A Progressive Hedging Approach to Multistage Stochastic Generation and Transmission Investment Planning Yixian Liu Ramteen Sioshansi Integrated Systems Engineering Department The Ohio State University 9th Annual Trans-Atlantic INFRADAY Conference Washington, DC 30 October, 2015
Disclaimer The following are my own views and not necessarily those of the Electricity Advisory Committee, the U.S. Department of Energy, or anyone else.
Overview Introduction Progressive Hedging Computational Results
Background How to satisfy electricity demands with minimum costs in the long run? Incentives for new investments: new technologies, increasing electricity demand, power plant retirements Perspective: a centralized model where generation and transmission investments are planned together Scope: consider long investment periods, multiple electricity-generating technologies, and uncertainties
Challenges There are uncertainties in the long run and in the operating stage These uncertainties occur at different scales The modeling of multiple stages and uncertainties increases the number of scenarios dramatically Tractability issues as the problem gets larger
Model Features Stochastic, multi-stage, multi-scale model investment stage: investment decisions made yearly or at coarser scale operating stage: operating decisions made hourly or at finer scale
Multi-Scale Uncertainty Investment stage: demand growth, fuel prices, environmental regulations, investment costs Operating stage: demand pattern, generation availability, wind speeds, solar insolation
Model Outline Decision variables: investment decisions + operating decisions Objective function: min E[investment cost + operating cost] Main constraints: nonanticipativity: decisions made in current stage cannot be dependent on future scenario information energy-balance: supply = demand capacity: operating decisions are limited to existing capacities ramping: the rate a generator changes its output is limited storage: define storage level, charging and discharging of storage devices power-flow: cannot exceed thermal limits of transmission lines Complete Formulation
Model Tractability Two problems Need many operating stages to capture fine-scale uncertainties Need many investment-stage scenarios to capture large-scale uncertainties Solutions Representative operating-stage periods Decomposition of investment-stage problem
Operating-Stage Periods Problem: Model is intractable if solving dispatch decisions for every hour in the operating stage Standard Solution: Use representative hours, based on LDC, to represent operating stage Loses correlations between load, wind, and solar Cannot model intertemporal constraints (e.g., storage, ramping) Our Solution: Use representative days with intact correlation structures and intertemporal constraints in operating stage
Representative Days Each representative day contains one day s hourly load, solar, and wind data at all buses Clustering method that respects the correlation among variables, locations, and time is used Figure: There are 72 data points in a representative day for each bus
Clustering Methods Method 1 Hierarchical Clustering using Dynamic Time Warping Dynamic Time Warping: measures similarity between two time series, which may vary in time Method 2 Step 1: Use k-means clustering, with Euclidean distance as a metric, to find a starting set of clusters Step 2: Apply Method 1 within each cluster to find representative days
Clustering Test Dataset: One year s hourly wind, solar, and load data for three cities in Texas Model: An investment model with one investment stage and 20 years operation Method: Run the model with original dataset and with representative days from Methods 1 and 2, separately Compare investment decisions and total cost Investigate how decisions change as a function of model inputs
Investment Results
Clustering Results The two clustering methods perform pretty equally well overall Method 2 takes less time to implement: Method 1 takes about 15 minutes of wall time in R studio as opposed to 2 minutes for Method 2 30 clusters (representative days) gives a good approximation of original dataset
Progressive Hedging Computational Results Investment-Stage Scenarios Problem: Model may need many investment-stage scenarios to capture coarse-grain uncertainties Solution: Progressive hedging algorithm [Rockafellar and Wets, 1991] Suppose we have the following stochastic problem: min X p s f s (X(s)) s S s.t. X(s) C s X(s) is implementable // X(s) is admissible (1)
Decomposition Progressive Hedging Computational Results The scenario-s problem: Penalized scenario-s problem: min f s(x(s)) + X(s) s.t. X(s) C s min f s(x(s)) (2) X(s) s.t. X(s) C s [ W X(s) + ρ 2 X(s) ˆX 2] (3) W : Lagrange multiplier vector ρ: positive penalty parameter, introduced to attain convergence stability in an algorithmic sense ˆX= s S p sx(s): average of X(s) s
Progressive Hedging Algorithm Progressive Hedging Computational Results
Lower Bound Progressive Hedging Computational Results A feasible solution gives an upper bound [Gade et al., 2013] show that duals of the non-anticipativity constraints in two-stage stochastic MIPs define implicit lower bounds p s [min f s (X(s)) + W X(s)] s We show a similar bound exists for multi-stage stochastic problems Allows us to assess the quality of a progressive hedging solution LB can be obtained with approximately the same effort as one PH iteration
Performance of PHA Progressive Hedging Computational Results Table: Performance of PHA with Different Number of Representative Days in Operating Stage Full Problem PHA CPU Time Objective CPU Time Upper Bound Lower Bound Days Variables Constraints [s] [$ billion] [s] [$ billion] [$ billion] 3 1337856 2248967 2968 95.096 1161 95.122 95.095 9 3992064 6727943 16535 100.988 5898 101.011 100.988 Tested on a three-region system Four investment periods, each operating stage lasts 10 years 128 scenarios in total
Investment Decisions Progressive Hedging Computational Results The investment decisions from the original model and the decomposed model are similar (maximum and average absolute differences < 2% and 0.3%) Table: Investment Capacities [MW] Investment No Decomposition Decomposed Model Period Bus 1 Bus 2 Bus 3 Bus 1 Bus 2 Bus 3 1 37717 3495 7781 37672 3502 7791 2 1985 2427 3634 1982 2423 3632 3 4509 2621 6728 4528 2621 6731 4 8402 949 2082 8407 951 2042
Progressive Hedging Computational Results Upper Bounds and Lower Bounds 3 Representative Days 9 Representative Days
Conclusions The proposed multi-stage, multi-scale stochastic model can be effectively solved using PHA By using representative days, inter-temporal constraints can be captured in long-term investment decisions Future Work Comprehensive numerical case study Model generation and transmission investments in a market framework Model transmission network with DC power flow
References Rockafellar, R.T., Wets, R.J.B. (1991) Scenarios and policy aggregation in optimization under uncertainty Mathematics of Operations Research 16 (1) 119 147. Gade, D., Hackebeil, G., Ryan, S., Watson, J., Wets, R., Woodruff, D. (2014) Obtaining Lower Bounds from the Progressive Hedging Algorithm for Stochastic Mixed-Integer Programs Mathematical Programming, submitted for publication.
