Day-Ahead Energy and Reserve Dispatch Problem under Non-Probabilistic Uncertainty
Abstract
:1. Introduction
Interpretation of Lower and Upper Previsions—Walley’s Integration
2. Model
2.1. Recap
2.1.1. Linear Programming under Uncertainty
2.1.2. Reformulating LP Problem as a Decision Problem
Maximality Criterion
Maximinity (Γ-Maximin) Criterion
2.1.3. Interval Model
Maximin Solutions in Interval Case
Maximal Solutions in Interval Case
2.1.4. -Contamination Model
Maximin and Maximal Solutions in Contamination Case
2.2. Day-Ahead Energy and Reserve Dispatch Problem
2.3. Use Case—Data and DA-Problem Modelling
2.4. Interval Model—DA-Energy and Reserve Dispatch Problem
- are unit metrics,
- are the optimisation variables,
- are given via Table 1,
- represent the set of generators, wind farms, and demands, respectively.
2.4.1. Maximin Solutions in Interval Case
2.4.2. Maximal Solutions in Interval Case
2.5. -Contamination Case—DA-Energy and Reserve Dispatch Problem
2.5.1. Maximin Solutions in -Contamination Case
2.5.2. Maximal Solutions in -Contamination Case
- (for the spacing, we assume ),
- is the maximin solution,
- , and
- is the mean value from the given probabilistic model.
3. Numerical Maximin and Maximal Solutions for the DA-Energy and Reserve Dispatch Problem
3.1. Interval Case
- ,
- are the optimisation variables,
- are given via Table 1, and
- represent the set of generators, wind farms, and demands, respectively.
3.1.1. Numerical Maximin Solution in Interval Case
- ,
- , and
- .
3.1.2. Numerical Maximal Solutions in Interval Case
- ,
- ,
- .
3.2. -Contamination Case
3.2.1. Numerical Maximin Solutions in -Contamination Case
- (),
- ,
- ,
- , and
- .
3.2.2. Numerical Maximal Solutions in -Contamination Case
- (),
- ,
- ,
- , and
- .
4. Discussion
4.1. Numerical Results
- Case (i) where the epsilon is very close to zero. In this case, the contamination solution coincides with the purely probabilistic uncertain model (normal distribution) solution. When then the imprecise part of contamination model is zero and we have a pure precise case. In this case, the maximin and maximal solutions are the same as in the solutions proposed in Section 3.2.1. In this case, the cost is equal to which is almost half of the interval case solution discussed in Section 3.1.1.
- Case (ii) where the epsilon is very close to zero. In this case, the contamination solution coincides with the interval solution. When then the precise part of the contamination model is zero and we have a pure imprecise case. In this case, the maximin and maximal solutions are equal to the interval solutions discussed in Section 3.1.1 and Section 3.1.2. We have discussed that maximal solutions are more optimal and riskier than the maximin solution. The most optimal solution has the highest risk, and the cost value is 21130, while the lowest risky solution has the cost value of .
- Case (iii) where the epsilon is between zero and one. In this case the contamination maximin (maximal) solution(s) given in Section 3.2.1, is defined based on the selected value. The cost value in this case is between and . For instance, if a decision maker chooses , e.g., , then he trusts less () to the probabilistic model and to the imprecise model. The cost value with is equal to which is lower than the interval case and higher than the pure probabilistic case.
4.2. Comparison
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
DA | day-ahead |
LP | linear programming |
S-LP | symbolic linear programming |
LPUU | LP under uncertainty |
sup | supremum |
inf | infimum |
DRO | distributionally robust optimisation |
set of generators (connected to busn) | |
generator | |
set of loads (connected to bus n) | |
load | |
set of wind farms (connected to bus n) | |
wind farm | |
production cost of generator g [€/MWh] | |
cost of energy generation | |
the generation costs in $/MWh | |
the energy generation from the generators in MWh | |
the cost of procuring upward/downward reserve capacity | |
the upward/downward reserve procurement costs in $/MW | |
the reserve procurement of the generators in MW | |
the real-time cost of balancing actions | |
participation factor of each generator per-unit uncertain renewable energy deviation | |
day-ahead forecast | |
the balancing action in real-time of the generators in MW | |
scheduling decision parameter | |
the set of decision variables |
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Generators | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
($/MWh) | 13.32 | 13.32 | 20.7 | 20.93 | 26.11 | 10.52 | 10.52 | 6.02 | 5.47 | 7 | 10.52 | 10.9 | |
($/MW) | 1.68 | 1.68 | 3.3 | 4.07 | 1.89 | 5.48 | 5.48 | 4.98 | 5.53 | 8 | 3.45 | 5.11 | |
($/MW) | 2.32 | 2.32 | 4.67 | 3.93 | 3.11 | 3.52 | 3.52 | 5.02 | 4.97 | 6.00 | 2.52 | 2.9 | |
(MW) | 127.7 | 127.7 | 294 | 496.5 | 50.4 | 130.2 | 130.2 | 336 | 336 | 252 | 260.4 | 294 | |
(MW) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
(MW) | 48 | 48 | 84 | 216 | 42 | 36 | 36 | 60 | 60 | 48 | 72 | 48 | |
Wind Farms | 1 | 2 | 3 | 4 | |||||||||
W (MW) | 500 | 500 | 300 | 300 | |||||||||
Loads | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13–17 |
d (MW) | 84 | 75 | 139 | 58 | 55 | 106 | 97 | 132 | 135 | 150 | 205 | 150 | ... |
...245 | 77 | 258 | 141 | 100 |
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Shariatmadar, K.; Arrigo, A.; Vallée, F.; Hallez, H.; Vandevelde, L.; Moens, D. Day-Ahead Energy and Reserve Dispatch Problem under Non-Probabilistic Uncertainty. Energies 2021, 14, 1016. https://doi.org/10.3390/en14041016
Shariatmadar K, Arrigo A, Vallée F, Hallez H, Vandevelde L, Moens D. Day-Ahead Energy and Reserve Dispatch Problem under Non-Probabilistic Uncertainty. Energies. 2021; 14(4):1016. https://doi.org/10.3390/en14041016
Chicago/Turabian StyleShariatmadar, Keivan, Adriano Arrigo, François Vallée, Hans Hallez, Lieven Vandevelde, and David Moens. 2021. "Day-Ahead Energy and Reserve Dispatch Problem under Non-Probabilistic Uncertainty" Energies 14, no. 4: 1016. https://doi.org/10.3390/en14041016