Optimal Demand Response - PDF Free Download

Optimal Demand Response Libin Jiang Steven Low Computing + Math Sciences Electrical Engineering Caltech June 2011

Outline o Motivation o Demand response model o Some results

Wind power over land (outside Antartica): 70 170 TW World power demand: 16 TW Solar power over land: 340 TW Source: M. Jacobson, 2011

Why renewable integration? Global energy demand (16 TW) Global electricity demand (2.2 TW) Global wind power potential (72 TW) Global installed wind capacity (128 GW) Source: Cristina Archer, 2010

US electricity flow 2009 quadrillion Btu Fossil : 67% Conversion loss: 63% Plant use: 2% T&D losses: 2.6% Nuclear: 22% Gross gen: 37% Renewable: 11% US total energy use: 94.6 quads For electricity gen: 41% End use: 33% Source: EIA Annual Energy Review 2009

Renewables are exploding o Renewables in 2009 n 26% of global electricity capacity n 18% of global electricity generation n Developing countries have >50% of world s renewable capacity n In both US & Europe, more than 50% of added capacity is renewable o Grid-connected PV has been doubling/yr for the past decade, 100x since 2000 o Wind capacity has grown by 27%/yr from 2004 09 Source: Renewable Energy Global Status Report, Sept 2010

Key challenges: uncertainty High Levels of Wind and Solar PV Will Present an Operating Challenge! Source: Rosa Yang

Today s demand response o Load side management has been practiced for a long time n E.g., direct load control o They are simple and small n Centralized and static n Invoked rarely o Hottest days in summer n Small number of participants o Usually industrial, commercial o Because n Simple system is sufficient n Lack of sensing, control, & 2-way communication infrastructure

Tomorrow s demand response o Benefits n Adapt elastic demand to uncertain supply n Reduce peak, shift load o Much more scalable n Distributed n Real-time dynamic n Larger user participation It d be cheaper to use photons than electrons to deal with power shortages!

Source: Steven Chu, GridWeek 2009

Outline o Motivation o Demand response model o Some results

Features to capture Wholesale markets n Day ahead, real-time balancing Renewable generation n Non-dispatchable Demand response n Real-time control (through pricing) day ahead balancing renewable utility utility users users

Model: user Each user has 1 appliance (wlog) n Operates appliance with probability π i (t) n Attains utility u i (x i (t)) when consumes x i (t) x i (t)! x i (t)! x i (t) " x i (t) # X i t Demand at t: D(t) :=! i! i x i (t) "! i = 1 wp " (t) i # $ 0 wp 1! " i (t)

Model: LSE (load serving entity) Power procurement n Renewable power: ( P ( t) ) 0 ( t), c = o Random variable, realized in real-time P r r r capacity energy n Day-ahead power: (t), c d ( (t)), c ( o!x(t)) o Control, decided a day ahead n Real-time balancing power: o P ( t) = D( t) P ( t) P ( t) b r d P b ( P ( )) ( t), c t b b Use as much renewable as possible Optimally provision day-ahead power Buy sufficient real-time power to balance demand

Key assumption Simplifying assumption n No network constraints

Questions Day-ahead decision should LSE buy from day- n How much power ahead market? Real-time decision (at t-) x i (t) n How much should users consume, given realization of wind power P r (t) and? How to compute these decisions distributively? How does closed-loop system behave? δ i available info: decision: u i t 24hrs ( ), π, F ( ) i r t-! i, P r (t), x i (t)

Our approach Real-time (at t-) n Given and realizations of, choose optimal x i (t) = x i ( P r(t),! i ;P r (t),! i ) to max social welfare, through DR Day-ahead n Choose optimal that maximizes expected optimal social welfare available info: decision: u i t 24hrs ( ), π, F ( ) i r t-! i, P r (t), x i (t)

Outline o Motivation o Demand response model o Some results n T=1 case: distributed alg n General T: distributed alg n Impact of uncertainty

T=1 case Each user has 1 appliance (wlog) n Operates appliance with probability π i (t) n Attains utility u i (x i (t)) when consumes x i (t) x i (t)! x i (t)! x i (t) " x i (t) # X i t Demand at t: D(t) :=! i! i x i (t) "! i = 1 wp " (t) i # $ 0 wp 1! " i (t)

Welfare function Supply cost c(, x) = c ( d P ) d + c ( o!(x)) 0 + c ( b!(x)" P ) d +!(x) := "! i x i # P r i excess demand Welfare function (random) W, x ( ) =! i u i! (x i )" c, x i ( ) user utility supply cost

