# Short-Term Load Dispatching Method for a Diversion Hydropower Plant with Multiple Turbines in One Tunnel Using a Two-Stage Model

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}. The river has a theoretical hydropower reserve of 33.71 GW with 99.2% of the hydropower reserve concentrated in the Sichuan Province, southwestern China. This reserve is equal to an annual power generation of 151.636 TWh, accounting for approximately 24% of the total generation of Sichuan Province. The mainstream has a natural drop of 3830 m, and a development scheme of 21 cascade reservoirs has been planned along the mainstream, which have a total capacity of 30 GW and place the river in third place among China’s 13 major hydropower bases in terms of installed capacity [3]. Simultaneously, a series of hydropower plants with multiple turbines in one tunnel [4] have been developed and are in development in southwestern China due to the influence of the geological environment, exploits, and economic conditions. Hydropower plant with multiple turbines in one tunnel (HPMTT) can be interpreted as multi-hydroturbine sets sharing a common penstock [5]. A hydropower plant with multiple turbines in one tunnel (HPMTT) with a long diversion system is illustrated in Figure 1 The Jinping-II, with the largest installed capacity in the midstream of the Yalong River, has the largest and longest hydraulic tunnels in the world and is a common example of a HPMTT. Eight Francis turbine generator sets, each with a unit capacity of 600 MW, were installed and the four headrace tunnels have a total installed capacity of 4800 MW. The Tianshengqiao-II reservoir (TSQII) is a huge hydropower station in the Hongshui River, and includes six large-size units with a total capacity of 1320 MW in the three tunnels. The Lubuge hydropower plant is the “window” of hydropower construction opening to the outside world in China and includes three hydroturbine sets sharing a common penstock. Table 1 lists the characteristics of the three hydropowerplants.

## 2. Two-Stage Model

#### 2.1. Influencing Factors of STLD for the HPMTT Problem

#### 2.1.1. Hydraulic and Electrical Connections of HPMTT

#### 2.1.2. Penstock Head Loss

_{f}is the frictional head loss, m; $\lambda $ is frictional head loss coefficient; $l$ is the length of the tunnel, m; R is hydraulic radius, m; v is flow velocity, m/s; and g is the gravity acceleration, m/s.

_{j}is the local head loss, m; and $\xi $ is local head loss coefficient.

#### 2.2. Unit On/Off Model

#### 2.2.1. Objective Function

- (1)
- Startup/Shutdown water consumption:$${f}_{1}=\mathrm{min}{\displaystyle \sum _{n=1}^{N}{\displaystyle \sum _{i=1}^{{M}_{n}}{\displaystyle \sum _{t=1}^{T}({W}_{n,i,t,on}+{W}_{n,i,t,off})}}}$$$$\begin{array}{l}{W}_{n,i,t,on}={y}_{n,i,t}\times (1-{y}_{n,i,t-1})\times {W}_{n,i,on},\\ {W}_{n,i,t,off}={y}_{n,i,t-1}\times (1-{y}_{n,i,t})\times {W}_{n,i,off}.\end{array}$$
- (2)
- Number of units:$$m={f}_{2}=\mathrm{min}{\displaystyle \sum _{t=1}^{T}{m}_{t}}$$
_{n}are the index and number of units for the tunnel n; ${f}_{1}$ is total water consumption of startup/shutdown cost, m^{3}/s; y_{n,i,t}is the on/off state of unit i for tunnel n in period t (on = 1 and off = 0); W_{n,i,t,on}and W_{n,i,t,off}are the startup water consumption and shutdown water consumption of unit i for tunnel n in period t, m^{3}; W_{n,i,on}and W_{n,i,off}are the startup water consumption and shutdown water consumption of unit i for tunnel n_{,}with a given value, m^{3}; m and ${f}_{2}$ are the total number of units over the scheduling periods; and m_{t}is the active number of units in period t.

