# Stochastic Unit Commitment Based on Multi-Scenario Tree Method Considering Uncertainty

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Background

#### 1.2. Literature Review and Motivation

#### 1.3. Contribution and Paper Organization

- ▪
- The MSTM is developed to control the load and wind power uncertainties. The MSTM helps to satisfy the power system balance and minimize the total operating costs.
- ▪
- The application of the MSTM based on multiple scenarios to increase the accuracy of stochastic models when simulating the effect of the prediction errors is described. This represents a possible future realization of the random process.
- ▪
- The proposed solution procedure of the stochastic UC problem can obtain an approximate value for the probability variable with different accuracy requirements, thereby solving the optimal UC problem with the best unit state combination, considering the forecasting error factors.

## 2. Implementation of Stochastic Approach

#### 2.1. Forecast Uncertainty Modeling

#### 2.2. Multi-Scenario Tree Method

- ▪
- Initial Input: this module includes the forecasted load and wind data for each hour of day-ahead.
- ▪
- Scenario Generation: this module utilizes MCS to generate the random variables of the load and wind scenarios, which are structures branching from the load and wind forecasting error information.
- ▪
- Forced Outages: this module generates the reserve required by the forced outages of a thermal generator to realize a reliable and economical electrical supply.
- ▪
- Reserve Scenarios: this module uses the outputs of the load and wind scenarios and forced outages module, together with the forecasting errors, to calculate the additional reserve requirement.
- ▪
- Net Load Scenarios: this module represents each load and wind scenario for 24 h to generate net load scenarios. Depending on the source considered, net load is calculated as follows:
- -
- Load Only: Basic reserve + Load + Load error reserve,
- -
- Wind Only: Basic reserve + Wind + Wind error reserve,
- -
- Load and Wind both considered: Basic reserve + Load scenario + Wind scenario + Load error reserve + Wind error reserve,

- ▪
- Output Module: this module collates and converts the information from the other modules into input data.

## 3. Proposed Stochastic UC Problem

#### 3.1. Formulation

#### 3.1.1. Objective Functions

_{g}s

_{gst}, is given by the following linear function of time in the optimization problem:

#### 3.1.2. System Constraints

#### 3.1.3. Thermal Unit Constraints

#### 3.2. Uncertainty Constraints

## 4. Solution Procedure of Stochastic UC Problem

- Step 1
- Determine the uncertain parameters when forecasting the load and wind power based on an initial configuration using the available system information.
- Step 2
- Obtain the initial input data for the load and wind power using the MAPE. The data, in general, evolve over time according to a multivariate stochastic process, which represents branching trees comprising probability scenarios.
- Step 3
- Generate each load and wind power scenario considering random forecasting errors of the MAPE using MCS.
- Step 4
- Formulate the stochastic UC problem:
- (i)
- The additional reserve requirement in Labels (21), WLR
_{st}, is calculated without including forced outages, and, then, the scheduled reserve in the UC problem is put together to give the load and wind forecasting error for 24 h. - (ii)
- When the set of additional reserve requirements has been computed using the MSTM, the total amount of reserve required can be calculated with the basic reserve for forced outages from Labels (22) for each scenario.

- Step 5
- Check the power system balance, including the reserve requirements in Labels (23) and (24). If it is satisfied, go to Step 6; otherwise, return to Step 2.
- Step 6
- Compute the net load scenario from Labels (25). The two random scenarios can be merged into a single scenario for which a better UC solution becomes available.
- Step 7
- Solve the hourly UC problem. This procedure for each stochastic value with forecasting error is repeated to obtain the lowest operating cost for a number of scenarios sequentially.
- (i)
- Primary unit scheduling: Determine the order in which each generator is committed according to the average production cost.
- (ii)
- Maximum uptime/downtime repair: Performs repair operations to meet minimum up/down constraints in primary unit scheduling.
- (iii)
- Spinning reserve repair: The estimated spinning reserve in the primary unit scheduling process is reduced because of capacity generation. This is the process of repairing the reduced spinning reserve.
- (iv)
- Shutdown repair process: This is the process of adjusting units so that they have sufficient time to be effectively decommitted. In this process, the shutdown ramp rate constraint should also be considered.
- (v)
- Unit substitution process: After the minimum uptime/downtime repair process is performed, this process substitutes units to achieve cost-effective scheduling.
- (vi)
- Shutdown excess generation: This is a procedure for obtaining efficient unit scheduling considering the increased generation cost caused by minimum uptime/downtime repair and spinning reserve repair.

