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A Demand Response Calculus with Perfect Batteries Dan-Cristian - - PowerPoint PPT Presentation
A Demand Response Calculus with Perfect Batteries Dan-Cristian - - PowerPoint PPT Presentation
A Demand Response Calculus with Perfect Batteries Dan-Cristian Tomozei Joint work with Jean-Yves Le Boudec CCW, Sedona AZ, 07/11/2012 Demand Response by Quantity = distribution network operator may interrupt / modulate power elastic
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Network Calculus? Service curve?
Voltalis:
At most 30 mn of interruption total per day
“Service curve” contract 𝐻𝑣𝑏𝑠𝑏𝑜𝑢𝑓𝑓𝑒 𝑓𝑜𝑓𝑠𝑧 𝑒𝑓𝑚𝑗𝑤𝑓𝑠𝑓𝑒 𝑗𝑜 (𝑡, 𝑢) ≥ 𝛾 𝑢 − 𝑡 , ∀0 ≤ 𝑡 ≤ 𝑢 𝛾 𝑢 = superadditive function.
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Real Situation: Unexpected Consumption Peaks
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Today
Aggregate demand is predictable Operators foresee “reserve” (primary, secondary, tertiary)
E.g., gas turbines
Reserve is expensive (capacity) / rare event demand response
Delay (“buffer”) demand until the peak has passed ~ virtual energy storage
Tomorrow?
High penetration of renewables
Large (unaffordable) reserve requirements
E.g., fleet of e-cars DR exploits load flexibility
Is Demand Response a good solution?
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Formally:
Consumption:
𝑉 𝑢 = 𝑣 𝑡 𝑒𝑡
𝑢
Allowed consumption (control):
𝐻 𝑢 = 𝑡 𝑒𝑡
𝑢
Demand Response imposes:
0 ≤ 𝑉 𝑢 − 𝑉 𝑡 ≤ 𝐻 𝑢 − 𝐻 𝑡 , ∀0 ≤ 𝑡 ≤ 𝑢
Demand Response by Quantity
[Le Boudec, Tomozei – ISGT-EU’11]
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“power” (Watts) “energy” (Watt-hours)
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Inelastic (non-dispatchable) loads
Lamps, TVs, Microwaves, …
Elastic (dispatchable) loads
Heating, A/C (TCLs)
Make it dispatchable!
Inelastic load 𝑀 𝑢 = ℓ 𝑡 𝑒𝑡
𝑢
Use a large enough battery!
Inelastic load = lights out?
Load sees no constraints
Grid
Actual consumption (constrained!) “Energy buffer”
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The Perfect Battery
Battery may be charged (𝑣 𝑢 > ℓ(𝑢)) or discharged (𝑣 𝑢 < ℓ(𝑢)) Load ℓ 𝑢 is given Problem is to determine a power schedule 𝑣(𝑢), subject to 0 ≤ 𝑣 𝑢 ≤ (𝑢) and within battery constraints
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System Equations for the Perfect Battery
1. 𝑀 𝑢 ≤ 𝐶0 + 𝑉(𝑢) no underflow 2. 𝑉 𝑢 − 𝑀 𝑢 + 𝐶0 ≤ 𝐶 no overflow 3. 𝑉 𝑢 − 𝑉 𝑡 ≤ 𝐻 𝑢 − 𝐻 𝑡 , ∀𝑡 ≤ 𝑢 power constraint where 𝑉 𝑢 , 𝑀 𝑢 , 𝐻 𝑢 are cumulative functions such as 𝑉 𝑢 = 𝑣 𝑡 𝑒𝑡
𝑢
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Constraints:
