Vasilis Zois CS @ USC

1. Profit – Optimal & Stability Aware Load Curtailment in Smart Grids VasilisZois CS @ USC

2. Introduction • Dynamic and sophisticated demand control • Direct control over household appliances • Curtailment Reasons • Reactive Curtailment • Loss of power generation • Renewable sources don’t work at full capacity • Proactive • Maximize profits • Reduced power consumption overweigh customer compensation • Customer Satisfaction • Discounted plan  Valuation Function • Plan connected to customer load elasticity

3. Previous Work • Dynamic pricing • Direct control achieved by monetary incentives • Cost & valuation functions • Convex cost functions • Concave valuation functions • Optimal Curtailment • Component failure as subject of attack • Quantify severity by the amount of the curtailed power • Frequency stability • Locally measured frequency • Centralized approach • Physical constraints • Low computational cost

4. Model Analysis • Physical power systems model • Graph G= (V,E) • Vertices  Buses that generate or consume power • Edges  Transmission line i with capacity ci • Power flow model • Voltage at each bus is fixed • Cost model of power supply • with marginal cost • As power production increases cost increases rapidly • Valuation model of provided power • with marginal cost • Single valuation function for aggregated customer in bus i • Law of diminishing marginal returns

5. Optimization Framework • =0 • Optimization problem hardness • Power grid normal operation • Phase difference • and • Theorem 1: If the supply cost functions are convex and the valuation functions are concave, then both reactive and proactive load curtailment problems are convex after linearization.

6. Curtailment problems • Reactive curtailment • Fixed amount of supply reduction • Match the supply loss while minimizing compensation • Proactive curtailment • Supply reduction • Savings outweigh curtailment costs

7. Experiments Overview • Curtailment Period • Fixed (e.g 15 minutes) • Optimization at the beginning • Cost savings and profits for one period • Comparison of valuation functions • Linear vs concave • Effect of line capacity in optimization

8. Reactive curtailment experiments • Concave function • Line capacities limit load shedding on specific busses • Linear function • Same curtailment for different capacities • Comparison • Better distribution of curtailment with concave function

9. Proactive curtailment experiments • Setup • Cost functions • Variable α and β • Load Shedding • Supply reduction on each bus changes • Total supply reduction decreases

10. Proactive curtailment experiments (2) • Capacity effect • Profits always increase in contrast to power supply • Comparison • Higher profit than in reactive curtailment by optimizing supply reduction

11. Curtailment limits • Additional constraints • Limit curtailed load on each bus • Preserved convexity of optimization problem • Effect of limits • Reduced profits • Limited power reduction • Limit is not reached

12. Computational Cost • Fast response • Critical in reactive curtailment • Primary control within 5- 30s • Experiments • 14,57 or 118 bus systems • Average time from 100 iterations

13. Thank you! Questions? https://publish.illinois.edu/incentive-pricing/publications/