## 5. Numerical Studies

#### 5.1. Dataset for Test System

- Case 1
- Considering only the uncertainty of the load, input data were generated for 10 random scenarios with a load forecasting error of 2–4% deviance from the predicted values in the day-ahead scheduling.
- Case 2
- This case considered only a wind forecasting error of 15–20% to solve the stochastic UC problem based on each of the different scenarios.
- Case 3
- Combining the load and wind forecasting errors, the scenarios were generated to simulate the load and wind fluctuations.

#### 5.2. Simulation Results of UC Performance

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

Constants | |

π_{s} | probability of occurrence scenario s |

FC_{g} | generator fuel cost function of generator g |

S_{g} | start-up and shut down cost of generator g |

a_{i}, b_{i}, c_{i} | fitted parameters of fuel cost coefficients for each unit |

SU_{g} | start-up cost of generator g |

SD_{g} | shut-down cost of generator g |

$\underset{\_}{P}g,\overline{P}g$ | minimum and maximum capacity of generator g |

$\overline{R}g$ | maximum ramping rate of generator g |

UT_{i}, DT_{i} | minimum required down and up times for thermal generator g |

RU_{g} | ramping-up rate of generator g |

SUR_{g} | start-up ramping rate of generator g |

RD_{g} | ramping-down rate of generator g |

SDR_{g} | shut-down ramping rate of generator g |

V_{c} | cut-in wind speed in m/s |

V_{o} | cut-out wind speed in m/s |

V_{r} | rated wind speed in m/s |

Variables | |

${P}_{t}^{For}$ | forecasted value at t-hour |

${P}_{t}^{Act}$ | actual value at t-hour |

n | population size |

p_{gst} | power output of generator g in scenario s at time t |

u_{gst} | commitment of generator g in scenario s at time t |

s_{gst} | start-up and shut down cost of generator g in scenario s at time t |

x_{gst} | shut-down of generator g in scenario s at time t |

W_{gst} | wind power of generator g in scenario s at time t |

D_{st} | load in scenario s at time t |

${\overline{p}}_{gst}$ | maximum capacity of generator g in scenario s at time t |

R_{st} | basic reserve requirement in scenario s at time t |

R_{gst} | basic reserve requirement of generator g in scenario s at time t |

US_{gst} | up-spinning reserve of thermal generator g in scenario s at time t |

$\underset{\_}{R}st,\overline{R}st$ | minimum and maximum basic reserve requirement in scenario s at time t |

SU_{gst} | start-up cost for thermal generator g in scenario s at time t |

W_{gst} | wind power of generator g in scenario s at time t |

v_{gst} | wind speed of wind turbine generator g in scenario s at time t in m/s |

WLR_{st} | additional reserve requirement in scenario s at time t |

${W}_{gst}^{rated}$ | rated output of wind power |

D_{forecasted,st} | forecasted load in scenario s at time t |

W_{forecasted,gst} | forecasted wind power of generator g in scenario s at time t |

W_{error,gst} | wind power forecasting error of generator g in scenario s at time t |

D_{error,st} | load forecasting error in scenario s at time t |

R_{total,st} | total reserve requirement in scenario s at time t |

D_{net,st} | net load in scenario s at time t |

Indices | |

$g$ | Index of generator |

$s$ | Index of scenario |

$t$ | Index of time |

$st$ | Index of scenario s at time t |

$gst$ | Index of generator g in scenario s at time t |

$i$ | Index of thermal generator |

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Case | Descriptions | Basic Reserve |
---|---|---|

