# A Risk-Averse Approach for Distribution Grid Expansion Planning

^{*}

## Abstract

**:**

## 1. Introduction

- 1.
- A stochastic investment and planning model for distribution grids with risk-based explicit metrics that allow utilities and network planners to explore the trade-offs between reliability and resilience when selecting the best portfolio of conductors and DER investments. This model is able to capture the system operation, in particular the multistage aspects of time-coupling constraints related with energy storage management.
- 2.
- A tailored novel temporal decomposition framework that renders a tractable and effective solution approach for the model and accommodates the minimization of expected value and CVaR of the operational costs, including energy not served.

## 2. Mathematical Formulation: Incorporating Risk Aversion into Distribution Grid Planning

#### 2.1. Costs and Risk-Averse Modeling

#### 2.2. Investment in Lines

#### 2.3. Investment in Storage Devices

#### 2.4. Operation of the Grid

#### 2.5. Operation of Storage Devices

## 3. Solution Methodology: Solving the Risk-Averse Multistage Problem

Algorithm 1: Compute ${\chi}_{md}^{\u2020}\phantom{\rule{3.33333pt}{0ex}}\forall d\in \mathcal{D},m\in \mathcal{M}$. |

Algorithm 2: Solution algorith. |

## 4. Case Study

^{®}Xeon

^{®}E5-2680 processors @ 2.40GHz and 64 GB of RAM, using Julia 1.1, JuMP and solved via CPLEX 12.9.

#### 4.1. Investment Results

#### 4.2. Discussion: Out-of-Sample Analysis

#### 4.2.1. Reliability

#### 4.2.2. Risk Aversion

#### 4.3. Discussion: Risk-Aversion Parameter $\lambda $

#### 4.4. Discussion: Imposing Investment Budget Constraint

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Sets | |

$\mathcal{D}$ | Set of typical days. |

${\psi}^{N}$ | Set of indices of all buses (including substations and nonsubstations). |

${\psi}^{SS}$ | Set of indices of buses that are substations. |

E | Set of stages. |

H | Set of indices of all storage devices (including existing and candidates). |

${H}^{C}$ | Set of indices of candidate storage devices |

$\mathcal{L}$ | Set of indices of all lines (including existing and candidates). |

${\mathcal{L}}^{C}$ | Set of indices of candidate lines. |

${\mathcal{L}}^{E}$ | Set of indices of existing lines. |

$\mathcal{M}$ | Set of indices of scenario tree nodes. |

${\mathcal{M}}_{d}$ | Set of indices of scenario tree nodes that belong to day d. |

${\mathcal{N}}_{m}^{+}$ | Set of indices of scenario tree nodes that are “kids” of scenario tree node m. |

${T}_{m}$ | Set of time periods of each scenario tree node. |

Indices | |

d | Index of typical days. |

e | Index of stages. |

$\mathrm{e}\left(m\right)$ | Index of the stage to which scenario tree node m belongs. |

h | Index of storage devices. |

l | Index of lines. |

n | Index of buses. |

m | Index of scenario tree nodes. |

${m}^{-}$ | Index of the scenario tree node that is the parent of scenario tree node m. |

t | Index of time periods. |

Parameters | |

${\alpha}^{CVaR}$ | CVaR parameter. |

${\overline{\beta}}_{lj}^{L}$ | Maximum amount of flow in line l associated with jth piecewise linear function used to linearize losses. |

${\overline{\beta}}_{nj}^{Tr}$ | Maximum amount of substation injection in bus n associated with jth piecewise linear function used to linearize losses. |

${\gamma}_{lj}^{L}$ | Slope of jth piecewise linear function used to linearize losses multiplied by respective impedance ${Z}_{l}^{L}$. |

${\gamma}_{nj}^{Tr}$ | Slope of jth piecewise linear function used to linearize losses multiplied by respective impedance ${Z}_{nj}^{Tr}$. |

$\lambda $ | Risk aversion user-defined parameter (between 0 and 1). |

${\pi}_{md}$ | Probability of transition to scenario tree node m in day d. |

${C}_{l}^{fix}$ | Fixed investment cost of candidate line l. |

${C}^{Imb}$ | Cost of imbalance. |

${C}^{L}$ | Cost of losses. |

${C}_{h}^{SD,fix}$ | Fixed investment cost of candidate storage device h. |

${C}_{h}^{SD,var}$ | Variable investment cost of candidate storage device h. |

${C}_{nmd}^{Tr}$ | Injection cost in substation n at scenario tree node m and day d. |

${\overline{F}}_{l}$ | Maximum capacity of existing line l. |

${\overline{G}}_{n}^{Tr}$ | Limit of injection in substation n. |

M | Sufficiently large number. |

${n}^{J}$ | Number of piecewise linear functions used to linearize losses. |

${\overline{P}}_{h}^{in}$ | Maximum charging of storage device h per stage. |

${\overline{P}}_{h}^{out}$ | Maximum discharging of storage device h per stage. |

$pf$ | Power factor. |

${r}^{len}$ | Length of line l. |

$\overline{S}$ | Number of time periods to fully charge storage devices. |

${SOC}_{h{m}^{0}}$ | Initial and final stored energy in storage device h. |

$\underline{V}$ | Minimum voltage. |

$\overline{V}$ | Maximum voltage. |

${\overline{x}}_{h}^{SD}$ | Maximum investment in storage device h. |

${y}_{lmtd}$ | Parameter that determines if line l is available (being equal to 1) or unavailable (being equal to 0). |

${Z}_{l}^{L}$ | Impedance of line l. |

Decision variables | |

${\beta}_{ljmtd}^{L}$ | Amount of flow in line l associated with jth piecewise linear function used to linearize losses. |

