# Stochastic Second-Order Conic Programming for Optimal Sizing of Distributed Generator Units and Electric Vehicle Charging Stations

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## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

#### 1.2. Literature Review

#### 1.3. Paper Contribution and Organization

- To consider the uncertainties of the system assets (PV, WT, and EVCS) simultaneously, an integrated scenario method is introduced, and a scenario-based stochastic program is conducted to optimize the PV, WT, and EVCS capacities.
- The stochastic program is formulated as an SOCP model, to solve the AC power flow constrained problem; this is a convex problem that can be solved in polynomial time using a global optimal solution.

## 2. Scenario Generation of DG Units and EVCS

#### 2.1. Modeling of Uncertainty

#### 2.1.1. Output Uncertainty Modeling for WTs

#### 2.1.2. Output Uncertainty of PVs

#### 2.1.3. Modeling for EV Charging Demand

#### 2.1.4. Integrated Scenario Generation

- (1)
- Select the number of clusters k. In this step, the elbow method was used to determine the optimal number of clusters.
- (2)
- The initial centroids that are the central data point for each cluster are randomly selected.
- (3)
- The distance between the centroids and observation (i.e., each data point) and each observation is assigned to the nearest centroid.
- (4)
- The centroids are recalculated to be the center of the mass of observations within each cluster.
- (5)
- Repeat steps (3) and (4) until the clusters no longer charge.
- (6)
- The centroid data were taken as representative scenarios, and the probability of each representative scenario was assigned according to the number of observations within each cluster divided by the total number of sampling data points. This can be expressed as

## 3. Proposed Optimization Formulation and Preliminaries for Optimization Model

#### 3.1. Stochastic SOCP Program

#### 3.2. Objective Function

#### 3.3. Constraints

## 4. Results and Discussion

- Case 1 (${C}_{A}$ = 1, ${C}_{B}$ = 0): Line loss minimization (active power loss);
- Case 2 (${C}_{A}$ = 0, ${C}_{B}$ = 1): Voltage deviation minimization;
- Case 3 (${C}_{A}$ = 1, ${C}_{B}$ = 1): Line loss and voltage deviation minimization (multi-objective function).