#### 2.2.2. Constraints

- (1)
- Unit number constraints:$$0\le {m}_{t}\le {\overline{m}}_{t}$$$${\overline{m}}_{t}={\displaystyle \sum _{n=1}^{{N}_{t}}{M}_{n,t}}$$
_{t}is the maximum number of effective tunnels in period t; and M_{n,t}is the maximum number of effective units of tunnel n in period t. - (2)
- System power balance constraints:$$0\le {D}_{t}\le {\displaystyle \sum _{n=1}^{{N}_{t}}{\displaystyle \sum _{i=1}^{{M}_{n,t}}{C}_{n,i}}}$$
_{t}is the system load demand in period t, MW; and C_{n,i}is the installed capacity of unit i for tunnel n, MW. - (3)
- Combining vibration zones limits:$$({D}_{t}-{\overline{pz}}_{t})({D}_{t}-{\underset{\_}{pz}}_{t})>0$$
- (4)
- Minimum uptime/downtime constrains:$$\{\begin{array}{l}{T}_{n,i,t,on}\le {T}_{n,i,t,up}\\ {T}_{n,i,t,off}\le {T}_{n,i,t,down}\end{array}$$
_{n,i,t,up}and T_{n,i,t,down}are the continuously uptime/downtime of unit i for tunnel n in period t, h; and T_{n,i,t,on}and T_{n,i,t,off}are the online/offline durations that unit i for tunnel n had been continuously up/down until period t, h.

#### 2.3. Load Distribution Model

#### 2.3.1. Objective Function

_{t}is the total power release in period t, m

^{3}/s; q

_{i,t}is the power release of unit i in period t, m

^{3}/s; and ${f}_{1}^{*}$ is the optimal water consumption of startup/shutdown cost in the first stage, m

^{3}.

#### 2.3.2. Constraints

- (1)
- Water balance constraints$${V}_{t}={V}_{t-1}+(I{N}_{t}-{u}_{t})\times \Delta t$$
_{t}is the water volume at the end of period t, m^{3}; and IN_{t}is the inflow in period t, m^{3}/s. - (2)
- Load balance constraints$$\sum _{t=1}^{T}{\displaystyle \sum _{i=1}^{{m}_{t}}{p}_{i,t}}}={D}_{t$$
_{i,t}is power output of unit i in period t, MW. - (3)
- Power output constraints$${\underset{\_}{P}}_{i,t}\le {p}_{i,t}\le {\overline{P}}_{i,t}$$
- (4)
- Reservoir storage volume limits$$\underset{\_}{V}\le {V}_{t}\le \overline{V}$$
^{3}. - (5)
- Water release limits$${\underset{\_}{Q}}_{i}\le {q}_{i,t}\le {\overline{Q}}_{i}$$
^{3}/s. - (6)
- Initial reservoir level limits$${z}_{0}={z}_{beg}$$
_{0}and z_{beg}are the initial reservoir level and the initial value of reservoir level, m. - (7)
- Vibration zones limits$$({p}_{i,t}-{\overline{ps}}_{i,t})({p}_{i,t}-{\underset{\_}{ps}}_{i,t})>0$$

## 3. Model Solution

#### 3.1. Solution Approach

#### 3.2. Search Process of Single-Period Feasible Solution Space

#### 3.3. Initial Feasible Solution Generation of Multiperiod

_{tk}, the kth element of the initial solution space S

_{t}in period t. Set t = 1 and k = 1. Ensure s

_{11}is the first element of the initial feasible solution ${S}_{1}^{\u2034}$.

_{t1}as the first element of the initial feasible solution ${S}_{t}^{\u2034}$. Otherwise, continue to Step 6.

_{t}, continue to Step 5. If not, replace s

_{tk}in period t with the kth element of the initial solution space S

_{t}and return to Step 3.

#### 3.4. Optimization Process Based on Progressive Optimality Algorithm of Multiperiod

_{1}and f

_{2}. Set t = 1, k = 1.

_{t}. If the new startup mode meets Constraint Group (4), continue to step 3; if not, continue to step 4.

