# A Decision Support System to Enhance Electricity Grid Resilience against Flooding Disasters

^{1}

^{2}

^{*}

## Abstract

**:**

## Highlights

- A decision support system framework based on network cost minimization is proposed to divert flood waters from flood-susceptible utility poles, thereby enhancing electricity grid resilience.
- This optimization framework is evaluated in three different watersheds in the United States using state-of-the-art mathematical optimization platforms, i.e., JuMP/Julia interface and the Gurobi solver.
- The results of this proposed optimization framework could provide adequate flood diversion capacity to prevent failure of utility poles.

## Abstract

## 1. Introduction

## 2. Methodology

#### 2.1. Study Areas and Hydrologic Analyses

#### 2.2. Pole Failure Model

- (1)
- Utility pole–soil interactions can be modeled as a small retaining wall subjected to a passive pressure earth force (Equation (2)), which contributes to the total resisting moment. This total resisting moment opposes the moment created by the force applied by stormwater Equation (1) and the WSE.
- (2)
- The governing drag force formula,$${\mathrm{F}}_{\mathrm{d}}={\mathrm{C}}_{\mathrm{d}}\mathrm{A}\mathsf{\rho}\frac{{\mathrm{V}}^{2}}{2}$$
_{d}is the drag coefficient, A is the cross-sectional area, and ρ is the density of water. We also considered the momentum equation for the flow in motion: $\mathrm{F}=\mathsf{\rho}\mathrm{QV}$ assuming the velocity is dissipated after hitting the pole (as a conservative assumption), which resulted in more conservative values compared to drag forces, and thus selected in this paper. - (3)
- (4)
- Full setting depth is determined by the commonly accepted rule, 10% of the utility pole length plus two feet [35].
- (5)
- All utility poles are directly buried in the soil with no embedment foundation, based on observations made in Google Street View.
- (6)
- The version of the Rankine passive earth pressure force of the soil (Equation (11)) was used based on the assumption that the soils in each watershed are predominantly granular [36].$${P}_{p}=0.5\left({K}_{p}\right)\gamma {H}^{2}$$

- (7)
- In Rankine theory, it is assumed that the structure being modeled as a retaining wall is completely vertical and has a smooth surface. Therefore, factors such as wall–soil friction and retaining wall sloping are negligible.
- (8)
- Equation (3) assumes that there is no angle of incline and only takes into consideration the angle of friction. Kp is the Rankine passive pressure coefficient, which can be calculated from the following equation.$${K}_{p}={\mathrm{tan}}^{2}\left(45+\frac{{\varphi}^{\prime}}{2}\right)=\frac{1+\mathrm{sin}{\varphi}^{\prime}}{1-\mathrm{sin}{\varphi}^{\prime}}$$
- (9)
- Based on [36], it is assumed the resisting force is applied at approximately 2/3 of the utility pole (retaining wall) burial depth (measured from the ground line downward), or a distance 1/3 from the bottom of the utility pole (measured upward).

_{t}) is the sum of the maximum allowable moment at the utility pole groundline (M

_{gl}) (Equation (5)) [35] and the product of the passive pressure occurring at a depth of (2/3)H (M

_{p}).

_{s}is the fiber stress (psi) of the wood utility pole, according to the American National Standards Institute (ANSI). This value varies based on the class and species of the pole (Table 2). The diameter (d) of the utility pole at the groundline can also be expressed as

_{0}and C

_{1}are the circumferences of the pole at the top and 6 feet from the bottom, respectively. The total length (L) of the pole has a setting depth (E) associated with it, and the constant 6 denotes 6 ft subtracted from the total length of the pole. Equations (5)–(7) follow the methodology used by Keshavarzian [35].

#### 2.3. Minimum Cost Network Flow Optimization Problem

_{i}and Cost

_{p}are the cost of the vertical and horizontal pipe, respectively. Near-horizontal pipes follow the mild slope of the natural terrain. The Manning’s equation was used to calculate the flow capacity of the subterranean pipes network. Links in the subterranean network consist of:

- (1)
- pipe extending vertically downward from surface nodes to connect to horizontal pipe and
- (2)
- horizontal pipe conveying stormwater to the network outfall, or discharge point.

_{i}is a binary decision variable that functions as either 0 or 1, depending on the existence of drainage at a node. U

_{p}is a second binary decision variable, that dictates if there will be flow through an underground pipe or not. Flow rate units were calculated in cubic feet per second (cfs).