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Shahab, B.; Amini, M.H. A decentralized trading algorithm for an electricity market with generation uncertainty. Appl. Energy
**2018**, 218, 520–532. [Google Scholar] - Luo, L.; Gu, W.; Wu, Z.; Zhou, S. Joint planning of distributed generation and electric vehicle charging stations considering real-time charging navigation. Appl. Energy
**2019**, 242, 1274–1284. [Google Scholar] [CrossRef] - Wang, S.; Dong, Z.Y.; Chen, C.; Fan, H.; Luo, F. Expansion Planning of Active Distribution Networks with Multiple Distributed Energy Resources and EV Sharing System. IEEE Trans. Smart Grid
**2020**, 11, 602–611. [Google Scholar] [CrossRef] - Stott, B.; Alsac, O. Fast Decoupled Load Flow. IEEE Trans. Power Appar. Syst.
**1974**, PAS-93, 859–869. [Google Scholar] [CrossRef] - Alsac, O.; Bright, J.; Prais, M.; Stott, B. Further developments in LP-based optimal power flow. IEEE Trans. Power Syst.
**1990**, 5, 697–711. [Google Scholar] [CrossRef] - Purchala, K.; Meeus, L.; Van Dommelen, D.; Belmans, R. Usefulness of DC power flow for active power flow analysis. In Proceedings of the IEEE Power Engineering Society General Meeting, San Francisco, CA, USA, 16 June 2005; pp. 454–459. [Google Scholar]
- Stott, B.; Jardim, J.; Alsac, O. DC Power Flow Revisited. IEEE Trans. Power Syst.
**2009**, 24, 1290–1300. [Google Scholar] [CrossRef] - Javadi, M.S.; Gouveia, C.S.; Carvalho, L.M.; Silva, R. Optimal Power Flow Solution for Distribution Networks using Quadratically Constrained Programming and McCormick Relaxation Technique. In Proceedings of the 2021 IEEE International Conference on Environment and Electrical Engineering and 2021 IEEE Industrial and Commercial Power Systems Europe (EEEIC/I&CPS Europe), Bari, Italy, 7–10 September 2021; pp. 1–6. [Google Scholar]
- Mamun, K.A.; Islam, F.R.; Haque, R.; Chand, A.A.; Prasad, K.A.; Goundar, K.K.; Prakash, K.; Maharaj, S. Systematic Modeling and Analysis of On-Board Vehicle Integrated Novel Hybrid Renewable Energy System with Storage for Electric Vehicles. Sustainability
**2022**, 14, 2538. [Google Scholar] [CrossRef] - Kandil, S.M.; Farag, H.E.; Shaaban, M.; El-Sharafy, M.Z. A combined resource allocation framework for PEVs charging stations, renewable energy resources and distributed energy storage systems. Energy
**2018**, 143, 961–972. [Google Scholar] [CrossRef] - Ahmadian, A.; Aliakbar-Golkar, M. Fuzzy load modeling of plug-in electric vehicles for optimal storage and DG planning in active distribution network. IEEE Trans. Veh. Technol.
**2016**, 66, 3622–3631. [Google Scholar] [CrossRef] - De Quevedo, P.M.; Muñoz-Delgado, G.; Contreras, J. Impact of electric vehicles on the expansion planning of distribution systems considering renewable energy, storage, and charging stations. IEEE Trans. Smart Grid
**2017**, 10, 794–804. [Google Scholar] [CrossRef] - Zheng, Y.; Song, Y.; Hill, D.J.; Meng, K. Online Distributed MPC-Based Optimal Scheduling for EV Charging Stations in Distribution Systems. IEEE Trans. Ind. Inform.
**2019**, 15, 638–649. [Google Scholar] [CrossRef] - Erdinc, O.; Tascikaraoglu, A.; Paterakis, N.G.; Dursun, I.; Sinim, M.C.; Catalao, J.P.S. Comprehensive Optimization Model for Sizing and Siting of DG Units, EV Charging Stations, and Energy Storage Systems. IEEE Trans. Smart Grid
**2018**, 9, 3871–3882. [Google Scholar] [CrossRef] - Amini, M.H.; Moghaddam, M.P.; Karabasoglu, O. Simultaneous allocation of electric vehicles’ parking lots and distributed renewable resources in smart power distribution networks. Sustain. Cities Soc.
**2017**, 28, 332–342. [Google Scholar] [CrossRef] - Wang, G.; Xu, Z.; Wen, F.; Wong, K.P. Traffic-Constrained Multiobjective Planning of Electric-Vehicle Charging Stations. IEEE Trans. Power Deliv.
**2013**, 28, 2363–2372. [Google Scholar] [CrossRef] - Liu, Z.; Wen, F.; Ledwich, G. Optimal siting and sizing of distributed generators in distribution systems considering uncertainties. IEEE Trans. Power Deliv.
**2011**, 26, 2541–2551. [Google Scholar] [CrossRef] - Amer, A.; Azab, A.; Azzouz, M.A.; Awad, A.S.A. A Stochastic Program for Siting and Sizing Fast Charging Stations and Small Wind Turbines in Urban Areas. IEEE Trans. Sustain. Energy
**2021**, 12, 1217–1228. [Google Scholar] [CrossRef] - Fan, V.H.; Dong, Z.; Meng, K. Integrated distribution expansion planning considering stochastic renewable energy resources and electric vehicles. Appl. Energy
**2020**, 278, 115720. [Google Scholar] [CrossRef] - Thangaraju, I. Optimal allocation of distributed generation and electric vehicle charging stations-based SPOA2B approach. Int. J. Intell. Syst.
**2022**, 37, 2061–2088. [Google Scholar] [CrossRef] - Muñoz-Delgado, G.; Contreras, J.; Arroyo, J.M. Joint expansion planning of distributed generation and distribution networks. IEEE Trans. Power Syst.
**2014**, 30, 2579–2590. [Google Scholar] [CrossRef] - Yuwei, C.; Xiang, J.; Li, Y. SOCP relaxations of optimal power flow problem considering current margins in radial networks. Energies
**2018**, 11, 3164. [Google Scholar] - Zhang, H.; Moura, S.J.; Yonghua, S.; Qi, W.; Song, Y. A Second-Order Cone Programming Model for Planning PEV Fast-Charging Stations. IEEE Trans. Power Syst.
**2018**, 33, 2763–2777. [Google Scholar] [CrossRef] [Green Version] - Farivar, M.; Low, S.H. Branch flow model: Relaxations and convexification—Part I. IEEE Trans. Power Syst.
**2013**, 28, 2554–2564. [Google Scholar] [CrossRef] - Luo, L.; Gu, W.; Zhang, X.-P.; Cao, G.; Wang, W.; Zhu, G.; You, D.; Wu, Z. Optimal siting and sizing of distributed generation in distribution systems with PV solar farm utilized as STATCOM (PV-STATCOM). Appl. Energy
**2018**, 210, 1092–1100. [Google Scholar] [CrossRef] - Feijoo, A.E.; Cidras, J.; Dornelas, J.L.C. Wind speed simulation in wind farms for steady-state security assessment of electrical power systems. IEEE Trans. Energy Convers.
**1999**, 14, 1582–1588. [Google Scholar] [CrossRef] - Sheather, S.J.; Jones, M.C. A reliable data-based bandwidth selection method for kernel density estimation. J. R. Stat. Soc. Ser. B Methodol.
**1991**, 53, 683–690. [Google Scholar] [CrossRef] - NEOS. Stochastic Linear Programming. Available online: https://neos-guide.org/content/stochastic-linear-programming (accessed on 16 February 2022).

