a local relaxation approach for the siting of electrical substations
DESCRIPTION
A Local Relaxation Approach for the Siting of Electrical Substations. Walter Murray and Uday Shanbhag Systems Optimization Laboratory Department of Management Science and Engineering Stanford University, CA 94305. SSO - Review. Service area. Washington State. SSO - Review. Colour: - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/1.jpg)
A Local Relaxation Approach for the Siting of Electrical
Substations
Walter Murray and Uday Shanbhag
Systems Optimization Laboratory
Department of Management Science and Engineering
Stanford University, CA 94305
![Page 2: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/2.jpg)
![Page 3: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/3.jpg)
![Page 4: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/4.jpg)
![Page 5: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/5.jpg)
![Page 6: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/6.jpg)
![Page 7: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/7.jpg)
![Page 8: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/8.jpg)
SSO - ReviewSSO - ReviewService area
Washington State
![Page 9: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/9.jpg)
SSO - ReviewSSO - ReviewColour:
•Black –substation
•Other –Kw Load
Service area: each grid block is 1/2 mile by 1/2 mile
![Page 10: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/10.jpg)
SSO - ReviewSSO - Review“Model distribution lines and
substation locations and– Determine the optimal substation
capacity additions To serve a known load at a minimum
cost”
Service area: each grid block is 1/2 mile by 1/2 mile
![Page 11: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/11.jpg)
SSO - ReviewSSO - Review
More substations:Higher capital costLower transmission cost
Characteristics:
Capital costs:$4,000,000 for a 28 MW substation
Cost of losses:$3,000 per kw of losses
Service area: each grid block is 1/2 mile by 1/2 mile
![Page 12: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/12.jpg)
VariablesVariables
![Page 13: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/13.jpg)
Problem of InterestProblem of Interest
![Page 14: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/14.jpg)
Admittance MatrixAdmittance Matrix
![Page 15: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/15.jpg)
A Multiscale ProblemA Multiscale Problem
![Page 16: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/16.jpg)
SSO AlgorithmSSO AlgorithmDETERMINE INITIAL DISCRETE
FEASIBLE SOLUTION
INITIAL NUMBER OF SS
DETERMINE SEARCH DIRECTION
DETERMINE SEARCH STEP TO GET IMPROVED SOLN
FINAL NUMBER AND POSITIONS OF SUBSTATIONS
WHILE # OF SSNOT CONVERGED
ADJUST # OF SS
WHILE IMPROVED SOLUTIONCAN BE FOUND
UPDATE POSITIONS OF SS
![Page 17: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/17.jpg)
Finding an Initial Feasible SolutionFinding an Initial Feasible SolutionGlobal RelaxationGlobal Relaxation
Continuous relaxation
ModifiedObjective
![Page 18: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/18.jpg)
Finding an Initial Feasible SolutionFinding an Initial Feasible SolutionGlobal RelaxationGlobal Relaxation
![Page 19: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/19.jpg)
Search DirectionSearch Direction
SubstationPositions
CandidatePositions
Good Neighbor
19 K
![Page 20: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/20.jpg)
Search DirectionSearch DirectionLocal RelaxationLocal Relaxation
QP Subproblem
![Page 21: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/21.jpg)
Center of Gravity
Center of Gravity
Search StepSearch StepCenter of GravityCenter of Gravity
![Page 22: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/22.jpg)
Optimal Number of SubstationsOptimal Number of Substations
![Page 23: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/23.jpg)
Sample Load DistributionsSample Load Distributions
Gaussian DistributionGaussian Distribution Snohomish PUD DistributionSnohomish PUD Distribution
![Page 24: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/24.jpg)
Comparison with MINLP Comparison with MINLP SolversSolvers
Note: Note: n n and and z* z* represent the number of substations and the optimal cost. represent the number of substations and the optimal cost. In the SBB column, In the SBB column, z z represents the cost for early termination (1000 b&b) nodes.represents the cost for early termination (1000 b&b) nodes.