_{hydraulic capacity}is expressed as the upper limit of Equations (10) and (11), and can be used to size the vertical pipes connected to nodes and the horizontal underground pipes that will be used in the model. There are two conservation of flow constraints, one for surface network nodes and another for nodes in the underground network. Both of the conservation of flow constraints (Equations (12) and (13)) incorporate the Q

_{stream}

_{,}Q

_{drained,}and Q

_{pipe}decision variables seen above. Q

_{stream}refers to only the surface stream network. Q

_{drained}is used in Equations (12) and (13) and is denoted in the surface network (Equation (12)) as Q

_{drained}

_{(i)}to determine the amount of stormwater to be conveyed into the vertical pipe at certain surface nodes so that utility pole failure is mitigated. In Equation (13), Q

_{drained}

_{(p)}signifies stormwater in the underground network being conveyed downward through a vertical pipe to the horizontal pipe. Once Q

_{drained}

_{(p)}reaches the horizontal pipe, it becomes Q

_{pipe}, and is carried to the underground network’s node of discharge (outfall).

_{rainfall}becomes Q

_{stream}in every subwatershed and flows into every downstream subwatershed.

_{(i)}) and in the underground pipe network (Discharge

_{(p)}). For all networks in this study, discharge only occurs at the downstream terminal node.

## 3. Results

## 4. Discussions and Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Whittier, NC watershed with labeled nodes shown in yellow square text boxes (35.399467°, −83.292734°).

**Figure 2.**Leadville, CO watershed with labeled nodes shown in yellow square text boxes (39.222713°, −106.356648°).

**Figure 3.**London, Arkansas watershed with labeled nodes shown in yellow square text boxes (35.345107°, −93.329761°).

**Figure 5.**Methodology workflow to generate the decision support system to mitigate flood water impact on utility poles.

**Table 1.**Bulk densities assigned to each watershed based on dominant soil texture and vegetation [38].

Watershed | Dominant Soil Texture | Amount of Vegetation | ^{1} Bulk Density (g/cm^{3}) |
---|---|---|---|

Whittier, NC | Sandy loam | plentiful | 1.40 |

Leadville, CO | Gravely, sandy loam | sparse | 1.63 |

London, AR | Sandy loam | plentiful | 1.40 |

^{1}Bulk densities reported in [38] are in g/cm

^{3}. During calculations these were converted to lb/ft

^{3}. 1 g/cm

^{3}= 62.43 lb/ft

^{3}.

**Table 2.**Utility pole attributes for each watershed [42].

Watershed | Pole Class | ^{1} Length Range (ft) | ^{1} Assumed Length (ft) | ^{2} Top Circ. (in) | ^{2} Circ. 6 ft from Pole Bottom (in) | ^{3} Fiber Stress (lb/in^{2}) | ^{1} Setting Depth (ft) |
---|---|---|---|---|---|---|---|

Whittier, NC | 7 | 20–45 | 30 | 15 | 21.00 | 8000 | 5 |

Leadville, CO | 7 | 20–45 | 35,45 | 15 | 22.25, 24.75 | 8000 | 5.5, 6.5 |

London, AR | 7 | 20–45 | 45 | 15 | 24.75 | 8000 | 6.5 |

^{1}1 ft = 0.305 m.

^{2}1 in = 2.54 cm.

^{3}1 lb/in

^{2}= 6.89 kPa.

**Table 3.**Calculations template of parameters used to determine overturn flow rates as the model constraints for a flood-susceptible utility pole in Whittier, NC watershed.

Utility Pole * | Storm Return Period (year) | ^{1} Flow Rate Colliding with Pole (ft^{3}/s) | ^{2} WSE (ft) | ^{3} Applied Moment (ft-lb) | ^{3} Resisting Moment(ft-lb) | Pass/Fail? | ^{1} Overturn Flow Rate (ft^{3}/s) |
---|---|---|---|---|---|---|---|

1 | 2 | 233.97 | 0.13 | 2109.61 | 26,686.93 | Pass | 2959.91 |

1 | 5 | 297.41 | 0.18 | 4719.74 | 26,686.93 | Pass | 1681.73 |

1 | 10 | 328.01 | 0.2 | 6378.83 | 26,686.93 | Pass | 1372.35 |

1 | 25 | 387.83 | 0.39 | 17389.02 | 26,686.93 | Pass | 595.22 |

1 | 50 | 425.5 | 0.46 | 24688.61 | 26,686.93 | Pass | 459.96 |

1 | 100 | 468.72 | 0.54 | 35167.58 | 26,686.93 | Fail | 355.70 |

1 | 500 | 569.1 | 0.61 | 58565.07 | 26,686.93 | Fail | 259.34 |

^{1}1 ft

^{3}/s = 0.028 m

^{3}/s.

