The best-deterministic method for the stochastic unit commitment - PowerPoint PPT Presentation

The best-deterministic method for the stochastic unit commitment problem Boris Defourny Joint work with Hugo P. Simao, Warren B. Powell Feb 21, 2013 DIMACS Workshop on Energy Infrastructure: Designing for Stability and Resilience

The challenge of using wind energy [MW] This is 5-min data of energy injected by a wind farm 80 80 60 60 40 40 20 20 0 0 14:00 16:00 18:00 20:00 22:00 14:00 16:00 18:00 20:00 22:00 0:00 0:00 2:00 2:00 4:00 4:00 6:00 6:00 8:00 8:00 10:00 12:00 14:00 10:00 12:00 14:00 Wind: complex to forecast; high-dimensional process. Net power injections at each bus (node) from generators and loads, including wind bus: f ( v , θ )= p active power balance g ( v , θ )= q reactive power balance voltage magnitude and angle at each bus, assuming steady state @ 60Hz Loads Generators Generating units must balance variations from stochastic injections (load:- and wind:+) in real-time. The control relies on frequency changes and on signals sent by the system operator. 60.05 Hz 60 Hz 59.95 Hz B. Defourny (Princeton) The best-deterministic method for the stochastic unit commitment problem DIMACS 2/21/2013 2/28

Decision time lag for steam turbines F.P. de Mello, J.C. Wescott, Steam Plant Startup and Control in System Restoration, IEEE Trans Power Syst 9(1) 1994 Start-up of a gas-fired steam turbine Steam units need time to start up and after a 7-hour shutdown. be online (spin at required frequency). They must be committed to produce power in advance. Initial period where 150 the unit is committed to produce power Output [MW] time lag 100 50 00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 00:00 Time Ignition (estimate) B. Defourny (Princeton) The best-deterministic method for the stochastic unit commitment problem DIMACS 2/21/2013 3/28

Aggregated cost curves say: Do not wait too long Cost-based offer curve of dispatchable units Total Marginal Cost [$/MWh] Capacity [GW] Units that can be started up on Data: EIA-860 & Ventyx Velocity Suite short notice Units to be committed in advance must-run units Cumulative Capacity [MW] Assumptions for this graph: No transmission constraints. No startup costs. Not plotted: Pumped Storage , Hydro, Wind, Solar. We are plotting curves from cost estimates, not bids. B. Defourny (Princeton) The best-deterministic method for the stochastic unit commitment problem DIMACS 2/21/2013 4/28

Offer dynamics for peaker units daily bids of a combustion turbine bidding a single price-quantity block, year 2010 Price [USD/MWh] Quantity [MW] 125 130 120 100 75 110 50 100 J F M A M J J A S O N D J F M A M J J A S O N D Natural Gas Price [USD/MMBtu] Temperature [F] 6 90 winter winter 5 65 summer 4 40 3 15 J F M A M J J A S O N D J F M A M J J A S O N D B. Defourny (Princeton) The best-deterministic method for the stochastic unit commitment problem DIMACS 2/21/2013 5/28

Multistage stochastic unit commitment Stochastic formulation with startup decision time lags δ j (12h, 6h, 3h, 1h,…) given {W tj } : random process for variable energy resource j in J VER {L t } : random demand process 0-1 indicator of # periods # units output of unit j startup at time t 𝐾 𝑈 𝔽 { minimize c tj start v t- δ j, tj + c tj p ttj } startup & energy cost 𝑢=1 𝑘=1 startup cost energy unit-cost 𝐾 energy balance subject to 𝑘=1 p ttj = L t a.s., for each t (assuming NO demand-side flexibility) constraints for dispatchable j ∈ J D , for each t: 0-1 shutdown indicator v t- δ j , tj - w t- δ j , tj = u t- δ j , tj – u t- δ j -1, t-1, j decision for lagged startup decisions time t’ 0-1 online u t- δ j , tj P j ≤ p ttj ≤ u t- δ j , tj P j capacity constraints t t’ j state unit j -R j down ≤ p ttj – p t-1, t-1, j ≤ R j up ramping constraints (simplified statement) … F t -measurable constraints for variable energy resources j in J VER [curtailment] p ttj ≤ u t- δ j , tj W tj a.s., for each t u t- δ j , tj , v t- δ j , tj , w t- δ j , tj ∈ {0,1}. B. Defourny (Princeton) The best-deterministic method for the stochastic unit commitment problem DIMACS 2/21/2013 6/28

Multistage stochastic unit commitment updated information Locked commitments for slow-start units D-1 12:00 samples in high-dimensional uncertainty space D-1 D D D 18:00 0:00 1:00 … recommitments and redispatching B. Defourny (Princeton) The best-deterministic method for the stochastic unit commitment problem DIMACS 2/21/2013 7/28

