unclassified resilient distribution tool (rdt) · 2016. 5. 5. · arthur k. barnes harsha...
TRANSCRIPT
![Page 1: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/1.jpg)
Los Alamos National Laboratory
4/14/2016 | 1
UNCLASSIFIED
logo/management
Operate by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA
Resilient Distribution Tool (RDT)
4/14/2016
Optimal storm-hardening of distribution systems
with microgrids and conventional assets
Emre Yamangil
Russell Bent
Arthur K. Barnes
Harsha Natarajan
![Page 2: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/2.jpg)
Los Alamos National Laboratory
4/14/2016 | 2
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Hardening/Resilience options
• Asset hardening
• System design
• System operations
• Repair scheduling
• Emergency operations
User-Specified Inputs
• Base network model
• Resilience metrics
• Suggested upgrades
• Costs
• Threats & scenarios
Capabilities
• Assess current resilience posture
• Optimize over user-suggested upgrades to improve resilience considering budget
Goal: Develop new tools to design
resilient distribution systems
![Page 3: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/3.jpg)
Los Alamos National Laboratory
4/14/2016 | 3
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Resilience Design Process Flow—End Goal
![Page 4: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/4.jpg)
Los Alamos National Laboratory
4/14/2016 | 4
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Resilience Design Process Flow—Today
![Page 5: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/5.jpg)
Los Alamos National Laboratory
4/14/2016 | 5
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Flexibility for the user
User’s base network model
User-defined metrics, e.g. critical load service
User suggests upgrades
User-defined costs
User-defined threat and scenarios
Resilience Design Process—System Model
![Page 6: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/6.jpg)
Los Alamos National Laboratory
4/14/2016 | 6
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Resilience Design Process—Direct Impacts
Flexibility for the user
User’s base network model
User-defined metrics, e.g. critical load service
User suggests upgrades
User-defined costs
User-defined threat and scenarios
Source: Y. Sa, Reliability analysis of electric distribution lines
Ph.D. dissertation, McGill University, Montreal, Canada, 2002
![Page 7: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/7.jpg)
Los Alamos National Laboratory
4/14/2016 | 7
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Resilience Design Process—Design Network
Hardening/Resilience options
Asset hardening
System design
System operations
Repair scheduling
Emergency operations
Capabilities
– Assess current resilience posture
– Optimize over user-suggested
upgrade to improve resilience
considering budget
![Page 8: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/8.jpg)
Los Alamos National Laboratory
4/14/2016 | 8
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Implementation of “Withstand”—Scenario
Based Decomposition
𝑅𝑒𝑠𝑖𝑙𝑖𝑒𝑛𝑡𝐷𝑒𝑠𝑖𝑔𝑛 𝑆
𝑠 ← 𝑐ℎ𝑜𝑜𝑠𝑒𝑆𝑐𝑒𝑛𝑎𝑟𝑖𝑜 𝑆
𝜎 → 𝑠𝑜𝑙𝑣𝑒𝑀𝐼𝑃 𝑠
𝑤ℎ𝑖𝑙𝑒 ~𝐹𝑒𝑎𝑠𝑖𝑏𝑙𝑒 𝜎, 𝑆\s
𝑠 → 𝑠 ∪ 𝑐ℎ𝑜𝑜𝑠𝑒𝑆𝑐𝑒𝑛𝑎𝑟𝑖𝑜 𝑆\s
𝜎 → 𝑠𝑜𝑙𝑣𝑒𝑀𝐼𝑃 𝑠
Solve over all
damage scenarios
Select 1 scenario
Design network for
damage scenario 1
Is solution feasible for
remaining scenarios
If NOT, add an infeasible
scenario to the set under
consideration
Find a new solution
Iterate until solution is
feasible for all scenarios
![Page 9: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/9.jpg)
Los Alamos National Laboratory
4/14/2016 | 9
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Adaptation to Practical Distribution Systems
• Objective is to run on a 1,000+ bus practical
distribution systems consisting of multiple feeders
• Key insights to enable this are:
• Multi-commodity or linearized unbalanced powerflow (Low et al)
• Sparsity in the number of Boolean & integer variables. This
requires engineering judgement to select good candidates for new
lines, hardening paths and distributed generation placement
• Pre-compute switching states
• Hardened paths have zero probability of damage
![Page 10: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/10.jpg)
Los Alamos National Laboratory
4/14/2016 | 10
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Candidate Feeder Upgrades
Sub1
Sub2
Critical Load
DG Candidate
Line
Candidate
Upgrade
Path
![Page 11: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/11.jpg)
Los Alamos National Laboratory
4/14/2016 | 11
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Case Study 1: Small 12.47 kV Feeder
1.6 km / 1 mi
1 k
m /
0.6
mi
Sub
1 km2 / 0.4 mi2
5 MW / 2.5 MVar
total load
3Φ
1Φ
300 buses /
300 lines / 100 loads
![Page 12: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/12.jpg)
Los Alamos National Laboratory
4/14/2016 | 12
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Case Study 1 ResultsDamage & Resilience
Requirements
20% Damage prob.
50% /
98%
Normal / Critical
load served
500
kW
Generator
increments
Computation
30 min. Exact search
20 Disaster
scenarios
Results (Exact)
40 Lines
hardened
0 Lines built
1 MW / 1 Total gen.
power / # gen.
$3.5M Capex
Computation
3 h
(timeout)
Exact search
20 Disaster
scenarios
Results (Exact)
40 Lines
hardened
0 Lines built
4 MW / 7 Total gen.
power / # gen.
$12.5M Capex
Damage & Resilience
Requirements
20% Damage prob.
60% /
98%
Normal / Critical
load served
500
kW
Generator
increments
![Page 13: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/13.jpg)
Los Alamos National Laboratory
4/14/2016 | 13
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Case Study 1 Results
Computation
3 h
(timeout)
Exact search
20 Disaster
scenarios
Results (Exact)
40 Lines
hardened
0 Lines built
4 MW / 7 Total gen.
power / # gen.
$12.5M Capex
Damage & Resilience
Requirements
20% Damage prob.
60% /
98%
Normal / Critical
load served
500
kW
Generator
increments
Computation
3 min. VN search
20 Disaster
scenarios
Results (Exact)
55 Lines
hardened
0 Lines built
1.5 MW /
2
Total gen.
power / # gen.
$5M Capex
Damage & Resilience
Requirements
20% Damage prob.
60% /
98%
Normal / Critical
load served
500
kW
Generator
increments
![Page 14: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/14.jpg)
Los Alamos National Laboratory
4/14/2016 | 14
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Case Study 2: Larger 12.47 kV Feeder
2.5 km / 1.5 mi
2 k
m /
1.2
5 m
i
Sub
2.5 km2 / 1 mi2
8 MW / 4 MVar
total load
3Φ
1Φ
900 buses /
900 lines / 100 loads
![Page 15: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/15.jpg)
Los Alamos National Laboratory
4/14/2016 | 15
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Case Study 2 ResultsDamage & Resilience
Requirements
20% Damage prob.
50% /
98%
Normal / Critical
load served
500
kW
Generator
increments
Computation
3 h Exact
20 Disaster
scenarios
Results (Exact)
35 Lines
hardened
0 Lines built
0 MW / 0 Total gen.
power / # gen.
$500k Capex
Computation
10 min. VN search
20 Disaster
scenarios
Results (Exact)
40 Lines
hardened
0 Lines built
0 MW / 0 Total gen.
power / # gen.
$600k Capex
Damage & Resilience
Requirements
20% Damage prob.
50% /
98%
Normal / Critical
load served
500
kW
Generator
increments
![Page 16: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/16.jpg)
Los Alamos National Laboratory
4/14/2016 | 16
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Development Roadmap
Existing
Commodity flow
constraints: neglect
voltage & reactive power
flow
C++ HPC implementation
Test on feeder made of
IEEE 37-bus test case
copied 3-times
In Progress
Extension to distribution
systems with 1000+
buses and multiple
substations
Java HPC implementation
Pulls system data from
DEW commercial
software
AWS
Implementation
within Amazon
Web Services
User interface via
web-based GIS
frontend
Access via
RESTful API for
research use
NCRECA OMF
Implementation as
study module
within OMF web-
based analysis
frameworkOpen-source
code release
![Page 17: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/17.jpg)
Los Alamos National Laboratory
4/14/2016 | 17
UNCLASSIFIED NOTE
simple, text only,
statement layout.
Thanks & Questions
• T
![Page 18: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/18.jpg)
Los Alamos National Laboratory
4/14/2016 | 18
UNCLASSIFIED NOTE
simple statement
layout. I
photograph that
supports your
statement. Use
only a high
resolution
photograph.
