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Vasilis Zois
CS @ USC
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 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
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 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
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 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
» 𝐴𝑇 𝐵 sin 𝐴θ − 𝑝 = 0
 Cost model of power supply
– 𝑓𝑖 (𝑥) with marginal cost 𝑀𝐶𝑖 =
𝑑𝑓𝑖 𝑥
𝑑𝑥
>0
– As power production increases cost increases rapidly
 Valuation model of provided power
– 𝑢𝑖 𝑥 with marginal cost 𝑀𝑉𝑖 =
𝑑𝑢𝑖 𝑥
𝑑𝑥
– Single valuation function for aggregated customer in bus i
– Law of diminishing marginal returns
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𝑘
𝑔
max
𝑙 𝑔
𝑧 ,𝑧 ,𝜃
𝑛−𝑘
𝑔
𝑔
𝑓𝑖 (𝑝𝑖 ) − 𝑓𝑖 𝑝𝑖 + 𝑧𝑖
𝑖=1
𝑢𝑖 −𝑝𝑖𝑙 − 𝑢𝑖 −𝑝𝑖𝑙 − 𝑧𝑖𝑙
−
𝑖=1
1. 𝐴𝑇 𝑠𝑖𝑛 𝐴𝜃 − 𝑝 + 𝑧 =0
2. 𝑝𝑙 ≤ 𝑝𝑙 + 𝑧 𝑙 ≤ 0
3. 0 ≤ 𝑝 𝑔 + 𝑧 𝑔 ≤ 𝑝 𝑔
𝜋
𝜋
4. − 2 ≤ Αθ ≤ 2
5. −𝑐 ≤ 𝐵𝑠𝑖𝑛(𝐴𝜃) ≤ 𝑐
 Optimization problem hardness
– Power grid normal operation
» Phase difference 𝐴𝜃 i ≈ 0
» 𝐴𝑇 𝐵 𝐴𝜃 − 𝑝 + 𝑧 = 0 and −𝑐 ≤ 𝐵(𝐴𝜃) ≤ 𝑐
– 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.
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 Reactive curtailment
– Fixed amount of supply reduction
– Match the supply loss while minimizing compensation
 Proactive curtailment
– Supply reduction
» Savings outweigh curtailment costs
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 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
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 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
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 Setup
– Cost functions 𝑓1 𝑥 =
𝑥2
𝑎
+ 0.15𝑥 + 𝛾1 , 𝑓2 𝑥 =
𝑥2
𝛽
+ 0.1𝑥 + 𝛾2
– Variable α and β
 Load Shedding
– Supply reduction on each bus changes
– Total supply reduction decreases
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 Capacity effect
– Profits always increase in contrast to power supply
 Comparison
– Higher profit than in reactive curtailment by optimizing supply
reduction
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 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
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 Fast response
– Critical in reactive curtailment
– Primary control within 5- 30s
 Experiments
– 14,57 or 118 bus systems
– Average time from 100 iterations
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Thank you!
Questions?
https://publish.illinois.edu/incentive-pricing/publications/
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