_{1}and ${f}_{2}^{\prime}$ < f

_{2}, replace the original generating scheme with a new generating scheme and set f

_{1}= ${f}_{1}^{\prime}$ and f

_{2}= ${f}_{2}^{\prime}$.

_{t}, continue to Step 5. If not, return to Step 2.

#### 3.5. Solution of Load Distribution Model

_{i,t}as the decision variable; water release Q

_{i}(p

_{i,t}) is the cost function. The vibration zone constraint (constraint (7)) is considered a rigid restriction in this paper so the power output has to permanently avoid using the penalty function over the entire time horizon. The penalty function can be expressed as ${Q}^{\prime}(p,h)=Q(p,h)+\Delta h$, where $\Delta h$ is the penalty value.

#### 3.6. Two-Phase Decomposition Approach for Solving STLD of HPMTT Problem

## 4. Case Study

#### 4.1. Introduction of the Engineering Background and Setting the Parameters

^{2}and a natural drop of 762 m. As a region with abundant water energy resources in China, 11 cascaded hydropower stations for the main upstream of Hongshui River have been put into operation. Combined with the Datengxia Hydropower Station under construction, the mainstream has a total cascade generating capacity of 13,645 MW. Among the plants, TSQII is second, containing six main generating units with three different tunnels and an installed capacity of 1320 MW. The #1 and #2 units, #3 and #4 units, and #5 and #6 units are located in tunnels A, B, and C, respectively. TSQII is a daily regulating storage reservoir and is very sensitive to the change of water head. The head change directly impacts vibration zones dynamically and cannot be overlooked. TSQII is one of the main power plants for the West-to-East Electricity Transmission Project and has a significant role in frequency control and peak load regulation tasks. TSQII clearly reflects the characteristics of a HPMTT, which has greatly increased the difficult of modeling and optimization. A real-time operation example that incorporates different typical load rates in the dry season is developed.

#### 4.2. Analysis of Water Consumption for Different Startup Modes

^{3}/s, and the optimal water consumption is 3.34 × 10

^{5}m

^{3}. Meanwhile, the average head loss of tunnel B is 4.14 m, and the average water consumption rate is 2.05 m

^{3}/(kWh) for a turbine in one tunnel, in which the optimal startup mode is #1 for tunnel A, #4 for tunnel B, and #6 for tunnel C. The total power release is 393.4 m

^{3}/s and the optimal water consumption is 3.54 × 10

^{5}m

^{3}. On the other hand, the average head loss of tunnel B is 19.62 m, and the average water consumption rate is 2.17 m

^{3}/(kWh) for two turbines in one tunnel, in which the optimal startup mode is #1 for tunnel A, and #3 and #4 for tunnel B. As a result, the single period water consumption decreased by 2 × 10

^{4}m

^{3}, and the total water consumption decreased by 1.92 × 10

^{6}m

^{3}. The scheduling results show a significant benefit for the daily regulating storage reservoir with high water head.

#### 4.3. Comparison of Homogeneous Dispatch and TSM

^{7}m

^{3}; the TSM fell into the vibration zone operation 0 times and the total water consumption is 2.70 × 10

^{7}m

^{3}. Further, in dry season with low-rate load, the homogeneous dispatch fell into the vibration zone operation 34 times and the total water consumption is 2.93 × 10

^{7}m

^{3}; the TSM fell into the vibration zone operation 0 times and the total water consumption is 2.01 × 10

^{7}m

^{3}. These compared results fully demonstrate that the TSM is efficient for solving STLD for HPMTT problem while considering various constrains, and can get higher quality solutions with lower total water consumption.

#### 4.4. On/Off Status of Units and Tunnel Analysis

^{3}. There are four different schedules in Table 5: Two units operating from period 1 to 31 and period 84 to 96, four units operating from period 33 to 34 and period 54 to 82, five units operating from period 35 to 53 during high load, and three units for the other periods. Taking period 32 to 35 as an example, the power generation increased from 466.1 MW to 876.1 MW to respond to system demands. The number of startup units increased from three units to five units in order to connect the startup mode between periods. The three units operating included #1 for tunnel A, #3 for tunnel B, and #6 for tunnel C. Furthermore, when the load demands sharply decrease, the number of startup units correspondingly decrease. Power generation decreased from 646.1 MW to 428.5 MW to respond to system demands from period 83 to 84, and the number of startup units decreased from three units to two units.