^{2}1 ft = 0.305 m.

^{3}1 ft-lb = 1.36 N-m.

Node | ^{1} Drainage Capacity Recommended (ft^{3}/s) | Links | ^{2} Underground Pipe Flow (ft^{3}/s) |
---|---|---|---|

1 | 1.5 | ||

2 | 2.4 | ||

3 | 3.4 | ||

4 | 4.5 | ||

5 | 5.7 | ||

6 | 6.7 | ||

7 | 75.00 | 7.11 | 75.00 |

8 | 8.10 | ||

9 | 9.10 | ||

10 | 10.11 | ||

11 | 75.00 | 11.13 | 150.00 |

12 | 12.13 | ||

13 | 75.00 | 13.14 | 225.00 |

14 | 75.00 | 15.17 | |

15 | 16.17 | ||

16 | 17.19 | ||

17 | 18.19 | ||

18 | 19.21 | ||

19 | 20.21 | ||

20 | 21.23 | 75.00 | |

21 | 75.00 | 22.23 | |

22 | 23.25 | 127.04 | |

23 | 52.04 | 24.25 | |

24 | 25.27 | 202.04 | |

25 | 75.00 | 26.27 | |

26 | 27.29 | 277.04 | |

27 | 75.00 | 28.29 | |

28 | 29.31 | 352.04 | |

29 | 75.00 | 30.31 | |

30 | 31.14 | 427.04 | |

31 | 75.00 | 14.32 | 727.04 |

32 |

^{1}Drainage capacity recommended is the amount of flow the model recommends be drained at a particular node to reach the optimal solution of the network optimization problem.

^{2}Underground pipe flow is the sum of surface drainage from a node and existing pipe flow from surface drainage of upstream nodes (notes 1 and 2 are typical for Table 4, Table 5 and Table 6). All blank areas indicate a flow of zero (typical for Table 4, Table 5 and Table 6).

Nodes | ^{1} Drainage Capacity Recommended (ft^{3}/s) | Links | ^{2} Underground Pipe Flow (ft^{3}/s) |
---|---|---|---|

1 | 1.3 | ||

2 | 2375 | 2.3 | 2375 |

3 | 2375 | 3.5 | 4750 |

4 | 4.5 | ||

5 | 2375 | 5.7 | 7125 |

6 | 6.7 | ||

7 | 2375 | 7.11 | 9500 |

8 | 8.10 | ||

9 | 9.10 | ||

10 | 10.11 | ||

11 | 2375 | 11.13 | 11,875 |

12 | 12.13 | ||

13 | 2375 | 13.17 | 14,250 |

14 | 14.16 | ||

15 | 15.16 | ||

16 | 16.17 | ||

17 | 2375 | 17.19 | 16,625 |

18 | 112.10 | 18.19 | 112.10 |

19 | 2375 | 19.25 | 19,112.10 |

20 | 20.22 | ||

21 | 21.22 | ||

22 | 22.24 | ||

23 | 23.24 | ||

24 | 24.25 | ||

25 | 2375 | 26.28 | |

26 | 27.28 | ||

27 | 28.32 | ||

28 | 30.31 | ||

29 | 29.31 | ||

30 | 31.32 | ||

31 | 32.34 | ||

32 | 33.34 | ||

33 | 35.37 | ||

34 | 36.37 | ||

35 | 37.39 | ||

36 | 38.39 | ||

37 | 39.40 | ||

38 | 34.40 | ||

39 | 40.42 | 2375 | |

40 | 2375 | 41.42 | |

41 | 42.44 | 4750 | |

42 | 2375 | 43.44 | |

43 | 44.52 | 7125 | |

44 | 2375 | 45.47 | |

45 | 46.47 | ||

46 | 47.49 | ||

47 | 48.49 | ||

48 | 49.51 | 178.75 | |

49 | 178.75 | 50.51 | |

50 | 51.52 | 2553.75 | |

51 | 2375 | 53.57 | |

52 | 2375 | 54.56 | |

53 | 55.56 | ||

54 | 56.57 | ||

55 | 57.58 | 952.81 | |

56 | 52.58 | 12,053.75 | |

57 | 952.81 | 58.59 | 15,381.57 |

58 | 2375 | 25.59 | 21,487.10 |

59 | 2336.96 | 59.61 | 39,205.63 |

60 | 60.61 | ||

61 | 2375 | 62.64 | |

62 | 63.64 | ||

63 | 64.68 | 484.43 | |

64 | 484.43 | 65.67 | |

65 | 66.67 | ||

66 | 67.68 | ||

67 | 68.69 | 484.43 | |

68 | 69.70 | 2195.88 | |

69 | 1711.44 | 61.70 | 41,580.63 |

70 |

^{1}Drainage capacity recommended is the amount of flow the model recommends be drained at a particular node to reach the optimal solution of the network optimization problem.

