# Transmission Expansion Planning under Uncertainty for Investment Options with Various Lead-Times

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Proposed Framework

#### 2.1. Multi-Stage Model

#### 2.2. Demand Uncertainty in Multi-Stage Model

#### 2.3. TEP Problems with Multi-Stage Model

## 3. Mathematical Formulation and Solution Procedure

#### 3.1. Model Formulation

#### 3.2. Stochastic Mixed Integer Linear Problem

#### 3.3. Solution Procedure

## 4. Case Studies

#### 4.1. Modified Garver’s Six-Bus System

#### 4.2. Simulation: Validity of the Proposed Framework

_{26}or L

_{46}was/were constructed only when the benefit of cheap generation was greater than the cost of construction and load shedding. Moreover, the two transmission lines had a longer construction lead time than other transmission lines.

_{23}and L

_{35}. Therefore, the optimal solution was exposed to less uncertainty than in Case 2. Only one circuit connecting the generating unit at bus 6 was built because circuit L

_{46}was sufficient to cover the uncertainty in this case. The total expected cost was also the least in this case, indicating that the proposed approach showed better performance in dealing with the risk of uncertainty. Figure 9 shows an RSD curve estimated from historical data and the calculated RSD values. The value of $RS{D}_{{D}_{yt}}^{Case3}$ was 10.6%, while the value of $RS{D}_{{D}_{yt}}^{Case2}$ was 15%.

#### 4.3. Simulation: Various RSD Curve Shapes

#### 4.4. Simulation: Various Lead Times

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Indices | |

$y$ | Years |

$l$ | Transmission lines |

$d$ | Demands |

$n$ | Nodes |

$g$ | Generating Units |

$p$ | Forecasting periods, i.e., lead times |

Sets | |

${\Omega}_{Y}$ | Years of stages |

${\Omega}_{N}$ | Nodes |

${\Omega}_{G}$ | Generating units |

${\Omega}_{D}$ | Demands |

${\Omega}_{L}$ | Transmission lines |

${\Omega}_{P}$ | Forecasting periods, i.e., lead times |

${\Psi}_{L}^{{y}_{t}-y}$ | Transmission investment candidates with lead time $({y}_{t}-y)$ |

${\Psi}_{L}^{({y}_{t}-{y}_{k})+}$ | Transmission investment candidates with a lead time longer than or equal to $({y}_{t}-{y}_{k})$ |

${\Psi}_{L}^{+}$ | Transmission line candidates |

${\Psi}_{D}^{n}$ | Demands located at node $n$ |

${\Psi}_{G}^{n}$ | Generating units located at node $n$ |

$i(l)$ | Sending-end node $n$ of Transmission line $l$ |

$j(l)$ | Receiving-end node $n$ of Transmission line $l$ |

Parameters | |

${\mu}_{E}$ | Expected demand growth |

${\sigma}_{E}$ | Standard deviation of ${\mu}_{E}$ |

${\mu}_{D}$ | Expected demand |

${\sigma}_{D}$ | Standard deviation of ${\mu}_{D}$ |

${I}_{l}$ | Annualized investment cost of transmission line $l$ ($/MW) |

$a$ | Discount rate |

$\alpha $ | Dimension factor |

${y}_{0}$ | Beginning year of the planning process |

${y}_{t}$ | Terminal year of the planning process |

${C}_{g}^{Gen}$ | Generation cost of generating unit $g$ ($) |

${C}^{Lsh}$ | Load shedding cost ($) |

${B}_{l}$ | Susceptance of transmission line $l$ (S) |

${f}_{l}^{\mathrm{max}}$ | Maximum transmission capacity of line $l$ (MW) |

${p}_{g}^{\mathrm{max}}$ | Maximum generation capacity of unit $g$ (MW) |

${r}_{d}^{\mathrm{max}}$ | Upper bound of load shedding for demand $d$ (MW) |

Binary Variable | |

${x}_{l}$ | Binary variable that is 1 for the construction of transmission line $l$, 0 otherwise |