![Page 25: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/25.jpg)
Time (scaled) vs. Number of Integers (scaled)Time (scaled) vs. Number of Integers (scaled)
Sc a
led
Tim
e
![Page 26: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/26.jpg)
Large-Scale SolutionsLarge-Scale Solutions
Note: Note: nn00 and and zz00 represent the initial number of substations and the initial cost. represent the initial number of substations and the initial cost.
![Page 27: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/27.jpg)
Uniform Load DistributionUniform Load Distribution
![Page 28: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/28.jpg)
Different Starting PointsDifferent Starting Points
![Page 29: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/29.jpg)
![Page 30: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/30.jpg)
Quality of SolutionQuality of SolutionInitial VoltageInitial Voltage
Load
Distributio
n
Initial Voltage
Most Load Nodes
Have Lower Voltages
![Page 31: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/31.jpg)
Final Voltage
Most Load Nodes Have High Voltages
Load
Distributio
n
Quality of SolutionQuality of SolutionFinal VoltageFinal Voltage
![Page 32: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/32.jpg)
Conclusions and CommentsConclusions and Comments
A very fast algorithm has been developed to find the optimal location in a large electrical network.
The algorithm is embedded in a GUI developed by Bergen Software Services International (BSSI).
Fast algorithm enables further embellishment of model to include Contingency constraints Varying impedance across network Varying substation sizes
![Page 33: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/33.jpg)
AcknowledgementsAcknowledgements
Robert H. Fletcher, Snohomish PUD, Washington
Patrick Gaffney, BSSI, Bergen, Norway.
![Page 34: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/34.jpg)
![Page 35: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/35.jpg)
Appendix
![Page 36: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/36.jpg)
Lower Bounds Lower Bounds Based on MIPs and Convex RelaxationsBased on MIPs and Convex Relaxations
Note: We obtain two sets of bounds. The first is based on a solution of mixed-integer Note: We obtain two sets of bounds. The first is based on a solution of mixed-integer linear programs and the second is based on solving a continuous relaxation (convex linear programs and the second is based on solving a continuous relaxation (convex QP).QP).
![Page 37: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/37.jpg)
Comparison with MINLP Comparison with MINLP SolversSolvers
Note: Note: n n and and z* z* represent the number of substations and the optimal cost. represent the number of substations and the optimal cost. In the SBB column, In the SBB column, z z represents the cost for early termination (1000 b&b) nodes.represents the cost for early termination (1000 b&b) nodes.
![Page 38: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/38.jpg)
SSO - ReviewSSO - Review– Varying sizes of substations– Transmission voltages– Contingency constraints:
Is the solution feasible if one substation fails?
Complexities:
Constraints:
Load-flow equations (Kirchoff’s laws)Voltage boundsVoltages at substations specifiedCurrent at loads is specified
Service area: each grid block is 1/2 mile by 1/2 mile
![Page 39: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/39.jpg)
Cost function:
SSO - ReviewSSO - Review
New equipmentLosses in the networkMaintenance costs
Constraints:
Load and voltage constraintsReliability and substation capacity constraintsDecision variables:
Installation / upgrading of substations
Characteristics:
![Page 40: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/40.jpg)
VariablesVariables
![Page 41: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/41.jpg)
Admittance Matrix : YAdmittance Matrix : Y
![Page 42: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/42.jpg)
Admittance MatrixAdmittance Matrix
![Page 43: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/43.jpg)
![Page 44: A Local Relaxation Approach for the Siting of Electrical Substations](https://reader035.vdocuments.site/reader035/viewer/2022062309/568144a5550346895db16eaa/html5/thumbnails/44.jpg)
A Local Relaxation Approach for the Siting of Electrical
Substations
Multiscale Optimization Methods and Applications
University of Florida at Gainesville
February 26th – 28th, 2004
Walter Murray and Uday Shanbhag
Systems Optimization Laboratory
Department of Management Science and Engineering
Stanford University, CA 94305