Welfare function Supply cost c(, x) = c ( d P ) d + c ( o!(x)) 0 + c ( b!(x)" P ) d +!(x) := "! i x i # P r excess demand i c d P ( ) c d (!(x)) d c 0 b (!(x)" ) +!(x)

Optimal operation Welfare function (random) W, x ( ) =! i u i! (x i )" c, x i ( ) Optimal real-time demand response x ( ) := arg max Optimal day-ahead procurement := arg max EW, x x W (, x) given realization of P r,! i ( ( )) Overall problem: max E maxw (, x) x

Algorithm 1 (real-time DR) max E maxw (, x) x real-time DR Active user i computes n Optimal consumption x i LSE computes n Real-time price µ b n Optimal day-ahead power to use n Optimal real-time balancing power y o y b

Algorithm 1 (real-time DR) Active user i : ( ( ( ) k! µ )) b xi x i k+1 = x i k +! u i ' x i k x i inc if marginal utility > real-time price LSE : ( ( ( ) " y k k o " y )) b + µ b k+1 = µ b k +!! x k inc if total demand > total supply Decentralized Iterative computation at t-

Algorithm 1 (real-time DR) Theorem: Algorithm 1 Socially optimal n Converges to welfare-maximizing DR x = x n Real-time price aligns marginal cost of realtime power with individual marginal utility µ b = c b ' ( y ) ' b = u i ( x ) i ( P ) d Incentive compatible n x i max i s surplus given price µ b

Algorithm 1 (real-time DR) Theorem: Algorithm 1 Marginal costs, optimal day-ahead and balancing power consumed: c b ' ( y ) ' b = c o ( y ) o + µ o ( ) µ o =!W!

Algorithm 2 (day-ahead procurement) Optimal day-ahead procurement max EW (, x ( P )) d LSE: P m+1 d = ( P m d +! m ( µ m m o! c d '( P ))) d + calculated from Monte Carlo simulation of Alg 1 (stochastic approximation)

Algorithm 2 (day-ahead procurement) Optimal day-ahead procurement max EW (, x ( P )) d LSE: P m+1 d = ( P m d +! m ( µ m m o! c d '( P ))) d + Given! m, P r m : ( m ) µ m o =!W! µ b m = µ o m + c o ' ( m y ) o

Algorithm 2 (day-ahead procurement) Theorem Algorithm 2 converges a.s. to optimal for appropriate stepsize! k

Outline o Motivation o Demand response model o Some results n T=1 case: distributed alg n General T: distributed alg n Impact of uncertainty

General T case Each user has 1 appliance (wlog) n Operates appliance with probability π i (t) n Attains utility u i (x i (t)) when consumes x i (t) x i (t)! x i (t)! x i (t) " x i (t) # X i t Coupling across time è Need state Demand at t: D(t) :=! i! i x i (t) "! i = 1 wp " (t) i # $ 0 wp 1! " i (t)

Time correlation o Example: EV charging n Time-correlating constraint: o Day-ahead decision and real-time decisions T t= 1 x () t R, i i i Day-ahead t- t = 1,2,, T available info: decision: u i (), F () (), t r t P t P t R t r(), d (), i() x i ( t), i Remaining demand o (1+T)-period dynamic programming

Algorithm 3 (T>1) o Main idea n Solve deterministic problem in each step using conditional expectation of P r (distributed) n Apply decision at current step o One day ahead, decide by solving, x ( ) max W P ( τ), x( τ); P( τ) st.. x ( τ) R P d T d r i i τ= t τ= 1 o At time t-, decide x (t) by solving T ( ) T max W P ( τ), x( τ); P( τ t) st.. x ( τ) R( t) x d r i i τ= t τ= t T

Algorithm 3 (T>1) Theorem: performance o Algorithm 3 is optimal in special cases o J A3! J! 1 2 T P r 2-500 -1000 Welfare with heuristic algorithm Welfare with idealized algorithm Welfare -1500-2000 -2500-3000 1 2 3 4 ρ: penetration of renewable energy

Effect of renewable on welfare Renewable power: P r (a, b) := a!µ r + b!v r mean zero-mean RV Optimal welfare of (1+T)-period DP W P r (a, b) ( )

Effect of renewable on welfare P r (a, b) := a!µ r + b!v r Theorem o Cost increases in var of o o W P r (a, b) ( ) W P r (s, s) ( ) increases in a, decreases in b increases in s P r