Acknowledgment This material is based upon work supported by the National Science Foundation under Grant No. CBET-1029337
Thanks!
Mathematical Formulation Notation Formulation Indices n t r d l O(l) D(l) ω τ Ω t (ω) index for regions in the network index for investment periods index for representative days index for hours in representative days index for transmission lines in the network index for the sending-end node of transmission line l index for the receiving-end node of transmission line l index for scenarios index for technologies set of parameters defining scenario ω in investment period t
Mathematical Formulation Notation Formulation Parameters D t,n,r,d (ω) c τ,t,n (ω) ct,l L (ω) c τ,t,n (ω) X ES,n f max l,es B l Ft,n,r,d τ R τ N r η h α(ω) electricity demand in period t in region n at hour d in representative day r per-mw cost of building technology τ in region n in period t per-mw cost of increasing capacity of transmission line l per-mwh cost of energy from technology τ in region n in scenario ω existing storage capacity in region n at the beginning of the planning horizon existing capacity of transmission line l at the beginning of the planning horizon susceptance of transmission line l capacity factor for technology τ in period t in region n at hour d in representative day r ramping factor for technology τ weight on representative day r round-trip efficiency of storage device number of hours of storage probability of scenario ω
Mathematical Formulation Notation Formulation Variables Xt,n τ (ω) Y t,l (ω) Pt,n,r,d ST (ω) P STD t,n,r,d (ω) P STC t,n,r,d (ω) P τ t,n,r,d (ω) f t,l,r,d (ω) θ t,n,r,d (ω) UD t,n,r,d (ω) MW of technology τ added in region n in investment period t MW added to transmission line l in investment period t ending state of charge of storage in period t in region n at hour d in representative day r power discharged from storage in period t in region n at hour d in representative day r power charged into storage in period t in region n at hour d in representative day r MW produced by technology τ in period t in region n at hour d in representative day r power flow through transmission line l in period t at hour d in representative day r voltage angle in period t in region n at hour d in representative day r unsatisfied demand in period t in region n at hour d in representative day r
Mathematical Formulation Notation Formulation min α(ω) c t,n τ (ω)xτ t,n (ω) + c l,t L (ω)y l,t (ω) + Nr c t,n τ (ω)pτ t,n,r,d ω τ t n t l τ t n r d s.t. 0 X τ t,n (ω) Xτ,max n, ω, τ, t, n 0 Y t,l (ω) Y l max, ω, t, n X τ t,n (ω k ) = Xτ t,n (ω k ) : Ωm(ω k ) = Ωm(ω k ), m < t, τ, ω, t, n Y t,l (ω k ) = Y t,l (ω k ) : Ωm(ω k ) = Ωm(ω k ), m < t, ω, t, n c t,n τ (ω)xτ t,n (ω) + c l,t L (ω)y t,l (ω) cmax t, ω, t τ n l P t,n,r,d τ (ω) + PSTD t,n,r,d (ω) PSTC t,n,r,d (ω) + UD t,n,r,d (ω) τ f t,l,r,d (ω) + f t,l,r,d (ω) = D tnrd (ω), ω, t, n, r, d l O(l)=n l D(l)=n P t,n,r,d ST (ω) = PST t,n,r,d 1 (ω) PSTD t,n,r,d (ω) + ηpstc t,n,r,d (ω), ω, t, n, r, d P t,n,r,0 ST (ω) = 1 h X ES,n ST t + Xm,n ST (ω), ω, t, n, r 2 m=0
Mathematical Formulation Notation Formulation P t,n,r,24 ST (ω) = 1 h X ES,n ST t + Xm,n ST (ω), ω, t, n, r 2 m=0 0 P t,n,r,d ST (ω) h X ES,n ST t + Xm,n ST (ω), ω, t, n, r, d m=0 0 P t,n,r,d STC (ω), PSTD t,n,r,d (ω) XST ES,n + t Xm,n ST (ω), ω, t, n, r, d m=0 0 P t,n,r,d τ (ω) Fτ t,n,r,d X ES,n τ t + Xm,n τ (ω), ω, τ, t, n, r, d m=0 Rτ X ES τ t + Xm,n τ (ω) P t,n,r,d τ (ω) Pτ t,n,r,d 1 (ω) Rτ X ES τ t + Xm,n τ (ω), ω, τ, t, n, r, d m=0 m=0 f t,l,r,d (ω) = B l (θ t,o(l),r,d (ω) θ t,d(l),r,d (ω)), ω, t, l, r, d f l,es max t Y m,l (ω) f t,l,r,d (ω) f max t l,es + Y m,l (ω), ω, t, l, r, d m=0 m=0 π θ t,n,r,d (ω) π, ω, t, n, r, d Back