# Modeling and Optimization of the Medium-Term Units Commitment of Thermal Power

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

#### 2.1. Objective Function

_{i}and h

_{j}are the accumulated operating hours of installed capacity for plants i, j, respectively, in the selected period.

$i,j$ | plant i and plant j, $i,j=1,2,\cdots m$; |

$T$ | Number of time steps. $t=1,2,\cdots ,T$; |

${h}_{i}$ | Accumulated operating hours of installed capacity of plant i in the selected period; |

${h}_{i}^{P}$ | Operating hours of installed capacity of plant i accumulated in pre-calculation period; |

${h}_{i}^{T}$ | Operating hours of installed capacity of plant i accumulated in the calculation period; |

${h}_{i}^{G}$ | Extra award hours assigned to plant i; |

${N}_{i}$ | Installed capacity of plant i; |

$\overline{{C}_{i}}$ | Average operating capacity of plant i in the calculation period; |

${r}_{i}$ | Given load factor of plant i; |

${C}_{i}^{t}$ | Operating capacity of plant i in period t. It is also the decision variable to be solved. |

#### 2.2. Constraints

- (1)
- Unit number constraints:

- (2)
- System power balance constraints:

- (3)
- Peak and valley duration constraints:

## 3. Model Solution

#### 3.1. Solution Approach

^{2}, so the total number of unit combinations will reach [(3 + 1)

^{2}]

^{30}$\approx $ 1.329$\times $10

^{36}for a horizon of one month with 30 periods, and the overhead of optimization will be computationally expensive. Considering the complex constraints, especially Constraint Group 3, the solving process can be divided into two procedures:

#### 3.2. Initial Feasible Solution Generation

- Step 1:
- Construct an integer array ks of length T. Set ks[1] = 1 and t = 1. Make sure that the combination of boot capacity in period 1 is the first element of the solution space ${S}_{t}^{\prime}$.
- Step 2:
- Set t = t + 1. If t > T, go to Step 6. Otherwise, set ks[1] = 1 and set the combination of boot capacity in period t as the first element of the solution space ${S}_{t}^{\prime}$.
- Step 3:
- Verify whether the start-up mode lying in the interval of [0,1] satisfies the constraints of Equation (3). If it does satisfy them, go to Step 2; otherwise, go to Step 4.
- Step 4:
- Set ks[t] = ks[t] + 1. If ks[t] is greater than the number of elements in solution space ${S}_{t}^{\prime}$, go to Step 5. If not, replace the combination of boot capacity in period t with the ks[t]th element of the solution space ${S}_{t}^{\prime}$ and go to Step 3.
- Step 5:
- Set t = t + 1, and go to Step 1.
- Step 6:
- Output the result.

#### 3.3. Optimization Process of Progressive Optimality Algorithm (POA)

- Step 1:
- Set the generating scheme to an initial feasible solution generated by the heuristic search and an initial objective function value $value$. Set t = 1 and k = 1.
- Step 2:
- Obtain a generating scheme by replacing the generating scheme in period t with the kth element of the solution space ${S}_{t}$. If the new generating scheme meets Constraint Group 3, go to Step 3. If not, go to Step 4.
- Step 3:
- Calculate the objective function value $valu{e}^{\prime}$ according to Equation (5). If $valu{e}^{\prime}<value$, adjust the results by using the equalization operation and combination operation and set $valu{e}^{\prime}=value$.
- Step 4:
- Set k = k + 1. If k is greater than the number of elements in solution space ${S}_{t}$, go to Step 5. If not, go to Step 2.
- Step 5:
- Set t = t + 1 and k = 1. If t < T, go to Step 2. If not, it means one iteration has been performed, then check whether the objective value has been improved over the previous iteration. If so, set the initial solution as the acquired result and go to Step 2. If not, go to Step 6.
- Step 6:
- Output the result.

#### 3.4. Equalization Operation and Combination Operation

#### 3.5. Solution Architecture

- Step 1:
- Obtain the initial feasible solution by heuristic search and calculate the objective function $value$;
- Step 2:
- Obtain the optimal results by using POA and calculate the objective function $valu{e}^{\prime}$ (the detailed process is described in Section 3.3);
- Step 3:
- Judge whether the current isopleths of the objective function value are equal to zero, if not, use the two strategies to move the optimal results from the constraint boundary to the feasible region internal area to search again. Otherwise, go to Step 4.
- Step 4:
- Output optimal scheduling.