Continuous Variables | |

${p}_{g}$ | Power produced by generating unit $g$ (MW) |

${r}_{d}$ | Load shedding of demand $d$ (MW) |

${f}_{l}$ | Power flow through transmission line $l$ (MW) |

${\theta}_{n}$ | Voltage phase angle at node $n$ (rad) |

Stochastic Variables | |

${D}^{peak}$ | Stochastic variable for peak demand |

${D}^{node}$ | Stochastic variable for node demand |

## References

- Li, J.; Ma, Y.; Mu, G.; Feng, X.; Yan, G.; Guo, G.; Zhang, T. Optimal configuration of energy storage system coordinating wind turbine to participate power system primary frequency regulation. Energies
**2018**, 11, 1396. [Google Scholar] [CrossRef] - Oprea, S.-V.; Bara, A.; Majstrovic, G. Aspects referring wind energy integration from the power system point of view in the region of southeast Europe. Study case of Romania. Energies
**2018**, 11, 251. [Google Scholar] [CrossRef] - Fang, X.; Krishnan, V.; Hodge, B.M. Strategic offering for wind power producers considering energy and flexible ramping products. Energies
**2018**, 11, 1239. [Google Scholar] [CrossRef] - Jo, K.H.; Kim, M.K. Improved genetic algorithm-based unit commitment considering uncertainty integration method. Energies
**2018**, 11, 1387. [Google Scholar] [CrossRef] - Orfanos, G.A.; Georgilakis, P.S.; Hatziargyriou, N.D. Transmission expansion planning of systems with increasing wind power integration. IEEE Trans. Power Syst.
**2013**, 28, 1355–1362. [Google Scholar] [CrossRef] - Choi, J.; Tran, T.; El-Keib, A.A.; Thomas, R.; Oh, H.; Billinton, A. A method for transmission system expansion planning considering probabilistic reliability criteria. IEEE Trans. Power Syst.
**2005**, 20, 1606–1615. [Google Scholar] [CrossRef] - Choi, J.; Mount, T.M.; Thomas, R.J. Transmission expansion planning using contingency criteria. IEEE Trans. Power Syst.
**2007**, 22, 2249–2261. [Google Scholar] [CrossRef] - Moreira, A.; Street, A.; Arroyo, J.M. An adjustable robust optimization approach for contingency-constrained transmission expansion planning. IEEE Trans. Power Syst.
**2015**, 30, 2013–2022. [Google Scholar] [CrossRef] - Yu, H.; Chung, C.Y.; Wong, K.P.; Zhang, J.H. A chance constrained transmission network expansion planning method with consideration of load and wind farm uncertainties. IEEE Trans. Power Syst.
**2009**, 24, 1568–1576. [Google Scholar] [CrossRef] - Teh, J.; Lai, C.M.; Cheng, Y.H. Composite reliability evaluation for transmission network planning. AIMS Energy
**2018**, 6, 170–186. [Google Scholar] [CrossRef] - Hemmati, R.; hooshmand, R.A.; Kodabakhshian, A. State-of-the-art of transmission expansion planning: Comprehensive review. Renew. Sustain. Ener. Rev.
**2013**, 23, 312–319. [Google Scholar] [CrossRef] - Lee, C.W.; Ng, S.K.; Zhong, J.; Wu, F.F. Transmission expansion planning from past to future. In Proceedings of the 2006 IEEE PES Power Systems Conference and Exposition, Atlanta, GA, USA, 29 Octomber–1 November 2006. [Google Scholar]
- Vinasco, G.; Rider, M.J.; Romero, R. A strategy to solve multistage transmission expansion planning problem. IEEE Trans. Power Syst.
**2011**, 26, 2574–2576. [Google Scholar] [CrossRef] - Escobar, A.H.; Gallego, R.A.; Romero, R. Multistage and coordinated planning of the expansion of transmission systems. IEEE Trans. Power Syst.
**2004**, 19, 735–744. [Google Scholar] [CrossRef] - Olsina, F.; Garcés, F.; Haubrich, H.J. Modeling long-term dynamics of electricity markets. Energy Policy
**2006**, 34, 1411–1433. [Google Scholar] [CrossRef] - Newham, N. Power System Investment Planning Using Stochastic Dual Dynamic Programming. Ph.D. Thesis, University of Canterbury, Christchurch, New Zealand, April 2008. [Google Scholar]
- Lumbreras, S.; Ramos, A. The new challenges to transmission expansion planning. Survey of recent practice and literature review. Electr. Power Syst. Res.
