# 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 |

## References

- Kim, H.Y.; Kim, M.K. Optimal generation rescheduling for meshed AC/HIS grids with multi-terminal voltage source converter high voltage direct current and battery energy storage system. Energy
**2016**, 119, 309–321. [Google Scholar] [CrossRef] - Ma, H.; Wang, B.; Gao, W.; Liu, D.; Sun, Y.; Liu, Z. Optimal Scheduling of a Regional Integrated Energy System with Energy Storage Systems for Service Regulation. Energies
**2018**, 11, 195. [Google Scholar] - Ji, B.; Yuan, X.; Chen, Z.; Tian, H. Improved gravitational search algorithm for unit commitment considering uncertainty of wind power. Energy
**2014**, 67, 52–62. [Google Scholar] [CrossRef] - Tolba, M.A.; Rezk, H.; Tulsky, V.; Diab, A.A.Z.; Abdelaziz, A.Y.; Vanin, A. Impact of Optimum Allocation of Renewable Distributed Generations on Distribution Networks Based on Different Optimization Algorithms. Energies
**2018**, 11, 245. [Google Scholar] - Zhou, J.; Xu, Y.; Zheng, Y.; Zhang, Y. Optimization of Guide Vane Closing Schemes of Pumped Storage Hydro Unit Using an Enhanced Multi-Objective Gravitational Search Algorithm. Energies
**2015**, 109, 765–780. [Google Scholar] - Bai, Y.; Zhong, H.; Xia, Q.; Kang, C.; Xie, L. A decomposition method for network-constrained unit commitment with AC power flow constraints. Energy
**2015**, 88, 595–603. [Google Scholar] [CrossRef] - Jorge, V.; Smith, A. A seeded memetic algorithm for large unit commitment problems. J. Heurist.
**2002**, 8, 173–195. [Google Scholar] - Roque, L.A.; Fontes, D.B.; Fontes, F.A. A Metaheuristic Approach to the Multi-Objective Unit Commitment Problem Combining Economic and Environmental Criteria. Energies
**2017**, 10, 2029. [Google Scholar] - Wind Power Integration in Liberalised Electricity Markets (Wilmar) Project. Available online: http://www.wilmar.risoe.dk (accessed on 2 October 2017).
- Ruiz, P.A.; Philbrick, C.R.; Zak, E.; Cheung, K.W.; Sauer, P.W. Uncertainty managementin the unit commitment problem. IEEE Trans. Power Syst.
**2009**, 24, 642–651. [Google Scholar] [CrossRef] - Ruiz, P.A.; Philbrick, C.R.; Sauer, P.W. Wind power day-ahead uncertainty management through stochastic unit commitment policies. In Proceedings of the Power Systems Conference and Exhibition, Seattle, WA, USA, 15–18 March 2009. [Google Scholar]
- Wu, L.; Shahidehpour, M.; Li, T. Stochastic security-constrained unit commitment. IEEE Trans. Power Syst.
**2007**, 22, 800–811. [Google Scholar] [CrossRef] - Wu, L.; Shahidehpour, M.; Li, Z. Comparison of scenario-based and interval optimization approaches to stochastic SCUC. IEEE Trans. Power Syst.
**2012**, 27, 913–921. [Google Scholar] [CrossRef] - Zhang, Y.; Wang, J.; Ding, T.; Wang, X. Conditional value at risk-based stochastic unit commitment considering the uncertainty of wind power generation. IET Gener. Trans. Distrib.
**2018**, 12, 482–489. [Google Scholar] [CrossRef] - Bertsimas, D.; Litvinov, E.; Sun, X.A.; Zhao, J.; Zheng, T. Adaptive robust optimization for the security constrained unit commitment problem. IEEE Trans. Power Syst.
**2013**, 28, 52–63. [Google Scholar] [CrossRef] - Jiang, R.; Wang, J.; Guan, Y. Robust unit commitment with wind power and pumped storage hydro. IEEE Trans. Power Syst.
**2012**, 27, 800–810. [Google Scholar] [CrossRef] - Wang, Q.; Guan, Y.; Wang, J. A Chance-Constrained Two-Stage Stochastic Program for Unit Commitment With Uncertain Wind Power Output IEEE Trans. Power Syst. Power Systems. IEEE Trans. Power Syst.
**2012**, 27, 206–215. [Google Scholar] [CrossRef] - Gooi, H.B.; Mendes, D.P.; Bell, K.R.W.; Kirschen, D.S. Optimal Scheduling of Spinning Reserve. IEEE Trans. Power Syst.
**1999**, 14, 1485–1492. [Google Scholar] [CrossRef] - Morales, J.M.; Conejo, A.J.; Perez-Ruiz, J. Economic Valuation of Reserves in Power Systems with High Penetration of Wind Power. IEEE Trans. Power Syst.
**2009**, 24, 900–910. [Google Scholar] [CrossRef] - Kim, M.-K. Short-term price forecasting of Nordic power market by combination Levenberg-Marquardt and Cuckoo search algorithms. IET Gener Trans. Distrib.
**2015**, 9, 1553–1653. [Google Scholar] [CrossRef] - Jordan, G.; Piwko, R. The Value of Wind Power Forecasting; National Renewable Energy Lab. (NREL): Golden, CO, USA, 2011.
- Constantinescu, E.M.; Zavala, V.M.; Rocklin, M.; Lee, S.; Anitescu, M. Unit Commitment with Wind Power Generation: Integrating Wind Forecast Uncertainty and Stochastic Programming; ANL/MCS-TM-309; Argonne National Lab. (ANL): Argonne, IL, USA, 2009.
- Safari, A.; Shayanfar, H.A.; Jahani, R. Optimal Unit Commitment of Power System using Fast Messy Genetic Algorithm. Int. J. Tech. Phys. Probl. Eng.
**2010**, 2, 22–27. [Google Scholar] - Pappala, V.S.; Rohrig, I.E.K.; Dobschinski, J. A Stochastic Model for the Optimal Operation of a Wind-Thermal Power System. IEEE Trans. Power Syst.
**2009**, 24, 940–950. [Google Scholar] [CrossRef] - llinois Institute of Technology Power Group Test Case. Available online: http://www.motor.ece.iit.edu/Data/JEASIEEE118.doc (accessed on 7 July 2017).
- Kazemi, M.; Siano, P.; Sarno, D.; Goudarzi, A. Evaluating the impact of sub-hourly unit commitment method on spinning reserve in presence of intermittent generators. Energy
**2016**, 113, 338–354. [Google Scholar] [CrossRef]

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