Power Grid Sizing via Convex Programming

Peng Du, Shih-Hung Weng, Xiang Hu, Chung-Kuan Cheng

University of California, San Diego

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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

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Problem Formulation

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• 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

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• 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(w1)+I(w2)+…+I(wr)=1 where W={w1,w2,…,wr}.

Convex Programming Reduction

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SDP Formulation

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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.

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Evaluate the Effective Resistances

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Evaluate the Derivative of Effective Resistances

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Experimental Results (Regular Grids)

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Experimental Results (Regular Grids)

The effective resistances before (blue plane) and after (green plane) optimization.

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Experimental Results (Regular Grids)

(R,C) MaxR(uni) MaxR(opt) Improvement

(4,4) 44.57 36.00 19.23%

(7,5) 142.21 100.00 29.68%

(10,10) 542.10 324.89 40.07%

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Experimental Results (Practical Power Grid)

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Experimental Results (Practical Power Grid)

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Voltage Map of Layer M1

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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.

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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.