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 - ReviewSSO - ReviewService area
Washington State
SSO - ReviewSSO - ReviewColour:
•Black –substation
•Other –Kw Load
Service area: each grid block is 1/2 mile by 1/2 mile
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
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
VariablesVariables
Problem of InterestProblem of Interest
Admittance MatrixAdmittance Matrix
A Multiscale ProblemA Multiscale Problem
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
Finding an Initial Feasible SolutionFinding an Initial Feasible SolutionGlobal RelaxationGlobal Relaxation
Continuous relaxation
ModifiedObjective
Finding an Initial Feasible SolutionFinding an Initial Feasible SolutionGlobal RelaxationGlobal Relaxation
Search DirectionSearch Direction
SubstationPositions
CandidatePositions
Good Neighbor
19 K
Search DirectionSearch DirectionLocal RelaxationLocal Relaxation
QP Subproblem
Center of Gravity
Center of Gravity
Search StepSearch StepCenter of GravityCenter of Gravity
Optimal Number of SubstationsOptimal Number of Substations
Sample Load DistributionsSample Load Distributions
Gaussian DistributionGaussian Distribution Snohomish PUD DistributionSnohomish PUD Distribution
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.
Time (scaled) vs. Number of Integers (scaled)Time (scaled) vs. Number of Integers (scaled)
Sc a
led
Tim
e
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.
Uniform Load DistributionUniform Load Distribution
Different Starting PointsDifferent Starting Points
Quality of SolutionQuality of SolutionInitial VoltageInitial Voltage
Load
Distributio
n
Initial Voltage
Most Load Nodes
Have Lower Voltages
Final Voltage
Most Load Nodes Have High Voltages
Load
Distributio
n
Quality of SolutionQuality of SolutionFinal VoltageFinal Voltage
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
AcknowledgementsAcknowledgements
Robert H. Fletcher, Snohomish PUD, Washington
Patrick Gaffney, BSSI, Bergen, Norway.
Appendix
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).
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.
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
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:
VariablesVariables
Admittance Matrix : YAdmittance Matrix : Y
Admittance MatrixAdmittance Matrix
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