A New Lower Bound from Progressive Hedging
Dinakar Gade, Sarah M. Ryan, Jean-Paul Watson, David L.
Woodruff
ICSP 2013, Bergamo
Sabre Holdings
Iowa State University
Sandia National Laboratories
University of California Davis
12 July 2013
1 / 20
Outline
1
Stochastic Mixed Integer Program
2
Progressive Hedging and Bounds
3
Stochastic Unit Commitment
4
Some Numerical Results
1 / 20
Two-Stage Stochastic Mixed Integer Program
˜
min c> x + E[f (x, ξ)]
s.t. Ax ≥ b,
(1)
(2)
where ξ˜ is a random vector defined on a probability space (Ξ, A, P) and for a
˜ f (x, ξ) is defined as:
particular realization ξ of ξ,
f (x, ξ) = min g(ξ)> y
s.t. W y ≥ r(ξ) − T (ξ)x.
p1
× Rn1 −p1
x ∈ Z+
p2
y(ξ) ∈ Z+
× Rn2 −p2
n1
c ∈ Q , g(ξ) ∈ Qn2
A ∈ Qm1 ×n1 , b ∈ Qm2
W ∈ Qm2 ×n2
T (ξ) ∈ Qm2 ×n1 , r(ξ) ∈ Qm2
(3)
(4)
First-stage variables
Second-stage variables
First- and second-stage costs
First-stage constraint coefficients and right-hand-sides
Recourse matrix
Technology matrix and right-hand-sides
2 / 20
Scenario Formulation of Two-Stage SMIP
min
X
h
i
pξ c> x(ξ) + g(ξ)> y(ξ)
(5)
ξ∈Ξ
s.t. x(ξ) − x̂ = 0,
ξ∈Ξ
(6)
Ax(ξ) ≥ b,
ξ∈Ξ
(7)
W y(ξ) ≥ r(ξ) − T (ξ)x,
ξ∈Ξ
(8)
n1 −p1
ξ∈Ξ
(9)
n2 −p2
ξ∈Ξ
(10)
x(ξ) ∈
y(ξ) ∈
x̂ ∈
pξ
x̂
(6)
X(Ξ)
Zp+1
Zp+2
Zp+1
×R
×R
×R
n1 −p1
(11)
Probability of scenario ξ
Variable to model non-anticipativity
Non-anticipativity constraints
Feasible set for scenario ξ defined by (7)-(10)
3 / 20
The Progressive Hedging Algorithm
0. Initialization: Let ν ← 0 and wν (ξ) ← 0, ∀ξ ∈ Ξ. Compute for each ξ ∈ Ξ
(xν+1 (ξ), y ν+1 (ξ)) ∈
arg min
c> x(ξ) + g(ξ)> y(ξ).
(x(ξ),y(ξ))∈X(ξ)
1. Iteration Counter Increment: ν ← ν + 1.
P
2. Aggregation: Compute x̂ν ← ξ∈Ξ pξ xν (ξ).
3. Price Update: wν (ξ) ← wν−1 (ξ) + ρ(xν (ξ) − x̂ν ), ∀ξ ∈ Ξ.
4. Decomposition: Compute for each scenario ξ ∈ Ξ
xν+1 (ξ), y ν+1 (ξ) ∈
arg min c(ξ)> x(ξ) + g(ξ)> y(ξ)
(x(ξ),y(ξ))∈X(ξ)
+ wν (ξ)> x(ξ) +
ρ
||x(ξ) − x̂ν ||.
2
We have a price-based penalty term and a proximal term added to the
objective of each scenario subproblem.
5. Termination: If all the 1st-stage scenario solutions xν+1 (ξ) agree, then stop.
Otherwise, return to Step 1.
4 / 20
The Progressive Hedging Algorithm
0. Initialization: Let ν ← 0 and wν (ξ) ← 0, ∀ξ ∈ Ξ. Compute for each ξ ∈ Ξ
(xν+1 (ξ), y ν+1 (ξ)) ∈
arg min
c> x(ξ) + g(ξ)> y(ξ).
(x(ξ),y(ξ))∈X(ξ)
1. Iteration Counter Increment: ν ← ν + 1.
P
2. Aggregation: Compute x̂ν ← ξ∈Ξ pξ xν (ξ).
3. Price Update: wν (ξ) ← wν−1 (ξ) + ρ(xν (ξ) − x̂ν ), ∀ξ ∈ Ξ.
4. Decomposition: Compute for each scenario ξ ∈ Ξ
xν+1 (ξ), y ν+1 (ξ) ∈
arg min c(ξ)> x(ξ) + g(ξ)> y(ξ)
(x(ξ),y(ξ))∈X(ξ)
+ wν (ξ)> x(ξ) +
ρ
||x(ξ) − x̂ν ||.
