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: Black – substation Other – Kw Load Service area: each grid block is 1/2 mile by 1/2 mile

SSO - 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 - Review More substations: Higher capital cost Lower 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

Variables

Problem of Interest

Admittance Matrix

A Multiscale Problem

SSO Algorithm DETERMINE 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 SS NOT CONVERGED ADJUST # OF SS WHILE IMPROVED SOLUTION CAN BE FOUND UPDATE POSITIONS OF SS

Finding an Initial Feasible Solution Global Relaxation Continuous relaxation Modified Objective

Finding an Initial Feasible Solution Global Relaxation

Search Direction Substation Positions Candidate Positions Good Neighbor

Search Direction Local Relaxation QP Subproblem

Center of Gravity Search Step Center of Gravity

Optimal Number of Substations

Sample Load Distributions Gaussian Distribution Snohomish PUD Distribution

Comparison with MINLP Solvers Note: n and z* represent the number of substations and the optimal cost. In the SBB column, z represents the cost for early termination (1000 b&b) nodes.

Time (scaled) vs. Number of Integers (scaled) Scaled Time

Large-Scale Solutions Note: n 0 and z 0 represent the initial number of substations and the initial cost.

Uniform Load Distribution

Different Starting Points

Quality of Solution Initial Voltage Load Distribution Initial Voltage Most Load Nodes Have Lower Voltages

Final Voltage Most Load Nodes Have High Voltages Load Distribution Quality of Solution Final Voltage

Conclusions 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

Acknowledgements  Robert H. Fletcher, Snohomish PUD, Washington  Patrick Gaffney, BSSI, Bergen, Norway.

Appendix

Lower Bounds Based on MIPs and Convex Relaxations 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 QP).

Comparison with MINLP Solvers Note: n and z* represent the number of substations and the optimal cost. In the SBB column, z represents the cost for early termination (1000 b&b) nodes.

SSO - 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 bounds Voltages at substations specified Current at loads is specified Service area: each grid block is 1/2 mile by 1/2 mile

Cost function: SSO - Review New equipment Losses in the network Maintenance costs Constraints: Load and voltage constraints Reliability and substation capacity constraints Decision variables: Installation / upgrading of substations Characteristics:

Variables

Admittance Matrix : Y

Admittance Matrix

A Local Relaxation Approach for the Siting of Electrical Substations Multiscale Optimization Methods and Applications University of Florida at Gainesville February 26 th – 28 th, 2004 Walter Murray and Uday Shanbhag Systems Optimization Laboratory Department of Management Science and Engineering Stanford University, CA 94305