**Figure 6.**Solution flowchart of medium-term optimal commitment of thermal units (MOCTU) by improved progressive optimality algorithm (IPOA).

## 4. Simulation Results

^{®}Core™2 Duo CPU, operating at 2.93 GHz, with 2 GB of memory. A real scheduling for 2013 in the YNPG is used to test the validity and computational efficiency of the proposed method.

- 1)
- Given load rate of plant $i$, ${r}_{i}=0.8$;
- 2)
- Minimum load rate of thermal power, $\underset{\_}{r}=0.7$;
- 3)
- Maximum load rate of thermal power, $\overline{r}=0.9$;
- 4)
- Minimum duration periods of peak in the operating capacity process of plant $i$ during the calculation period, ${T}_{i}^{\text{up}}=7d$;
- 5)
- Minimum duration periods of valley in the operating capacity process of plant $i$ during the calculation period, ${T}_{i}^{\text{down}}=3d$;
- 6)
- Maximum unit number of plant k in period t, set $\overline{{n}_{i}^{t}}$ as the number of installed units
- 7)
- Minimum unit number of plant i in period t, $\underset{\_}{{n}_{i}^{t}}=1(i=A~H)$, except $\underset{\_}{{n}_{i}^{t}}=0$ while $i=I$;
- 8)
- Actual capacity utilization hours of plant i in the pre-calculation period, ${h}_{i}^{P}=0$;
- 9)
- Extra award hours of plant i, ${h}_{i}^{G}=0$;
- 10)
- System load demands in each period and the first ten days’ boot process ($p=10d$) are all given.

Thermal plant | Units (number × capacity, MW) | Capacity (MW) |
---|---|---|

A | 4 × 600 | 2400 |

B | 6 × 300 | 1800 |

C | 4 × 300 | 1200 |

D | 2 × 200 + 2 × 300 | 1000 |

E | 2 × 300 | 600 |

F | 2 × 300 | 600 |

G | 2 × 300 | 600 |

H | 2 × 300 | 600 |

I | 1 × 135 | 135 |

Items | Heuristic search | POA | First adjustment | IPOA |
---|---|---|---|---|

Plant A | 547.2 | 422.4 | 422.4 | 422.4 |

Plant B | 297.6 | 416.0 | 416.0 | 422.4 |

Plant C | 369.6 | 422.4 | 422.4 | 422.4 |

Plant D | 240.0 | 422.4 | 422.4 | 422.4 |

Plant E | 489.6 | 422.4 | 422.4 | 422.4 |

Plant F | 489.6 | 432.0 | 432.0 | 422.4 |

Plant G | 499.2 | 422.4 | 422.4 | 422.4 |

Plant H | 528.0 | 432.0 | 432.0 | 422.4 |

Plant I | 422.4 | 422.4 | 422.4 | 422.4 |

Average value | 431.5 | 423.8 | 423.8 | 422.4 |

Max-min difference | 307.2 | 16.0 | 16.0 | 0 |

Objective value (h^{2}) | 10283 | 23.0 | 23.0 | 0 |

^{2}and 23.0 h

^{2}. Although the results have been significantly improved, the optimal solution is still not found. The reason is that the results obtained from POA run into local optima. The results by POA in Figure 7a illustrate that it has reached the system’s power constraint boundary in many periods (10-08, 10-09, 10-30) and Figure 8a shows that two peak values of Plant F’s boot mode reach the duration periods of peak in the operating capacity constraint, and it cannot be optimized any further by POA. Table 2, Figure 7b and Figure 8b show that the equalization operation and combination operation can change the structure of solutions and make the solution leave the constraint boundary under the condition of the unchanged objective function value (23.0 h

^{2}). After the first combination operation, the boot process of Plant F combines two peaks (Figure 8a) into one peak (Figure 8b) and makes the duration periods of the peak change from 7 d (T

^{up}= 7) to 14 d, leaving the constraint boundary. After applying IPOA, the second unit’s boot time of Plant F is adjusted from 18 October (Figure 8b) to 19 October (Figure 8c), and the capacity utilization hours change from 432 h to 422.4 h.

