Collaborative Optimization of Post-Disaster Damage Repair and Power System Operation
Abstract
:1. Introduction
2. Model Description
3. Mathematical Model
- All repair teams are capable of repairing any type of damage. Once a damaged component is repaired, repair team will leave for the next one immediately, until all the tasks are completed.
- The repair time and resource to fix a fault is known and certain; the vehicle speed of repair teams is fixed.
- During the process of damage repair and system restoration, no extra new equipment damage occurs.
- The repair expense of fault points and the outage unit restoration cost are fixed, which have nothing to do with the repair moment or repair teams.
- Generators are not damaged by disasters because they are often located indoors.
3.1. Objective Function
3.2. Constraints
3.2.1. Constraints of Damage Repair
- Constraints of damage repair routing
- Constraints of repair resources
- Constraints of damaged component states
3.2.2. Constraints of Power System Operation
- Constraints of power system safe and stable operation
- The restoration characteristic of generators
- Constraints of component operation states
3.2.3. Coupling Constraints
3.2.4. Transformation of Complex Nonlinear Constraint
4. Solution
4.1. Lagrange Relaxation of the MISOCP Model
- The damage repair routing problem
- The power system operation optimization problem
4.2. The Acceleration Strategy
4.3. The Algorithm Flow
- The original co-optimization model is transformed to the modified MISOCP model by linearizing some complex constraints.
- Set the initial values of Lagrange multipliers.
- The MISOCP model is decomposed into the upper sub problem of damage repair routing and the lower sub problem of power system operation optimization.
- Solve the two sub problems alternately in each iteration. If the LR convergence criterion is satisfied, output the result and end the calculation or else go to step 5.
- If the acceleration convergence condition is satisfied, implement the acceleration strategy and output the result. Otherwise, update the values of Lagrange multipliers and go to step 4.
5. Case Study
5.1. The Advantage Analysis of the Proposed Co-Optimization Model
5.2. The Effect Analysis of the Acceleration Algorithm
5.3. The Impacts of Damage Repair Resources Allocation and Adequacy on Restoration
5.4. The Impacts of Weight Factors in the Objective Function on Restoration
6. Conclusions
- The proposed model can support the formulation of reasonable damage repair scheme, the plan of unit output and the decisions of optimal transmission switching, to minimize the power outage loss with lower cost of damage repair and power system operation.
- Lagrange relaxation decomposes the original complex model into two small-scale sub problems and the acceleration strategy is implemented to realize the fast solution. For utilities, the work of maintenance department and system dispatching department could be separated and Lagrange multipliers help to coordinate their work. Consequently, the work efficiency will be improved.
- The adequacy and allocation of damage repair resource can greatly influence the restoration effect and the load loss level. The sufficient resource reserve of repair centers will significantly decrease loss in economy due to power outage.
- To reduce the power outage loss and realize the enhancement of post-disaster power system resilience, it is significant to highlight the fast restoration of outage loads according to their importance.
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Nomenclature
Indices and sets | |
g | Index for generation units |
i/j | Index for buses |
x/y/z | Index for damaged components |
c | Index for repair team |
t | Index for time |
b | Index for repair center (starting point) |
d | Index for repair center (return point) |
DC | Set of damaged components |
DB | Set of damaged buses |
DL | Set of damaged transmission lines |
Parameters | |
α | Weight factor |