Two-stage stochastic unit commitment Stochastic MILP formulation in the day-ahead paradigm: Time lags δ j valued in {12h, 0h} only (slow- and fast- start). 𝐾 𝑈 𝔽 { minimize c tj start v t- δ j, tj + c tj p t- δ j, tj } 𝑢=1 𝑘=1 𝐾 subject to 𝑘=1 p ttj = L t a.s., for each t constraints for dispatchable j ∈ J D : v 0tj – w 0tj = u 0tj – u 0, t-1, j j in slow-start units: u 0tj P j ≤ p ttj ≤ u 0tj P j lock the day-ahead startups v ttj -w ttj = u ttj – u t-1, t-1, j j in fast-start units: u ttj P j ≤ p ttj ≤ u ttj P j do not lock day-ahead startups -R j down ≤ p ttj – p t-1, t-1, j ≤ R j up constraints for variable energy resources j in J VER p ttj ≤ u ttj W tj a.s., for each t u 0tj , v 0tj , w 0tj (j slow), u ttj , v ttj , w ttj (j fast ) ∈ {0,1}. Each u 0tj (j slow start) is implemented as a here-and-now decision. B. Defourny (Princeton) The best-deterministic method for the stochastic unit commitment problem DIMACS 2/21/2013 8/28

Two-stage stochastic unit commitment updated information Locked commitments for slow-start units D-1 12:00 samples in high-dimensional scenario space 0:00-23:55 (i.e. whole day D) Perfect dispatch over day D (since whole day is visible) B. Defourny (Princeton) The best-deterministic method for the stochastic unit commitment problem DIMACS 2/21/2013 9/28

Deterministic unit commitment output of variable energy source load W tj , L t are set to forecasts W 0tj , L 0t . W 0tj 𝐾 𝑈 minimize c tj start v 0tj + c tj p 0tj 𝑢=1 𝑘=1 𝐾 subject to 𝑘=1 p 0tj = L 0t 𝐾 D –p 0tj ) ≥ S 0t S 0t 𝑘=1 (u 0tj P j reserve requirements constraints for dispatchable j ∈ J D : v 0tj - w 0tj = u 0tj – u 0, t-1, j decision for time t u 0tj P j ≤ p 0tj ≤ u 0tj P j 0 t j unit j -R j down ≤ p 0tj – p 0, t-1,j ≤ R j up F 0 -measurable constraints for variable energy resources j in J VER p 0tj ≤ u 0tj W 0tj W 0tj u 0tj , v 0tj , w 0tj ∈ {0,1}. Each u 0tj (j slow start) is implemented as here-and-now decision. B. Defourny (Princeton) The best-deterministic method for the stochastic unit commitment problem DIMACS 2/21/2013 10/28

Practical complexity of stochastic unit commitment 2006 2010 1968 1990 Convex PJM completes a 6-year effort Early stochastic parallel mixed-integer computing for multistage of deploying and integrating its stochastic security-constrained MILP unit linear programming solving stochastic (MILP) model for programs programming commitment is intractable (*) unit commitment (*) For generic convex programs, using the sample average approximation Abstract idealized setup: Reality : Dream : solve the 2-stage MILP model We have tools to reduce to 1-2% 1 st -stage decision probability of scenario k the optimality gap of the MILP 𝐿 min f( x )+ g( x , y k , 𝝄 k ) SP: p k 𝑙=1 P( 𝝄 ): min f( x )+ g( x , y , 𝝄 ) s.t. x ∈ 𝒴 , y k ∈ 𝒵 ( x , 𝝄 k ) k=1,…, K. s.t. x ∈ 𝒴 , y ∈ 𝒵 ( x , 𝝄 ). scenario k 2 nd -stage decisions B. Defourny (Princeton) The best-deterministic method for the stochastic unit commitment problem DIMACS 2/21/2013 11/28

Best-Deterministic Approximation • Let v*, S be the optimal value and first-stage solution set of the stochastic program. Let x* ∈ S. • Let v(x) be the optimal value of the stochastic program when the first- stage decision is fixed to x. We have v(x*)=v* for all x* ∈ S. v(x) can be evaluated by optimizing separately over each scenario. Let S’( 𝝄 ) be the optimal first-stage solution set of the stochastic program • with its probability distribution degenerated to 𝝄 . Let x’( 𝝄 ) ∈ S’( 𝝄 ). • Value of the Stochastic Solution [Birge 1982]: ))-v(x*) where 𝝄 = VSS = v(x’( 𝝄 𝐿 𝝄 k p k 𝑙=1 J.R. Birge, The value of the stochastic solution in stochastic linear programs with fixed recourse, Math. prog. 24, 314-325, 1982. • Value of the stochastic solution over the best-deterministic solution: VSS BD = inf 𝝄∈𝚶 [v(x’( 𝝄 ))- v(x*)] for 𝚶 : space easy to cover. • Best-deterministic approximation: )) and then implement x’( 𝝄 *). Try to find 𝝄 * ∈ arg min 𝝄∈𝚶 v(x’( 𝝄 B. Defourny (Princeton) The best-deterministic method for the stochastic unit commitment problem DIMACS 2/21/2013 12/28