replace the
photo currently in
the background,
right click and
select “Change
picture…”
Problem Formulation &
Solution Methods
![Page 19: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/19.jpg)
Los Alamos National Laboratory
4/14/2016 | 19
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Implementation of “Withstand”—Algorithm
• Baseline Standard
– CPLEX 12.6—commercial mixed integer program solver
• Decomposition Algorithms (cutting planes)
– Danzig-Wolfe
– Benders
– Disjunctive
– Logic
– Scenario Biggest computational gains
![Page 20: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/20.jpg)
Los Alamos National Laboratory
4/14/2016 | 20
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Implementation of “Withstand”—
Decomposition AlgorithmC
on
str
ain
tsFirst Stage
VariablesFirst
Scenario
Variables
Second
Scenario
Variables
Third
Scenario
Variables
…
Scenario-based decomposition
strategies exploit the separable
structure of the problem over
scenarios when the first stage
variables are fixed
![Page 21: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/21.jpg)
Los Alamos National Laboratory
4/14/2016 | 21
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Design Network—Optimization
minimize 𝑖𝑗∈𝐸 𝑐𝑖𝑗𝑥𝑖𝑗 + 𝑖,𝑗∈𝐸 𝜅𝑖𝑗𝜏𝑖𝑗 + 𝑖∈𝑁,𝑘∈ 𝑝𝑖 𝜁𝑖𝑘𝑧𝑖𝑘 + 𝑖∈𝑁 𝜇𝑖𝑢𝑖 + 𝑖𝑗∈𝐸 𝛼𝑖𝑗𝑡𝑖𝑗
s.t. −x𝑖𝑗𝑠 𝑄𝑖𝑗𝑘 ≤ 𝑘∈𝑝𝑖𝑗 𝑓𝑖𝑗
𝑠𝑘 ≤ 𝑥𝑖𝑗𝑠 𝑄𝑖𝑗𝑘
− 1 − 𝜏𝑖𝑗𝑠 𝑄𝑖𝑗
𝑘 ≤ 𝑘∈𝑝𝑖𝑗 𝑓𝑖𝑗𝑘𝑠 ≤ 1 − 𝜏𝑖𝑗
𝑠 𝑄𝑖𝑗𝑘
−𝛽𝑖𝑗 𝑘∈𝑝𝑖,𝑗
𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗≤ 𝑓𝑖𝑗𝑘′𝑠 −
𝑘∈𝑝𝑖,𝑗𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗≤ 𝛽𝑖𝑗
𝑘∈𝑝𝑖,𝑗𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗
𝑥𝑖𝑗𝑠 ≤ 𝑥𝑖𝑗, 𝜏𝑖𝑗
𝑠 ≤ 𝜏𝑖𝑗, 𝑡𝑖𝑗𝑠 ≤ 𝑡𝑖𝑗, 𝑧𝑖𝑗
𝑠𝑘 ≤ 𝑧𝑖𝑘, 𝑢𝑖𝑠 ≤ 𝑢𝑖
𝑧𝑖𝑘 ≤ 𝑀𝑖
𝑘𝑢𝑖, 𝑥𝑖𝑗𝑠 = 𝑡𝑖𝑗
𝑠 , 𝑥𝑖𝑗𝑠 ≤ 𝑥𝑖𝑗
𝑠, 𝜏𝑖𝑗𝑠 ≤ 𝑥𝑖𝑗
𝑠
3 − 𝑥𝑖𝑗𝑠 − 𝜏𝑖𝑗
𝑠≥ 𝜏𝑖𝑗𝑠 ≥ 𝑥𝑖𝑗
𝑠 + 𝜏𝑖𝑗𝑠− 1
liks = 𝑦𝑖
𝑠𝑑𝑖𝑘
0 ≤ 𝑔𝑖𝑠𝑘 ≤ 𝑧𝑖
𝑘𝑠 + 𝑔𝑖𝑘+
giks − 𝑙𝑖
𝑘𝑠 − 𝑗∈𝑁 𝑓𝑖𝑗𝑘𝑠 = 0
0 ≤ 𝑧𝑖𝑘𝑠 ≤ 𝑢𝑖
𝑠𝑍𝑖𝑘
𝑖𝑗∈𝑠 𝑥𝑖𝑗𝑠+ 1 − 𝜏𝑖𝑗 ≤ 𝑠 − 1
𝑖∈𝐶𝐿,𝑘∈𝑝𝑖 𝑙𝑖𝑘𝑠 ≥ 𝜆 𝑖∈𝐶𝐿,𝑘∈𝑝𝑖 𝑑𝑖
𝑘
𝑖∈𝑁∖𝐿,𝑘∈𝑝𝑖 𝑙𝑖𝑘𝑠 ≥ 𝛾 𝑖∈𝑁∖𝐿,𝑘∈𝑝𝑖 𝑑𝑖
𝑘
𝑥, 𝑦, 𝜏, 𝑢, 𝑡 ∈ {0,1}
Key Features
• Least cost design for a set of scenarios
• Three-phase real power flows
• Enforces radial operations
• Enforces phase balance
• Discrete variables for load shedding
(per scenario), line switching (per
scenario), capital construction (first
stage)
• Understand the boundaries of
tractability
• Optimality vs. computation tradeoff
![Page 22: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/22.jpg)
Los Alamos National Laboratory
4/14/2016 | 22
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
minimize 𝑖𝑗∈𝐸 𝑐𝑖𝑗𝑥𝑖𝑗 + 𝑖,𝑗∈𝐸 𝜅𝑖𝑗𝜏𝑖𝑗 + 𝑖∈𝑁,𝑘∈ 𝑝𝑖 𝜁𝑖𝑘𝑧𝑖𝑘 + 𝑖∈𝑁 𝜇𝑖𝑢𝑖 + 𝑖𝑗∈𝐸 𝛼𝑖𝑗𝑡𝑖𝑗
s.t. 𝑥𝑖𝑗𝑠 ≤ 𝑥𝑖𝑗 , 𝜏𝑖𝑗
𝑠 ≤ 𝜏𝑖𝑗 , 𝑡𝑖𝑗𝑠 ≤ 𝑡𝑖𝑗 , 𝑧𝑖
𝑘𝑠 ≤ 𝑧𝑖𝑘, 𝑢𝑖𝑠 ≤ 𝑢𝑖
−𝑥𝑖𝑗0𝑠 𝑄𝑖𝑗𝑘 ≤
𝑘∈𝑝𝑖𝑗
𝑓𝑖𝑗𝑠𝑘 ≤ 𝑥𝑖𝑗1
𝑠 𝑄𝑖𝑗𝑘
𝑥𝑖𝑗0𝑠 + 𝑥𝑖𝑗1
𝑠 ≤ 𝑥𝑖𝑗𝑠
− 1 − 𝜏𝑖𝑗𝑠 𝑄𝑖𝑗
𝑘 ≤ 𝑘∈𝑝𝑖𝑗 𝑓𝑖𝑗𝑘𝑠 ≤ 1 − 𝜏𝑖𝑗
𝑠 𝑄𝑖𝑗𝑘
−𝛽𝑖𝑗 𝑘∈𝑝𝑖,𝑗
𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗≤ 𝑓𝑖𝑗𝑘′𝑠 −
𝑘∈𝑝𝑖,𝑗𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗≤ 𝛽𝑖𝑗
𝑘∈𝑝𝑖,𝑗𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗
𝑥𝑖𝑗𝑠 = 𝑡𝑖𝑗
𝑠 when x is damaged
liks = 𝑦𝑖
𝑠𝑑𝑖𝑘
0 ≤ 𝑔𝑖𝑠𝑘 ≤ 𝑧𝑖
𝑘𝑠 + 𝑔𝑖𝑘+
giks − 𝑙𝑖
𝑘𝑠 − 𝑗∈𝑁 𝑓𝑖𝑗𝑘𝑠 = 0
0 ≤ 𝑧𝑖𝑘𝑠 ≤ 𝑢𝑖
𝑠𝑍𝑖𝑘
𝑖𝑗∈𝑠 𝑥𝑖𝑗𝑠+ 1 − 𝜏𝑖𝑗 ≤ 𝑠 − 1
𝜏𝑖𝑗𝑠 ≥ 𝑥𝑖𝑗
𝑠 + 𝜏𝑖𝑗𝑠− 1, 𝑥𝑖𝑗
𝑠 ≤ 𝑥𝑖𝑗𝑠
𝑖∈𝐶𝐿,𝑘∈𝑝𝑖 𝑙𝑖𝑘𝑠 ≥ 𝜆 𝑖∈𝐶𝐿,𝑘∈𝑝𝑖 𝑑𝑖
𝑘
𝑖∈𝑁∖𝐿,𝑘∈𝑝𝑖 𝑙𝑖𝑘𝑠 ≥ 𝛾 𝑖∈𝑁∖𝐿,𝑘∈𝑝𝑖 𝑑𝑖
𝑘
𝑥, 𝑦, 𝜏, 𝑢, 𝑡 ∈ {0,1}
Optimization Model: Multi-Commodity Flow
![Page 23: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/23.jpg)
Los Alamos National Laboratory
4/14/2016 | 23
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Optimization Model: Multi-Commodity Flowminimize 𝑖𝑗∈𝐸 𝑐𝑖𝑗𝑥𝑖𝑗 + 𝑖,𝑗∈𝐸 𝜅𝑖𝑗𝜏𝑖𝑗 + 𝑖∈𝑁,𝑘∈ 𝑝𝑖 𝜁𝑖
𝑘𝑧𝑖𝑘 + 𝑖∈𝑁 𝜇𝑖𝑢𝑖 + 𝑖𝑗∈𝐸 𝛼𝑖𝑗𝑡𝑖𝑗
s.t. 𝑥𝑖𝑗𝑠 ≤ 𝑥𝑖𝑗 , 𝜏𝑖𝑗
𝑠 ≤ 𝜏𝑖𝑗 , 𝑡𝑖𝑗𝑠 ≤ 𝑡𝑖𝑗 , 𝑧𝑖
𝑘𝑠 ≤ 𝑧𝑖𝑘, 𝑢𝑖𝑠 ≤ 𝑢𝑖
−𝑥𝑖𝑗0𝑠 𝑄𝑖𝑗𝑘 ≤
𝑘∈𝑝𝑖𝑗
𝑓𝑖𝑗𝑠𝑘 ≤ 𝑥𝑖𝑗1
𝑠 𝑄𝑖𝑗𝑘
𝑥𝑖𝑗0𝑠 + 𝑥𝑖𝑗1
𝑠 ≤ 𝑥𝑖𝑗𝑠
− 1 − 𝜏𝑖𝑗𝑠 𝑄𝑖𝑗
𝑘 ≤ 𝑘∈𝑝𝑖𝑗 𝑓𝑖𝑗𝑘𝑠 ≤ 1 − 𝜏𝑖𝑗
𝑠 𝑄𝑖𝑗𝑘
−𝛽𝑖𝑗 𝑘∈𝑝𝑖,𝑗
𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗≤ 𝑓𝑖𝑗𝑘′𝑠 −
𝑘∈𝑝𝑖,𝑗𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗≤ 𝛽𝑖𝑗
𝑘∈𝑝𝑖,𝑗𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗
𝑥𝑖𝑗𝑠 = 𝑡𝑖𝑗
𝑠 when x is damaged
liks = 𝑦𝑖
𝑠𝑑𝑖𝑘
0 ≤ 𝑔𝑖𝑠𝑘 ≤ 𝑧𝑖
𝑘𝑠 + 𝑔𝑖𝑘+
giks − 𝑙𝑖
𝑘𝑠 − 𝑗∈𝑁 𝑓𝑖𝑗𝑘𝑠 = 0
0 ≤ 𝑧𝑖𝑘𝑠 ≤ 𝑢𝑖
𝑠𝑍𝑖𝑘
𝑖𝑗∈𝑠 𝑥𝑖𝑗𝑠+ 1 − 𝜏𝑖𝑗 ≤ 𝑠 − 1
𝜏𝑖𝑗𝑠 ≥ 𝑥𝑖𝑗
𝑠 + 𝜏𝑖𝑗𝑠− 1, 𝑥𝑖𝑗
𝑠 ≤ 𝑥𝑖𝑗𝑠
𝑖∈𝐶𝐿,𝑘∈𝑝𝑖 𝑙𝑖𝑘𝑠 ≥ 𝜆 𝑖∈𝐶𝐿,𝑘∈𝑝𝑖 𝑑𝑖
𝑘
𝑖∈𝑁∖𝐿,𝑘∈𝑝𝑖 𝑙𝑖𝑘𝑠 ≥ 𝛾 𝑖∈𝑁∖𝐿,𝑘∈𝑝𝑖 𝑑𝑖
𝑘
𝑥, 𝑦, 𝜏, 𝑢, 𝑡 ∈ {0,1}
Minimize expansion cost
![Page 24: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/24.jpg)
Los Alamos National Laboratory
4/14/2016 | 24
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Optimization Model: Multi-Commodity Flowminimize 𝑖𝑗∈𝐸 𝑐𝑖𝑗𝑥𝑖𝑗 + 𝑖,𝑗∈𝐸 𝜅𝑖𝑗𝜏𝑖𝑗 + 𝑖∈𝑁,𝑘∈ 𝑝𝑖 𝜁𝑖
𝑘𝑧𝑖𝑘 + 𝑖∈𝑁 𝜇𝑖𝑢𝑖 + 𝑖𝑗∈𝐸 𝛼𝑖𝑗𝑡𝑖𝑗
s.t. 𝑥𝑖𝑗𝑠 ≤ 𝑥𝑖𝑗 , 𝜏𝑖𝑗
𝑠 ≤ 𝜏𝑖𝑗 , 𝑡𝑖𝑗𝑠 ≤ 𝑡𝑖𝑗 , 𝑧𝑖
𝑘𝑠 ≤ 𝑧𝑖𝑘, 𝑢𝑖𝑠 ≤ 𝑢𝑖
−𝑥𝑖𝑗0𝑠 𝑄𝑖𝑗𝑘 ≤
𝑘∈𝑝𝑖𝑗
𝑓𝑖𝑗𝑠𝑘 ≤ 𝑥𝑖𝑗1
𝑠 𝑄𝑖𝑗𝑘
𝑥𝑖𝑗0𝑠 + 𝑥𝑖𝑗1
𝑠 ≤ 𝑥𝑖𝑗𝑠
− 1 − 𝜏𝑖𝑗𝑠 𝑄𝑖𝑗
𝑘 ≤ 𝑘∈𝑝𝑖𝑗 𝑓𝑖𝑗𝑘𝑠 ≤ 1 − 𝜏𝑖𝑗
𝑠 𝑄𝑖𝑗𝑘
−𝛽𝑖𝑗 𝑘∈𝑝𝑖,𝑗
𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗≤ 𝑓𝑖𝑗𝑘′𝑠 −
𝑘∈𝑝𝑖,𝑗𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗≤ 𝛽𝑖𝑗
𝑘∈𝑝𝑖,𝑗𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗
𝑥𝑖𝑗𝑠 = 𝑡𝑖𝑗
𝑠 when x is damaged
liks = 𝑦𝑖
𝑠𝑑𝑖𝑘
0 ≤ 𝑔𝑖𝑠𝑘 ≤ 𝑧𝑖
𝑘𝑠 + 𝑔𝑖𝑘+
giks − 𝑙𝑖
𝑘𝑠 − 𝑗∈𝑁 𝑓𝑖𝑗𝑘𝑠 = 0
0 ≤ 𝑧𝑖𝑘𝑠 ≤ 𝑢𝑖
𝑠𝑍𝑖𝑘
𝑖𝑗∈𝑠 𝑥𝑖𝑗𝑠+ 1 − 𝜏𝑖𝑗 ≤ 𝑠 − 1
𝜏𝑖𝑗𝑠 ≥ 𝑥𝑖𝑗
𝑠 + 𝜏𝑖𝑗𝑠− 1, 𝑥𝑖𝑗
𝑠 ≤ 𝑥𝑖𝑗𝑠
𝑖∈𝐶𝐿,𝑘∈𝑝𝑖 𝑙𝑖𝑘𝑠 ≥ 𝜆 𝑖∈𝐶𝐿,𝑘∈𝑝𝑖 𝑑𝑖
𝑘
𝑖∈𝑁∖𝐿,𝑘∈𝑝𝑖 𝑙𝑖𝑘𝑠 ≥ 𝛾 𝑖∈𝑁∖𝐿,𝑘∈𝑝𝑖 𝑑𝑖
𝑘
𝑥, 𝑦, 𝜏, 𝑢, 𝑡 ∈ {0,1}
Auxiliary variables for linking
first and second stage.