**2016**, 134, 19–29. [Google Scholar] [CrossRef] - Rabih, J. Robust Transmission network expansion planning with uncertain renewable generation and load. IEEE Trans. Power Syst.
**2013**, 28, 4558–4567. [Google Scholar] - Silva, I.J.; Rider, M.J.; Romero, R.; Murari, C.A.F. Transmission network expansion planning considering uncertainty in demand. IEEE Trans. Power Syst.
**2006**, 21, 1565–1573. [Google Scholar] [CrossRef] - Botterud, A.; Ilic, M.D.; Wangensteen, I. Optimal investments in power generation under centralized and decentralized decision making. IEEE Trans. Power Syst.
**2005**, 20, 254–263. [Google Scholar] [CrossRef] - Roh, J.H.; Shahidehpour, M.; Wu, L. Market-based generation and transmission planning with uncertainties. IEEE Trans. Power Syst.
**2009**, 24, 1587–1598. [Google Scholar] - Zhao, J.H.; Dong, Z.Y.; Lindsay, P.; Wong, K.P. Flexible Transmission expansion planning with uncertainties in an electricity market. IEEE Trans. Power Syst.
**2009**, 24, 479–488. [Google Scholar] [CrossRef] - Yang, N.; Wen, F. A chance constrained programming approach to transmission system expansion planning. Elec. Power Syst. Res.
**2005**, 75, 171–177. [Google Scholar] [CrossRef] - Botterud, A.; Korpas, M. A stochastic dynamic model for optimal timing of investments in new generation capacity in restructured power systems. Electr. Power Energy Syst.
**2007**, 29, 163–174. [Google Scholar] [CrossRef] - Kagiannas, A.G.; Askounis, D.T.; Psarras, J. Power generation planning: A survey from monopoly to competition. Electr. Power Energy Syst.
**2004**, 26, 413–421. [Google Scholar] [CrossRef] - Wu, F.F.; Zheng, F.L.; Wen, F.S. Transmission investment and expansion planning in a restructured electricity market. Energy
**2006**, 31, 954–966. [Google Scholar] [CrossRef] - TYNDP 2018 Scenario Report. Available online: https://docstore.entsoe.eu/Documents/TYNDP%20documents/14475_ENTSO_ScenarioReport_Main.pdf (accessed on 1 September 2018).
- PJM Regional Transmission Planning Process. Available online: https://pjm.com/-/media/documents/manuals/archive/m14b/m14bv39-regional-transmission-planning-process-09-29-2017.ashx (accessed on 1 September 2018).
- Blanco, G.; Olsina, F.; Garcés, F.; Rehtanz, C. Real option valuation of FACTS investments based on the least square Monte Carlo method. IEEE Trans. Power Syst.
**2011**, 26, 1389–1398. [Google Scholar] [CrossRef] - Konstantelos, I.; Strbac, G. Valuation of flexible transmission investment options under uncertainty. IEEE Trans. Power Syst.
**2015**, 30, 1047–1055. [Google Scholar] [CrossRef] - Li, J.; Li, Z.; Liu, F.; Ye, H.; Zhang, X.; Mei, S.; Chang, N. Robust coordinated transmission and generation expansion planning considering ramping requirements and construction periods. IEEE Trans. Power Syst.
**2018**, 33, 268–280. [Google Scholar] [CrossRef] - Romero, R.; Monticelli, A.; Garcia, A.; Haffner, S. Test systems and mathematical models for transmission network expansion planning. IEEE Proc.-Gener. Transm. Distrib.
**2002**, 149, 27–36. [Google Scholar] [CrossRef] - Birge, J.R.; Louveaux, F. Introduction to Stochastic Programming; Springer: New York, NY, USA, 1997. [Google Scholar]
- Villasana, R.; Garver, L.L.; Salon, S.J. Transmission network planning using linear programming. IEEE Trans. Power Apparatus Syst.
**1985**, PAS-104, 349–356. [Google Scholar] [CrossRef] - Sharifnia, A.; Aashtiani, H.Z. Transmission network planning: A method for synthesis of minimum-cost secure networks. IEEE Trans. Power Apparatus Syst.
**1985**, PAS-104, 2026–2034. [Google Scholar] [CrossRef] - Garver, L.L. Transmission network estimation using linear programming. IEEE Trans. Power Apparatus Syst.
**1970**, PAS-89, 1688–1697. [Google Scholar] [CrossRef]

