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Satisfiability of Elastic Demand in the smart grid

Jean-Yves Le Boudec, Dan- Cristian Tomozei EPFL Energy Systems Day Isaac Newton Institute Cambridge, 2012 March 12. Satisfiability of Elastic Demand in the smart grid. Contents. Introduction A Model of Elastic Demand Stability Results. Swiss Future Supply is Uncertain.

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Satisfiability of Elastic Demand in the smart grid

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  1. Jean-Yves Le Boudec, Dan-CristianTomozei EPFL Energy Systems Day Isaac Newton Institute Cambridge, 2012 March 12 Satisfiability of Elastic Demand in the smart grid 1

  2. Contents • Introduction • A Model of ElasticDemand • StabilityResults 2

  3. Swiss Future SupplyisUncertain • Source: ASE (Association des Entreprises Electriques Suisses, ) Winter semesterforecasts 2011-2012 3

  4. Smart Grid Vision : Larger Network Source: ELECTRICITY SUPPLY SYSTEMS OF THE FUTURE White paper on behalf of the CIGRE Technical Committee TC Chair: Klaus Froelich 2011 4

  5. Smart Grid Vision : Smaller, More AutonomousNetworks • Flexible services as alternative to blackout, includingacrossslack bus • Islandedoperationpossible Source: ELECTRICITY SUPPLY SYSTEMS OF THE FUTURE White paper on behalf of the CIGRE Technical Committee TC Chair: Klaus Froelich 2011 EPFL Smart Grid 5

  6. Flexible Services • Flexible services = distribution network operatormayinterrupt / modulate power • elasticloads support gracefuldegradation • Thermal load (Voltalis), washingmachines (Romande Energie«commandecentralisée»)e-cars, VoltalisBluepod switches off thermal load for 60 mn 6

  7. Our ProblemStatement • Do elastic services work ? • Delays • Returningload • ProblemStatementIs there a control mechanismthatcanstabilizedemand? • A very course (but fundamental) first step • Weleave out for now the details of signals and algorithms 7

  8. 2. A MODEL OF ELASTIC DEMAND 8

  9. Macroscopic Model of Cho and Meyn [1], non elastic demand,mapped to discrete time Step 1: Day-ahead market Step 2: Real-time market Actual demand Actualsupply • Forecast demand: • Forecastsupply: control random deterministic Wenowadd the effect of elasticdemand / flexible serviceSomedemandcanbe «frustrated» (delayed)

  10. Our Macroscopic Model with Elastic Demand Control Ramping Constraint Randomness Supply Natural Demand Evaporation Expressed Demand Satisfied Demand Returning Demand Reserve (Excess supply) Frustrated Demand Backlogged Demand 10

  11. BackloggedDemand • We assume backloggeddemandissubject to twoprocesses: update and re-submit • Update term (evaporation): with or is the evaporation rate (proportion lost per time slot) • Re-submissionterm (time slots) is the averagedelay Control Randomness Supply Natural Demand Expressed Demand Evaporation Satisfied Demand Returning Demand Frustrated Demand Reserve (Excess supply) Backlogged Demand 11

  12. Macroscopic Model, continued S. Meyn “Dynamic Models and Dynamic Markets for Electric Power Markets” • Assumption : ARIMA(0, 1, 0)typical for deviation from forecast • 2-d Markov chain on continuous state space

  13. The Control Problem • Control variable: production bought one time slot ago in real time market • Controller seesonlysupplyand expresseddemand • Our Problem:keepbacklog stable • Ramp-up and ramp-down constraints 13

  14. ThresholdBasedPolicies Forecastsupplyisadjusted to forecastdemand R(t) := reserve = excess of demand over supply • Threshold policy: • ifincreasesupply to come as close toas possible (consideringramp up constraint) • elsedecreasesupply to come as close to as possible (consideringrampdown constraint) 14

  15. Simulation r* • Linearized system: 1 iseigenvalue 15

  16. 3. StabilityResults 16

  17. Findings • If evaporationis positive, system is stable (ergodic, positive recurrent Markov chain) for anythreshold • If evaporationisnegative, system unstable for anythreshold • Delay does not play a role in stability • Nor do ramp-up / ramp down constraints or size of reserve 17

  18. More Detailed Findings Evaporation Backlogged Demand • Case 1: 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: 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)

  19. Evaporation • Negativeevaporationmeans:delaying a loadmakes the returningloadlargerthan the original one. • Couldthishappen ? Q. Doeslettingyour house cool down nowimplyspending more heat in total compared to keepingtemperatureconstant ? • return of the load: Q. Doeslettingyour house cool down nowimplyspending more heatlater ? A. Yes(you will need to heat up your house later -- delayed load) 19

  20. heatprovided to building • Assume the house model of [6] leakiness outside inertia efficiency achieved E, total energyprovided 20

  21. WhenDelayedHeatingisLessHeat • With constant coefficient of performance , total energyprovidedisless if let building cool down and warm up again • Assume somedemandisfrustrated(second scenario)update process replaces backloggeddemand by whatisneeded to recover the targettemperature • Update processdecreasesbacklog, evaporationis positive Evaporation Backlogged Demand 21

  22. The Sign of Evaporation • Resistive heating system:evaporationis positive. This iswhyVoltalisbluepodisaccepted by users • If heat = heatpump, coefficient of performance maybe variablenegativeevaporationis possible • Electric vehicle: delayed charge may have to befaster, less efficient, negativeevaporationis possible 22

  23. Conclusions • A first model of adaptive applianceswith volatile demand and supply • Suggeststhatnegativeevaporationmakes system unstableExistingdemand-response positive experience (withVoltalis/PeakSaver) might not carry over to otherloads • Model suggeststhat large backlogs are possibleBackloggedloadis a new threat to gridoperationNeed to measure and forecastbackloggedload 23

  24. Questions ? [1] Cho, Meyn – Efficiency and marginal cost pricing in dynamic competitive markets with friction, Theoretical Economics, 2010 [2] Le Boudec, Tomozei, Satisfiability of Elastic Demand in the Smart Grid, Energy 2011 and ArXiv.1011.5606 [3] Le Boudec, Tomozei, Demand Response Using Service Curves, IEEE ISGT-EUROPE, 2011 [4] Le Boudec, Tomozei, A Demand-Response Calculus with Perfect Batteries, WoNeCa, 2012 [5] Papavasiliou, Oren - Integration of Contracted Renewable Energy and Spot Market Supply to Serve Flexible Loads, 18th World Congress of the International Federation of Automatic Control, 2011 [6] David MacKay, Sustainable Energy – Without the Hot Air, UIT Cambridge, 2009

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