Useful for decomposition.
![Page 25: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/25.jpg)
Los Alamos National Laboratory
4/14/2016 | 25
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Optimization Model: Multi-Commodity Flowminimize 𝑖𝑗∈𝐸 𝑐𝑖𝑗𝑥𝑖𝑗 + 𝑖,𝑗∈𝐸 𝜅𝑖𝑗𝜏𝑖𝑗 + 𝑖∈𝑁,𝑘∈ 𝑝𝑖 𝜁𝑖
𝑘𝑧𝑖𝑘 + 𝑖∈𝑁 𝜇𝑖𝑢𝑖 + 𝑖𝑗∈𝐸 𝛼𝑖𝑗𝑡𝑖𝑗
s.t. 𝑥𝑖𝑗𝑠 ≤ 𝑥𝑖𝑗 , 𝜏𝑖𝑗
𝑠 ≤ 𝜏𝑖𝑗 , 𝑡𝑖𝑗𝑠 ≤ 𝑡𝑖𝑗 , 𝑧𝑖
𝑘𝑠 ≤ 𝑧𝑖𝑘, 𝑢𝑖𝑠 ≤ 𝑢𝑖
−𝑥𝑖𝑗0𝑠 𝑄𝑖𝑗𝑘 ≤
𝑘∈𝑝𝑖𝑗
𝑓𝑖𝑗𝑠𝑘 ≤ 𝑥𝑖𝑗1
𝑠 𝑄𝑖𝑗𝑘
𝑥𝑖𝑗0𝑠 + 𝑥𝑖𝑗1
𝑠 ≤ 𝑥𝑖𝑗𝑠
− 1 − 𝜏𝑖𝑗𝑠 𝑄𝑖𝑗
𝑘 ≤ 𝑘∈𝑝𝑖𝑗 𝑓𝑖𝑗𝑘𝑠 ≤ 1 − 𝜏𝑖𝑗
𝑠 𝑄𝑖𝑗𝑘
−𝛽𝑖𝑗 𝑘∈𝑝𝑖,𝑗
𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗≤ 𝑓𝑖𝑗𝑘′𝑠 −
𝑘∈𝑝𝑖,𝑗𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗≤ 𝛽𝑖𝑗
𝑘∈𝑝𝑖,𝑗𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗
𝑥𝑖𝑗𝑠 = 𝑡𝑖𝑗
𝑠 when x is damaged
liks = 𝑦𝑖
𝑠𝑑𝑖𝑘
0 ≤ 𝑔𝑖𝑠𝑘 ≤ 𝑧𝑖
𝑘𝑠 + 𝑔𝑖𝑘+
giks − 𝑙𝑖
𝑘𝑠 − 𝑗∈𝑁 𝑓𝑖𝑗𝑘𝑠 = 0
0 ≤ 𝑧𝑖𝑘𝑠 ≤ 𝑢𝑖
𝑠𝑍𝑖𝑘
𝑖𝑗∈𝑠 𝑥𝑖𝑗𝑠+ 1 − 𝜏𝑖𝑗 ≤ 𝑠 − 1
𝜏𝑖𝑗𝑠 ≥ 𝑥𝑖𝑗
𝑠 + 𝜏𝑖𝑗𝑠− 1, 𝑥𝑖𝑗
𝑠 ≤ 𝑥𝑖𝑗𝑠
𝑖∈𝐶𝐿,𝑘∈𝑝𝑖 𝑙𝑖𝑘𝑠 ≥ 𝜆 𝑖∈𝐶𝐿,𝑘∈𝑝𝑖 𝑑𝑖
𝑘
𝑖∈𝑁∖𝐿,𝑘∈𝑝𝑖 𝑙𝑖𝑘𝑠 ≥ 𝛾 𝑖∈𝑁∖𝐿,𝑘∈𝑝𝑖 𝑑𝑖
𝑘
𝑥, 𝑦, 𝜏, 𝑢, 𝑡 ∈ {0,1}
Line capacity constraints.
Capacity is 0 when line is
unavailable or open.
![Page 26: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/26.jpg)
Los Alamos National Laboratory
4/14/2016 | 26
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Optimization Model: Multi-Commodity Flowminimize 𝑖𝑗∈𝐸 𝑐𝑖𝑗𝑥𝑖𝑗 + 𝑖,𝑗∈𝐸 𝜅𝑖𝑗𝜏𝑖𝑗 + 𝑖∈𝑁,𝑘∈ 𝑝𝑖 𝜁𝑖
𝑘𝑧𝑖𝑘 + 𝑖∈𝑁 𝜇𝑖𝑢𝑖 + 𝑖𝑗∈𝐸 𝛼𝑖𝑗𝑡𝑖𝑗
s.t. 𝑥𝑖𝑗𝑠 ≤ 𝑥𝑖𝑗 , 𝜏𝑖𝑗
𝑠 ≤ 𝜏𝑖𝑗 , 𝑡𝑖𝑗𝑠 ≤ 𝑡𝑖𝑗 , 𝑧𝑖
𝑘𝑠 ≤ 𝑧𝑖𝑘, 𝑢𝑖𝑠 ≤ 𝑢𝑖
−𝑥𝑖𝑗0𝑠 𝑄𝑖𝑗𝑘 ≤
𝑘∈𝑝𝑖𝑗
𝑓𝑖𝑗𝑠𝑘 ≤ 𝑥𝑖𝑗1
𝑠 𝑄𝑖𝑗𝑘
𝑥𝑖𝑗0𝑠 + 𝑥𝑖𝑗1
𝑠 ≤ 𝑥𝑖𝑗𝑠
− 1 − 𝜏𝑖𝑗𝑠 𝑄𝑖𝑗
𝑘 ≤ 𝑘∈𝑝𝑖𝑗 𝑓𝑖𝑗𝑘𝑠 ≤ 1 − 𝜏𝑖𝑗
𝑠 𝑄𝑖𝑗𝑘
−𝛽𝑖𝑗 𝑘∈𝑝𝑖,𝑗
𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗≤ 𝑓𝑖𝑗𝑘′𝑠 −
𝑘∈𝑝𝑖,𝑗𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗≤ 𝛽𝑖𝑗
𝑘∈𝑝𝑖,𝑗𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗
𝑥𝑖𝑗𝑠 = 𝑡𝑖𝑗
𝑠 when x is damaged
liks = 𝑦𝑖
𝑠𝑑𝑖𝑘
0 ≤ 𝑔𝑖𝑠𝑘 ≤ 𝑧𝑖
𝑘𝑠 + 𝑔𝑖𝑘+
giks − 𝑙𝑖
𝑘𝑠 − 𝑗∈𝑁 𝑓𝑖𝑗𝑘𝑠 = 0
0 ≤ 𝑧𝑖𝑘𝑠 ≤ 𝑢𝑖
𝑠𝑍𝑖𝑘
𝑖𝑗∈𝑠 𝑥𝑖𝑗𝑠+ 1 − 𝜏𝑖𝑗 ≤ 𝑠 − 1
𝜏𝑖𝑗𝑠 ≥ 𝑥𝑖𝑗
𝑠 + 𝜏𝑖𝑗𝑠− 1, 𝑥𝑖𝑗
𝑠 ≤ 𝑥𝑖𝑗𝑠
𝑖∈𝐶𝐿,𝑘∈𝑝𝑖 𝑙𝑖𝑘𝑠 ≥ 𝜆 𝑖∈𝐶𝐿,𝑘∈𝑝𝑖 𝑑𝑖
𝑘
𝑖∈𝑁∖𝐿,𝑘∈𝑝𝑖 𝑙𝑖𝑘𝑠 ≥ 𝛾 𝑖∈𝑁∖𝐿,𝑘∈𝑝𝑖 𝑑𝑖
𝑘
𝑥, 𝑦, 𝜏, 𝑢, 𝑡 ∈ {0,1}
Phase imbalance tolerance
![Page 27: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/27.jpg)
Los Alamos National Laboratory
4/14/2016 | 27
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Optimization Model: Multi-Commodity Flowminimize 𝑖𝑗∈𝐸 𝑐𝑖𝑗𝑥𝑖𝑗 + 𝑖,𝑗∈𝐸 𝜅𝑖𝑗𝜏𝑖𝑗 + 𝑖∈𝑁,𝑘∈ 𝑝𝑖 𝜁𝑖
𝑘𝑧𝑖𝑘 + 𝑖∈𝑁 𝜇𝑖𝑢𝑖 + 𝑖𝑗∈𝐸 𝛼𝑖𝑗𝑡𝑖𝑗
s.t. 𝑥𝑖𝑗𝑠 ≤ 𝑥𝑖𝑗 , 𝜏𝑖𝑗
𝑠 ≤ 𝜏𝑖𝑗 , 𝑡𝑖𝑗𝑠 ≤ 𝑡𝑖𝑗 , 𝑧𝑖
𝑘𝑠 ≤ 𝑧𝑖𝑘, 𝑢𝑖𝑠 ≤ 𝑢𝑖
−𝑥𝑖𝑗0𝑠 𝑄𝑖𝑗𝑘 ≤
𝑘∈𝑝𝑖𝑗
𝑓𝑖𝑗𝑠𝑘 ≤ 𝑥𝑖𝑗1
𝑠 𝑄𝑖𝑗𝑘
𝑥𝑖𝑗0𝑠 + 𝑥𝑖𝑗1
𝑠 ≤ 𝑥𝑖𝑗𝑠
− 1 − 𝜏𝑖𝑗𝑠 𝑄𝑖𝑗
𝑘 ≤ 𝑘∈𝑝𝑖𝑗 𝑓𝑖𝑗𝑘𝑠 ≤ 1 − 𝜏𝑖𝑗
𝑠 𝑄𝑖𝑗𝑘
−𝛽𝑖𝑗 𝑘∈𝑝𝑖,𝑗
𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗≤ 𝑓𝑖𝑗𝑘′𝑠 −
𝑘∈𝑝𝑖,𝑗𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗≤ 𝛽𝑖𝑗
𝑘∈𝑝𝑖,𝑗𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗
𝑥𝑖𝑗𝑠 = 𝑡𝑖𝑗
𝑠 when x is damaged
liks = 𝑦𝑖
𝑠𝑑𝑖𝑘
0 ≤ 𝑔𝑖𝑠𝑘 ≤ 𝑧𝑖
𝑘𝑠 + 𝑔𝑖𝑘+
giks − 𝑙𝑖
𝑘𝑠 − 𝑗∈𝑁 𝑓𝑖𝑗𝑘𝑠 = 0
0 ≤ 𝑧𝑖𝑘𝑠 ≤ 𝑢𝑖
𝑠𝑍𝑖𝑘
𝑖𝑗∈𝑠 𝑥𝑖𝑗𝑠+ 1 − 𝜏𝑖𝑗 ≤ 𝑠 − 1
𝜏𝑖𝑗𝑠 ≥ 𝑥𝑖𝑗
𝑠 + 𝜏𝑖𝑗𝑠− 1, 𝑥𝑖𝑗
𝑠 ≤ 𝑥𝑖𝑗𝑠
𝑖∈𝐶𝐿,𝑘∈𝑝𝑖 𝑙𝑖𝑘𝑠 ≥ 𝜆 𝑖∈𝐶𝐿,𝑘∈𝑝𝑖 𝑑𝑖
𝑘
𝑖∈𝑁∖𝐿,𝑘∈𝑝𝑖 𝑙𝑖𝑘𝑠 ≥ 𝛾 𝑖∈𝑁∖𝐿,𝑘∈𝑝𝑖 𝑑𝑖
𝑘
𝑥, 𝑦, 𝜏, 𝑢, 𝑡 ∈ {0,1}
Links damaged lines with
hardening variables
![Page 28: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/28.jpg)
Los Alamos National Laboratory
4/14/2016 | 28
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Optimization Model: Multi-Commodity Flowminimize 𝑖𝑗∈𝐸 𝑐𝑖𝑗𝑥𝑖𝑗 + 𝑖,𝑗∈𝐸 𝜅𝑖𝑗𝜏𝑖𝑗 + 𝑖∈𝑁,𝑘∈ 𝑝𝑖 𝜁𝑖
𝑘𝑧𝑖𝑘 + 𝑖∈𝑁 𝜇𝑖𝑢𝑖 + 𝑖𝑗∈𝐸 𝛼𝑖𝑗𝑡𝑖𝑗
s.t. 𝑥𝑖𝑗𝑠 ≤ 𝑥𝑖𝑗 , 𝜏𝑖𝑗
𝑠 ≤ 𝜏𝑖𝑗 , 𝑡𝑖𝑗𝑠 ≤ 𝑡𝑖𝑗 , 𝑧𝑖
𝑘𝑠 ≤ 𝑧𝑖𝑘, 𝑢𝑖𝑠 ≤ 𝑢𝑖
−𝑥𝑖𝑗0𝑠 𝑄𝑖𝑗𝑘 ≤
𝑘∈𝑝𝑖𝑗
𝑓𝑖𝑗𝑠𝑘 ≤ 𝑥𝑖𝑗1
𝑠 𝑄𝑖𝑗𝑘
𝑥𝑖𝑗0𝑠 + 𝑥𝑖𝑗1
𝑠 ≤ 𝑥𝑖𝑗𝑠
− 1 − 𝜏𝑖𝑗𝑠 𝑄𝑖𝑗
𝑘 ≤ 𝑘∈𝑝𝑖𝑗 𝑓𝑖𝑗𝑘𝑠 ≤ 1 − 𝜏𝑖𝑗
𝑠 𝑄𝑖𝑗𝑘
−𝛽𝑖𝑗 𝑘∈𝑝𝑖,𝑗
𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗≤ 𝑓𝑖𝑗𝑘′𝑠 −
𝑘∈𝑝𝑖,𝑗𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗≤ 𝛽𝑖𝑗