Bus No. | Max. Generation (MW) | Generation Cost ($/MW) | Load (MW) | Load Shedding Cost ($/MW) |
---|---|---|---|---|

1 | 150 | 21 | 80 | 150 |

2 | 0 | 0 | 240 | 150 |

3 | 360 | 17 | 40 | 150 |

4 | 0 | 0 | 160 | 150 |

5 | 0 | 0 | 240 | 150 |

6 | 600 | 10 | 0 | 150 |

From–To | Lead Time (year) | Susceptance (S) | ${\mathit{f}}_{\mathit{l}}^{\mathbf{max}}\left(\mathbf{MW}\right)$ | Cost (10^{5}$) |
---|---|---|---|---|

1–2 | 5 | 250 | 100 | 10 |

1–4 | 5 | 167 | 80 | 10 |

1–5 | 5 | 500 | 100 | 10 |

2–3 | 5 | 500 | 100 | 10 |

2–4 | 5 | 250 | 100 | 10 |

2–6 | 10 | 333 | 100 | 20 |

3–5 | 5 | 500 | 100 | 10 |

4–6 | 10 | 333 | 100 | 20 |

Case | Added Circuits | Investment Cost (10^{5}$) | Total Expected Cost (10^{5}$) | |||
---|---|---|---|---|---|---|

L_{23} | L_{35} | L_{26} | L_{46} | |||

1 | 1 | 1 | 0 | 0 | 20 | 863 |

2 | 0 | 1 | 2 | 0 | 50 | 854 |

3 | 1 | 1 | 0 | 1 | 40 | 839 |

Type | Added Circuits | $\mathit{R}\mathit{S}{\mathit{D}}_{\mathit{y}1}(\%)$ | $\mathit{R}\mathit{S}{\mathit{D}}_{\mathit{y}\mathit{t}}(\%)$ | |||
---|---|---|---|---|---|---|

L_{23} | L_{35} | L_{26} | L_{46} | |||

Linear | 1 | 1 | 0 | 1 | 7.5 | 10.6 |

Square Root | 0 | 1 | 2 | 0 | 10.1 | 14.3 |

Quadratic | 1 | 1 | 0 | 1 | 5.5 | 7.8 |

From-To | 1–2, 1–4, 1–5, 2–3, 2–4, 3–5 | 2–6, 4–6 | |

Lead Time (year) | Scenario 1 | 1 | 10 |

Scenario 2 | 3 | 10 | |

Scenario 3 | 5 | 10 |

Scenario | Added Circuits | ${\mathit{R}\mathit{S}\mathit{D}}_{{\mathit{D}}_{\mathit{y}\mathit{t}}}(\%)$ | Total Expected Cost (10^{5}$) | |||

L_{23} | L_{35} | L_{26} | L_{46} | |||

S1 | 0 | 1 | 2 | 0 | 13.6 | 851 |

S2 | 1 | 1 | 0 | 1 | 11.4 | 841 |

S3 | 1 | 1 | 0 | 1 | 10.6 | 839 |

© 2018 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**

Kim, W.-W.; Park, J.-K.; Yoon, Y.-T.; Kim, M.-K.
Transmission Expansion Planning under Uncertainty for Investment Options with Various Lead-Times. *Energies* **2018**, *11*, 2429.
https://doi.org/10.3390/en11092429

**AMA Style**

Kim W-W, Park J-K, Yoon Y-T, Kim M-K.
Transmission Expansion Planning under Uncertainty for Investment Options with Various Lead-Times. *Energies*. 2018; 11(9):2429.
https://doi.org/10.3390/en11092429

**Chicago/Turabian Style**

Kim, Wook-Won, Jong-Keun Park, Yong-Tae Yoon, and Mun-Kyeom Kim.
2018. "Transmission Expansion Planning under Uncertainty for Investment Options with Various Lead-Times" *Energies* 11, no. 9: 2429.
https://doi.org/10.3390/en11092429