**Figure 9.**Results of thermal plants: (

**a**) Plant A; (

**b**) Plant B; (

**c**) Plant C; (

**d**) Plant D; (

**e**) Plant E; (

**f**) Plant F; (

**g**) Plant G; (

**h**) Plant H; (

**i**) Plant I.

^{2}. Moreover, installed capacity utilization hours of Plants C, D and F are more 29.52, 19.36 and 9.68 h than other plants, respectively, and the calculation precision meets the demand of practical engineering.

Date | Plant A | Plant B | Plant C | Plant D | Plant E | Plant F | Plant G | Plant H | Plant I |
---|---|---|---|---|---|---|---|---|---|

1st | 1200 | 900 | 600 | 700 | 300 | 300 | 300 | 300 | 135 |

2nd | 1200 | 900 | 600 | 700 | 300 | 300 | 300 | 300 | 135 |

3rd | 1200 | 900 | 300 | 700 | 300 | 300 | 300 | 300 | 135 |

4th | 1200 | 900 | 300 | 700 | 300 | 300 | 300 | 300 | 135 |

5th | 1200 | 900 | 300 | 700 | 300 | 300 | 300 | 300 | 135 |

6th | 1200 | 900 | 600 | 700 | 300 | 300 | 300 | 300 | 135 |

7th | 1200 | 900 | 600 | 700 | 600 | 300 | 300 | 300 | 135 |

8th | 1800 | 900 | 600 | 700 | 600 | 300 | 300 | 300 | 135 |

9th | 1800 | 900 | 600 | 700 | 600 | 300 | 300 | 300 | 135 |

10th | 2400 | 600 | 600 | 700 | 600 | 300 | 300 | 300 | 135 |

11th | 2400 | 600 | 600 | 700 | 600 | 300 | 300 | 300 | 135 |

12th | 2400 | 600 | 1200 | 700 | 600 | 300 | 300 | 300 | 135 |

13th | 2400 | 600 | 1200 | 700 | 600 | 300 | 300 | 300 | 135 |

14th | 2400 | 900 | 1200 | 700 | 600 | 300 | 300 | 300 | 135 |

15th | 2400 | 900 | 1200 | 700 | 300 | 300 | 300 | 300 | 135 |

16th | 2400 | 1200 | 1200 | 700 | 300 | 300 | 300 | 300 | 135 |

17th | 1800 | 1200 | 1200 | 700 | 300 | 300 | 300 | 300 | 135 |

18th | 1800 | 1500 | 1200 | 400 | 300 | 300 | 300 | 300 | 135 |

19th | 1800 | 1800 | 1200 | 400 | 300 | 600 | 600 | 600 | 0 |

20th | 1800 | 1800 | 1200 | 400 | 300 | 600 | 600 | 600 | 0 |

21st | 1800 | 1800 | 1200 | 400 | 300 | 600 | 600 | 600 | 0 |

22nd | 1800 | 1800 | 900 | 500 | 300 | 600 | 600 | 600 | 0 |

23rd | 1800 | 1800 | 300 | 800 | 300 | 600 | 600 | 600 | 0 |

24th | 1800 | 1800 | 300 | 800 | 300 | 600 | 600 | 600 | 0 |

25th | 1800 | 1800 | 300 | 800 | 300 | 600 | 600 | 600 | 0 |

26th | 1800 | 1800 | 900 | 800 | 300 | 600 | 600 | 600 | 0 |

27th | 1200 | 1800 | 1200 | 800 | 600 | 600 | 600 | 600 | 0 |

28th | 1200 | 1800 | 1200 | 1000 | 600 | 600 | 600 | 600 | 135 |

29th | 1200 | 1800 | 1200 | 1000 | 600 | 600 | 600 | 600 | 135 |

30th | 1200 | 1800 | 1200 | 1000 | 600 | 600 | 600 | 600 | 135 |

31st | 1200 | 1800 | 1200 | 1000 | 600 | 600 | 600 | 600 | 135 |

Items | ${h}_{i}^{G}$ | Heuristic Search | IPOA | ||
---|---|---|---|---|---|

${h}_{i}^{p}+{h}_{i}^{T}$ | ${h}_{i}$ | ${h}_{i}^{p}+{h}_{i}^{T}$ | ${h}_{i}$ | ||

