# Optimal Scheduling of Power System Incorporating the Flexibility of Thermal Units

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## Abstract

**:**

## 1. Introduction

- (1)
- Considering both subjective and objective factors, a comprehensive evaluation index system for TPU flexibility is designed which covers the technical and economic characteristics of TPU.
- (2)
- A power system optimal scheduling model that considers the flexibility of each unit is proposed. The model takes both the economics and flexibility of the system operation into account which compensates for the greater amount of net load fluctuations and makes the scheduling plan more transparent.
- (3)
- The model presented in this article can increase the market competitiveness of flexible units and thus increase the power system flexibility.

## 2. Flexible Regulation Characteristics of the TPU

#### 2.1. Technical Characteristics

_{max}is the maximum capacity, P

_{min}is the minimum stable generation level, T

_{mut}is the minimum up time, T

_{sdt}is the shut-down time, T

_{mdt}is the minimum down time, T

_{sut}is the start-up time, V

_{rur}is the maximum ramp up speed, and V

_{rdr}is the maximum ramp down speed of the TPU.

#### 2.1.1. Adjustable Capacity

#### 2.1.2. Ramping Rate

#### 2.1.3. Adjustment Period

_{min}and ends at the next moment the TPU starts move to P

_{min}:

#### 2.2. Economic Characteristics

#### 2.2.1. Operation Costs

_{coal}is the unit coal price of the current season. C

_{cost}is loss cost of the unit. C

_{oil}is the cost of fuel consumption. C

_{env}is the additional environmental cost.

#### 2.2.2. Start-up Costs

_{c}is the cost of the cold start, F

_{c}is the fuel cost, C

_{f}is the fixed cost, and C

_{t}is the cost of maintaining the unit temperature.

#### 2.3. Selection of Flexibility Indexes of TPU

## 3. TPU Flexibility Evaluation Index System

#### 3.1. Analytic Hierarchy Process

- Establish a hierarchy, as shown in Figure 4.
- Construct a comparison judgment matrix (A).$$A={({a}_{ij})}_{n\times n}=\left[\begin{array}{cccc}{a}_{11}& {a}_{12}& \cdots & {a}_{1n}\\ {a}_{21}& {a}_{22}& \cdots & {a}_{2n}\\ \vdots & \cdots & \cdots & \vdots \\ {a}_{n1}& {a}_{n1}& \cdots & {a}_{nn}\end{array}\right]$$
_{ij}is the pairwise comparison rating between indicator i and indicator j. The pairwise comparison scale is shown in Table 1 [29]. - Consistency test of the judgment matrix.The quantitative indicator that measures the degree of inconsistency is called the consistency indicator (CI):$$CI=\frac{{\lambda}_{\mathrm{max}}-n}{n-1}$$
_{max}is the largest eigenvalue of the comparison judgment matrix, and the expert’s judgment becomes inconsistent with an increase in the number of indicators. In order to measure the CI standard, Satty et al. proposed the use of the random consistency ratio (CR < 0.1) to correct the consistency index [29]. - Calculate the weight of the indicator.The formulas for calculating the weight of each index using the weight vector calculation method of the product square root method are as follows:$${r}_{i}=\sqrt[n]{({\displaystyle \prod _{j=1}^{n}{a}_{ij}})}i=1,2,\cdots ,n$$$${p}_{j}={r}_{j}/{\displaystyle \sum _{k=1}^{n}{r}_{k},j=1,2,\cdots ,n}$$
_{i}is the geometric mean of the elements of each row in the judgment matrix A; p_{j}is the weight coefficient of index j.

#### 3.2. Entropy Method

_{ij}) was established, and represents the observation data of the j-th index of the evaluation object (i). For indicator j, the greater the difference in u

_{ij}, the more information that indicator j contains, which means the weighting factor of indicator j is greater. The steps for determining the index weight coefficient by the entropy method are as follows:

- Calculate the feature weight of the i-th evaluated object under the j-th index:$${h}_{ij}={u}_{ij}/{\displaystyle \sum _{i=1}^{m}{u}_{ij}}.$$
- Calculate the entropy value (e
_{j}) of the j-th indicator:$${e}_{j}=-k{\displaystyle \sum _{i=1}^{m}{h}_{ij}\mathrm{ln}\left({h}_{ij}\right)}.$$ - Calculate the difference coefficient matrix of the indicator:$${\beta}_{j}=1-{e}_{j}.$$
- Calculate the weight coefficient:$${q}_{j}={\beta}_{j}/{\displaystyle \sum _{k=1}^{n}{\beta}_{k},j=1,2,\cdots ,n}.$$

#### 3.3. Construction of the Comprehensive Weighting Model

- This paper uses the commonly used Lagrangian multiplier method for comprehensive empowerment:$${z}_{j}=\sqrt{{p}_{j}{q}_{j}}/{\displaystyle \sum _{k}^{n}\sqrt{{p}_{k}{q}_{k}}}$$
_{j}is the calculated comprehensive weight coefficient. - The assembly model used in this paper is a linear “addition” integrated assembly model.$$Fle{x}_{i}={\displaystyle \sum _{j=1}^{n}{z}_{j}}{x}_{ij}$$
_{i}is the comprehensive flexibility evaluation score of TPU i; n is the total number of indicators; x_{ij}is the observation value of index j of TPU i.

