Power Grid Sizing via Convex Programming Peng Du, Shih-Hung Weng, Xiang Hu, Chung-Kuan Cheng University of California, San Diego 1

Outline Problem Formulation Convex Programming Reduction Optimizer of the Convex Programming Problem – Interior point and gradient descent methods – A Krylov space method to evaluate effective resistances – Close form of the derivative of effective resistances Experimental Results 2

Problem Formulation 3 Power Network N(V,E) Voltage source at node u Current loads in set W Variables: conductance g(e) for each edge e Objective: min max voltage drop at all nodes and over all possible current loads

Problem Formulation 4 D(v,g,I) : the voltage drop between nodes u and v for given conductance assignment g and current profile I. We assume the current profile satisfies the normalization constraint I(w 1 )+I(w 2 )+…+I(w r )=1 where W={w 1,w 2,…,w r }.

Convex Programming Reduction 5

SDP Formulation 6

Optimizer for Larger Cases Interior point method. Gradient descent method. Obstacles: – Evaluate the objective: effective resistances. – Evaluate the derivative of effective resistances relative to conductance. 7

Evaluate the Effective Resistances 8

Evaluate the Derivative of Effective Resistances 9

Experimental Results (Regular Grids) 10

Experimental Results (Regular Grids) The effective resistances before (blue plane) and after (green plane) optimization. 11

Experimental Results (Regular Grids) (R,C)MaxR(uni)MaxR(opt)Improvement (4,4) % (7,5) % (10,10) % 12

Experimental Results (Practical Power Grid) 13

Experimental Results (Practical Power Grid) 14

Voltage Map of Layer M1 15 The worst voltage drop is achieved near the origin which is the farthest point to the voltage source.

Resource Transformation between M6 and AP “Adjust Percentage” indicates the ratio of M6 resource obtained from AP. A monotonically decreasing curve means that with more resource to M6 we can reduce worst voltage drop. 16

Conclusion A power grid sizing method to minimize the maximum voltage drop over all test locations and current source distributions. We reduce the original problem into a convex programming problem whose objective is to minimize the maximum effective resistance. We adopt a Krylov space method to evaluate the effective resistances and devise a formula to update the derivative of effective resistance. Experimental results show that our method can achieve up to 40% improvement for regular 2D grids and 7.32% improvement for a practical power grid with only top two layers tunable.