Plant A | 0 | 240.00 | 240.00 | 422.40 | 422.40 |

Plant B | 0 | 297.60 | 297.60 | 422.40 | 422.40 |

Plant C | 30 | 399.12 | 369.12 | 451.92 | 421.92 |

Plant D | 20 | 508.96 | 488.96 | 441.76 | 421.76 |

Plant E | 0 | 422.40 | 422.40 | 422.40 | 422.40 |

Plant F | 10 | 499.28 | 489.28 | 432.08 | 422.08 |

Plant G | 0 | 547.20 | 547.20 | 422.40 | 422.40 |

Plant H | 0 | 499.20 | 499.20 | 422.40 | 422.40 |

Plant I | 0 | 528.00 | 528.00 | 422.40 | 422.40 |

Average value | - | - | 431.50 | - | 422.24 |

Max-min difference | - | - | 307.20 | - | 0.64 |

Objective value (h^{2}) | - | - | 10,277.489 | - | 0.057 |

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- China Electric Power Yearbook Editorial Board. China Electric Power Yearbook; China Electric Power Press: Beijing, China, 2011. [Google Scholar]
- China Electric Power Yearbook Editorial Board. China Electric Power Yearbook; China Electric Power Press: Beijing, China, 2012. [Google Scholar]
- China Electric Power Yearbook Editorial Board. China Electric Power Yearbook; China Electric Power Press: Beijing, China, 2013. [Google Scholar]
- Ministry of Planning and Statistics. National Power Industry Statistics Express; China Electricity Council: Beijing, China, 2013. [Google Scholar]
- Liu, L.; Zong, H.; Zhao, E.; Chen, C.; Wang, J. Can China realize its carbon emission reduction goal in 2020: From the perspective of thermal power development. Appl. Energy
**2014**, 124, 199–212. [Google Scholar] [CrossRef] - Mario, C.; Antonio, M.; Giuseppina, N. A Dynamic Fuzzy Controller to Meet Thermal Comfort by Using Neural Network Forecasted Parameters as the Input. Energies
**2014**, 7, 4727–4756. [Google Scholar] [CrossRef] - Padhy, N.P. Unit commitment—A bibliographical survey. IEEE Trans. Power Syst.
**2004**, 19, 1196–1205. [Google Scholar] [CrossRef] - Niknam, T.; Khodaei, A.; Fallahi, F. A new decomposition approach for the thermal unit commitment problem. Appl. Energy
**2009**, 86, 1667–1674. [Google Scholar] [CrossRef] - Senjyu, T.; Shimabukuro, K.; Uezato, K.; Funabashi, T. A fast technique for unit commitment problem by extended priority list. IEEE Trans. Power Syst.
**2003**, 18, 882–888. [Google Scholar] [CrossRef] - Baldwin, C.J.; Dale, K.M.; Dittrich, R.F. A study of the economic shutdown of generating units in daily dispatch. Trans. Am. Inst. Electr. Eng.
**1959**, 78, 1272–1284. [Google Scholar] [CrossRef] - Iguchi, M.; Yamashiro, S. An efficient scheduling method for weekly hydro-thermal unit commitment. In Proceedings of the 2002 IEEE Region 10 Conference on Computers, Communications, Control and Power Engineering, Beijing, China, 28–31 October 2002; Volume 3, pp. 1772–1777.
- Dudek, G. Genetic algorithm with integer representation of unit start-up and shut-down times for the unit commitment problem. Eur. Trans. Electr. Power
**2007**, 17, 500–511. [Google Scholar] [CrossRef] - Tong, S.K.; Shahidehpour, S.M.; Ouyang, Z. A heuristic short-term unit commitment. IEEE Trans. Power Syst.
**1991**, 6, 1210–1216. [Google Scholar] [CrossRef] - Hyeon, G.P.; Jae, K.L.; Kang, Y.C.; Jong, K.P. Unit commitment considering interruptible load for power system operation with wind power. Energies
**2014**, 7, 4281–4299. [Google Scholar] [CrossRef] - Liao, S.L.; Cheng, C.T.; Wang, J.; Feng, Z.K. A hybrid search algorithm for midterm optimal scheduling of thermal power plants. Math. Probl. Eng.
**2015**, 2015. [Google Scholar] [CrossRef] - Venkata, S.P.; Istvan, E.; Kurt, R.; Jan, D. A stochastic model for the optimal operation of a wind-thermal power system. IEEE Trans. Power Syst.
**2009**, 24, 940–949. [Google Scholar] [CrossRef] - Saadawi, M.M.; Tantawi, M.A.; Tawfik, E. A fuzzy optimization-based approach to large scale thermal unit commitment. Electr. Power Syst. Res.