𝑘∈𝑝𝑖,𝑗𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗
𝑥𝑖𝑗𝑠 = 𝑡𝑖𝑗
𝑠 when x is damaged
liks = 𝑦𝑖
𝑠𝑑𝑖𝑘
0 ≤ 𝑔𝑖𝑠𝑘 ≤ 𝑧𝑖
𝑘𝑠 + 𝑔𝑖𝑘+
giks − 𝑙𝑖
𝑘𝑠 − 𝑗∈𝑁 𝑓𝑖𝑗𝑘𝑠 = 0
0 ≤ 𝑧𝑖𝑘𝑠 ≤ 𝑢𝑖
𝑠𝑍𝑖𝑘
𝑖𝑗∈𝑠 𝑥𝑖𝑗𝑠+ 1 − 𝜏𝑖𝑗 ≤ 𝑠 − 1
𝜏𝑖𝑗𝑠 ≥ 𝑥𝑖𝑗
𝑠 + 𝜏𝑖𝑗𝑠− 1, 𝑥𝑖𝑗
𝑠 ≤ 𝑥𝑖𝑗𝑠
𝑖∈𝐶𝐿,𝑘∈𝑝𝑖 𝑙𝑖𝑘𝑠 ≥ 𝜆 𝑖∈𝐶𝐿,𝑘∈𝑝𝑖 𝑑𝑖
𝑘
𝑖∈𝑁∖𝐿,𝑘∈𝑝𝑖 𝑙𝑖𝑘𝑠 ≥ 𝛾 𝑖∈𝑁∖𝐿,𝑘∈𝑝𝑖 𝑑𝑖
𝑘
𝑥, 𝑦, 𝜏, 𝑢, 𝑡 ∈ {0,1}
Load switching
![Page 29: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/29.jpg)
Los Alamos National Laboratory
4/14/2016 | 29
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Optimization Model: Multi-Commodity Flowminimize 𝑖𝑗∈𝐸 𝑐𝑖𝑗𝑥𝑖𝑗 + 𝑖,𝑗∈𝐸 𝜅𝑖𝑗𝜏𝑖𝑗 + 𝑖∈𝑁,𝑘∈ 𝑝𝑖 𝜁𝑖
𝑘𝑧𝑖𝑘 + 𝑖∈𝑁 𝜇𝑖𝑢𝑖 + 𝑖𝑗∈𝐸 𝛼𝑖𝑗𝑡𝑖𝑗
s.t. 𝑥𝑖𝑗𝑠 ≤ 𝑥𝑖𝑗 , 𝜏𝑖𝑗
𝑠 ≤ 𝜏𝑖𝑗 , 𝑡𝑖𝑗𝑠 ≤ 𝑡𝑖𝑗 , 𝑧𝑖
𝑘𝑠 ≤ 𝑧𝑖𝑘, 𝑢𝑖𝑠 ≤ 𝑢𝑖
−𝑥𝑖𝑗0𝑠 𝑄𝑖𝑗𝑘 ≤
𝑘∈𝑝𝑖𝑗
𝑓𝑖𝑗𝑠𝑘 ≤ 𝑥𝑖𝑗1
𝑠 𝑄𝑖𝑗𝑘
𝑥𝑖𝑗0𝑠 + 𝑥𝑖𝑗1
𝑠 ≤ 𝑥𝑖𝑗𝑠
− 1 − 𝜏𝑖𝑗𝑠 𝑄𝑖𝑗
𝑘 ≤ 𝑘∈𝑝𝑖𝑗 𝑓𝑖𝑗𝑘𝑠 ≤ 1 − 𝜏𝑖𝑗
𝑠 𝑄𝑖𝑗𝑘
−𝛽𝑖𝑗 𝑘∈𝑝𝑖,𝑗
𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗≤ 𝑓𝑖𝑗𝑘′𝑠 −
𝑘∈𝑝𝑖,𝑗𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗≤ 𝛽𝑖𝑗
𝑘∈𝑝𝑖,𝑗𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗
𝑥𝑖𝑗𝑠 = 𝑡𝑖𝑗
𝑠 when x is damaged
liks = 𝑦𝑖
𝑠𝑑𝑖𝑘
0 ≤ 𝑔𝑖𝑠𝑘 ≤ 𝑧𝑖
𝑘𝑠 + 𝑔𝑖𝑘+
giks − 𝑙𝑖
𝑘𝑠 − 𝑗∈𝑁 𝑓𝑖𝑗𝑘𝑠 = 0
0 ≤ 𝑧𝑖𝑘𝑠 ≤ 𝑢𝑖
𝑠𝑍𝑖𝑘
𝑖𝑗∈𝑠 𝑥𝑖𝑗𝑠+ 1 − 𝜏𝑖𝑗 ≤ 𝑠 − 1
𝜏𝑖𝑗𝑠 ≥ 𝑥𝑖𝑗
𝑠 + 𝜏𝑖𝑗𝑠− 1, 𝑥𝑖𝑗
𝑠 ≤ 𝑥𝑖𝑗𝑠
𝑖∈𝐶𝐿,𝑘∈𝑝𝑖 𝑙𝑖𝑘𝑠 ≥ 𝜆 𝑖∈𝐶𝐿,𝑘∈𝑝𝑖 𝑑𝑖
𝑘
𝑖∈𝑁∖𝐿,𝑘∈𝑝𝑖 𝑙𝑖𝑘𝑠 ≥ 𝛾 𝑖∈𝑁∖𝐿,𝑘∈𝑝𝑖 𝑑𝑖
𝑘
𝑥, 𝑦, 𝜏, 𝑢, 𝑡 ∈ {0,1}
Power produced
![Page 30: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/30.jpg)
Los Alamos National Laboratory
4/14/2016 | 30
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Optimization Model: Multi-Commodity Flowminimize 𝑖𝑗∈𝐸 𝑐𝑖𝑗𝑥𝑖𝑗 + 𝑖,𝑗∈𝐸 𝜅𝑖𝑗𝜏𝑖𝑗 + 𝑖∈𝑁,𝑘∈ 𝑝𝑖 𝜁𝑖
𝑘𝑧𝑖𝑘 + 𝑖∈𝑁 𝜇𝑖𝑢𝑖 + 𝑖𝑗∈𝐸 𝛼𝑖𝑗𝑡𝑖𝑗
s.t. 𝑥𝑖𝑗𝑠 ≤ 𝑥𝑖𝑗 , 𝜏𝑖𝑗
𝑠 ≤ 𝜏𝑖𝑗 , 𝑡𝑖𝑗𝑠 ≤ 𝑡𝑖𝑗 , 𝑧𝑖
𝑘𝑠 ≤ 𝑧𝑖𝑘, 𝑢𝑖𝑠 ≤ 𝑢𝑖
−𝑥𝑖𝑗0𝑠 𝑄𝑖𝑗𝑘 ≤
𝑘∈𝑝𝑖𝑗
𝑓𝑖𝑗𝑠𝑘 ≤ 𝑥𝑖𝑗1
𝑠 𝑄𝑖𝑗𝑘
𝑥𝑖𝑗0𝑠 + 𝑥𝑖𝑗1
𝑠 ≤ 𝑥𝑖𝑗𝑠
− 1 − 𝜏𝑖𝑗𝑠 𝑄𝑖𝑗
𝑘 ≤ 𝑘∈𝑝𝑖𝑗 𝑓𝑖𝑗𝑘𝑠 ≤ 1 − 𝜏𝑖𝑗
𝑠 𝑄𝑖𝑗𝑘
−𝛽𝑖𝑗 𝑘∈𝑝𝑖,𝑗
𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗≤ 𝑓𝑖𝑗𝑘′𝑠 −
𝑘∈𝑝𝑖,𝑗𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗≤ 𝛽𝑖𝑗
𝑘∈𝑝𝑖,𝑗𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗
𝑥𝑖𝑗𝑠 = 𝑡𝑖𝑗
𝑠 when x is damaged
liks = 𝑦𝑖
𝑠𝑑𝑖𝑘
0 ≤ 𝑔𝑖𝑠𝑘 ≤ 𝑧𝑖
𝑘𝑠 + 𝑔𝑖𝑘+
giks − 𝑙𝑖
𝑘𝑠 − 𝑗∈𝑁 𝑓𝑖𝑗𝑘𝑠 = 0
0 ≤ 𝑧𝑖𝑘𝑠 ≤ 𝑢𝑖
𝑠𝑍𝑖𝑘
𝑖𝑗∈𝑠 𝑥𝑖𝑗𝑠+ 1 − 𝜏𝑖𝑗 ≤ 𝑠 − 1
𝜏𝑖𝑗𝑠 ≥ 𝑥𝑖𝑗
𝑠 + 𝜏𝑖𝑗𝑠− 1, 𝑥𝑖𝑗
𝑠 ≤ 𝑥𝑖𝑗𝑠
𝑖∈𝐶𝐿,𝑘∈𝑝𝑖 𝑙𝑖𝑘𝑠 ≥ 𝜆 𝑖∈𝐶𝐿,𝑘∈𝑝𝑖 𝑑𝑖
𝑘
𝑖∈𝑁∖𝐿,𝑘∈𝑝𝑖 𝑙𝑖𝑘𝑠 ≥ 𝛾 𝑖∈𝑁∖𝐿,𝑘∈𝑝𝑖 𝑑𝑖
𝑘
𝑥, 𝑦, 𝜏, 𝑢, 𝑡 ∈ {0,1}
Nodal flow balance
![Page 31: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/31.jpg)
Los Alamos National Laboratory
4/14/2016 | 31
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Optimization Model: Multi-Commodity Flowminimize 𝑖𝑗∈𝐸 𝑐𝑖𝑗𝑥𝑖𝑗 + 𝑖,𝑗∈𝐸 𝜅𝑖𝑗𝜏𝑖𝑗 + 𝑖∈𝑁,𝑘∈ 𝑝𝑖 𝜁𝑖
𝑘𝑧𝑖𝑘 + 𝑖∈𝑁 𝜇𝑖𝑢𝑖 + 𝑖𝑗∈𝐸 𝛼𝑖𝑗𝑡𝑖𝑗
s.t. 𝑥𝑖𝑗𝑠 ≤ 𝑥𝑖𝑗 , 𝜏𝑖𝑗
𝑠 ≤ 𝜏𝑖𝑗 , 𝑡𝑖𝑗𝑠 ≤ 𝑡𝑖𝑗 , 𝑧𝑖
𝑘𝑠 ≤ 𝑧𝑖𝑘, 𝑢𝑖𝑠 ≤ 𝑢𝑖
−𝑥𝑖𝑗0𝑠 𝑄𝑖𝑗𝑘 ≤
𝑘∈𝑝𝑖𝑗
𝑓𝑖𝑗𝑠𝑘 ≤ 𝑥𝑖𝑗1
𝑠 𝑄𝑖𝑗𝑘
𝑥𝑖𝑗0𝑠 + 𝑥𝑖𝑗1
𝑠 ≤ 𝑥𝑖𝑗𝑠
− 1 − 𝜏𝑖𝑗𝑠 𝑄𝑖𝑗
𝑘 ≤ 𝑘∈𝑝𝑖𝑗 𝑓𝑖𝑗𝑘𝑠 ≤ 1 − 𝜏𝑖𝑗
𝑠 𝑄𝑖𝑗𝑘
−𝛽𝑖𝑗 𝑘∈𝑝𝑖,𝑗
𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗≤ 𝑓𝑖𝑗𝑘′𝑠 −
𝑘∈𝑝𝑖,𝑗𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗≤ 𝛽𝑖𝑗
𝑘∈𝑝𝑖,𝑗𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗
𝑥𝑖𝑗𝑠 = 𝑡𝑖𝑗
𝑠 when x is damaged
liks = 𝑦𝑖
𝑠𝑑𝑖𝑘
0 ≤ 𝑔𝑖𝑠𝑘 ≤ 𝑧𝑖
𝑘𝑠 + 𝑔𝑖𝑘+
giks − 𝑙𝑖
𝑘𝑠 − 𝑗∈𝑁 𝑓𝑖𝑗𝑘𝑠 = 0
0 ≤ 𝑧𝑖𝑘𝑠 ≤ 𝑢𝑖
𝑠𝑍𝑖𝑘
𝑖𝑗∈𝑠 𝑥𝑖𝑗𝑠+ 1 − 𝜏𝑖𝑗 ≤ 𝑠 − 1
𝜏𝑖𝑗𝑠 ≥ 𝑥𝑖𝑗
𝑠 + 𝜏𝑖𝑗𝑠− 1, 𝑥𝑖𝑗
𝑠 ≤ 𝑥𝑖𝑗𝑠
𝑖∈𝐶𝐿,𝑘∈𝑝𝑖 𝑙𝑖𝑘𝑠 ≥ 𝜆 𝑖∈𝐶𝐿,𝑘∈𝑝𝑖 𝑑𝑖
𝑘
𝑖∈𝑁∖𝐿,𝑘∈𝑝𝑖 𝑙𝑖𝑘𝑠 ≥ 𝛾 𝑖∈𝑁∖𝐿,𝑘∈𝑝𝑖 𝑑𝑖
𝑘
𝑥, 𝑦, 𝜏, 𝑢, 𝑡 ∈ {0,1}
Links generation construction
and capacity
![Page 32: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/32.jpg)
Los Alamos National Laboratory
4/14/2016 | 32
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Optimization Model: Multi-Commodity Flowminimize 𝑖𝑗∈𝐸 𝑐𝑖𝑗𝑥𝑖𝑗 + 𝑖,𝑗∈𝐸 𝜅𝑖𝑗𝜏𝑖𝑗 + 𝑖∈𝑁,𝑘∈ 𝑝𝑖 𝜁𝑖
𝑘𝑧𝑖𝑘 + 𝑖∈𝑁 𝜇𝑖𝑢𝑖 + 𝑖𝑗∈𝐸 𝛼𝑖𝑗𝑡𝑖𝑗
s.t. 𝑥𝑖𝑗𝑠 ≤ 𝑥𝑖𝑗 , 𝜏𝑖𝑗