2
We have a price-based penalty term and a proximal term added to the
objective of each scenario subproblem.
5. Termination: If all the 1st-stage scenario solutions xν+1 (ξ) agree, then stop.
Otherwise, return to Step 1.
4 / 20
The Progressive Hedging Algorithm
0. Initialization: Let ν ← 0 and wν (ξ) ← 0, ∀ξ ∈ Ξ. Compute for each ξ ∈ Ξ
(xν+1 (ξ), y ν+1 (ξ)) ∈
arg min
c> x(ξ) + g(ξ)> y(ξ).
(x(ξ),y(ξ))∈X(ξ)
1. Iteration Counter Increment: ν ← ν + 1.
P
2. Aggregation: Compute x̂ν ← ξ∈Ξ pξ xν (ξ).
3. Price Update: wν (ξ) ← wν−1 (ξ) + ρ(xν (ξ) − x̂ν ), ∀ξ ∈ Ξ.
4. Decomposition: Compute for each scenario ξ ∈ Ξ
xν+1 (ξ), y ν+1 (ξ) ∈
arg min c(ξ)> x(ξ) + g(ξ)> y(ξ)
(x(ξ),y(ξ))∈X(ξ)
+ wν (ξ)> x(ξ) +
ρ
||x(ξ) − x̂ν ||.
2
We have a price-based penalty term and a proximal term added to the
objective of each scenario subproblem.
5. Termination: If all the 1st-stage scenario solutions xν+1 (ξ) agree, then stop.
Otherwise, return to Step 1.
4 / 20
The Progressive Hedging Algorithm
0. Initialization: Let ν ← 0 and wν (ξ) ← 0, ∀ξ ∈ Ξ. Compute for each ξ ∈ Ξ
(xν+1 (ξ), y ν+1 (ξ)) ∈
arg min
c> x(ξ) + g(ξ)> y(ξ).
(x(ξ),y(ξ))∈X(ξ)
1. Iteration Counter Increment: ν ← ν + 1.
P
2. Aggregation: Compute x̂ν ← ξ∈Ξ pξ xν (ξ).
3. Price Update: wν (ξ) ← wν−1 (ξ) + ρ(xν (ξ) − x̂ν ), ∀ξ ∈ Ξ.
4. Decomposition: Compute for each scenario ξ ∈ Ξ
xν+1 (ξ), y ν+1 (ξ) ∈
arg min c(ξ)> x(ξ) + g(ξ)> y(ξ)
(x(ξ),y(ξ))∈X(ξ)
+ wν (ξ)> x(ξ) +
ρ
||x(ξ) − x̂ν ||.
2
We have a price-based penalty term and a proximal term added to the
objective of each scenario subproblem.
5. Termination: If all the 1st-stage scenario solutions xν+1 (ξ) agree, then stop.
Otherwise, return to Step 1.
4 / 20
The Progressive Hedging Algorithm
0. Initialization: Let ν ← 0 and wν (ξ) ← 0, ∀ξ ∈ Ξ. Compute for each ξ ∈ Ξ
(xν+1 (ξ), y ν+1 (ξ)) ∈
arg min
c> x(ξ) + g(ξ)> y(ξ).
(x(ξ),y(ξ))∈X(ξ)
1. Iteration Counter Increment: ν ← ν + 1.
P
2. Aggregation: Compute x̂ν ← ξ∈Ξ pξ xν (ξ).
3. Price Update: wν (ξ) ← wν−1 (ξ) + ρ(xν (ξ) − x̂ν ), ∀ξ ∈ Ξ.
4. Decomposition: Compute for each scenario ξ ∈ Ξ
xν+1 (ξ), y ν+1 (ξ) ∈
arg min c(ξ)> x(ξ) + g(ξ)> y(ξ)
(x(ξ),y(ξ))∈X(ξ)
+ wν (ξ)> x(ξ) +
ρ
||x(ξ) − x̂ν ||.
2
We have a price-based penalty term and a proximal term added to the
objective of each scenario subproblem.
5. Termination: If all the 1st-stage scenario solutions xν+1 (ξ) agree, then stop.
Otherwise, return to Step 1.
