Demand Response ith Stochastic Reneables and Inertial Thermal Loads: A Study of Some Fundamental Issues in Modeling and Optimization Gaurav Sharma, Le ie and P. R. Kumar Dept. of Electrical and Computer Engineering Texas A&M University Variability of Wind Goal: Use reneable energy ind Problem: Highly variable stochastic Email: prk.tamu@gmail.com Web: http://cesg.tamu.edu/faculty/p-r-kumar/ NINTH ANNUAL CARNEGIE MELLON CONFERENCE ON THE ELECTRICITY INDUSTRY CMU February 4, 24 /4 2/4 Demand Response Adjust demand to match supply Inertial thermal loads building air conditioners Air conditioner can be sitched off for a short hile ithout loss of comfort Traditionally under thermostatic control Limited capability of demand response Reneable energy may not be enough to satisfy load requirements Need to turn on air-conditioner T max T max T min T min 3/4 So there are itations to demand response 4/4 Variability of poer demand not met by reneables Several Questions Residual poer demand not met by reneables Time Prefer less variability so that operating reserve requirements are less To hat extent can demand of inertial loads be met by reneable sources? Ho does flexibility of load requirements, such as comfort level settings, influence ho much reneable poer can be used? Ho much flexibility can be extracted from thermal inertial loads for maximum utilization of variable generation such as ind? Time 5/4 6/4
Several Questions 2 To hat extent can operating reserve required be minimized? Ho beneficial is demand pooling? Can e come up ith quantitative ansers? Ho can demand pooling be done? What are the communication requirements? Ho much information exchange is needed beteen suppliers and consumers? Several Questions 3 What are the privacy implications? Does it require intrusive sensing? Ho distributed can the solution be? Ho tractable (computational complexity) is the solution? Ho robust is the solution? Ho implementable is it? Role of model features, cost functions, stochasticity assumptions, convexity, asymptotics, etc 7/4 8/4 Load Aggregator Load service entity a.k.a. Load aggregator Acts as coordinator Features of problem Possibly monitors temperature of A/Cs Possibly controls poer to cool A/Cs Appears as aggregated load 9/4 /4 Wind model Temperature dynamics Model ind as a finite state Markov process Wind poer Time x i (t) Temperature of load i Poer given to load i Time Even simpler model for illustration On-Off process W Wind poer Time ON OFF /4 Inertial thermal load (A/C) dynamics ẋ i (t) =h i (t) P i (t)» Rate of change in temperature = Ambient heating Poer for cooling. 2/4
User specified comfort range Range of comfortable temperature [, ] Either: Enforce hard constraint Or: Penalize the violations t t Allo but penalize the violation 3/4 Stochastic control problem: Comfort violation probability Minimize the probability of leaving a user specified comfort range [, ] Wind process Temperature dynamics Cost function P i (t) Markov process T T Z T ẋ i (t) =h i I(x i (t) > )dt i P i (t) 4/4 Optimal policy in comfort violation probability model Temperature Wind poer Load 2 Load Time Theorem: Provide poer to the coolest load that is above the temperature range. Issue: Unfair, temperatures of some loads ill remain higher than others Possible solution: Minimize the variance of comfort violation 5/4 Stochastic control problem: Variance minimization Stochastic control problem: Z T Cost function Theorem: Optimal policy synchronizes loads Load 2 Temperature max Load min Wind poer T T [(x i (t) i ) + ] 2 dt Loads ill remain synchronized after this time instant Time 6/4 Requirement for reserves (of nonreneable poer) Temperatures can go very high occasionally Cost function for reducing operating reserves Desire lo operating reserve requirements Temperature Load 2 Load More variability Less variability Prefer this Hard constraints require reliable non-reneable source Use non-reneable poe Need to turn on air-conditioner, maintain constraint but no ind available 7/4 t Impose a quadratic cost on non-reneable poer usage Z ( Pi n (t)) 2 dt t 8/4
Stochastic control problem: Reduction of variability ith temperature constraint Stochastic control Problem: Wind process P i (t) Markov process Temperature dynamics ẋ i (t) =h i Pi (t) Non-reneable poer P n i (t) P n i (t) Temperature constraint x i (t) 2 [, ], 8i Z T Quadratic cost to reduce variability [ T T i P n i (t)] 2 dt 9/4 Theorem: Optimal policy still synchronizes loads Optimal solution: Reduction of variability ith temperature constraint Loads ill remain synchronized after this time instant Temperature Load 2 Counter-intuitive?? Load Question: Is there some modification in the model or cost function hich leads to de-synchronization? 