^{2}Underground pipe flow is the sum of surface drainage from a node and existing pipe flow from surface drainage of upstream nodes (notes 1 and 2 are typical for Table 4, Table 5 and Table 6). All blank areas indicate a flow of zero (typical for Table 4, Table 5 and Table 6).

Nodes | ^{1} Drainage Capacity Recommended (ft^{3}/s) | Links | ^{2} Underground Pipe Flow (ft^{3}/s) |
---|---|---|---|

1 | 1.3 | ||

2 | 2.3 | ||

3 | 3.5 | ||

4 | 4.5 | ||

5 | 5.7 | ||

6 | 6.7 | ||

7 | 7.9 | ||

8 | 8.9 | ||

9 | 9.11 | ||

10 | 10.11 | ||

11 | 1490 | 11.15 | 1490 |

12 | 12.14 | ||

13 | 13.14 | ||

14 | 14.15 | ||

15 | 1490 | 15.19 | 2980 |

16 | 16.18 | ||

17 | 17.18 | ||

18 | 18.19 | ||

19 | 1490 | 19.21 | 4470 |

20 | 20.21 | ||

21 | 1490 | 21.23 | 5960 |

22 | 22.23 | ||

23 | 1490 | 23.27 | 7450 |

24 | 24.26 | ||

25 | 25.26 | ||

26 | 1490 | 26.27 | 1490 |

27 | 1490 | 27.29 | 10,430 |

28 | 28.29 | ||

29 | 1490 | 29.31 | 11,920 |

30 | 30.31 | ||

31 | 1490 | 31.33 | 13,410 |

32 | 32.33 | ||

33 | 1490 | 33.36 | 14,900 |

34 | 34.36 | ||

35 | 35.36 | ||

36 | 1349.02 | 36.37 | 16,249.02 |

37 | 1490 | 38.40 | |

38 | 39.40 | ||

39 | 40.42 | ||

40 | 41.42 | ||

41 | 42.44 | ||

42 | 43.44 | ||

43 | 44.46 | 1490 | |

44 | 1490 | 45.46 | |

45 | 46.48 | 2891.27 | |

46 | 1401.27 | 47.48 | 843.03 |

47 | 843.03 | 48.52 | 5222.30 |

48 | 1490 | 49.51 | |

49 | 50.51 | ||

50 | 51.52 | ||

51 | 52.54 | 6714.30 | |

52 | 1490 | 53.54 | |

53 | 54.56 | 8202.30 | |

54 | 1490 | 55.56 | 1490 |

55 | 1490 | 56.57 | 11,184.30 |

56 | 1490 | 37.57 | 17,739.02 |

57 | 1490 | 57.59 | 30,413.33 |

58 | 58.59 | ||

59 | 1490 | 59.61 | 31,903.33 |

60 | 60.61 | ||

61 | 1490 | 61.62 | 33,393.33 |

62 |

^{1}Drainage capacity recommended is the amount of flow the model recommends be drained at a particular node to reach the optimal solution of the network optimization problem.

^{2}Underground pipe flow is the sum of surface drainage from a node and existing pipe flow from surface drainage of upstream nodes (notes 1 and 2 are typical for Table 4, Table 5 and Table 6). All blank areas indicate a flow of zero (typical for Table 4, Table 5 and Table 6).

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## Share and Cite

**MDPI and ACS Style**

Violante, M.; Davani, H.; Manshadi, S.D.
A Decision Support System to Enhance Electricity Grid Resilience against Flooding Disasters. *Water* **2022**, *14*, 2483.
https://doi.org/10.3390/w14162483

**AMA Style**

Violante M, Davani H, Manshadi SD.
A Decision Support System to Enhance Electricity Grid Resilience against Flooding Disasters. *Water*. 2022; 14(16):2483.
https://doi.org/10.3390/w14162483

**Chicago/Turabian Style**

Violante, Michael, Hassan Davani, and Saeed D. Manshadi.
2022. "A Decision Support System to Enhance Electricity Grid Resilience against Flooding Disasters" *Water* 14, no. 16: 2483.
https://doi.org/10.3390/w14162483