4 / 20
The Progressive Hedging Algorithm
0. Initialization: Let ν ← 0 and wν (ξ) ← 0, ∀ξ ∈ Ξ. Compute for each ξ ∈ Ξ
(xν+1 (ξ), y ν+1 (ξ)) ∈
arg min
c> x(ξ) + g(ξ)> y(ξ).
(x(ξ),y(ξ))∈X(ξ)
1. Iteration Counter Increment: ν ← ν + 1.
P
2. Aggregation: Compute x̂ν ← ξ∈Ξ pξ xν (ξ).
3. Price Update: wν (ξ) ← wν−1 (ξ) + ρ(xν (ξ) − x̂ν ), ∀ξ ∈ Ξ.
4. Decomposition: Compute for each scenario ξ ∈ Ξ
xν+1 (ξ), y ν+1 (ξ) ∈
arg min c(ξ)> x(ξ) + g(ξ)> y(ξ)
(x(ξ),y(ξ))∈X(ξ)
+ wν (ξ)> x(ξ) +
ρ
||x(ξ) − x̂ν ||.
2
We have a price-based penalty term and a proximal term added to the
objective of each scenario subproblem.
5. Termination: If all the 1st-stage scenario solutions xν+1 (ξ) agree, then stop.
Otherwise, return to Step 1.
4 / 20
Advantages of Progressive Hedging
Can be used for multi-stage stochastic mixed-integer programs
For the convex/LP case it has good convergence properties
Can be easily parallelized
Unlike cutting plane-based decomposition algorithms (e.g. Benders’ or
L-shaped methods) there is no master program bottleneck during
parallelization
Better properties than the classical subgradient method used in
Lagrangian relaxation/dual decomposition
New! A lower bound on optimal cost can be obtained with
approximately the same effort as one PH iteration.
5 / 20
Advantages of Progressive Hedging
Can be used for multi-stage stochastic mixed-integer programs
For the convex/LP case it has good convergence properties
Can be easily parallelized
Unlike cutting plane-based decomposition algorithms (e.g. Benders’ or
L-shaped methods) there is no master program bottleneck during
parallelization
Better properties than the classical subgradient method used in
Lagrangian relaxation/dual decomposition
New! A lower bound on optimal cost can be obtained with
approximately the same effort as one PH iteration.
5 / 20
New Lower Bound
Proposition 1: The price system w(ξ) defines implicit lower
bounds
Let z ∗ be the optimal objective
P function value of the stochastic program.
n
Let w(ξ) ∈ R be such that ξ∈Ξ pξ w(ξ) = 0 (component-wise). Let
Dξ (w(ξ)) :=
Then D(w) :=
min
c> x(ξ) + g(ξ)> y(ξ) + w(ξ)> x(ξ) .
(x(ξ),y(ξ))∈X(ξ)
ξ∈Ξ pξ Dξ (w(ξ))
≤ z∗.
P
ν (ξ)
P
PH weights satisfy
ξ∈Ξ pξ w
= 0 for every ν.
Every so often, use the current weights to compute D(w).
6 / 20
New Lower Bound
Proposition 1: The price system w(ξ) defines implicit lower
bounds
Let z ∗ be the optimal objective
P function value of the stochastic program.
n
Let w(ξ) ∈ R be such that ξ∈Ξ pξ w(ξ) = 0 (component-wise). Let
Dξ (w(ξ)) :=
Then D(w) :=
min
c> x(ξ) + g(ξ)> y(ξ) + w(ξ)> x(ξ) .
(x(ξ),y(ξ))∈X(ξ)
ξ∈Ξ pξ Dξ (w(ξ))
≤ z∗.
P
ν (ξ)
P
PH weights satisfy
ξ∈Ξ pξ w
= 0 for every ν.
Every so often, use the current weights to compute D(w).
6 / 20
New Lower Bound
Proposition 1: The price system w(ξ) defines implicit lower
bounds
Let z ∗ be the optimal objective
P function value of the stochastic program.
n
Let w(ξ) ∈ R be such that ξ∈Ξ pξ w(ξ) = 0 (component-wise). Let
Dξ (w(ξ)) :=
Then D(w) :=
min
c> x(ξ) + g(ξ)> y(ξ) + w(ξ)> x(ξ) .
(x(ξ),y(ξ))∈X(ξ)
ξ∈Ξ pξ Dξ (w(ξ))
≤ z∗.