𝑠 ≤ 𝜏𝑖𝑗 , 𝑡𝑖𝑗𝑠 ≤ 𝑡𝑖𝑗 , 𝑧𝑖
𝑘𝑠 ≤ 𝑧𝑖𝑘, 𝑢𝑖𝑠 ≤ 𝑢𝑖
−𝑥𝑖𝑗0𝑠 𝑄𝑖𝑗𝑘 ≤
𝑘∈𝑝𝑖𝑗
𝑓𝑖𝑗𝑠𝑘 ≤ 𝑥𝑖𝑗1
𝑠 𝑄𝑖𝑗𝑘
𝑥𝑖𝑗0𝑠 + 𝑥𝑖𝑗1
𝑠 ≤ 𝑥𝑖𝑗𝑠
− 1 − 𝜏𝑖𝑗𝑠 𝑄𝑖𝑗
𝑘 ≤ 𝑘∈𝑝𝑖𝑗 𝑓𝑖𝑗𝑘𝑠 ≤ 1 − 𝜏𝑖𝑗
𝑠 𝑄𝑖𝑗𝑘
−𝛽𝑖𝑗 𝑘∈𝑝𝑖,𝑗
𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗≤ 𝑓𝑖𝑗𝑘′𝑠 −
𝑘∈𝑝𝑖,𝑗𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗≤ 𝛽𝑖𝑗
𝑘∈𝑝𝑖,𝑗𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗
𝑥𝑖𝑗𝑠 = 𝑡𝑖𝑗
𝑠 when x is damaged
liks = 𝑦𝑖
𝑠𝑑𝑖𝑘
0 ≤ 𝑔𝑖𝑠𝑘 ≤ 𝑧𝑖
𝑘𝑠 + 𝑔𝑖𝑘+
giks − 𝑙𝑖
𝑘𝑠 − 𝑗∈𝑁 𝑓𝑖𝑗𝑘𝑠 = 0
0 ≤ 𝑧𝑖𝑘𝑠 ≤ 𝑢𝑖
𝑠𝑍𝑖𝑘
𝑖𝑗∈𝑠 𝑥𝑖𝑗𝑠+ 1 − 𝜏𝑖𝑗 ≤ 𝑠 − 1
𝜏𝑖𝑗𝑠 ≥ 𝑥𝑖𝑗
𝑠 + 𝜏𝑖𝑗𝑠− 1, 𝑥𝑖𝑗
𝑠 ≤ 𝑥𝑖𝑗𝑠
𝑖∈𝐶𝐿,𝑘∈𝑝𝑖 𝑙𝑖𝑘𝑠 ≥ 𝜆 𝑖∈𝐶𝐿,𝑘∈𝑝𝑖 𝑑𝑖
𝑘
𝑖∈𝑁∖𝐿,𝑘∈𝑝𝑖 𝑙𝑖𝑘𝑠 ≥ 𝛾 𝑖∈𝑁∖𝐿,𝑘∈𝑝𝑖 𝑑𝑖
𝑘
𝑥, 𝑦, 𝜏, 𝑢, 𝑡 ∈ {0,1}
Enforces radial operation
![Page 33: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/33.jpg)
Los Alamos National Laboratory
4/14/2016 | 33
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Optimization Model: Multi-Commodity Flowminimize 𝑖𝑗∈𝐸 𝑐𝑖𝑗𝑥𝑖𝑗 + 𝑖,𝑗∈𝐸 𝜅𝑖𝑗𝜏𝑖𝑗 + 𝑖∈𝑁,𝑘∈ 𝑝𝑖 𝜁𝑖
𝑘𝑧𝑖𝑘 + 𝑖∈𝑁 𝜇𝑖𝑢𝑖 + 𝑖𝑗∈𝐸 𝛼𝑖𝑗𝑡𝑖𝑗
s.t. 𝑥𝑖𝑗𝑠 ≤ 𝑥𝑖𝑗 , 𝜏𝑖𝑗
𝑠 ≤ 𝜏𝑖𝑗 , 𝑡𝑖𝑗𝑠 ≤ 𝑡𝑖𝑗 , 𝑧𝑖
𝑘𝑠 ≤ 𝑧𝑖𝑘, 𝑢𝑖𝑠 ≤ 𝑢𝑖
−𝑥𝑖𝑗0𝑠 𝑄𝑖𝑗𝑘 ≤
𝑘∈𝑝𝑖𝑗
𝑓𝑖𝑗𝑠𝑘 ≤ 𝑥𝑖𝑗1
𝑠 𝑄𝑖𝑗𝑘
𝑥𝑖𝑗0𝑠 + 𝑥𝑖𝑗1
𝑠 ≤ 𝑥𝑖𝑗𝑠
− 1 − 𝜏𝑖𝑗𝑠 𝑄𝑖𝑗
𝑘 ≤ 𝑘∈𝑝𝑖𝑗 𝑓𝑖𝑗𝑘𝑠 ≤ 1 − 𝜏𝑖𝑗
𝑠 𝑄𝑖𝑗𝑘
−𝛽𝑖𝑗 𝑘∈𝑝𝑖,𝑗
𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗≤ 𝑓𝑖𝑗𝑘′𝑠 −
𝑘∈𝑝𝑖,𝑗𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗≤ 𝛽𝑖𝑗
𝑘∈𝑝𝑖,𝑗𝑓𝑖𝑗𝑘𝑠
𝑝𝑖𝑗
𝑥𝑖𝑗𝑠 = 𝑡𝑖𝑗
𝑠 when x is damaged
liks = 𝑦𝑖
𝑠𝑑𝑖𝑘
0 ≤ 𝑔𝑖𝑠𝑘 ≤ 𝑧𝑖
𝑘𝑠 + 𝑔𝑖𝑘+
giks − 𝑙𝑖
𝑘𝑠 − 𝑗∈𝑁 𝑓𝑖𝑗𝑘𝑠 = 0
0 ≤ 𝑧𝑖𝑘𝑠 ≤ 𝑢𝑖
𝑠𝑍𝑖𝑘
𝑖𝑗∈𝑠 𝑥𝑖𝑗𝑠+ 1 − 𝜏𝑖𝑗 ≤ 𝑠 − 1
𝜏𝑖𝑗𝑠 ≥ 𝑥𝑖𝑗
𝑠 + 𝜏𝑖𝑗𝑠− 1, 𝑥𝑖𝑗
𝑠 ≤ 𝑥𝑖𝑗𝑠
𝑖∈𝐶𝐿,𝑘∈𝑝𝑖 𝑙𝑖𝑘𝑠 ≥ 𝜆 𝑖∈𝐶𝐿,𝑘∈𝑝𝑖 𝑑𝑖
𝑘
𝑖∈𝑁∖𝐿,𝑘∈𝑝𝑖 𝑙𝑖𝑘𝑠 ≥ 𝛾 𝑖∈𝑁∖𝐿,𝑘∈𝑝𝑖 𝑑𝑖
𝑘
𝑥, 𝑦, 𝜏, 𝑢, 𝑡 ∈ {0,1}
Resilience criteria—minimum
amount of load served
Is generalized to a chance
constraint
![Page 34: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/34.jpg)
Los Alamos National Laboratory
4/14/2016 | 34
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Optimization Model: LinDist Flow
• Baran and Wu 89, Gan and Low 14
• Assumption
• Derivation of a linearized model of the power flow physics
• Adds voltage to the models
• Justification
• Situations where multi-commodity flow assumptions do not hold
• Loopy operations
• Capacitor placement
• Non-firm generation
• Contrived cases
• Long lines are cheaper than short lines
• Initial network does not satisfy voltage constraints
![Page 35: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/35.jpg)
Los Alamos National Laboratory
4/14/2016 | 35
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Optimization Model: LinDist Flow
𝑣𝑗𝑎 = 𝑣𝑖
𝑎 − 2(𝑟𝑖𝑗𝑎𝑎𝑃𝑖𝑗𝑎 + 𝑥𝑖𝑗
𝑎𝑎𝑄𝑖𝑗𝑎 + 𝑟𝑖𝑗
𝑎𝑏𝑃𝑖𝑗𝑏 + 𝑥𝑖𝑗
𝑎𝑏𝑄𝑖𝑗𝑏 + 𝑟𝑖𝑗
𝑎𝑐𝑃𝑖𝑗𝑐 + 𝑥𝑖𝑗
𝑎𝑐𝑄𝑖𝑗𝑐 )
𝑣𝑗𝑏 = 𝑣𝑖
𝑏 − 2(𝑟𝑖𝑗𝑏𝑎𝑃𝑖𝑗𝑎 + 𝑥𝑖𝑗
𝑏𝑎𝑄𝑖𝑗𝑎 + 𝑟𝑖𝑗
𝑏𝑏𝑃𝑖𝑗𝑏 + 𝑥𝑖𝑗
𝑏𝑏𝑄𝑖𝑗𝑏 + 𝑟𝑖𝑗
𝑏𝑐𝑃𝑖𝑗𝑐 + 𝑥𝑖𝑗
𝑏𝑐𝑄𝑖𝑗𝑐 )
𝑣𝑗𝑐 = 𝑣𝑖
𝑐 − 2(𝑟𝑖𝑗𝑐𝑎𝑃𝑖𝑗𝑎 + 𝑥𝑖𝑗
𝑐𝑎𝑄𝑖𝑗𝑎 + 𝑟𝑖𝑗
𝑐𝑏𝑃𝑖𝑗𝑏 + 𝑥𝑖𝑗
𝑐𝑏𝑄𝑖𝑗𝑏 + 𝑟𝑖𝑗
𝑐𝑐𝑃𝑖𝑗𝑐 + 𝑥𝑖𝑗
𝑐𝑐𝑄𝑖𝑗𝑐 )
g(q)iks − 𝑙(𝑞)𝑖
𝑘𝑠 − 𝑗∈𝑁 𝑓(𝑞)𝑖𝑗𝑘𝑠 = 0
𝑣−𝑘 ≤ 𝑣𝑘 ≤ 𝑣+
𝑘
![Page 36: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/36.jpg)
Los Alamos National Laboratory
4/14/2016 | 36
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Optimization Model: LinDist Flow
𝑣𝑗𝑎 = 𝑣𝑖
𝑎 − 2(𝑟𝑖𝑗𝑎𝑎𝑃𝑖𝑗𝑎 + 𝑥𝑖𝑗
𝑎𝑎𝑄𝑖𝑗𝑎 + 𝑟𝑖𝑗
𝑎𝑏𝑃𝑖𝑗𝑏 + 𝑥𝑖𝑗
𝑎𝑏𝑄𝑖𝑗𝑏 + 𝑟𝑖𝑗
𝑎𝑐𝑃𝑖𝑗𝑐 + 𝑥𝑖𝑗
𝑎𝑐𝑄𝑖𝑗𝑐 )
𝑣𝑗𝑏 = 𝑣𝑖
𝑏 − 2(𝑟𝑖𝑗𝑏𝑎𝑃𝑖𝑗𝑎 + 𝑥𝑖𝑗
𝑏𝑎𝑄𝑖𝑗𝑎 + 𝑟𝑖𝑗
𝑏𝑏𝑃𝑖𝑗𝑏 + 𝑥𝑖𝑗
𝑏𝑏𝑄𝑖𝑗𝑏 + 𝑟𝑖𝑗
𝑏𝑐𝑃𝑖𝑗𝑐 + 𝑥𝑖𝑗
𝑏𝑐𝑄𝑖𝑗𝑐 )
𝑣𝑗𝑐 = 𝑣𝑖
𝑐 − 2(𝑟𝑖𝑗𝑐𝑎𝑃𝑖𝑗𝑎 + 𝑥𝑖𝑗
𝑐𝑎𝑄𝑖𝑗𝑎 + 𝑟𝑖𝑗
𝑐𝑏𝑃𝑖𝑗𝑏 + 𝑥𝑖𝑗
𝑐𝑏𝑄𝑖𝑗𝑏 + 𝑟𝑖𝑗
𝑐𝑐𝑃𝑖𝑗𝑐 + 𝑥𝑖𝑗
𝑐𝑐𝑄𝑖𝑗𝑐 )
g(q)iks − 𝑙(𝑞)𝑖
𝑘𝑠 − 𝑗∈𝑁 𝑓(𝑞)𝑖𝑗𝑘𝑠 = 0
𝑣−𝑘 ≤ 𝑣𝑘 ≤ 𝑣+
𝑘
Ohm’s Law approximation
![Page 37: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/37.jpg)
Los Alamos National Laboratory
4/14/2016 | 37
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Optimization Model: LinDist Flow
𝑣𝑗𝑎 = 𝑣𝑖
𝑎 − 2(𝑟𝑖𝑗𝑎𝑎𝑃𝑖𝑗𝑎 + 𝑥𝑖𝑗
𝑎𝑎𝑄𝑖𝑗𝑎 + 𝑟𝑖𝑗
𝑎𝑏𝑃𝑖𝑗𝑏 + 𝑥𝑖𝑗
