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