^{3}. Correspondingly, to meet the peak load regulation task of the hydropower plant, four units operated from period 41 to 56 and period 71 to 85. During the other periods, the number of startup units was less than 4 and included a turbine in one-tunnel mode in order to make full use of water energy and reduce water consumption.

#### 4.5. Simulation Results and Analysis for Dry Season with High-Rate Load

#### 4.6. Simulation Results and Analysis for Dry Season with Low-Rate Load

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Feng, Z.K.; Niu, W.J.; Cheng, C.T. Optimizing electrical power production of hydropower system by uniform progressive optimality algorithm based on two-stage search mechanism and uniform design. J. Clean. Prod.
**2018**, 190, 432–442. [Google Scholar] [CrossRef] - Cheng, C.T.; Liao, S.L.; Tang, Z.T.; Zhao, M.Y. Comparison of particle swarm optimization and dynamic programming for large scale hydro unit load dispatch. Energy Conv. Manag.
**2009**, 50, 3007–3014. [Google Scholar] [CrossRef] - Han, J.C.; Huang, G.H.; Zhang, H.; Zhuge, Y.S.; He, L. Fuzzy constrained optimization of eco-friendly reservoir operation using self-adaptive genetic algorithm: A case study of a cascade reservoir system in the Yalong River, China. Ecohydrology
**2012**, 5, 768–778. [Google Scholar] [CrossRef] - Rezghi, A.; Riasi, A. The interaction effect of hydraulic transient conditions of two parallel pump-turbine units in a pumped-storage power plant with considering “S-shaped” instability region: Numerical simulation. Renew. Energy
**2018**, 118, 896–908. [Google Scholar] [CrossRef] - Xu, B.B.; Wang, F.F.; Chen, D.Y.; Zhang, H. Hamiltonian modeling of multi-hydro-turbine governing systems with sharing common penstock and dynamic analyses under shock load. Energy Conv. Manag.
**2016**, 108, 478–487. [Google Scholar] [CrossRef] - Arul, R.; Ravi, G.; Velusami, S. An improved harmony search algorithm to solve economic load dispatch problems with generator constraints. Electr. Eng.
**2014**, 96, 55–63. [Google Scholar] [CrossRef] - Alvarez, G.E.; Marcovecchio, M.G.; Aguirre, P.A. Security-constrained unit commitment problem including thermal and pumped storage units: An MILP formulation by the application of linear approximations techniques. Electr. Power Syst. Res.
**2018**, 154, 67–74. [Google Scholar] [CrossRef] - Fersi, M.; Triki, A. Investigation on redesigning strategies for water-hammer control in pressurized-piping systems. J. Press. Vessel Technol.
**2019**, 141, 021301. [Google Scholar] [CrossRef] - Ghidaoui, M.S.; Zhao, M.; McInnis, D.A.; Axworthy, D.H. A review of water hammer theory and practice. Appl. Mech. Rev.
**2005**, 58, 49–76. [Google Scholar] [CrossRef] - Nemati, M.; Braun, M.; Tenbohlen, S. Optimization of unit commitment and economic dispatch in microgrids based on genetic algorithm and mixed integer linear programming. Appl. Energy
**2018**, 210, 944–963. [Google Scholar] [CrossRef] - Kumar, D.N.; Reddy, M.J. Ant colony optimization for multi-purpose reservoir operation. Water Resour. Manag.
**2006**, 20, 879–898. [Google Scholar] [CrossRef] - Dubey, H.M.; Pandit, M.; Panigrahi, B.K. Ant lion optimization for short-term wind integratedhydrothermal power generation scheduling. Int. J. Electr. Power Energy Syst.
**2016**, 83, 158–174. [Google Scholar] [CrossRef] - Lu, P.; Zhou, J.Z.; Wang, C.; Qiao, Q.; Mo, L. Short-term hydro generation scheduling of Xiluodu and Xiangjiaba cascade hydropower stations using improved binary-real coded bee colony optimization algorithm. Energy Conv. Manag.
**2015**, 91, 19–31. [Google Scholar] [CrossRef] - Rasoulzadeh-Akhijahani, A.; Mohammadi-Ivatloo, B. Short-term hydrothermal generation scheduling by a modified dynamic neighborhood learning based particle swarm optimization. Int. J. Electr. Power Energy Syst.
