Shuai Q, Gu T, Zhang H.
Sustainable Development Strategies in Power Systems: Day-Ahead Stochastic Scheduling with Multi-Sources and Customer Directrix Load Demand Response. *Sustainability*. 2024; 16(6):2589.
https://doi.org/10.3390/su16062589

**Chicago/Turabian Style**

Liu, Jiacheng, Shan Huang, Qiang Shuai, Tingyun Gu, and Houyi Zhang.
2024. "Sustainable Development Strategies in Power Systems: Day-Ahead Stochastic Scheduling with Multi-Sources and Customer Directrix Load Demand Response" *Sustainability* 16, no. 6: 2589.
https://doi.org/10.3390/su16062589