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Satisfiability of Elastic Demand in the smart grid

Jean-Yves Le Boudec, Dan- Cristian Tomozei EPFL May 26, 2011 . Satisfiability of Elastic Demand in the smart grid. Demand Management. Variable Supply Variable Demand Frequency Response becomes insufficient Demand management required in future grid.

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Satisfiability of Elastic Demand in the smart grid

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  1. Jean-Yves Le Boudec, Dan-CristianTomozei EPFL May 26, 2011 Satisfiability of Elastic Demand in the smart grid 1

  2. Demand Management • Variable Supply • Variable Demand • FrequencyResponsebecomesinsufficient • Demand management required in future grid 2

  3. Demand Management must Be Simple, Adaptive and Distributed • Global, optimal schedules • But they are • inflexible • complex 3

  4. Adaptive Appliances • SomeDemandcanbedelayed ! • DSO provides best effort service withstatisticalguarantees [Keshav and Rosenberg 2010] VoltalisBluepodswitches off thermal load for 30 mn Programmabledishwasher PeakSaver cycles AC for 15mn 4

  5. Our ProblemStatement • Is elasticdemandfeasible ? • Weleave out (for now) the details of signals and algorithms • ProblemStatementIs there a control mechanismthatcanstabilizedemand ? • Instabilitycanbegenerated by • Delays in demand • Higherreturningdemand • A very course (but fundamental) first step 5

  6. Macroscopic Model • Step 1: Day-ahead market • Forecast demand : • Forecast supply : • Step 2: Real-time market • Actual demand : • Actual supply : • Deterministic processes : • Random processes : • Control : • Ramp up, ramp down

  7. Macroscopic Model Control Ramping Constraint Randomness Supply Natural Demand Evaporation Expressed Demand Satisfied Demand Returning Demand Reserve (Excess supply) Frustrated Demand Backlogged Demand

  8. Macroscopic Model – Normal Assumption • Assumption : (M – D) = ARIMA(0, 1, 0) • 2-d Markov chain on continuous state space

  9. The Control Problem Control Randomness • Control variable: G(t-1)production bought one second ago in real time market • Controller seesonlysupply Ga(t) and expresseddemandEa(t) • Our Problem:keepbacklog Z(t) stable • Ramp-up and ramp-down constraints ξ ≤ G(t) ⎼ G(t-1) ≤ ζ Supply Natural Demand Expressed Demand Evaporation Satisfied Demand Returning Demand Frustrated Demand Reserve (Excess supply) Backlogged Demand 9

  10. ThresholdBasedPolicies Forecastsupplyisadjusted to forecastdemand R(t) := reserve = excess of demand over supply • Thresholdpolicy: • if R(t) < r* increasesupply as much as possible (consideringramp up constraint) • else set R(t)=r* 10

  11. r* 11

  12. Findings • If evaporationμis positive, system is stable (ergodic, positive recurrent Markov chain) for anythreshold r* • If evaporationisnegative, system unstable for anythreshold r* • Delay does not play a role in stability • Nor do ramp-up / ramp down constraints or size of reserve 12

  13. More Detailed Findings Evaporation Backlogged Demand • Case 1: Positive evaporation Postponing a task = discount • Theorem 1: The Markov chain (R,Z) is Harris recurrent and ergodic. It converges to the unique steady state probability distribution, for any threshold and any strictly positive ramp-up constraint. • Case 2: Negative evaporation Postponing a task = penalty • Theorem 2: The Markov chain (R,Z) is non-positive, for any threshold. Method of Proof: quadraticLyapunov (case 1) or logarithmic L. (case 2)

  14. Evaporation • Evaporation = dropped fraction of delayeddemand • Negativeevaporationmeans:delaying a demandmakes the returningdemandlargerthan the original one • Couldthishappen ?Doeslettingyour house cool down nowimpliesspending more heatlater ? (vs keeping constant temperature) • Do not confuse with the sum of returningdemand + currentdemand, whichisalwayslargerthancurrentdemand) 14

  15. Evaporation: HeatingAppliances • Assume the house model of [MacKay 2009]thendelayedheatingislessheating • If heat = energy, thenevaporationis positive. This iswhyVoltalisbluepodisaccepted by users • If heat = heatpump, coefficient of performance maybe variableDelayedheatingwith air heatpumpmay have negativeevaporation heatprovided to building leakiness outside inertia 15

  16. Batteries • Thermal lossis non linear, delayedloading causes negativeevaporation (chargingathigherintensity) 16

  17. Conclusions • A first model of adaptive applianceswith volatile demand and supply • Suggeststhatnegativeevaporationmakes system unstable, • thusdetailedanalysisisrequired to avoidit • Model canbeused to quantify more detailedquantities • E.g. amount of backlog, optimal reserve 17

  18. Questions ? [1] Cho, Meyn – Efficiency and marginal cost pricing in dynamic competitive markets with friction [2] Papavasiliou, Oren - Integration of Contracted Renewable Energy and Spot Market Supply to Serve Flexible Loads [3] Operational Requirements and Generation Fleet Capability at 20% RPS (CAISO - 31 August, 2010)

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