${\beta}_{njmtd}^{Tr}$ | Amount of substation injection in bus n associated with jth piecewise linear function used to linearize losses. |

${\Delta}_{nmtd}^{+}$ | Positive imbalance in bus n. |

${\Delta}_{nmtd}^{-}$ | Negative imbalance in bus n. |

${\psi}_{md}$ | CVaR auxiliary variable. |

${c}_{md}^{Total}$ | Total cost of scenario tree node m. |

${f}_{lmtd}$ | Flow in line l. |

${g}_{nmtd}^{Tr}$ | Injection via susbtation n. |

${p}_{hmtd}^{in}$ | Charging of storage device h. |

${p}_{hmtd}^{out}$ | Discharging of storage device h. |

$SO{C}_{hmtd}$ | State of charge of storage device h. |

$SO{C}_{hmd}^{\u2020}$ | Auxiliary variable associated with the state of charge of storage device h. |

${u}_{md}$ | CVaR auxiliary variable that represents the value at risk. |

${v}_{nmtd}$ | Voltage in bus n. |

${x}_{lmd}^{fix}$ | Binary investment in line l. |

${x}_{lmd}^{fix,acu}$ | Accumulated binary investment in line l at scenario tree node m. |

${x}_{lmd}^{fix\u2020}$ | Auxiliary variable associated with binary investment in line l at scenario tree node m. |

${x}_{hmd}^{SD,fix}$ | Binary investment in storage device h. |

${x}_{hmd}^{SD,var}$ | Continuous investment in storage device h. |

${x}_{hmd}^{SD,fix,acu}$ | Accumulated binary investment in storage device h at scenario tree node m. |

${x}_{hmd}^{SD,var,acu}$ | Accumulated continuous investment in storage device h at scenario tree node m. |

${x}_{hmd}^{SD,fix,\u2020}$ | Auxiliary variable associated with binary investment in storage device h at scenario tree node m. |

${x}_{hmd}^{SD,var,\u2020}$ | Auxiliary variable associated with continuous investment in storage device h at scenario tree node m. |

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**Figure 2.**54-bus system—Existing line segments are the solid lines and candidate lines are the dashed lines. Buses 2, 19, 20, and 26 are candidates to receive investment in storage.

Risk Aversion | |||
---|---|---|---|

$\mathit{\lambda}=0$ | $\mathit{\lambda}=0.5$ | $\mathit{\lambda}=1$ | |

Total investment ($) | 271,618.59 | 602,996.97 | 698,766.82 |

Number of new lines | 12 | 17 | 17 |

Number of storageunits at node 2 | 0 | 45 | 66 |

Number of storageunits at node 19 | 0 | 30 | 46 |

Number of storageunits at node 20 | 28 | 136 | 161 |

Number of storageunits at node 26 | 13 | 71 | 105 |

Computing time (s) | 6813.41 | 6216.52 | 6202.456 |

**Table 2.**Out-of-sample analysis—Reliability metrics: Average annual energy not served, SAIFI, and SAIDI.

No Investment | $\mathit{\lambda}=0$ | $\mathit{\lambda}=0.5$ | $\mathit{\lambda}=1$ | |
---|---|---|---|---|

Average of annual energynot served (kWh) | 22,083.08 | 576.58 | 19.32 | 15.06 |

SAIFI | 1.7361 | 0.0595 | 0.0097 | 0.0073 |

SAIDI (h) | 2.9862 | 0.1802 | 0.0271 | 0.0241 |

**Table 3.**Out-of-sample analysis—Risk aversion metrics (CVaR${}_{5\%}$, CVaR${}_{1\%}$ and worst case) of annual energy not served.

No Investment | $\mathit{\lambda}=0$ | $\mathit{\lambda}=0.5$ | $\mathit{\lambda}=1$ | |
---|---|---|---|---|

CVaR${}_{\mathbf{5}\%}$ of annual | ||||

energy not served | 40,822.89 | 2975.84 | 151.93 | 114.88 |

(kWh) | ||||

CVaR${}_{\mathbf{1}\%}$ of annual | ||||

energy not served | 47,425.44 | 4476.05 | 400.03 | 276.97 |

(kWh) | ||||

Worst case of annual | ||||

energy not served | 59,454.96 | 9234.41 | 831.60 | 740.66 |

(kWh) |

Risk Aversion | |||
---|---|---|---|

$\mathit{\lambda}=0$ | $\mathit{\lambda}=0.5$ | $\mathit{\lambda}=1$ | |

Total investment ($) | 27,1618.59 | 300,000.00 | 300,000.00 |

Number of new lines | 12 | 12 | 12 |

Number of storageunits at node 2 | 0 | 0 | 7 |

Number of storageunits at node 19 | 0 | 0 | 0 |

Number of storageunits at node 20 | 28 | 70 | 63 |

Number of storageunits at node 26 | 13 | 0 | 0 |

Computing time (s) | 6813.41 | 9012.78 | 8467.41 |

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**MDPI and ACS Style**

Moreira, A.; Heleno, M.; Valenzuela, A. A Risk-Averse Approach for Distribution Grid Expansion Planning. *Energies* **2021**, *14*, 8482.
https://doi.org/10.3390/en14248482

**AMA Style**

Moreira A, Heleno M, Valenzuela A. A Risk-Averse Approach for Distribution Grid Expansion Planning. *Energies*. 2021; 14(24):8482.
https://doi.org/10.3390/en14248482

**Chicago/Turabian Style**

Moreira, Alexandre, Miguel Heleno, and Alan Valenzuela. 2021. "A Risk-Averse Approach for Distribution Grid Expansion Planning" *Energies* 14, no. 24: 8482.
https://doi.org/10.3390/en14248482