EVCS Code | Arrival Time | Charging Amount (kW) |
---|---|---|

EVCS 1 | 12.1 | 25.47 |

EVCS 2 | 13.7 | 17.57 |

$\vdots $ | $\vdots $ | $\vdots $ |

EVCS 839 | 19.7 | 25.6 |

Parameter | Meaning |
---|---|

${I}_{ij}$ | The current of branch $ij$. |

${Z}_{ij}/{R}_{ij}$$/{X}_{ij}$ | The impedance/resistance/reactance of branch $ij$. |

${V}_{0}$ | The nominal voltage. |

${V}_{i}$$,{V}_{j}$ | The $i/j$-th bus voltage. |

${V}_{1,j}/{V}_{2,j}$ | The auxiliary variables for linearization of objective function. |

${S}_{ij}/{S}_{jk}$ | The complex power in branch $ij/jk$. |

${P}_{ij}/{P}_{jk}$ | The active power in branch $ij/jk$. |

${Q}_{ij}/{Q}_{jk}$ | The reactive power in branch $ij/jk$. |

${S}_{j}/{P}_{j}/{Q}_{j}$ | The injected complex/active/reactive power at the $j$-th bus. |

${v}_{i}/{v}_{i}/{l}_{jj}$ | The auxiliary variables for SOCP relaxation. |

${P}_{Gi}/{Q}_{Gi}$ | The active/reactive power generated at $j$-th bus. |

$\underset{\xaf}{{P}_{Gj}},\overline{{P}_{Gj}}/\underset{\xaf}{{Q}_{Gj}},\overline{{Q}_{Gj}}$ | The under and upper bound of ${P}_{Gi}/{Q}_{Gi}$. |

${P}_{Dj}/{Q}_{Dj}$ | The active/reactive power consumed at $j$-th bus. |

$\underset{\xaf}{{P}_{Dj}}$$,\overline{{P}_{Dj}}$$/\underset{\xaf}{{Q}_{Dj}},\overline{{Q}_{Dj}}$ | The under/upper bound of ${P}_{Dj}/{Q}_{Dj}$. |

$Ca{p}_{Dj}^{type}$$/Ca{p}_{Gj}^{type}$ | The capacity of power-consuming/generating asset. |

$\overline{{P}_{Dj}^{Type}}$$/\overline{{P}_{Gj}^{Type}}$ | The upper bound of active power-consumed/generating asset. |