**2015**, 67, 350–367. [Google Scholar] [CrossRef] - Kumar, D.N.; Reddy, M.J. Multipurpose reservoir operation using particle swarm optimization. J. Water Resour. Plan. Manag.
**2007**, 133, 192–201. [Google Scholar] [CrossRef] - Lee, H.; Maravelias, C.T. Discrete-time mixed-integer programming models for short-term scheduling in multipurpose environments. Comput. Chem. Eng.
**2017**, 107, 171–183. [Google Scholar] [CrossRef] - Guedes, L.S.M.; Maia, P.D.M.; Lisboa, A.C.; Vieira, D.A.G.; Saldanha, R.R. A unit commitment algorithm and a compact MILP model for short-term hydro-power generation scheduling. IEEE Trans. Power Syst.
**2017**, 32, 3381–3390. [Google Scholar] [CrossRef] - Borghetti, A.; D’Ambrosio, C.; Lodi, A.; Martello, S. An MILP approach for short-term hydro scheduling and unit commitment with head-dependent reservoir. IEEE Trans. Power Syst.
**2008**, 23, 1115–1124. [Google Scholar] [CrossRef] - Li, X.; Li, T.J.; Wei, J.H.; Wang, G.Q.; Yeh, W.W.G. Hydro unit commitment via mixed integer linear programming: A case study of the three gorges project, China. IEEE Trans. Power Syst.
**2014**, 29, 1232–1241. [Google Scholar] [CrossRef] - Pérez, D.J.; Wilhelmi, J.R.; Arévalo, L.A. Optimal short-term operation schedule of a hydropower plant in a competitive electricity market. Energy Convers. Manag.
**2010**, 51, 2955–2966. [Google Scholar] [CrossRef] - Bhullar, S.; Ghosh, S. Optimal integration of multi distributed generation sources in radial distribution networks using a hybrid algorithm. Energies
**2018**, 11, 628. [Google Scholar] [CrossRef] - Rajan, C.C.A. Hydro-thermal unit commitment problem using simulated annealing embedded evolutionary programming approach. Int. J. Electr. Power Energy Syst.
**2011**, 33, 939–946. [Google Scholar] [CrossRef] - Finardi, E.C.; Scuzziato, M.R. Hydro unit commitment and loading problem for day-ahead operation planning problem. Int. J. Electr. Power Energy Syst.
**2013**, 44, 7–16. [Google Scholar] [CrossRef] - Liao, S.L.; Li, Z.F.; Li, G.; Wang, J.Y.; Wu, X.Y. Modeling and optimization of the medium-term units commitment of thermal power. Energies
**2015**, 8, 12848–12864. [Google Scholar] [CrossRef] - Moradi, H.; Alasty, A.; Vossoughi, G. Nonlinear dynamics and control of bifurcation to regulate the performance of a boiler-turbine unit. Energy Conv. Manag.
**2013**, 68, 105–113. [Google Scholar] [CrossRef] - Zhu, H.; Huang, G.H. Dynamic stochastic fractional programming for sustainable management of electric power systems. Int. J. Electr. Power Energy Syst.
**2013**, 53, 553–563. [Google Scholar] [CrossRef] - Yan, D.L.; Wang, W.Y.; Chen, Q.J. Nonlinear modeling and dynamic analyses of the hydro–turbine governing system in the load shedding transient regime. Energies
**2018**, 11, 1244. [Google Scholar] [CrossRef] - Tijsseling, A.S. Water hammer with fluid–structure interaction in thick-walled pipes. Comput. Struct.
**2007**, 85, 844–851. [Google Scholar] [CrossRef] [Green Version] - Yang, W.J.; Yang, J.D.; Guo, W.C.; Zeng, W.; Wang, C.; Saarinen, L.; Norrlund, P. A mathematical model and its application for hydro power units under different operating conditions. Energies
**2015**, 8, 10260–10275. [Google Scholar] [CrossRef] - Gabl, R.; Gems, B.; Birkner, F.; Aufleger, M. Adaptation of an existing intake structure caused by increased sediment level. Water
**2018**, 10, 1066. [Google Scholar] [CrossRef] - Bermudez, M.; Cea, L.; Puertas, J.; Conde, A.; Martin, A.; Baztan, J. Hydraulic model study of the intake-outlet of a pumped-storage hydropower plant. Eng. Appl. Comp. Fluid Mech.
**2017**, 11, 483–495. [Google Scholar] [CrossRef] [Green Version] - Khan, L.; Wicklein, E.; Rashid, M.; Ebner, L.; Richards, N. Computational fluid dynamics modeling of turbine intake hydraulics at a hydropower plant. J. Hydraul. Res.
**2004**, 42, 61–69. [Google Scholar] [CrossRef]