𝑎𝑏𝑄𝑖𝑗𝑏 + 𝑟𝑖𝑗
𝑎𝑐𝑃𝑖𝑗𝑐 + 𝑥𝑖𝑗
𝑎𝑐𝑄𝑖𝑗𝑐 )
𝑣𝑗𝑏 = 𝑣𝑖
𝑏 − 2(𝑟𝑖𝑗𝑏𝑎𝑃𝑖𝑗𝑎 + 𝑥𝑖𝑗
𝑏𝑎𝑄𝑖𝑗𝑎 + 𝑟𝑖𝑗
𝑏𝑏𝑃𝑖𝑗𝑏 + 𝑥𝑖𝑗
𝑏𝑏𝑄𝑖𝑗𝑏 + 𝑟𝑖𝑗
𝑏𝑐𝑃𝑖𝑗𝑐 + 𝑥𝑖𝑗
𝑏𝑐𝑄𝑖𝑗𝑐 )
𝑣𝑗𝑐 = 𝑣𝑖
𝑐 − 2(𝑟𝑖𝑗𝑐𝑎𝑃𝑖𝑗𝑎 + 𝑥𝑖𝑗
𝑐𝑎𝑄𝑖𝑗𝑎 + 𝑟𝑖𝑗
𝑐𝑏𝑃𝑖𝑗𝑏 + 𝑥𝑖𝑗
𝑐𝑏𝑄𝑖𝑗𝑏 + 𝑟𝑖𝑗
𝑐𝑐𝑃𝑖𝑗𝑐 + 𝑥𝑖𝑗
𝑐𝑐𝑄𝑖𝑗𝑐 )
g(q)iks − 𝑙(𝑞)𝑖
𝑘𝑠 − 𝑗∈𝑁 𝑓(𝑞)𝑖𝑗𝑘𝑠 = 0
𝑣−𝑘 ≤ 𝑣𝑘 ≤ 𝑣+
𝑘
Reactive power conservation
![Page 38: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/38.jpg)
Los Alamos National Laboratory
4/14/2016 | 38
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Optimization Model: LinDist Flow
𝑣𝑗𝑎 = 𝑣𝑖
𝑎 − 2(𝑟𝑖𝑗𝑎𝑎𝑃𝑖𝑗𝑎 + 𝑥𝑖𝑗
𝑎𝑎𝑄𝑖𝑗𝑎 + 𝑟𝑖𝑗
𝑎𝑏𝑃𝑖𝑗𝑏 + 𝑥𝑖𝑗
𝑎𝑏𝑄𝑖𝑗𝑏 + 𝑟𝑖𝑗
𝑎𝑐𝑃𝑖𝑗𝑐 + 𝑥𝑖𝑗
𝑎𝑐𝑄𝑖𝑗𝑐 )
𝑣𝑗𝑏 = 𝑣𝑖
𝑏 − 2(𝑟𝑖𝑗𝑏𝑎𝑃𝑖𝑗𝑎 + 𝑥𝑖𝑗
𝑏𝑎𝑄𝑖𝑗𝑎 + 𝑟𝑖𝑗
𝑏𝑏𝑃𝑖𝑗𝑏 + 𝑥𝑖𝑗
𝑏𝑏𝑄𝑖𝑗𝑏 + 𝑟𝑖𝑗
𝑏𝑐𝑃𝑖𝑗𝑐 + 𝑥𝑖𝑗
𝑏𝑐𝑄𝑖𝑗𝑐 )
𝑣𝑗𝑐 = 𝑣𝑖
𝑐 − 2(𝑟𝑖𝑗𝑐𝑎𝑃𝑖𝑗𝑎 + 𝑥𝑖𝑗
𝑐𝑎𝑄𝑖𝑗𝑎 + 𝑟𝑖𝑗
𝑐𝑏𝑃𝑖𝑗𝑏 + 𝑥𝑖𝑗
𝑐𝑏𝑄𝑖𝑗𝑏 + 𝑟𝑖𝑗
𝑐𝑐𝑃𝑖𝑗𝑐 + 𝑥𝑖𝑗
𝑐𝑐𝑄𝑖𝑗𝑐 )
g(q)iks − 𝑙(𝑞)𝑖
𝑘𝑠 − 𝑗∈𝑁 𝑓(𝑞)𝑖𝑗𝑘𝑠 = 0
𝑣−𝑘 ≤ 𝑣𝑘 ≤ 𝑣+
𝑘
Voltage limits
![Page 39: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/39.jpg)
Los Alamos National Laboratory
4/14/2016 | 39
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Algorithm Overview
• Exact Algorithms
• CPLEX 12.6
• Scenario-based Decomposition
• Heuristics
• Greedy
• Union of single scenario solutions
• Based on industry algorithms
• Variable Neighborhood Search
• Ruin and Recreate—hybrid of exact methods and local search
• Iteratively relax variable assignments (ruin)
• Use exact method to find optimal variable assignments for relaxed variables,
given the fixed partial solution (recreate)
![Page 40: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/40.jpg)
Los Alamos National Laboratory
4/14/2016 | 40
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Scenario Based Decomposition (Review)
𝑅𝑒𝑠𝑖𝑙𝑖𝑒𝑛𝑡𝐷𝑒𝑠𝑖𝑔𝑛 𝑆
𝑠 ← 𝑐ℎ𝑜𝑜𝑠𝑒𝑆𝑐𝑒𝑛𝑎𝑟𝑖𝑜 𝑆
𝜎 → 𝑠𝑜𝑙𝑣𝑒𝑀𝐼𝑃 𝑠
𝑤ℎ𝑖𝑙𝑒 ~𝐹𝑒𝑎𝑠𝑖𝑏𝑙𝑒 𝜎, 𝑆\s
𝑠 → 𝑠 ∪ 𝑐ℎ𝑜𝑜𝑠𝑒𝑆𝑐𝑒𝑛𝑎𝑟𝑖𝑜 𝑆\s
𝜎 → 𝑠𝑜𝑙𝑣𝑒𝑀𝐼𝑃 𝑠
Solve over all
damage scenarios
Select 1 scenario
Design network for
damage scenario 1
Is solution feasible for
remaining scenarios
If NOT, add an infeasible
scenario to the set under
consideration
Find a new solution
Iterate until solution is
feasible for all scenarios
Outperformed other decomposition
strategies—second stage influences
feasibility, not optimality. Continuous
investment variables also adds
difficulty
![Page 41: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/41.jpg)
Los Alamos National Laboratory
4/14/2016 | 41
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Variable Neighborhood Search𝑅𝑒𝑠𝑖𝑙𝑖𝑒𝑛𝑡𝐷𝑒𝑠𝑖𝑔𝑛 𝑆,𝑚𝑎𝑥𝑇𝑖𝑚𝑒,𝑚𝑎𝑥𝑅𝑒𝑠𝑡𝑎𝑟𝑡𝑠,𝑚𝑎𝑥𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠
𝜎𝐿𝑃 ← 𝑆𝑜𝑙𝑣𝑒 𝑃𝐿𝑃 , 𝜎∗ ← 𝜎′, 𝑟𝑒𝑠𝑡𝑎𝑟𝑡 ← 𝑓𝑎𝑙𝑠𝑒
while 𝑡 < 𝑚𝑎𝑥𝑇𝑖𝑚𝑒 and 𝑖 < 𝑚𝑎𝑥𝑅𝑒𝑠𝑡𝑎𝑟𝑡𝑠
𝑗 ← 0
𝑛 ← 𝑥 ∈ 𝑋 ∶ 𝜎∗ 𝑥 − 𝜎𝐿𝑃 𝑥 ≠ 0𝐽 ← 𝜋1, 𝜋2, … 𝜋 𝐽 ∈ 𝑋 ∶ 𝜎
∗ 𝜋𝑖 − 𝜎𝐿𝑃 𝜋𝑖 ≤ 𝜎
∗ 𝜋𝑖+1 − 𝜎𝐿𝑃 𝜋𝑖+1
if (𝑟𝑒𝑠𝑡𝑎𝑟𝑡 = 𝑡𝑟𝑢𝑒)𝑖 ← 𝑖 + 1
𝑠𝑡𝑒𝑝 ←4𝑛
𝑑, 𝑘 ← 𝑋 − 𝑠𝑡𝑒𝑝
𝑠ℎ𝑢𝑓𝑓𝑙𝑒 𝐽
else
𝑠𝑡𝑒𝑝 ←𝑛
𝑑, 𝑘 ← 𝑋 − 𝑠𝑡𝑒𝑝
while 𝑡 < 𝑚𝑎𝑥𝑇𝑖𝑚𝑒 𝑎𝑛𝑑 𝑗 ≤ 𝑚𝑎𝑥𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠𝜎′← 𝑆𝑜𝑙𝑣𝑒(𝑃(𝜎∗, 𝐽 1, … 𝑘 )
if 𝑓 𝜎′ < 𝑓 𝜎∗
𝜎∗← 𝜎′, 𝑖 ← 0, 𝑟𝑒𝑠𝑡𝑎𝑟𝑡 ← 𝑓𝑎𝑙𝑠𝑒, 𝑗 ← 𝑚𝑎𝑥𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠
else
𝑗 ← 𝑗 + 1, 𝑘 ← 𝑘 −𝑠𝑡𝑒𝑝
2return 𝜎∗
Solve the LP relaxation
Intuition: LP relaxation guides the
search procedure
![Page 42: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/42.jpg)
Los Alamos National Laboratory
4/14/2016 | 42
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Variable Neighborhood Search𝑅𝑒𝑠𝑖𝑙𝑖𝑒𝑛𝑡𝐷𝑒𝑠𝑖𝑔𝑛 𝑆,𝑚𝑎𝑥𝑇𝑖𝑚𝑒,𝑚𝑎𝑥𝑅𝑒𝑠𝑡𝑎𝑟𝑡𝑠,𝑚𝑎𝑥𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠
𝜎𝐿𝑃 ← 𝑆𝑜𝑙𝑣𝑒 𝑃𝐿𝑃 , 𝜎∗ ← 𝜎′, 𝑟𝑒𝑠𝑡𝑎𝑟𝑡 ← 𝑓𝑎𝑙𝑠𝑒
while 𝑡 < 𝑚𝑎𝑥𝑇𝑖𝑚𝑒 and 𝑖 < 𝑚𝑎𝑥𝑅𝑒𝑠𝑡𝑎𝑟𝑡𝑠
𝑗 ← 0
𝑛 ← 𝑥 ∈ 𝑋 ∶ 𝜎∗ 𝑥 − 𝜎𝐿𝑃 𝑥 ≠ 0𝐽 ← 𝜋1, 𝜋2, … 𝜋 𝐽 ∈ 𝑋 ∶ 𝜎
∗ 𝜋𝑖 − 𝜎𝐿𝑃 𝜋𝑖 ≤ 𝜎
∗ 𝜋𝑖+1 − 𝜎𝐿𝑃 𝜋𝑖+1
if (𝑟𝑒𝑠𝑡𝑎𝑟𝑡 = 𝑡𝑟𝑢𝑒)𝑖 ← 𝑖 + 1