P
ν (ξ)
P
PH weights satisfy
ξ∈Ξ pξ w
= 0 for every ν.
Every so often, use the current weights to compute D(w).
6 / 20
New Lower Bound
Proposition 1: The price system w(ξ) defines implicit lower
bounds
Let z ∗ be the optimal objective
P function value of the stochastic program.
n
Let w(ξ) ∈ R be such that ξ∈Ξ pξ w(ξ) = 0 (component-wise). Let
Dξ (w(ξ)) :=
Then D(w) :=
min
c> x(ξ) + g(ξ)> y(ξ) + w(ξ)> x(ξ) .
(x(ξ),y(ξ))∈X(ξ)
ξ∈Ξ pξ Dξ (w(ξ))
≤ z∗.
P
ν (ξ)
P
PH weights satisfy
ξ∈Ξ pξ w
= 0 for every ν.
Every so often, use the current weights to compute D(w).
6 / 20
How Tight is this Bound?
Ordinary Lagrangian by dualizing non-anticipativity:
X
L(x, λ) :=
pξ (c> x(ξ)+g(ξ)> y(ξ)+λ(ξ)> x(ξ)−λ(ξ)> x̂), x(ξ), y(ξ)) ∈ X(ξ), ∀ξ ∈ Ξ
ξ∈Ξ
Primal: F (λ) = min L(x, λ)
(12)
Dual: zLD := sup F (λ)
(13)
x
λ
Theorem: [Carøe & Schultz 99]
zLD = min
X
pξ [c> x(ξ) + g(ξ)> y(ξ)]
(14)
ξ∈Ξ
s.t.
x(ξ), y(ξ) ∈ clconv(X(ξ)), ξ ∈ Ξ
x(ξ) − x̂ = 0, ξ ∈ Ξ
7 / 20
How Tight is this Bound?
Ordinary Lagrangian by dualizing non-anticipativity:
X
L(x, λ) :=
pξ (c> x(ξ)+g(ξ)> y(ξ)+λ(ξ)> x(ξ)−λ(ξ)> x̂), x(ξ), y(ξ)) ∈ X(ξ), ∀ξ ∈ Ξ
ξ∈Ξ
Primal: F (λ) = min L(x, λ)
(12)
Dual: zLD := sup F (λ)
(13)
x
λ
Theorem: [Carøe & Schultz 99]
zLD = min
X
pξ [c> x(ξ) + g(ξ)> y(ξ)]
(14)
ξ∈Ξ
s.t.
x(ξ), y(ξ) ∈ clconv(X(ξ)), ξ ∈ Ξ
x(ξ) − x̂ = 0, ξ ∈ Ξ
7 / 20
How Tight is this Bound?
Ordinary Lagrangian by dualizing non-anticipativity:
X
L(x, λ) :=
pξ (c> x(ξ)+g(ξ)> y(ξ)+λ(ξ)> x(ξ)−λ(ξ)> x̂), x(ξ), y(ξ)) ∈ X(ξ), ∀ξ ∈ Ξ
ξ∈Ξ
Primal: F (λ) = min L(x, λ)
(12)
Dual: zLD := sup F (λ)
(13)
x
λ
Theorem: [Carøe & Schultz 99]
zLD = min
X
pξ [c> x(ξ) + g(ξ)> y(ξ)]
(14)
ξ∈Ξ
s.t.
x(ξ), y(ξ) ∈ clconv(X(ξ)), ξ ∈ Ξ
x(ξ) − x̂ = 0, ξ ∈ Ξ
7 / 20
As Tight as Possible!
Proposition 2: Best PH bound equals best bound from dual
decomposition
Suppose PH is applied to (14). Then in the limit, one obtains a
solution (x̂∗ , w∗ (ξ)), where x̂∗ solves the primal L.R. and
{w∗ (ξ), ξ ∈ Ξ} solves the dual L.R. Moreover, in the limit, the PH
lower bound is equal to zLD .
Only the duality gap remains.
PH can be interpreted as a primal-dual algorithm.
Sequences of primal solutions {x̂ν }∞
ν=1 and dual solutions
ν
∞
{{w (ξ)}ν=1 , ξ ∈ Ξ} converge to a saddle point of the ordinary
Lagrangian.
8 / 20
As Tight as Possible!