2/4 Ho to induce desynchronization: Markov model for changes in Θ max Stochastic control problem: Stochastic variation of temperature constraints Suppose users occasionally change setting at the same time E.g. Super Bol Sundays @ game time. Wind process: P i (t) Markov process Temperature dynamics: ẋ i (t) =h i Pi (t) Pi n (t) E.g. 2 max max is a to state Markov process Comfort range t 2 max / h / l max 2/4 Non-reneable poer Pi n (t) Stochastic comfort level (t) Markov process, (t) 2 { max, 2 max} Temperature constraint: x i (t) 2 [, 2 max], 8i Maximum cooling rate: Pi n (t) =M If x i(t) > (t) Z Quadratic cost: T [ Pi n (t)] 2 dt T T i 22/4 Optimal de-synchronization and resynchronization It is optimal to break symmetry at high temperatures Hedges against the future eventuality that the thermostats are sitched lo Vector field of optimal solution Nature of optimal solution De-synchronization at high temperatures Re-synchronization at lo temperatures Temperature 2 max max Temperature load 2 min Time Temperature load 23/4 Vector field of temperature changes 24/4
July 3, 2, P. R. Kumar July 3, 2, P. R. Kumar Local concavity/convexity of optimal cost-to-go resulting from HJB equation A heuristic Approximate optimal policy Keep the temperatures apart Locally concave De-synchronization above temperature 9» Provide poer to load ith minimum temperature amongst all loads ith temperature higher than 2» Bring the temperatures together for loads in[ min, 2 ] Cost to go 8 7 6 5 Bring the temperatures together 4 3 Poer is assumed affine in [ min, 2 ] and [ 2, max ] Policy is a function of a fe parameters, optimize iteratively 2 8 6 4 Load 2 To loads optimal solution along x = x2 2 2 3 4 5 6 7 8 9 Load Non reneable poer min But optimal policy is difficult to compute 25/4 2 2max Temperature But requires intrusive sensing 26/4 July 3, 2, P. R. Kumar July 3, 2, P. R. Kumar Thermostatic control ith set points Z A possibly implementable architecture of a solution Load service entity - Senses ind ddpoer - Sets set ddpoints Zi Z2 ZN 27/4 28/4 July 3, 2, P. R. Kumar Control policy July 3, 2, P. R. Kumar Information flo in architecture Information from LSE to consumer Minimal information needed to be responsive? Wind not bloing Zi Wind bloing or not = Price signal LSE need not set thermostat set-points Only needs to set empirical distribution of set-points Not detailed actuation Ambient temperature rise t Cooling using ind No flo of state information from home to LSE Information and communication requirements Price signal to consumers Infrequent distribution signaling to consumers Cooling using non-reneable 29/4 (LSE also monitors total poer usage by consumers) 3/4
Overall optimization problem Stochastic Wind process: Temperature dynamics: Pi W t Overall optimization problem Stochastic Wind process: Temperature dynamics: Pi W t Comfort setting dynamics: 2 Cost: Min 4 T T Z T 2 P g i (t) + 3 [(x i (t) M i (t)) + ] 2 dt5 2 Cost: Min 4 T T Z T 2 P g i (t) + 3 [(x i (t) M i (t)) + ] 2 dt5 Grid poer variation cost User discomfort cost Grid poer variation cost User discomfort cost 3/4 32/4 Overall optimization problem Ho to choose {Z so as to minimize: 2,Z 2,...,Z N } Z 2 3 Min 4 T P g i T T (t) + [(x i (t) M i (t)) + ] 2 dt5 Difficult: Complex as N is large, high dimensional. Need to solve different problem for different N Solution: Study asymptotic it as N. Solution becomes explicit And asymptotic solution is also nearly optimal even for small N 33/4 Continuum it of Z-policy Continuum of loads in [,] u(x)= fraction of loads ith set-points less than x, = empirical distribution of set-points Cost function C [,] (u) = Z 2 Masses at set points Z 2 (z)u (z)dz +(h) 2 ( 2 + u 2 (z Z 2 (h) 2 ( 2 + u 2 (z)p({ z = z} \ { z+dz <z+ dz})) 34/4 Continuum it optimization problem Resulting variational problem: Solution to variational problem (Euler-Lagrange): Optimal solution of continuum it u (z) @F @u d @F dx @u =) 2(h)2 u(x)d(x) d dx (x) =) u(x) = (x) 2(h) 2 D(x) 2 Z Not so fast, singularity: Solution is given by: u (z) = @ 2 F = @u2 ( min, (z) 2(h) 2 D(z) If z< 2 If z = 2. 35/4 Optimal desynchronization of demand response 36/4
Z-policy: Finite population approximation from continuum it asymptotic Some simulation results Generate {Z i } N it to approximate continuum This appears to ork very ell even hen N is small Even N = 5 Distribution.8.6.4.2 Optimal infinite case distribution Approximation based finite distribution Optimal finite distribution Nonreneable cost Variation cost.8 Optimal grid cost.6 Grid cost for generated Z.4.2..5.5 2 2.5 3 3.5 4 4.5 5 Optimal temperature variation cost Time.8 Variation cost for generated Z x 4.6.4.2 2 3 4 5 6 7 8 9 Setpoint, Z Total cost.8.5.5 2 2.5 3 3.5 4 4.5 5 Optimal total cost Time x 4.6 Total cost for generated Z.4.2 37/4.5.5 2 2.5 3 3.5 4 4.5 5 Time x 4 38/4 Concluding remarks Attempt to develop an architecture and tractable solution for demand response Many extensions needed and feasible Response to comfort variations Availability of ind poer Generalize ind model, temperature dynamics, etc. Thank you 39/4 4/4