Base | No load forecasting error No wind power | 10% |

1 | Load forecasting error of 2–4% No wind power | 10% |

2 | No load forecasting error Wind forecasting error of 15–20% | 10% |

3 | Load forecasting error of 2–4% Wind forecasting error of 15–20% | 10% |

Scenario | Case 1 | Case 2 | Case 3 | |||
---|---|---|---|---|---|---|

Reserve (MW) | Operating Cost ($) | Reserve (MW) | Operating Cost ($) | Reserve (MW) | Operating Cost ($) | |

S1 | 16,959 | 3,614,894 | 15,779 | 3,248,309 | 17,855 | 3,335,855 |

S2 | 17,032 | 3,614,894 | 15,686 | 3,226,984 | 17,815 | 3,311,563 |

S3 | 16,885 | 3,583,911 | 15,754 | 3,229,614 | 17,723 | 3,310,324 |

S4 | 17,889 | 3,682,022 | 15,970 | 3,270,765 | 18,080 | 3,372,785 |

S5 | 16,856 | 3,565,803 | 15,577 | 3,137,272 | 18,110 | 3,385,998 |

S6 | 17,561 | 3,645,439 | 15,654 | 3,154,512 | 18,124 | 3,394,307 |

S7 | 16,763 | 3,539,429 | 15,325 | 3,109,556 | 17,578 | 3,303,129 |

S8 | 16,922 | 3,607,255 | 15,679 | 3,182,966 | 18,317 | 3,453,657 |

S9 | 17,738 | 3,675,453 | 15,361 | 3,129,913 | 18,136 | 3,400,132 |

S10 | 17,339 | 3,640,091 | 15,902 | 3,248,621 | 18,278 | 3,420,737 |

Average | 17,194 | 3,616,731 | 15,669 | 3,193,851 | 17,972 | 3,367,849 |

Penetration Level of Wind Power | Case 2 | Case 3 | ||||
---|---|---|---|---|---|---|

Reserve (MW) | Total Operating Cost ($) | Operating Cost per Reserve ($/MW) | Reserve (MW) | Total Operating Cost ($) | Operating Cost per Reserve ($/MW) | |

5% | 17,454 | 3,224,845 | 184.76 | 20,078 | 3,492,843 | 173.96 |

10% | 17,977 | 3,160,753 | 175.82 | 20,986 | 3,367,859 | 160.48 |

15% | 18,667 | 2,936,810 | 157.32 | 21,753 | 3,173,760 | 145.89 |

20% | 18,906 | 2,736,799 | 144.75 | 21,986 | 3,039,813 | 138.26 |

25% | 18,931 | 2,524,389 | 133.34 | 22,119 | 2,856,692 | 129.15 |

Unit | Time (h) | |||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |

1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |

2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |

3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |

4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 |

7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |

9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |

10 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

11 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |

13 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |

14 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

15 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |

16 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

17 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |

18 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |

19 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

20 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

21 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

22 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

23 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

24 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

25 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

26 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

27 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

28 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

29 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

30 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 |

31 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 |

32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |

33 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

34 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

35 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

36 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

37 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

38 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 |

39 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

40 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

41 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

42 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 |

43 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

44 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

45 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

46 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

47 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

48 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

49 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

50 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 |

51 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

52 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

53 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

54 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 |

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## Share and Cite

**MDPI and ACS Style**

Jo, K.-H.; Kim, M.-K.
Stochastic Unit Commitment Based on Multi-Scenario Tree Method Considering Uncertainty. *Energies* **2018**, *11*, 740.
https://doi.org/10.3390/en11040740

**AMA Style**

Jo K-H, Kim M-K.
Stochastic Unit Commitment Based on Multi-Scenario Tree Method Considering Uncertainty. *Energies*. 2018; 11(4):740.
https://doi.org/10.3390/en11040740

**Chicago/Turabian Style**

Jo, Kyu-Hyung, and Mun-Kyeom Kim.
2018. "Stochastic Unit Commitment Based on Multi-Scenario Tree Method Considering Uncertainty" *Energies* 11, no. 4: 740.
https://doi.org/10.3390/en11040740