Abstract
The high penetration of wind generation has prompted the development of stochastic hydrothermal unit commitment (SHTUC) models, which are more difficult to be solved than their thermal-based counterparts due to hydro generation constraints and inflow uncertainties. For handling the uncertainty, the problem is usually formulated as a two-stage stochastic model (2S-SHTUC), although multistage (MS-SHTUC) formulations have gained increasing attention due to their more realistic assumptions about on–off decisions over the planning horizon. Benders decomposition (BD) is one of the most common methodologies used for solving 2S-SHTUC and MS-SHTUC problems. To overcome the well-known slow convergence of the classical BD when applied to large problems, most authors use accelerating techniques. In this paper, we implement state-of-the-art stabilization methods tailored for speeding up the convergence of the classical BD: local branching and the level regularization. Our experiments are conducted in a real-life SHTUC problem with 11 thermal units, 16 hydro plants, 46 buses and 95 lines. The results show that 2S and MS-SHTUC can benefit from stabilization. The savings in computing times range from 69 to 95% for the 2S-SHTUC model and from 77 to 89% for the MS-SHTUC.
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Notes
We attempted to solve the deterministic equivalent of the cases without decomposition with a 1% gap and a 24 h time-limit. In our tests, Gurobi was not able to find a feasible solution to any instance.
We assume a zero cost when the current master problem’s solution is not feasible in the SP.
Abbreviations
- Ns, Mp :
-
Nodes in the subproblem/master problem tree
- \({\mathcal{L}}\) :
-
Leaf nodes
- \({\mathcal{P}}^{n}\) :
-
Set of nodes in the path of node n
- G, H :
-
Thermal and hydro plants
- UR h :
-
Hydro plants upstream of plant h
- HP h :
-
Hydro production function of plant h
- E :
-
Future cost function
- B, TL :
-
Buses and transmission lines
- n, m :
-
Node in the subproblem/master problem tree
- o :
-
Auxiliary index for nodes
- l :
-
Leaf node
- \(n_{n}^{ - } ,\;m_{m}^{ - }\) :
-
Parent nodes of nodes n and m
- g, h, w :
-
Thermal, hydro and wind plant
- u :
-
Auxiliary index for hydro plants
- b, j :
-
Indices for buses
- tl:
-
Transmission line
- tg g,n, z g,m, a g,m, b g,m :
-
Generation at node n, status, start-up and shut-down decision of thermal plant g at node m
- hg h,n, v h,n, q h,n, s h,n :
-
Generation, volume, turbine discharge and spillage of hydro h at node n
- \(\delta_{b,n}^{ + }\), \(\delta_{b,n}^{ - }\) :
-
Deficit and surplus at bus b and node n
- ω l :
-
Expected operation cost at leaf node l
- θ b ,n :
-
Voltage angle at bus b and node n
- CV g, UC g, DC g :
-
Variable, start-up and shut-down cost of thermal g
- \(\overline{{\mathbf{R}}}_{g}\), \(\underline{{\mathbf{R}}}_{g}\) :
-
Ramp-up/down rate limit of thermal unit g
- \(\overline{{{\mathbf{TG}}}}_{g}\), \(\underline{{{\mathbf{TG}}}}_{g}\) :
-
Generation limits of thermal unit g
- SU g, SD g :
-
Start-up/shut-down generation requirement of thermal unit g
- TU g, TD g :
-
Minimum-up and down-times of unit g
- A G :
-
Generator-bus incidence matrix
- A h,n :
-
Inflow realization to hydro plant h at node n
- \(\overline{{\mathbf{V}}}_{h}\), \(\underline{{\mathbf{V}}}_{h}\) :
-
Bounds on reservoir volume of plant h
- \(\overline{{\mathbf{Q}}}_{h}\), \(\overline{{\mathbf{S}}}_{h}\) :
-
Maximum turbined outflow (spillage) of plant h
- C x h,i :
-
Constant x (0,…,3) of constraint i of the piecewise linear production function of hydro h
- K h,i :
-
Constant of plant h in i-th linear constraint of the future cost function
- R i :
-
Right-hand side of the i-th linear constraint of the future cost function
- A H :
-
Hydro plant-bus incidence matrix
- D u,h :
-
Water travel time from plant u to h
- P n :
-
Absolute probability of node n
- CS :
-
Unitary cost of deficit and generation surplus
- T n :
-
Time associated with node n
- WG w,n :
-
Wind generation of wind farm w at node n
- A W :
-
Wind farm-bus incidence matrix
- \({\mathbf{L}}_{b,n}^{{}}\) :
-
Load at bus b and node n
- \({\mathbf{B}}\) :
-
Susceptance matrix
- A TL :
-
Transmission line-bus incidence matrix
- \(\underline{{{\mathbf{TL}}}}_{t,l}\), \(\overline{{{\mathbf{TL}}}}_{t,l}\) :
-
Transmission limits of line tl
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Colonetti, B., Finardi, E.C. Stochastic hydrothermal unit commitment models via stabilized benders decomposition. Electr Eng 103, 2197–2211 (2021). https://doi.org/10.1007/s00202-020-01206-0
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DOI: https://doi.org/10.1007/s00202-020-01206-0