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A Local Relaxation Approach for the Siting of Electrical Substations

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:

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A Local Relaxation Approach for the Siting of Electrical Substations

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  1. 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

  2. SSO - Review Service area Washington State

  3. SSO - Review • Colour: • Black – • substation • Other – • Kw Load Service area: each grid block is 1/2 mile by 1/2 mile

  4. 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

  5. SSO - Review Characteristics: More substations: Higher capital cost Lower transmission cost • Capital costs: • $4,000,000 for a 28 MW substation Service area: each grid block is 1/2 mile by 1/2 mile • Cost of losses: • $3,000 per kw of losses

  6. Variables

  7. Problem of Interest

  8. Admittance Matrix

  9. A Multiscale Problem

  10. DETERMINE SEARCH DIRECTION UPDATE POSITIONS OF SS DETERMINE SEARCH STEP TO GET IMPROVED SOLN ADJUST # OF SS WHILE IMPROVED SOLUTION CAN BE FOUND WHILE # OF SS NOT CONVERGED FINAL NUMBER AND POSITIONS OF SUBSTATIONS SSO Algorithm DETERMINE INITIAL DISCRETE FEASIBLE SOLUTION INITIAL NUMBER OF SS

  11. Finding an Initial Feasible SolutionGlobal Relaxation Modified Objective Continuous relaxation

  12. Finding an Initial Feasible SolutionGlobal Relaxation

  13. Search Direction Substation Positions Candidate Positions Good Neighbor

  14. Search DirectionLocal Relaxation QP Subproblem

  15. Search Step Center of Gravity Center of Gravity Center of Gravity

  16. Optimal Number of Substations

  17. Sample Load Distributions Snohomish PUD Distribution Gaussian Distribution

  18. 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.

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

  20. Large-Scale Solutions Note: n0and z0represent the initial number of substations and the initial cost.

  21. Uniform Load Distribution

  22. Different Starting Points

  23. Quality of SolutionInitial Voltage Initial Voltage Load Distribution Most Load Nodes Have Lower Voltages

  24. Quality of SolutionFinal Voltage Final Voltage Load Distribution Most Load Nodes Have High Voltages

  25. 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

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

  27. Appendix

  28. 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).

  29. 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.

  30. SSO - Review Complexities: • Varying sizes of substations • Transmission voltages • Contingency constraints: • Is the solution feasible if one substation fails? Constraints: Service area: each grid block is 1/2 mile by 1/2 mile • Load-flow equations (Kirchoff’s laws) • Voltage bounds • Voltages at substations specified • Current at loads is specified

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

  32. Variables

  33. Admittance Matrix : Y

  34. Admittance Matrix

  35. 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

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