Proposition 2: Best PH bound equals best bound from dual
decomposition
Suppose PH is applied to (14). Then in the limit, one obtains a
solution (x̂∗ , w∗ (ξ)), where x̂∗ solves the primal L.R. and
{w∗ (ξ), ξ ∈ Ξ} solves the dual L.R. Moreover, in the limit, the PH
lower bound is equal to zLD .
Only the duality gap remains.
PH can be interpreted as a primal-dual algorithm.
Sequences of primal solutions {x̂ν }∞
ν=1 and dual solutions
ν
∞
{{w (ξ)}ν=1 , ξ ∈ Ξ} converge to a saddle point of the ordinary
Lagrangian.
8 / 20
Stochastic Unit Commitment Problem
Stochastic Unit Commitment - A Tactical Planning Problem
Given a set of generators, a transmission network and their characteristics,
with uncertain loads, variable generation outputs and outages for the next
day, compute a minimum cost generator on/off schedule for a 24-hour period
(tomorrow) today, which hedges against the future uncertainty.
First Stage: Generator on/off schedule
Uncertainty: Load, renewables, outages, etc.
Second Stage: Power dispatch schedule, power flow schedule, voltage
angles, etc.
The modeled trade-off: First-stage (commitment, startup, shutdown)
costs and the expected second-stage (production) costs.
Commitment in second stage for fast-start units currently not considered
but can be easily incorporated.
9 / 20
Stochastic SCUC Formulation
Based on the deterministic formulation of [Carrion & Arroyo 06].
Other formulations exist e.g. [Ostrowski et al. 12], [Morales et al.
13], and many others.
Objective Function
Minimize E[Commitment + Startup + Shutdown + Production costs]
Constraints for each scenario:
Piecewise linearization of the (quadratic) production offer curves
Power balance constraints: Linear DC approximation of flow balance on
the network
Generation limits: Generator production levels must lie within bounds
Ramp up and down limits
Minimum up and down times
Non-anticipativity constraints on unit commitment binary variables.
10 / 20
Stochastic SCUC Formulation
Based on the deterministic formulation of [Carrion & Arroyo 06].
Other formulations exist e.g. [Ostrowski et al. 12], [Morales et al.
13], and many others.
Objective Function
Minimize E[Commitment + Startup + Shutdown + Production costs]
Constraints for each scenario:
Piecewise linearization of the (quadratic) production offer curves
Power balance constraints: Linear DC approximation of flow balance on
the network
Generation limits: Generator production levels must lie within bounds
Ramp up and down limits
Minimum up and down times
Non-anticipativity constraints on unit commitment binary variables.
10 / 20
Stochastic SCUC Formulation
Based on the deterministic formulation of [Carrion & Arroyo 06].
Other formulations exist e.g. [Ostrowski et al. 12], [Morales et al.
13], and many others.
Objective Function
Minimize E[Commitment + Startup + Shutdown + Production costs]
Constraints for each scenario:
Piecewise linearization of the (quadratic) production offer curves
Power balance constraints: Linear DC approximation of flow balance on
the network
Generation limits: Generator production levels must lie within bounds
Ramp up and down limits
Minimum up and down times
Non-anticipativity constraints on unit commitment binary variables.
10 / 20
Implementation Details
Models implemented in Coopr Pyomo and PySP.
Coopr: Common optimization python repository. Open-source
Python-based software for modeling deterministic and stochastic
optimization problems. https://software.sandia.gov/trac/coopr.
Allows interfacing with many solvers including CPLEX, Gurobi, etc.
Includes an implementation of Progressive Hedging with different
parameter settings and user-specified callbacks to allow for tuning and
customization.
Stochastic SCUC problem specific tuning strategies designed using ideas
from [Watson & Woodruff 11]
Small instances run on a desktop (3.1 GHz Intel i5 QuadCore, 8GB
RAM)
Large instances run on 2.4 GHz Opteron, 64 core, 512 GB RAM
workstation. We are using only a tiny fraction of the RAM.
11 / 20
Implementation Details
Models implemented in Coopr Pyomo and PySP.
Coopr: Common optimization python repository. Open-source
Python-based software for modeling deterministic and stochastic
optimization problems. https://software.sandia.gov/trac/coopr.
Allows interfacing with many solvers including CPLEX, Gurobi, etc.
Includes an implementation of Progressive Hedging with different
parameter settings and user-specified callbacks to allow for tuning and
customization.