## 4. Optimization Scheduling Model Considering TPU Flexibility

#### 4.1. Objective Function and Constraints

#### 4.1.1. Objective Function

_{th}is the conventional economic dispatch target and f

_{f}is the flexibility dispatch target; F

_{i,t}is the operating cost of unit i during t period, I

_{i,t}is the binary integer variable which reflects the opening and stopping state of the unit i in the period t, and U

_{i}is the starting cost of unit i. Flex

_{i}in Equation (18) represents the flexibility score of the ith TPU. Through the scheduling models of Equations (17) and (18), it is possible to achieve a higher total TPU flexibility supply for a small power generation cost. Equations (17) and (18) represent a dual target optimization problem, in which the direction of optimization of the two objective functions is opposite. In order to solve the dual target optimization problem, the linear model (Equations (19)) commonly used in engineering was utilized in this paper:

_{f}and f

_{th};

#### 4.1.2. Restrictions

_{Gi,t}is the active power of unit i in period t; P

_{Lt}is the predicted value of load in period t; and P

_{Wt}is the total predicted value of wind power in the scheduling range of period t.

_{Rt}is the spare capacity of t period t.

_{i}and DR

_{i}are the limits of the amounts of active power increase and decrease of unit i, respectively.

#### 4.2. System Flexibility Assessment

#### 4.2.1. System Flexibility Assessment Indicators

_{t}} and the downward adjustment flexibility demand {dnd

_{t}}, which is mainly related to the changes in the system’s net load (P

_{net}):

_{UFNS}and downward flexibility deficiency probability P

_{DFNS}) to evaluate the system’s flexibility:

_{t}and RD

_{t}are, respectively, the upward adjustment flexibility supply and the downward adjustment flexibility supply provided by TPU, during the system’s scheduling interval of Δt at time t.

#### 4.2.2. Power Generation Plan Flexibility Assessment Process

- (1)
- Input the unit parameters, the comprehensive flexibility evaluation value, and the load and wind data and determine the flexibility factor (α) in Equation (19).
- (2)
- Based on the wind power and load forecast data, determine the power generation plan, obtain the upward adjustment flexibility supply ({RU
_{t}}) and the downward adjustment flexibility supply ({RD_{t}}), and set the simulation frequency (M). - (3)
- Based on Equation (27), obtain the up-regulated flexibility requirement sequence ({upd
_{t}}) and down-regulated flexibility requirement sequence ({dnd_{t}}) during the scheduling period. - (4)
- Using the WP historical prediction error distribution within the scheduling range, use the Monte Carlo simulation to generate the prediction error timing ({ε
_{t}}). - (5)
- According to the WP prediction error sequence ({ε
_{t}}) generated in step 4, modify the up-regulated flexibility requirement sequence and down-regulated flexibility requirement sequence to obtain ({$up{d}_{t}^{\ast}$}) and ({$dn{d}_{t}^{\ast}$}). - (6)
- Set the intermediate variables γ
_{u}and γ_{d}to record the simulation results, and finally get the upward flexibility deficiency probability (P_{UFNS}) and downward flexibility deficiency probability (P_{DFNS}).

## 5. Case Study

#### 5.1. Unit Flexibility Evaluation

#### 5.2. The Impacts of Uncertainty on the System

#### 5.3. The Impacts of the TPU Flexibility on the System

_{UNFS}and P

_{DNFS}, are better than Strategy 2 compared to Strategy 3.

_{UNFS}and P

_{DNFS}were obtained, and the comparison of strategies S2, S3, and S4 are shown in Figure 9. P

_{UNFS}and P

_{DNFS}indicators experienced different degrees of decline. The Equation (19) provide two open source parameters to the dispatching department which can adjust the parameters according to actual conditions. When the flexibility of TPU was considered, the system’s indicators of P

_{UNFS}and P

_{DNFS}decreased, as shown in Figure 9.

#### 5.4. Sensitivity Analysis

_{UNFS}and P

_{DNFS}do not infinitely decrease. Convergence occurs near α = 0.4, which reaches the limit of the flexibility of the system.