ag | Generation cost quadric coefficient of unit g |
bg | Generation cost linear coefficient of unit g |
cg | Generation cost constant of unit g |
lpi,t/lqi,t | Active/reactive demand at bus i and time t under normal conditions |
eli | Economic loss of lost load at bus i per hour |
Gi/Bi | Conductance/susceptance from bus i to the ground |
Vimax/Vimin | Upper/lower limit of voltage magnitude at bus i |
θimax/θimin | Upper/lower limit of voltage angle at bus i |
Pgmax/Pgmin | Upper/lower limit of active power generation of unit g |
Qgmax/Qgmin | Upper/lower limit of reactive power generation of unit g |
Pgstart | Power required by unit g for start-up |
Rg | Ramp speed limit of unit g |
tsg | Start-up time of unit g |
tdg | The interval of unit g with positive start-up power requirement and zero ramping rate |
Sij | Upper limit of apparent power flow on line i→j |
Iijmax | Upper limit of current on line i→j |
rij/xij | Resistance/reactance of line i→j |
dx,y | Distance between damaged component x and y |
Vc | Average driving speed of team c |
Capc | Resource capacity of team c |
RESx | Repair resources required to fix damaged component x |
resd | Resource amount of repair center |
RTx,c | Repair time required to fix damaged component x |
Numc | Number of repair teams |
M | Value of big M |
T | Time horizon |
ωcrew | Wages of a repair team per hour |
ωroad | Driving cost of a repair team per km |
Variables | |
PGg,t/QGg,t | Active/reactive power generation of unit g at time t |
PLij,t/QLij,t | Active/reactive power flow on line i→j |
hij,t | Square of current magnitude on line i→j at time t |
LPi,t/LQi,t | Actual active/reactive load at bus i and time t |
Vi,t | Voltage magnitude at bus i and time t |
θi,t | Voltage phase at bus i and time t |
ALSij,t | Binary variable equals to 0 if line i→j is damaged or under repair and 1 else at time t |
LSij,t | Binary variable equals to 0 if line i→j is removed from the system and 1 else at time t |
BSi,t | Binary variable equals to 0 if bus i is damaged or under repair and 1 else at time t |
GSg,t | Binary variable equals to 1 if unit g is committed and 0 else at time t |
SSg,t | Binary variable equals to 1 if unit g is started up at time t and 0 else |
ATx,c | Arrival time of team c at damaged component x |
FTx,t | Binary variable equals to 1 if damaged component x is repaired at time t and 0 else |
Mx,y,c | Binary variable equals to 1 if team c moves from damaged component x to y and 0 else |
Nx,c | Binary variable equals to 1 if damaged component x is repaired by team c and 0 else |
Sx,t | Binary variable equals to 0 if damaged component x is damaged or under repair and 1 else at time t |
Appendix A
Numbers of Damaged Components | Bus Number | Line Number | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
3 | 14 | 52 | 53 | 14 | 17 | 29 | 32 | 40 | 70 | ||
Bus number | 3 | 0 | 345 | 840 | 870 | 375 | 450 | 60 | 120 | 735 | 975 |
14 | 345 | 0 | 915 | 900 | 45 | 135 | 225 | 195 | 1065 | 840 | |
52 | 840 | 915 | 0 | 90 | 885 | 1050 | 780 | 660 | 90 | 180 | |
53 | 870 | 900 | 90 | 0 | 855 | 1080 | 750 | 630 | 180 | 90 | |
Line number | 14 | 375 | 45 | 885 | 855 | 0 | 165 | 330 | 285 | 900 | 930 |
17 | 450 | 135 | 1050 | 1080 | 165 | 0 | 375 | 345 | 1050 | 1065 | |
29 | 60 | 225 | 780 | 750 | 330 | 375 | 0 | 90 | 750 | 870 | |
32 | 120 | 195 | 660 | 630 | 285 | 345 | 90 | 0 | 705 | 690 | |
40 | 735 | 1065 | 90 | 180 | 900 | 1050 | 750 | 705 | 0 | 270 | |
70 | 975 | 840 | 180 | 90 | 930 | 1065 | 870 | 690 | 270 | 0 |
Repair Center | Bus Number | Line Number | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
3 | 14 | 52 | 53 | 14 | 17 | 29 | 32 | 40 | 70 | |
1 | 780 | 885 | 90 | 82.5 | 810 | 1020 | 832.5 | 555 | 150 | 120 |
2 | 120 | 315 | 735 | 720 | 450 | 540 | 105 | 180 | 780 | 780 |
3 | 465 | 45 | 840 | 825 | 120 | 195 | 360 | 307.5 | 795 | 950 |
Damage Repair Requirements | Bus Number | Line Number | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
3 | 14 | 52 | 53 | 14 | 17 | 29 | 32 | 40 | 70 | |
