A L-shaped method for mid-term hydrogeneration scheduling under uncertainty

Pierre-Luc Carpentier, Michel Gendreau, Fabian Bastin

NSERC/Hydro-Quebec Industrial Research Chair on theStochastic Optimization of Electricity Generation

Department of Mathematics and Industrial EngineeringEcole Polytechnique de Montreal

ISCP2013, July 8-12th

Mid-term hydro scheduling

Controlled power system: Hydro plants, high dimensionalreservoir system, thermal plants, transmission network, energytrading.

Aim: Find marginal water value in hydroelectric reservoir (inputfor short-term model), water release targets, expected operatingcosts, ...

Sources of complexity:

I Time-coupling constraints: reservoirs dynamics, water delays

I Horizon: 6-60 months, weekly or monthly time steps

I Uncertainty: inflows, prices, load, wind, solar, failures, ...

I Hydro generation (non-linear & non convex constraints,forbidden zones, ...)

Mathematical formulation

Objective

minEξ1,...,ξT

[T∑t=1

ct(ut , ξt)−Ψ(vT )

]ut : control vector at t (release, power generation, ...)vt : state vector at t (storage)ξt : random vector at t (inflow, load, prices, ...)ct : cost function at t (convex, piecewise linear)Ψ: value function at TConstraints

I Physical: Water/energy balance, hydroelectric generation, ...

vt = ft(vt−1, ut , ξt) : dynamic

ut ∈ Ut(vt , ξt) : static

I Non-anticipativity: ut ∈ Ft(ξ1, ..., ξt)

Stochastic process ξt : t = 1, ...,TI Autocorrelation, continuous distribution, high dimensional.

Scenario tree approximation

T : scenario tree, ω ∈ Ω : scenarios (realizations)Traditional methods:

I Stage-wise decomposition : nested Benders

I Scenario-wise: progressive hedging

Challenges

I Exponential growth of problem size with the branching level.

I Coarse discretisation

I Stability of solutions, optimality gap?

Proposed two-stage decomposition

Definitions

I t = 1, ..., τ : stage 1

I t = τ + 1, ...,T : stage 2

I vτ ∈ Rn: storage at the end of stage 1.

I Q(vτ , ξ1, ..., ξτ ): expected cost-to-go at stage 2.

Assumption A1ξt : t = 1, ...,T has a memory loss at τ .

Consequences:

I T possess a special structure

I Eliminate redundancy at stage 2

I Q(vτ , ξ1, ..., ξτ ) = Q(vτ )

Examples

Ex. 1 T = 6, τ = 3 with two types of second-stage subtrees.

Ex. 2 T = 104, τ = 52, ξt ∈ R5, 3 outcomes per branching stage

BRANCHING STAGES SCENARIOS WITHOUT A1 WITH A1

12 105 230 MB 566 kB16 107 14 GB 3 MB20 109 754 GB 25 MB24 1011 61 TB 230 MB28 1013 3,710 TB 1.6 GB32 1015 105 TB 14 GB

Branching stages |Ω| MEM

20 109 754 GB24 1011 61 TB28 1013 3,710 TB

Two-stage decomposition scheme

Master problem (subtree T1)

(M) minET1

[τ∑

t=1

ct(xt , ξt) +Q(vτ )

]

Expected cost at the second stage

Q(·) :=Nσ∑σ=1

pσQσ(·)

σ : index for second stage subtrees

Second-stage subproblems σ (subtree T σ2 )

(Sσ(v)) Qσ(v) := min

ET σ

2

[T∑

t=τ+1

ct(xt , ξt)

]: vτ = v

Extended L-Shaped (ELS) algorithm

At each iteration k = 1, 2, ...Step 1. Solve* M to find a set V = v` : ` ∈ L1 of feasible statevector v` at each first-stage leaf ` ∈ L1*e.g. nested Benders, progressive hedging, directly.Step 2. Benders cuts generationFor each state vector v ∈ V and subtree σ = 1, ...,Nσ,

I Solve* Sσ(v)I Add a new cut h to M

v

Q(v)

Case study

Set-up8 reservoirs (42,144 hm3), 10 hydro plants (8.3 GW).Mean Selling price = 37.50 $ MWh−1.Load = 69.5 TWh, Thermal generation = 80 $ MWh−1.T = 104 weeks from January 1st , stochastic inflow with τ = 52.

Historical inflow (1962-2003)

1 14 27 39 52

0

2

4

6

8

10

12

14

WEEKS SINCE JANURARY 1st

TO

TA

L IN

FL

OW

(× 1

03 m

3 s

−1)

Subtrees construction method

HISTORICALINFLOW

1962-20038 dimensions

CALIBRATEMPAR(1)

GENERATE S1

SYNTHETIC SERIES

CONSTRUCT SUBTREE1

WITH M1 SCENARIOS

GENERATE S2

SYNTHETIC SERIES

CONSTRUCT SUBTREE2

WITH M2 SCENARIOS

FULL TREEM

1 X M

2 SCENARIOS

L-SHAPEDL-SHAPED

DETERMINISTIC EQUIVALENT

I Calibration MPAR(1) & generation of synthetic series usingStochastic Analysis Modeling and Simulation (SAMS)

I Subtrees construction using the probability metric-based method ofHeitsch and Romisch (2009) using SCENRED2/GAMS.

Results

SCEN DIRECT L-SHAPEDTIME RAM COST TIME RAM COST

(MB) (M$) (MB) (M$)100 41 sec 378 3.09469 5 min 81 3.09474529 2 min 1,761 169.26300 25 min 198 169.27106

1,600 10 min 4,625 147.81100 55 min 288 147.816315,041 - - - 2.7 hours 414 138.88103

10,000 - - - 5.6 hours 572 217.5730050,176 - - - 24 hours 1,084 73.92699

Solution stability

N = 15 replicationsRight figure: Cumulative marginal water value =

∑j∈J λj

λj : marginal water value for reservoir j ($ hm−3)

Solution stability

N = 15 replications|Ω| = tree with 10,000 scenarios (output of SCENRED2)

Right figure: Cumulative marginal water value =∑

j∈J λjλj : marginal water value for reservoir j ($ hm−3)

Summary

Traditional methods

I Exponential growth of memory requirement with branchinglevel

I Needs lots of RAM (4,6 GB for 1,600 scenarios)

I Coarse discretization → unstable solutions, large optimalitygap

Extended L-Shaped method

I Dramatic decrease of memory requirements

I Allows to consider much larger scenario trees

I Memory loss → higher stability

I Slower than direct resolution for small scenario trees

I Tested on a small system (8 reservoirs)

I Industrial applications: Subproblems could be solved inparallel, use Nested Benders decomposition (NBD)