_{UNFS}and P

_{DNFS}, which have greater uncertainty adaptability.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

Unit 1 | Unit 2 | Unit 3 | Unit 4 | Unit 5 | Unit 6 | Unit 7 | Unit 8 | Unit 9 | Unit 10 | |
---|---|---|---|---|---|---|---|---|---|---|

Pmax (MW) | 455 | 455 | 130 | 130 | 162 | 80 | 85 | 55 | 55 | 55 |

Pmin (MW) | 150 | 150 | 20 | 20 | 25 | 20 | 25 | 10 | 10 | 10 |

a ($/h) | 1000 | 970 | 700 | 680 | 450 | 370 | 480 | 660 | 665 | 670 |

b ($/MWh) | 16.19 | 17.26 | 16.60 | 16.50 | 19.70 | 22.26 | 27.74 | 25.92 | 27.27 | 27.79 |

c (10^{−2} $/(MW)^{2}h) | 0.048 | 0.031 | 0.2 | 0.211 | 0.398 | 0.712 | 0.079 | 0.413 | 0.222 | 0.173 |

Min up (h) | 8 | 8 | 5 | 5 | 6 | 3 | 3 | 1 | 1 | 1 |

Min down (h) | 8 | 8 | 5 | 5 | 6 | 3 | 3 | 1 | 1 | 1 |

Hot start cost ($) | 4500 | 5000 | 550 | 560 | 900 | 170 | 260 | 30 | 30 | 30 |

Cold start cost | 9000 | 10000 | 1100 | 1120 | 1800 | 340 | 520 | 60 | 60 | 60 |

Cold start hours | 5 | 5 | 4 | 4 | 4 | 2 | 2 | 0 | 0 | 0 |

Initial status (h) | 8 | 8 | −5 | −5 | −6 | −3 | −3 | −1 | −1 | −1 |

Time | Demand (MW) | Time | Demand (MW) | Time | Demand (MW) | Time | Demand (MW) |
---|---|---|---|---|---|---|---|

1 | 700 | 7 | 1150 | 13 | 1400 | 19 | 1200 |

2 | 750 | 8 | 1200 | 14 | 1300 | 20 | 1400 |

3 | 850 | 9 | 1300 | 15 | 1200 | 21 | 1300 |

4 | 950 | 10 | 1400 | 16 | 1050 | 22 | 1100 |

5 | 1000 | 11 | 1450 | 17 | 1000 | 23 | 900 |

6 | 1100 | 12 | 1500 | 18 | 1100 | 24 | 800 |

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**Figure 3.**Comparison of five flexibility indicators of different units: (

**a**) 50 MW gas unit and 100 MW coal-fired unit; (

**b**) 100 MW and 300 MW coal-fired units.

**Figure 7.**The results of the comparison of the S1 and S2 strategies. (

**a**) The comparison of the upward adjustment flexibility supply (RU) and the up-regulated flexibility requirement (UPD) of S1; (

**b**) comparison of the downward adjustment flexibility supply (RD) and the down-regulated flexibility requirement DND of S1; (

**c**) comparison of RU and UPD of S2; (

**d**) comparison of RD and DND of S2.

**Figure 9.**Comparison of strategies of S2, S3 and S4. (

**a**) Comparison of P

_{UNFS}; (

**b**) Comparison of P

_{DNFS}.

Numerical Values | Verbal Scale | Explanation |
---|---|---|

1 | Equal importance of both elements | Two elements contribute equally |

3 | Moderate importance of one indicator over another | Experience and judgment favor one indicator over another |

5 | Strong importance of one indicator over another | An indicator is strongly favored |

7 | Very strong importance of one indicator over another | An indicator is very strongly dominant |

9 | Extreme importance of one indicator over another | An indicator is favored by at least an order of magnitude |

2, 4, 6, 8 | Intermediate values | Used to compromise between two judgments |

Index | S | V | T | C | U |
---|---|---|---|---|---|

Subjective weight | 0.228 | 0.210 | 0.327 | 0.084 | 0.150 |

Objective weight | 0.195 | 0.199 | 0.198 | 0.180 | 0.227 |

Comprehensive weight | 0.222 | 0.209 | 0.322 | 0.076 | 0.171 |

Index | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

P_{max} | 455 | 455 | 130 | 130 | 162 | 80 | 85 | 55 | 55 | 55 |

Flex | 126.50 | 126.61 | 60 | 60.14 | 69.15 | 66.15 | 60.89 | 143.58 | 143.51 | 143.46 |

Strategies | S1 | S2 | S3 | S4 |
---|---|---|---|---|

k | 0 | 10% | 10% | 10% |

α | 0 | 0 | 0.2 | 0.35 |

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## Share and Cite

**MDPI and ACS Style**

Guo, T.; Gao, Y.; Zhou, X.; Li, Y.; Liu, J.
Optimal Scheduling of Power System Incorporating the Flexibility of Thermal Units. *Energies* **2018**, *11*, 2195.
https://doi.org/10.3390/en11092195

**AMA Style**

Guo T, Gao Y, Zhou X, Li Y, Liu J.
Optimal Scheduling of Power System Incorporating the Flexibility of Thermal Units. *Energies*. 2018; 11(9):2195.
https://doi.org/10.3390/en11092195

**Chicago/Turabian Style**

Guo, Tong, Yajing Gao, Xiaojie Zhou, Yonggang Li, and Jiaomin Liu.
2018. "Optimal Scheduling of Power System Incorporating the Flexibility of Thermal Units" *Energies* 11, no. 9: 2195.
https://doi.org/10.3390/en11092195