**2004**, 72, 245–252. [Google Scholar] [CrossRef] - Dang, C.Y.; Li, M.Q. A floating point genetic algorithm for solving the unit commitment problem. Eur. J. Oper. Res.
**2007**, 181, 1670–1395. [Google Scholar] [CrossRef] - Simonovic, S.P. Reservoir systems analysis: Closing gap between theory and practice. J. Water Resour. Plan. Manag.
**1992**, 118, 262–280. [Google Scholar] [CrossRef] - Yang, G.; James, M.C.; Ming, N.; Rui, B. Economic modeling of compressed air energy storage. Energies
**2013**, 6, 2221–2241. [Google Scholar] [CrossRef] - Rogelio, P.M.; Juan, A.M.; Miguel, J.P.; Lourdes, A.B.; Juan, M.M.S. A novel modeling of molten-salt heat storage systems in thermal solar power plants. Energies
**2014**, 7, 6721–6740. [Google Scholar] [CrossRef] - Wei, S.Y.; Xu, F.; Min, Y. Study and modeling on maintenance strategy for a thermal power plant in the new market environment. In Proceedings of the IEEE International Conference on Electric Utility Deregulation, Restructuring and Power Technologies, Hong Kong, China, 5–8 April 2004; pp. 200–204.
- Janusz, B.; Andrzej, O. Modelling of thermal power plants reliability. In Proceedings of the International Conference on Power Engineering, Energy and Electrical Drives, Istanbul, Turkey, 13–17 May 2013; pp. 1038–1043.
- Turgeon, A. Optimal short-term hydro scheduling from the principle of progressive optimality. Water Resour. Res.
**1981**, 17, 481–486. [Google Scholar] [CrossRef] - Nanda, J.; Bijwe, P.R.; Kothari, D.P. Application of progressive optimality algorithm to optimal hydrothermal scheduling considering deterministic and stochastic data. Int. J. Electr. Power Energy Syst.
**1986**, 8, 61–64. [Google Scholar] [CrossRef] - Nanda, J.; Bijwe, P.R. Optimal hydrothermal scheduling with cascaded plants using progressive optimality algorithm. IEEE Trans. Power Syst.
**1981**, 100, 2093–2099. [Google Scholar] [CrossRef] - Lucas, N.J.D.; Perera, P.J. Short-term hydroelectric scheduling using the progressive optimality algorithm. Water Resour. Res.
**1985**, 21, 1456–1458. [Google Scholar] [CrossRef] - Cheng, C.T.; Shen, J.J.; Wu, X.Y.; Chau, K.W. Short-term hydro scheduling with discrepant objectives using multi-step progressive optimality algorithm. J. Am. Water Resour. Assoc.
**2012**, 48, 464–479. [Google Scholar] [CrossRef] - Labadie, J.W. Optimal operation of multireservoir systems: State-of-the-art review. J. Water Resour. Plan. Manag.
**2004**, 130, 93–111. [Google Scholar] [CrossRef] - Howson, H.R.; Sancho, N.G. A new algorithm for the solution of multi-stage dynamic programming problems. Math. Program.
**1975**, 8, 104–116. [Google Scholar] [CrossRef] - Marino, M.A.; Loaiciga, H.A. Dynamic model for multireservoir operation. Water Resour. Res.
**1985**, 21, 619–630. [Google Scholar] [CrossRef] - Marino, M.A.; Loaiciga, H.A. Quadratic model for reservoir management: Application to the Central Valley Project. Water Resour. Res.
**1985**, 21, 631–641. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Liao, S.; Li, Z.; Li, G.; Wang, J.; Wu, X.
Modeling and Optimization of the Medium-Term Units Commitment of Thermal Power. *Energies* **2015**, *8*, 12848-12864.
https://doi.org/10.3390/en81112345

**AMA Style**

Liao S, Li Z, Li G, Wang J, Wu X.
Modeling and Optimization of the Medium-Term Units Commitment of Thermal Power. *Energies*. 2015; 8(11):12848-12864.
https://doi.org/10.3390/en81112345

**Chicago/Turabian Style**

Liao, Shengli, Zhifu Li, Gang Li, Jiayang Wang, and Xinyu Wu.
2015. "Modeling and Optimization of the Medium-Term Units Commitment of Thermal Power" *Energies* 8, no. 11: 12848-12864.
https://doi.org/10.3390/en81112345