𝑠𝑡𝑒𝑝 ←4𝑛
𝑑, 𝑘 ← 𝑋 − 𝑠𝑡𝑒𝑝
𝑠ℎ𝑢𝑓𝑓𝑙𝑒 𝐽
else
𝑠𝑡𝑒𝑝 ←𝑛
𝑑, 𝑘 ← 𝑋 − 𝑠𝑡𝑒𝑝
while 𝑡 < 𝑚𝑎𝑥𝑇𝑖𝑚𝑒 𝑎𝑛𝑑 𝑗 ≤ 𝑚𝑎𝑥𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠𝜎′← 𝑆𝑜𝑙𝑣𝑒(𝑃(𝜎∗, 𝐽 1, … 𝑘 )
if 𝑓 𝜎′ < 𝑓 𝜎∗
𝜎∗← 𝜎′, 𝑖 ← 0, 𝑟𝑒𝑠𝑡𝑎𝑟𝑡 ← 𝑓𝑎𝑙𝑠𝑒, 𝑗 ← 𝑚𝑎𝑥𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠
else
𝑗 ← 𝑗 + 1, 𝑘 ← 𝑘 −𝑠𝑡𝑒𝑝
2return 𝜎∗
Count differences between
current best solution and
relaxation
Intuition: n is a parameter used to
control the size of the neighborhood.
Larger differences between the LP
relaxation and the incumbent solution
indicate that a larger neighborhood
should be considered.
![Page 43: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/43.jpg)
Los Alamos National Laboratory
4/14/2016 | 43
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Variable Neighborhood Search𝑅𝑒𝑠𝑖𝑙𝑖𝑒𝑛𝑡𝐷𝑒𝑠𝑖𝑔𝑛 𝑆,𝑚𝑎𝑥𝑇𝑖𝑚𝑒,𝑚𝑎𝑥𝑅𝑒𝑠𝑡𝑎𝑟𝑡𝑠,𝑚𝑎𝑥𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠
𝜎𝐿𝑃 ← 𝑆𝑜𝑙𝑣𝑒 𝑃𝐿𝑃 , 𝜎∗ ← 𝜎′, 𝑟𝑒𝑠𝑡𝑎𝑟𝑡 ← 𝑓𝑎𝑙𝑠𝑒
while 𝑡 < 𝑚𝑎𝑥𝑇𝑖𝑚𝑒 and 𝑖 < 𝑚𝑎𝑥𝑅𝑒𝑠𝑡𝑎𝑟𝑡𝑠
𝑗 ← 0
𝑛 ← 𝑥 ∈ 𝑋 ∶ 𝜎∗ 𝑥 − 𝜎𝐿𝑃 𝑥 ≠ 0𝐽 ← 𝜋1, 𝜋2, … 𝜋 𝐽 ∈ 𝑋 ∶ 𝜎
∗ 𝜋𝑖 − 𝜎𝐿𝑃 𝜋𝑖 ≤ 𝜎
∗ 𝜋𝑖+1 − 𝜎𝐿𝑃 𝜋𝑖+1
if (𝑟𝑒𝑠𝑡𝑎𝑟𝑡 = 𝑡𝑟𝑢𝑒)𝑖 ← 𝑖 + 1
𝑠𝑡𝑒𝑝 ←4𝑛
𝑑, 𝑘 ← 𝑋 − 𝑠𝑡𝑒𝑝
𝑠ℎ𝑢𝑓𝑓𝑙𝑒 𝐽
else
𝑠𝑡𝑒𝑝 ←𝑛
𝑑, 𝑘 ← 𝑋 − 𝑠𝑡𝑒𝑝
while 𝑡 < 𝑚𝑎𝑥𝑇𝑖𝑚𝑒 𝑎𝑛𝑑 𝑗 ≤ 𝑚𝑎𝑥𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠𝜎′← 𝑆𝑜𝑙𝑣𝑒(𝑃(𝜎∗, 𝐽 1, … 𝑘 )
if 𝑓 𝜎′ < 𝑓 𝜎∗
𝜎∗← 𝜎′, 𝑖 ← 0, 𝑟𝑒𝑠𝑡𝑎𝑟𝑡 ← 𝑓𝑎𝑙𝑠𝑒, 𝑗 ← 𝑚𝑎𝑥𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠
else
𝑗 ← 𝑗 + 1, 𝑘 ← 𝑘 −𝑠𝑡𝑒𝑝
2return 𝜎∗
Order variables by difference
from LP relaxation
Intuition: Variables whose assignments
differ from the LP relaxation have more
potential to improve the incumbent
solution.
![Page 44: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/44.jpg)
Los Alamos National Laboratory
4/14/2016 | 44
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Variable Neighborhood Search𝑅𝑒𝑠𝑖𝑙𝑖𝑒𝑛𝑡𝐷𝑒𝑠𝑖𝑔𝑛 𝑆,𝑚𝑎𝑥𝑇𝑖𝑚𝑒,𝑚𝑎𝑥𝑅𝑒𝑠𝑡𝑎𝑟𝑡𝑠,𝑚𝑎𝑥𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠
𝜎𝐿𝑃 ← 𝑆𝑜𝑙𝑣𝑒 𝑃𝐿𝑃 , 𝜎∗ ← 𝜎′, 𝑟𝑒𝑠𝑡𝑎𝑟𝑡 ← 𝑓𝑎𝑙𝑠𝑒
while 𝑡 < 𝑚𝑎𝑥𝑇𝑖𝑚𝑒 and 𝑖 < 𝑚𝑎𝑥𝑅𝑒𝑠𝑡𝑎𝑟𝑡𝑠
𝑗 ← 0
𝑛 ← 𝑥 ∈ 𝑋 ∶ 𝜎∗ 𝑥 − 𝜎𝐿𝑃 𝑥 ≠ 0𝐽 ← 𝜋1, 𝜋2, … 𝜋 𝐽 ∈ 𝑋 ∶ 𝜎
∗ 𝜋𝑖 − 𝜎𝐿𝑃 𝜋𝑖 ≤ 𝜎
∗ 𝜋𝑖+1 − 𝜎𝐿𝑃 𝜋𝑖+1
if (𝑟𝑒𝑠𝑡𝑎𝑟𝑡 = 𝑡𝑟𝑢𝑒)𝑖 ← 𝑖 + 1
𝑠𝑡𝑒𝑝 ←4𝑛
𝑑, 𝑘 ← 𝑋 − 𝑠𝑡𝑒𝑝
𝑠ℎ𝑢𝑓𝑓𝑙𝑒 𝐽
else
𝑠𝑡𝑒𝑝 ←𝑛
𝑑, 𝑘 ← 𝑋 − 𝑠𝑡𝑒𝑝
while 𝑡 < 𝑚𝑎𝑥𝑇𝑖𝑚𝑒 𝑎𝑛𝑑 𝑗 ≤ 𝑚𝑎𝑥𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠𝜎′← 𝑆𝑜𝑙𝑣𝑒(𝑃(𝜎∗, 𝐽 1, … 𝑘 )
if 𝑓 𝜎′ < 𝑓 𝜎∗
𝜎∗← 𝜎′, 𝑖 ← 0, 𝑟𝑒𝑠𝑡𝑎𝑟𝑡 ← 𝑓𝑎𝑙𝑠𝑒, 𝑗 ← 𝑚𝑎𝑥𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠
else
𝑗 ← 𝑗 + 1, 𝑘 ← 𝑘 −𝑠𝑡𝑒𝑝
2return 𝜎∗
Compute best solution in
neighborhood J(1…k)
Intuition: Ruin and recreate
![Page 45: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/45.jpg)
Los Alamos National Laboratory
4/14/2016 | 45
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Variable Neighborhood Search𝑅𝑒𝑠𝑖𝑙𝑖𝑒𝑛𝑡𝐷𝑒𝑠𝑖𝑔𝑛 𝑆,𝑚𝑎𝑥𝑇𝑖𝑚𝑒,𝑚𝑎𝑥𝑅𝑒𝑠𝑡𝑎𝑟𝑡𝑠,𝑚𝑎𝑥𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠
𝜎𝐿𝑃 ← 𝑆𝑜𝑙𝑣𝑒 𝑃𝐿𝑃 , 𝜎∗ ← 𝜎′, 𝑟𝑒𝑠𝑡𝑎𝑟𝑡 ← 𝑓𝑎𝑙𝑠𝑒
while 𝑡 < 𝑚𝑎𝑥𝑇𝑖𝑚𝑒 and 𝑖 < 𝑚𝑎𝑥𝑅𝑒𝑠𝑡𝑎𝑟𝑡𝑠
𝑗 ← 0
𝑛 ← 𝑥 ∈ 𝑋 ∶ 𝜎∗ 𝑥 − 𝜎𝐿𝑃 𝑥 ≠ 0𝐽 ← 𝜋1, 𝜋2, … 𝜋 𝐽 ∈ 𝑋 ∶ 𝜎
∗ 𝜋𝑖 − 𝜎𝐿𝑃 𝜋𝑖 ≤ 𝜎
∗ 𝜋𝑖+1 − 𝜎𝐿𝑃 𝜋𝑖+1
if (𝑟𝑒𝑠𝑡𝑎𝑟𝑡 = 𝑡𝑟𝑢𝑒)𝑖 ← 𝑖 + 1
𝑠𝑡𝑒𝑝 ←4𝑛
𝑑, 𝑘 ← 𝑋 − 𝑠𝑡𝑒𝑝
𝑠ℎ𝑢𝑓𝑓𝑙𝑒 𝐽
else
𝑠𝑡𝑒𝑝 ←𝑛
𝑑, 𝑘 ← 𝑋 − 𝑠𝑡𝑒𝑝
while 𝑡 < 𝑚𝑎𝑥𝑇𝑖𝑚𝑒 𝑎𝑛𝑑 𝑗 ≤ 𝑚𝑎𝑥𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠𝜎′← 𝑆𝑜𝑙𝑣𝑒(𝑃(𝜎∗, 𝐽 1, … 𝑘 )
if 𝑓 𝜎′ < 𝑓 𝜎∗
𝜎∗← 𝜎′, 𝑖 ← 0, 𝑟𝑒𝑠𝑡𝑎𝑟𝑡 ← 𝑓𝑎𝑙𝑠𝑒, 𝑗 ← 𝑚𝑎𝑥𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠
else
𝑗 ← 𝑗 + 1, 𝑘 ← 𝑘 −𝑠𝑡𝑒𝑝
2return 𝜎∗ Update best solution
![Page 46: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/46.jpg)
Los Alamos National Laboratory
4/14/2016 | 46
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Variable Neighborhood Search𝑅𝑒𝑠𝑖𝑙𝑖𝑒𝑛𝑡𝐷𝑒𝑠𝑖𝑔𝑛 𝑆,𝑚𝑎𝑥𝑇𝑖𝑚𝑒,𝑚𝑎𝑥𝑅𝑒𝑠𝑡𝑎𝑟𝑡𝑠,𝑚𝑎𝑥𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠
𝜎𝐿𝑃 ← 𝑆𝑜𝑙𝑣𝑒 𝑃𝐿𝑃 , 𝜎∗ ← 𝜎′, 𝑟𝑒𝑠𝑡𝑎𝑟𝑡 ← 𝑓𝑎𝑙𝑠𝑒
while 𝑡 < 𝑚𝑎𝑥𝑇𝑖𝑚𝑒 and 𝑖 < 𝑚𝑎𝑥𝑅𝑒𝑠𝑡𝑎𝑟𝑡𝑠
𝑗 ← 0
𝑛 ← 𝑥 ∈ 𝑋 ∶ 𝜎∗ 𝑥 − 𝜎𝐿𝑃 𝑥 ≠ 0𝐽 ← 𝜋1, 𝜋2, … 𝜋 𝐽 ∈ 𝑋 ∶ 𝜎
∗ 𝜋𝑖 − 𝜎𝐿𝑃 𝜋𝑖 ≤ 𝜎
∗ 𝜋𝑖+1 − 𝜎𝐿𝑃 𝜋𝑖+1
if (𝑟𝑒𝑠𝑡𝑎𝑟𝑡 = 𝑡𝑟𝑢𝑒)𝑖 ← 𝑖 + 1
𝑠𝑡𝑒𝑝 ←4𝑛
𝑑, 𝑘 ← 𝑋 − 𝑠𝑡𝑒𝑝
𝑠ℎ𝑢𝑓𝑓𝑙𝑒 𝐽
else
𝑠𝑡𝑒𝑝 ←𝑛
𝑑, 𝑘 ← 𝑋 − 𝑠𝑡𝑒𝑝
while 𝑡 < 𝑚𝑎𝑥𝑇𝑖𝑚𝑒 𝑎𝑛𝑑 𝑗 ≤ 𝑚𝑎𝑥𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠𝜎′← 𝑆𝑜𝑙𝑣𝑒(𝑃(𝜎∗, 𝐽 1, … 𝑘 )
if 𝑓 𝜎′ < 𝑓 𝜎∗
𝜎∗← 𝜎′, 𝑖 ← 0, 𝑟𝑒𝑠𝑡𝑎𝑟𝑡 ← 𝑓𝑎𝑙𝑠𝑒, 𝑗 ← 𝑚𝑎𝑥𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠
else
𝑗 ← 𝑗 + 1, 𝑘 ← 𝑘 −𝑠𝑡𝑒𝑝
2return 𝜎∗ Increase neighborhood size
Intuition: When a better solution is not
found, increase the size of the
neighborhood
![Page 47: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/47.jpg)
Los Alamos National Laboratory
4/14/2016 | 47
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Variable Neighborhood Search𝑅𝑒𝑠𝑖𝑙𝑖𝑒𝑛𝑡𝐷𝑒𝑠𝑖𝑔𝑛 𝑆,𝑚𝑎𝑥𝑇𝑖𝑚𝑒,𝑚𝑎𝑥𝑅𝑒𝑠𝑡𝑎𝑟𝑡𝑠,𝑚𝑎𝑥𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠
𝜎𝐿𝑃 ← 𝑆𝑜𝑙𝑣𝑒 𝑃𝐿𝑃 , 𝜎∗ ← 𝜎′, 𝑟𝑒𝑠𝑡𝑎𝑟𝑡 ← 𝑓𝑎𝑙𝑠𝑒
while 𝑡 < 𝑚𝑎𝑥𝑇𝑖𝑚𝑒 and 𝑖 < 𝑚𝑎𝑥𝑅𝑒𝑠𝑡𝑎𝑟𝑡𝑠
𝑗 ← 0
𝑛 ← 𝑥 ∈ 𝑋 ∶ 𝜎∗ 𝑥 − 𝜎𝐿𝑃 𝑥 ≠ 0𝐽 ← 𝜋1, 𝜋2, … 𝜋 𝐽 ∈ 𝑋 ∶ 𝜎
∗ 𝜋𝑖 − 𝜎𝐿𝑃 𝜋𝑖 ≤ 𝜎
∗ 𝜋𝑖+1 − 𝜎𝐿𝑃 𝜋𝑖+1
if (𝑟𝑒𝑠𝑡𝑎𝑟𝑡 = 𝑡𝑟𝑢𝑒)𝑖 ← 𝑖 + 1
𝑠𝑡𝑒𝑝 ←4𝑛
𝑑, 𝑘 ← 𝑋 − 𝑠𝑡𝑒𝑝
𝑠ℎ𝑢𝑓𝑓𝑙𝑒 𝐽
else
𝑠𝑡𝑒𝑝 ←𝑛
𝑑, 𝑘 ← 𝑋 − 𝑠𝑡𝑒𝑝
while 𝑡 < 𝑚𝑎𝑥𝑇𝑖𝑚𝑒 𝑎𝑛𝑑 𝑗 ≤ 𝑚𝑎𝑥𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠𝜎′← 𝑆𝑜𝑙𝑣𝑒(𝑃(𝜎∗, 𝐽 1, … 𝑘 )
if 𝑓 𝜎′ < 𝑓 𝜎∗
𝜎∗← 𝜎′, 𝑖 ← 0, 𝑟𝑒𝑠𝑡𝑎𝑟𝑡 ← 𝑓𝑎𝑙𝑠𝑒, 𝑗 ← 𝑚𝑎𝑥𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠
else
𝑗 ← 𝑗 + 1, 𝑘 ← 𝑘 −𝑠𝑡𝑒𝑝
2return 𝜎∗
Shuffle ordering after restart
Intuition: Consider different sets of
variables to relax
![Page 48: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/48.jpg)
Los Alamos National Laboratory
4/14/2016 | 48
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Variable Neighborhood Search𝑅𝑒𝑠𝑖𝑙𝑖𝑒𝑛𝑡𝐷𝑒𝑠𝑖𝑔𝑛 𝑆,𝑚𝑎𝑥𝑇𝑖𝑚𝑒,𝑚𝑎𝑥𝑅𝑒𝑠𝑡𝑎𝑟𝑡𝑠,𝑚𝑎𝑥𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠
𝜎𝐿𝑃 ← 𝑆𝑜𝑙𝑣𝑒 𝑃𝐿𝑃 , 𝜎∗ ← 𝜎′, 𝑟𝑒𝑠𝑡𝑎𝑟𝑡 ← 𝑓𝑎𝑙𝑠𝑒
while 𝑡 < 𝑚𝑎𝑥𝑇𝑖𝑚𝑒 and 𝑖 < 𝑚𝑎𝑥𝑅𝑒𝑠𝑡𝑎𝑟𝑡𝑠
𝑗 ← 0
𝑛 ← 𝑥 ∈ 𝑋 ∶ 𝜎∗ 𝑥 − 𝜎𝐿𝑃 𝑥 ≠ 0𝐽 ← 𝜋1, 𝜋2, … 𝜋 𝐽 ∈ 𝑋 ∶ 𝜎
∗ 𝜋𝑖 − 𝜎𝐿𝑃 𝜋𝑖 ≤ 𝜎
∗ 𝜋𝑖+1 − 𝜎𝐿𝑃 𝜋𝑖+1
if (𝑟𝑒𝑠𝑡𝑎𝑟𝑡 = 𝑡𝑟𝑢𝑒)𝑖 ← 𝑖 + 1
𝑠𝑡𝑒𝑝 ←4𝑛
𝑑, 𝑘 ← 𝑋 − 𝑠𝑡𝑒𝑝
𝑠ℎ𝑢𝑓𝑓𝑙𝑒 𝐽
else
𝑠𝑡𝑒𝑝 ←𝑛
𝑑, 𝑘 ← 𝑋 − 𝑠𝑡𝑒𝑝
while 𝑡 < 𝑚𝑎𝑥𝑇𝑖𝑚𝑒 𝑎𝑛𝑑 𝑗 ≤ 𝑚𝑎𝑥𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠𝜎′← 𝑆𝑜𝑙𝑣𝑒(𝑃(𝜎∗, 𝐽 1, … 𝑘 )
if 𝑓 𝜎′ < 𝑓 𝜎∗
𝜎∗← 𝜎′, 𝑖 ← 0, 𝑟𝑒𝑠𝑡𝑎𝑟𝑡 ← 𝑓𝑎𝑙𝑠𝑒, 𝑗 ← 𝑚𝑎𝑥𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠
else
𝑗 ← 𝑗 + 1, 𝑘 ← 𝑘 −𝑠𝑡𝑒𝑝
2return 𝜎∗
Adjust neighborhood
parameters
Intuition: Neighborhood size is based
on differences between LP relaxation
and incumbent solution.
![Page 49: UNCLASSIFIED Resilient Distribution Tool (RDT) · 2016. 5. 5. · Arthur K. Barnes Harsha Natarajan. Los Alamos National Laboratory 4/14/2016 | 2 UNCLASSIFIED NOTE standard slide](https://reader037.vdocuments.us/reader037/viewer/2022090811/611c7cc38d074c08b913e763/html5/thumbnails/49.jpg)
Los Alamos National Laboratory
4/14/2016 | 49
UNCLASSIFIED NOTE
standard slide
layout with large,
open, white
space. Try to keep
your slides simple,
and stick to one
thought per slide.
Case Study 1 ResultsDamage & Resilience
Requirements
20% Damage prob.
50% /
98%
Normal / Critical
load served
500
kW
Generator
increments
Computation
3-5 min. Greedy search
30 min. Exact search
20 Disaster
scenarios
Results (Exact)
40 Lines
hardened
0 Lines built
1 MW / 1 Total gen.
power / # gen.
$17M Capex
Computation
3-5 min. Greedy search
3 h Exact search
20 Disaster
scenarios
Results (Exact)
40 Lines
hardened
0 Lines built
4 MW / 7 Total gen.
power / # gen.
$26M Capex
Damage & Resilience
Requirements
20% Damage prob.
60% /
98%
Normal / Critical
load served
500
kW
Generator
increments