Balancing supply and demand in the presence of renewable generation via demand response for electric water heaters
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Balancing supply and demand in the presence of renewable generation via demand response for electric water heaters Adham I. Tammam1 · Miguel F. Anjos1,2
· Michel Gendreau3
© The Author(s) 2020
Abstract With the increasing presence of renewable energy sources in the electrical power grid, demand response via thermostatic appliances such as electric water heaters is a promising way to compensate for the significant variability in renewable power generation. We propose a multistage stochastic optimization model that computes the optimal day-ahead target profile of the mean thermal energy contained in a large population of heaters, given various possible wind power production and uncontrollable load scenarios. This optimal profile is calculated to make the variable net demand as even as possible. Keywords Demand response · Electric water heaters · Stochastic optimization · Renewable power generation
List of symbols Indices t n i, j
Time steps Nodes of the scenario tree States of the Markov chain / types of water extraction
Parameters K
Thermal conductivity per unit length of EWHs
B
Miguel F. Anjos [email protected] Adham I. Tammam [email protected] Michel Gendreau [email protected]
1
GERAD and Department of Mathematics and Industrial Engineering, Polytechnique Montreal, C.P. 6079, Succ. Centre-Ville, Montreal, QC H3C 3A7, Canada
2
School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom
3
CIRRELT and Department of Mathematics and Industrial Engineering, Polytechnique Montreal, C.P. 6079, Succ. Centre-Ville, Montreal, QC H3C 3A7, Canada
123
Annals of Operations Research
A Cp ρ V N ewh TL T env Θ = {1, 2,…, I} Λ = {λi j , i, j, = 1, . . . , I} pi T mix V˙ mix
r (ωn ) n Pn
Total surface area of all the EWHs Hot-water specific heat Water density Total volume of hot water in the EWHs Number of EWHs Inlet water temperature Environment temperature Set of water extraction values for the Markov chain θt Infinitesimal generator for the Markov chain θt Probability of occurrence of each state of the Markov chain Desired temperature for the end-user of hot water Desired flow of water for the end-user for extraction of type i Minimum amount of energy that can be stored in the EWHs Maximum amount of energy that can be stored in the EWHs Lower limit of the temperature of the population of EWHs Upper limit of the temperature of the population of EWHs Number of nodes in the scenario tree Number of time periods Discrete time step Scenario tree Value of the uncontrollable demand at node n of the scenario tree Wind power production at node n of the scenario tree Parent node of node n in the scenario tree Marginal probability of the occurrence of node n
Decision variables et x(et ) pn
Energy stored in the system at time step t Quantity of energy injected into the EHWs at time step t Net demand curve for the mean field controller
i
emin emax T min T max N T Δt {ξt }t∈T d(ωn )
Other variables
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