1. Voronoi Diagrams as they relate to the Planning and Optimization of Power Distribution Systems By Mark Dechant 1

2. The Nature Of Power 1 Power is Calculated By: P = V*I where V = I * R Power(P) is in Watts, Voltage (V) is in Volts, Current (I) is in Amperes, Resistance (R) is in Ohms. Like water through a hose V is considered pressure, I would be considered the volume of water, R is the resistance of the hose. A smaller diameter hose gives a higher resistance to the water going through it. 2

3. The Nature Of Power 2 From the water/hose analogy a smaller wire will be more resistive(more loss) to current than a larger wire. When power is constant, one can raise the voltage and use a smaller wire. When cost is considered smaller diameter, shorter wires are cheaper. 3

4. Power Generation Due to Cost and Efficiency, the Power Originating from the Power Plant is a very high Voltage ( >100,000 Volts) ie smaller wires. Kind of obvious >100000 at your outlet is undesirable. The Power used today is Alternating Current (AC). This enables the ability for transformers to be used to change the voltage value. 4

5. Distribution Transformers are used to decrease to voltage to levels suitable for the user. Because power is constant, current will increase and bigger wires are necessary. The Power from the Power Plant 1 st goes to the Sub- station transformers where the voltage is reduced. (small wires to bigger wires) From the Sub-station the Voltage may go directly to the user or come out as an intermediate voltage and be reduced further if the run is long. 5

6. Power Distribution Plant to House Power Plant Transmission Line Sub-station Pole Transformer My House 6

7. Distribution Costs Costs associated with distribution go as follows: Cost of Wire: -Thicker wire = more cost -Longer runs = more cost (more wire, greater loss) Cost of transformers: -Transformed Power - more power = bigger Transformer = greater cost -Transformer Efficiency – more loss = greater cost 7

8. Distribution Costs - Conclusion Given an area Power Distribution Costs can be minimized through: – Number, Size and Location of transformers – Optimal wire runs 8

9. Global Optimization The main idea with Global Optimization is to divide the region up into arbitrary small zones and Micro Optimize them with themselves. Next map these zones through a Voronoi algorithm using the transformers as the sites – this will have the effect of changing the their shape and size. Next Micro Optimize each Voronoi polygon with itself. Next find the Delaunay Triangulation, again using the transformers as the triangle points – this gives a list of neighboring zones which can be optimized with each other. Repeat Voronoi/Micro Optimization/Delaunay until cost is minimal. Important Note: It is assumed that studies have been done in advance to ascertain the power requirements to met prior to this planning activity. 9

10. Micro Optimization Micro Optimization involves a small zone of arbitrary size of users (call them loads from now on) and create a graph Usually this graph will have mini-zones or sub-graphs of connected components Each sub-graph will have their loads grouped into sub-sets through a k-means algorithm using Euclidian Distance. Each sub-graph will then get a transformer capable of supplying the required power Connecting of loads occurs through Minimum Spanning Tree(MST) Feasibility both technical and cost is calculated If not feasible add another transformer, k-means and MST Repeat with more transformers/MST/k-means until feasible 10

11. Micro Optimization - Zones The region is divided up into arbitrary zones containing sub graphs of connected users or loads. K-Means is used to place transformers, Minimum spanning tree is used to decide wire runs. 11

12. Micro Optimization K-Means K-Means is a clustering algorithm which clusters data according to an emergent set of averages – Euclidian Distance is used. The following equation is minimized: K-Means clusters the Users(Loads) around the Transformers which are located at the Mean Points. Notice that it look very Voronoi like. 12

13. Voronoi Graph The Voronoi Phase of the problem uses the Micro Optimization calculation previously as its input The sites for Voronoi are now the transformer locations Key to this is that after Voronoi the zones are now changed in both shape and size Micro Optimization is again performed for every Voronoi zone (more transformers may be added) 13

14. Voronoi Motivation Voronoi removes the inefficiencies of the hard and fast restriction of the somewhat regular zone mapping first arrive at by the 1 st go around with Micro Optimization. Voronoi also gives Delaunay for further optimization. 14

15. Delaunay Triangulation Zone to zone feasibility now has to be assessed Since we are only concerned with a zone and its neighbors, the Delaunay Triangulation will be used Again Micro Optimization may be employed between neighbors (the number of transformers can change – preferably less) 15

16. Delaunay Motivation Assessing each zone with every other takes too much time - O(N*N). Delaunay gives a list of neighbors. Delaunay is easily arrived at because we have Voronoi Neighbors 16

17. More Delaunay Notice on the Voronoi Diagram that there are situations as follows: 2 Transformers right next to each other 1 less Transformer after Neighbor/Neighbor Optimization 17

18. Incremental Results Before and After Voronoi/Delaunay run: Before After 18

19. Conclusion The application of Micro Optimization, Voronoi Map, and Delauney Triangulation provide a very good means of arriving at a power distribution plan for a large region. Because Delaunay and Voronoi graphs are compliments of each other calculations are kept to a minimum. Papers used are as follows: – Large-Scale Distribution Planning – Part 1 Simultaneous Network and Transformer Optimization Alejandro Navarro, Hugh Rudnick – Large-Scale Distribution Planning – Part 2 Macro Optimization With Voronoi’s Diagram and Tabu Search Alejandro Navarro, Hugh Rudnick – Creating Electrical Distribution Boundaries Using Computational Geometry Martin Held, Robert B. Williamson 19