Damage repair resource | 30 | 28 | 34 | 32 | 8 | 11 | 9 | 14 | 6 | 7 |
Damage repair time (h) | 12 | 13 | 12 | 14 | 8 | 7 | 9 | 10 | 6 | 8 |
Repair Center | Resource (Capability of Each Team) |
---|---|
1 | 85 (45; 45) |
2 | 60 (45; 45) |
3 | 60 (45; 45) |
Load Bus Number | Economic Loss of Lost Load ($/kWh) |
---|---|
1 | 0.110 |
2 | 0.110 |
3 | 0.110 |
4 | 3.816 |
5 | 0.110 |
6 | 10.000 |
8 | 0.110 |
9 | 0.110 |
10 | 0.110 |
12 | 0.110 |
13 | 0.110 |
14 | 0.110 |
15 | 0.110 |
16 | 6.979 |
17 | 0.110 |
18 | 0.110 |
19 | 0.110 |
20 | 3.816 |
23 | 0.110 |
25 | 10.000 |
27 | 3.816 |
28 | 0.110 |
29 | 0.110 |
30 | 0.110 |
31 | 0.110 |
32 | 6.979 |
33 | 0.110 |
35 | 0.110 |
38 | 0.110 |
41 | 6.979 |
42 | 0.110 |
43 | 0.110 |
44 | 0.110 |
47 | 0.110 |
49 | 0.110 |
50 | 0.110 |
51 | 0.110 |
52 | 3.816 |
53 | 0.110 |
54 | 0.110 |
55 | 3.816 |
56 | 0.110 |
57 | 0.110 |
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Repair Center | Bus Number | Line Number (Bus i–Bus j) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
3 | 14 | 52 | 53 | 14(13–15) | 17(1–17) | 29(18–19) | 32(21–22) | 40(28–29) | 70(54–55) | |
1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 |
2 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
3 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
Damaged Component | Bus Number | Line Number | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
3 | 14 | 52 | 53 | 14 | 17 | 29 | 32 | 40 | 70 | |||
Arrival moment (h) | Repair center 1 | Team 1 | - | - | 1.8 | - | - | - | - | - | - | 15.6 |
Team 2 | - | - | - | 12.2 | - | - | - | - | 2.4 | - | ||
Repair center 2 | Team 1 | - | - | - | - | - | - | 2.1 | 12.9 | - | - | |
Team 2 | 14.0 | - | - | - | - | - | - | - | - | - | ||
Repair center 3 | Team 1 | - | - | - | - | 14.2 | 3.9 | - | - | - | - | |
Team 2 | - | 0.9 | - | - | - | - | - | - | - | - |
Damaged Component | Bus Number | Line Number | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
3 | 14 | 52 | 53 | 14 | 17 | 29 | 32 | 40 | 70 | |||
Arrival moment (h) | Repair center 1 | Team 1 | - | - | - | 12.2 | - | - | - | - | - | 2.4 |
Team 2 | - | 1.8 | - | - | - | - | - | 15.6 | - | |||
Repair center 2 | Team 1 | - | - | - | - | - | - | 2.1 | 12.9 | - | - | |
Team 2 | 2.4 | - | - | - | - | - | - | - | - | - | ||
Repair center 3 | Team 1 | - | - | - | - | 2.4 | 13.7 | - | - | - | - | |
Team 2 | - | 0.9 | - | - | - | - | - | - | - | - |
Economic Index | Restoration with Co-Optimization | Restoration without Co-Optimization |
---|---|---|
Damage repair expense ($1000) | 53.32 | 47.90 |
System operation cost ($1000) | 1741.14 | 1750.21 |
Power outage loss ($1000) | 803.38 | 967.55 |
Model | The Original Model | The MISOCP Model | The MISOCP Model+ the Acceleration Algorithm |
---|---|---|---|
Computation time | Did not converge | 22.5 h | 4.6 h |
The objective function value | - | 9802.31 | 9828.24 |
Economic Index | The Original Resource Allocation Case | The Changed Resource Allocation Case |
---|---|---|
Damage repair expense ($1000) | 53.32 | 57.61 |
Power outage loss ($1000) | 803.38 | 865.62 |
α1 | α2 | α3 | Power Outage Loss ($1000) | System Operation Cost ($1000) | Damage Repair Expense ($1000) |
---|---|---|---|---|---|
1 | 1 | 10 | 803.38 | 1741.14 | 53.32 |
1 | 1 | 1 | 833.47 | 1695.92 | 49.25 |
10 | 1 | 1 | 1930.08 | 1361.93 | 54.79 |
1 | 10 | 1 | 883.23 | 1696.17 | 51.20 |
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Zhang, H.; Li, G.; Yuan, H. Collaborative Optimization of Post-Disaster Damage Repair and Power System Operation. Energies 2018, 11, 2611. https://doi.org/10.3390/en11102611
Zhang H, Li G, Yuan H. Collaborative Optimization of Post-Disaster Damage Repair and Power System Operation. Energies. 2018; 11(10):2611. https://doi.org/10.3390/en11102611
Chicago/Turabian StyleZhang, Han, Gengfeng Li, and Hanjie Yuan. 2018. "Collaborative Optimization of Post-Disaster Damage Repair and Power System Operation" Energies 11, no. 10: 2611. https://doi.org/10.3390/en11102611