Demand Response: 0 ≤ 𝑉 𝑢 − 𝑉 𝑡 ≤ 𝐻 𝑢 − 𝐻 𝑡 , ∀𝑡 ≤ 𝑢 Perfect battery constraints: 𝑀 𝑢 ≤ 𝐶0 + 𝑉 𝑢
𝑉 𝑢 − 𝑀 𝑢 + 𝐶0 ≤ 𝐶 Given (known) signals:
The load
𝑀 𝑢 = ℓ 𝑡 𝑒𝑡
𝑢
Allowed consumption
𝐻 𝑢 = 𝑡 𝑒𝑡
𝑢
To be determined:
Battery initial charge 𝐶0 Max battery capacity 𝐶 Schedule (consumption from grid)
𝑉 𝑢 = 𝑣 𝑡 𝑒𝑡
𝑢
Omniscient Problem
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Main Result
Theorem
There exists a feasible schedule if and only if
𝐶0 ≥ sup
𝑢
𝑀 𝑢 − 𝐻 𝑢 𝐶 ≥ sup
0≤𝑡≤𝑢
(𝑀 𝑢 − 𝑀 𝑡 − 𝐻 𝑢 + 𝐻(𝑡))
Moreover, if this is the case, then there exist a “minimal” and a “maximal”
schedule: 𝑉∗ 𝑢 = 0 ∨ sup
𝜐 ≥ 𝑢
𝐻 𝑢 − 𝐻 𝜐 + 𝑀 𝜐 − 𝐶0 𝑉∗ 𝑢 = 𝐻 𝑢 ∧ inf
𝑡 ≤ 𝑢 𝐻 𝑢 − 𝐻 𝑡 + 𝑀 𝑡 + 𝐶 − 𝐶0
𝑉∗ 𝑢 ≤ 𝑉 𝑢 ≤ 𝑉∗ 𝑢 , ∀𝑢 ≥ 0
The maximal schedule is causal & corresponds to the greedy policy
(maximizes battery charge)
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Service Curve Approach to Demand Response
Assume we do not know the control signal 𝐻 𝑢 Instead: service curve contract [Le Boudec, Tomozei, ISGT-EU’11] 𝐻 𝑢 − 𝐻 𝑡 ≥ 𝛾 𝑢 − 𝑡 , ∀0 ≤ 𝑡 ≤ 𝑢 𝛾 𝑢 = superadditive function. Example:
At most 30 mn of interruption total per day Or reduction to
𝑨 𝑛𝑏𝑦 2
for 60mn total per day
Similar theorem closed form condition on 𝐶, 𝐶0 + min/max schedule
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Service Curve + Arrival Curve
Assume we don’t know the load 𝑀(𝑢) either! Instead, 𝑀(𝑢) is constrained by a subadditive arrival curve: 𝑀 𝑢 − 𝑀 𝑡 ≤ 𝛽 𝑢 − 𝑡 , ∀0 ≤ 𝑡 ≤ 𝑢 Smallest arrival curve – obtained via min-plus deconvolution: 𝛽 𝑢 ≔ sup
𝑡≥0
𝑀 𝑡 + 𝑢 − 𝑀 𝑡 𝐻(𝑢) is well behaved (according to superadditive service curve): 𝐻 𝑢 − 𝐻 𝑡 ≥ 𝛾 𝑢 − 𝑡 , ∀0 ≤ 𝑡 ≤ 𝑢 Theorem For all (B ≥)𝐶0 ≥ 𝐶∗ ≔ sup
𝑡
{𝛽 𝑡 − 𝛾 𝑡 }, there exists a feasible
- nline (causal) schedule, valid for all loads and control signal
compatible with 𝛽 ⋅ and 𝛾 ⋅ respectively.
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Application: Transparent DR for data centers
Akamai data set [Qureshi et al, SIGCOMM 2009]
- Traffic at Akamai (millions of hits over 24 days)
- Measured power consumption of a desktop (SPEC)
- Uniform repartition of tasks => consumption of one server
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Empirical arrival curve
𝛽𝑛 𝑢 ≔ sup
0≤𝑡≤𝑈
𝑛𝑏𝑦−𝑢
𝑀 𝑡 + 𝑢 − 𝑀 𝑡
Intuitively = worst observed day
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Choosing a SC contract and a battery
Interruption time = 𝑢0 Maximum power = 𝑨𝑛𝑏𝑦 Required battery charge = 𝐶∗
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1h/24h Service interruption
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A run of the system using the greedy policy
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