**Figure 1.**Sketch of hydropower plant with multiple turbines in one tunnel (HPMTT) with long diversion system.

**Figure 2.**Flowchart of two-phase decomposition approach for solving short-term load dispatching (STLD).

Power Station | TSQII | Jinping II | Lubuge |
---|---|---|---|

Total installed capacity (MW) | 1320 | 4800 | 600 |

Length of tunnel $l$(km) | 9.77 | 16.67 | 9.38 |

Average head $\Delta h$(m) | 176 | 290 | 327.7 |

Number of tunnels $n$ | 3 | 4 | 1 |

Number of units $m$ | 2 | 2 | 3 |

Item | Value |
---|---|

Maximum water head (m) | 645.00 |

Minimum water head (m) | 637.00 |

Units (capacity × number, MW) | 220.0 × 6 |

Vibration zones (MW) | (80,190) |

Startup/Shutdown water consumption (m^{3}) | 1200 |

Discrete step of water level (m) | 0.1 |

Initial dam water level for dry season (m) | 642.18 |

Duration of online/offline of units | 4 |

Scheduling period (min) | 15 |

Parameter A of penstock head loss | 2.7 × 10^{−4} |

Tunnel | Unit | Turbine in One Tunnel | Two Turbines in One Tunnel | ||||||
---|---|---|---|---|---|---|---|---|---|

Load (MW) | Power Release (m ^{3}/s) | Head Loss (m) | Water Consumption Rate (m ^{3}/(kWh)) | Load (MW) | Power Release (m ^{3}/s) | Head Loss (m) | Water Consumption Rate (m ^{3}/(kWh)) | ||

A | #1 | 217.6 | 123.8 | 4.14 | 2.05 | 217.6 | 123.8 | 4.14 | 2.05 |

#2 | 0 | 0 | - | 0 | 0 | - | |||

B | #3 | 0 | 0 | 4.14 | - | 217.5 | 134.8 | 19.62 | 2.23 |

#4 | 217.5 | 123.8 | 2.05 | 217.5 | 134.8 | 2.23 | |||

C | #5 | 0 | 0 | 4.14 | - | 0 | 0 | 0 | - |

#6 | 217.5 | 123.8 | 2.05 | 0 | 0 | - | |||

Total value | 652.6 | 371.4 | - | 2.05 | 652.6 | 393.4 | - | 2.17 |

Item | Dry Season with High-Rate Load | Dry Season with Low-Rate Load | ||
---|---|---|---|---|

Homogeneous Dispatch | TSM | Homogeneous Dispatch | TSM | |

Number of fell into vibration zone | 51 | 0 | 34 | 0 |

Water consumption (m^{3}) | 3.75 × 10^{7} | 2.70 × 10^{7} | 2.93 × 10^{7} | 2.01 × 10^{7} |