$E{V}_{pen}$ | EV penetration rate. |

$E{V}_{min}$ | The minimum active power of the EVCS. |

${P}_{j}^{type}$$/{Q}_{j}^{type}$ | The active/reactive power of asset at $j$-th bus. |

${C}_{m}$ | The normalization coefficient of the $m$-th scenario. |

$p{f}^{type}$ | The power factor of each asset |

Scenario | PV | WT | EVCS | |||
---|---|---|---|---|---|---|

Scenario Coefficient | Probability (%) | Scenario Coefficient | Probability (%) | Scenario Coefficient | Probability (%) | |

1 | 0.5819 | 16.83 | 0.6192 | 19.65 | 0.4345 | 19.18 |

$2$ | 0.4278 | 44.35 | 0.4594 | 18.90 | 0.7838 | 9.59 |

$\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ |

Total | 1 | 1 | 1 |

Integrated Scenario | PV | WT | EVCS | Probability (%) |
---|---|---|---|---|

Scenario 1 | 0.5819 | 0.6192 | 0.4345 | 0.63 |

Scenario 2 | 0.5819 | 0.6192 | 0.7838 | 0.32 |

$\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ |

Scenario 245 | 0.4943 | 0.8227 | 0.5193 | 0.47 |

Total | 1 |

Parameter | Value | Parameter | Value |
---|---|---|---|

$E{V}_{min}$ (kW) | 350 | ${V}_{0}$ (kV) | 12 |

$Ca{p}^{max}$ (MW) | 12 | $p{f}^{EVCS}$ | 0.99 lag |

${V}_{min}$ (pu) | 0.95 | $p{f}^{PV}$ | 0.99 lag |

${V}_{max}$ (pu) | 1.05 | $p{f}^{WT}$ | 0.95 lag |

Asset | Case 1 | Case 2 | Case 3 |
---|---|---|---|

EVCS1 (kVA) | 685.7 | 685.7 | 685.7 |

EVCS2 (kVA) | 685.8 | 3051.5 | 3048.6 |

PV (kVA) | 7971.9 | 7858.8 | 10,377.4 |

WT (kVA) | 3921.6 | $0$ | 3743.1 |

Active power loss (Line loss) (kW) | 608 | 3737.1 | 706.7 |

Voltage deviation (pu) | 0.4905 | 0.1343 | 0.3393 |

Bus | Case 1 | Case 2 | Case 3 |
---|---|---|---|

7 (EVCS1) | 1.0241 | 1.0000 | 1.0207 |

8 (WT) | 1.0500 | 1.0037 | 1.0457 |

11 (PV) | 1.0500 | 1.0500 | 1.0500 |

13 (EVCS2) | 1.0399 | 1.0000 | 1.0000 |

Asset | With k-Means Clustering | Without k-Means Clustering | ||
---|---|---|---|---|

Capacity (pu) | Computation Time (s) | Capacity (pu) | Computation Time (s) | |

EVCS1 (kVA) | 685.7 | 5.37 | 5378.7 | 208.40 |

EVCS2 (kVA) | 3048.6 | 236,905 | ||

PV (kVA) | 10,377.4 | 265,649.4 | ||

WT (kVA) | 3743.1 | 11,619.9 |

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**MDPI and ACS Style**

Woo, H.; Son, Y.; Cho, J.; Choi, S.
Stochastic Second-Order Conic Programming for Optimal Sizing of Distributed Generator Units and Electric Vehicle Charging Stations. *Sustainability* **2022**, *14*, 4964.
https://doi.org/10.3390/su14094964

**AMA Style**

Woo H, Son Y, Cho J, Choi S.
Stochastic Second-Order Conic Programming for Optimal Sizing of Distributed Generator Units and Electric Vehicle Charging Stations. *Sustainability*. 2022; 14(9):4964.
https://doi.org/10.3390/su14094964

**Chicago/Turabian Style**

Woo, Hyeon, Yongju Son, Jintae Cho, and Sungyun Choi.
2022. "Stochastic Second-Order Conic Programming for Optimal Sizing of Distributed Generator Units and Electric Vehicle Charging Stations" *Sustainability* 14, no. 9: 4964.
https://doi.org/10.3390/su14094964