Period | Load (MW) | Tunnel A | Tunnel B | Tunnel C | Total | Duration of Unit | |||
---|---|---|---|---|---|---|---|---|---|

#1 | #2 | #3 | #4 | #5 | #6 | ||||

1 | 427.5 | 1 | 0 | 1 | 0 | 0 | 0 | 2 | 31 |

32 | 466.1 | 1 | 0 | 1 | 0 | 0 | 1 | 3 | 1 |

33 | 690.2 | 1 | 0 | 1 | 1 | 0 | 1 | 4 | 2 |

35 | 876.1 | 1 | 1 | 1 | 1 | 0 | 1 | 5 | 19 |

54 | 857.6 | 1 | 1 | 0 | 1 | 0 | 1 | 4 | 29 |

83 | 646.1 | 0 | 1 | 0 | 1 | 0 | 1 | 3 | 1 |

84 | 428.5 | 0 | 1 | 0 | 0 | 0 | 1 | 2 | 13 |

96 | 397.5 | 0 | 1 | 0 | 0 | 0 | 1 | 2 |

^{3}.

Period | Load (MW) | Tunnel A | Tunnel B | Tunnel C | Total | The Duration of Unit | |||
---|---|---|---|---|---|---|---|---|---|

#1 | #2 | #3 | #4 | #5 | #6 | ||||

1 | 71.3 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 36 |

37 | 268.6 | 0 | 1 | 1 | 0 | 0 | 0 | 2 | 2 |

39 | 466.4 | 0 | 1 | 1 | 0 | 1 | 0 | 3 | 2 |

41 | 685.3 | 1 | 1 | 1 | 0 | 1 | 0 | 4 | 16 |

57 | 479.9 | 1 | 0 | 1 | 0 | 1 | 0 | 3 | 1 |

58 | 436.5 | 1 | 0 | 0 | 0 | 1 | 0 | 2 | 2 |

60 | 78.3 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 7 |

67 | 243.1 | 1 | 0 | 0 | 0 | 1 | 0 | 2 | 2 |

69 | 444.3 | 1 | 0 | 1 | 0 | 1 | 0 | 3 | 2 |

71 | 670 | 1 | 0 | 1 | 1 | 1 | 0 | 4 | 15 |

86 | 652.6 | 1 | 0 | 1 | 0 | 1 | 0 | 3 | 2 |

88 | 438.5 | 1 | 0 | 1 | 0 | 0 | 0 | 2 | 3 |

91 | 77.6 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 6 |

96 | 72.3 | 1 | 0 | 0 | 0 | 0 | 0 | 1 |

^{3}.

Combinations of Generating Units | Capacity Combination/MW | Combined Vibration Zones/MW |
---|---|---|

One unit | 220 | (80,190) |

Two units | 440 | (160,190) ∪ (300,380) |

Three units | 660 | (520,570) |

Four units | 880 | (740,760) |

Five units | 1100 | Vibration-free Zone |

Six units | 1320 | Vibration-free Zone |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Liao, S.; Zhao, H.; Li, G.; Liu, B.
Short-Term Load Dispatching Method for a Diversion Hydropower Plant with Multiple Turbines in One Tunnel Using a Two-Stage Model. *Energies* **2019**, *12*, 1476.
https://doi.org/10.3390/en12081476

**AMA Style**

Liao S, Zhao H, Li G, Liu B.
Short-Term Load Dispatching Method for a Diversion Hydropower Plant with Multiple Turbines in One Tunnel Using a Two-Stage Model. *Energies*. 2019; 12(8):1476.
https://doi.org/10.3390/en12081476

**Chicago/Turabian Style**

Liao, Shengli, Hongye Zhao, Gang Li, and Benxi Liu.
2019. "Short-Term Load Dispatching Method for a Diversion Hydropower Plant with Multiple Turbines in One Tunnel Using a Two-Stage Model" *Energies* 12, no. 8: 1476.
https://doi.org/10.3390/en12081476