Stochastic SCUC problem specific tuning strategies designed using ideas
from [Watson & Woodruff 11]
Small instances run on a desktop (3.1 GHz Intel i5 QuadCore, 8GB
RAM)
Large instances run on 2.4 GHz Opteron, 64 core, 512 GB RAM
workstation. We are using only a tiny fraction of the RAM.
11 / 20
Implementation Details
Models implemented in Coopr Pyomo and PySP.
Coopr: Common optimization python repository. Open-source
Python-based software for modeling deterministic and stochastic
optimization problems. https://software.sandia.gov/trac/coopr.
Allows interfacing with many solvers including CPLEX, Gurobi, etc.
Includes an implementation of Progressive Hedging with different
parameter settings and user-specified callbacks to allow for tuning and
customization.
Stochastic SCUC problem specific tuning strategies designed using ideas
from [Watson & Woodruff 11]
Small instances run on a desktop (3.1 GHz Intel i5 QuadCore, 8GB
RAM)
Large instances run on 2.4 GHz Opteron, 64 core, 512 GB RAM
workstation. We are using only a tiny fraction of the RAM.
11 / 20
PH Bound Implementation
phboundextension computes a lower bound in any iteration.
Can compute a bound once every N iterations.
Can assign the lower bounding task to separate processors, one for
each (bundle of) scenario sub-problem(s).
12 / 20
PH Bound Implementation
phboundextension computes a lower bound in any iteration.
Can compute a bound once every N iterations.
Can assign the lower bounding task to separate processors, one for
each (bundle of) scenario sub-problem(s).
12 / 20
PH Bound Implementation
phboundextension computes a lower bound in any iteration.
Can compute a bound once every N iterations.
Can assign the lower bounding task to separate processors, one for
each (bundle of) scenario sub-problem(s).
12 / 20
Small-Sized Instances - Without Tuning or
Parallelization
Inst
5Bus6
5Bus24
Inst
Gen
Scen
Hrs
Bins
Vars
Cons
T(s)
Gen
5
5
Scen
5
10
Hrs
6
24
Bins
150
1,200
Vars
2,129
16,194
Cons
3,052
24,092
T(s)
11
95
Instance name
Generators
Scenarios
Scheduling horizon
Binaries in extensive form
Total variables in extensive form
Constraints in extensive form
Time in seconds for PH to find a solution
13 / 20
Solution Quality and Lower Bound Depend on ρ
ρ
1
2
5
15
30
Iterations
160
66
32
13
13
EC
399.668
399.968
399.968
404.153
433.235
LB
392.931
388.667
371.165
321.409
244.912
400
380
PH Lower Bound
360
340
opt
320
ρ = 1
300
ρ = 2
280
260
ρ = 5
240
ρ = 15
220
ρ = 30
200
0
30
60
90
120
150
180
Iteration
14 / 20
Bunching Scenarios Improves Convergence
Lower Bounds from phboundextension 400
390
Bound
380
370
360
Final Incumbent 350
No Scenario Bunching
2 Scenarios per bunch
340
0
20
40
60
80
Iteration Number
100
120
140
Figure 3: Lower bounds for 5Bus24 with and without scenario bunching
5/13/13 ARPA-­‐E Q4 Review 17 shows the progress of the lower bounds for these two settings. The final percentage gap
15 / 20
Large-Scale Instances - With Tuning and PH Parallel
Inst
WECC-240 10
WECC-240 25
Inst
Gen
Scen
Hrs
Bins
Vars
Cons
T(s)
Gen Scen Hrs
Bins
Vars
85
10
48
40,800
360,803
85
25
48
102,000 895,883
Similar results up to 100 scenarios
Cons
674,532
1,686,327
T(s)
360
420
Instance name
Generators
Scenarios
Scheduling horizon
Binaries in extensive form
Total variables in extensive form
Constraints in extensive form
Time in seconds for PH to find a solution
16 / 20
Convergence of Implicit Bounds for WECC-240: 10
Scenarios
phboundextension)WECCI240)10)Scenarios)
5
x 10
2.25
0.8%)gap)
2.2
Bound
2.15
2.1
2.05
Lower Bound
Final Incumbent
2
0
5/13/13)
10
20
30
40
50
60
Iteration Number
ARPAIE)Q4)Review)
70
80
90
100
19)
17 / 20
Convergence of Implicit Bounds for WECC-240: 25
Scenarios
phboundextension)WECCI240)25)Scenarios)
5
x 10
2.3
2.6%)gap)
2.25
Bound
2.2
2.15
2.1
Lower Bound
Final Incumbent
2.05
10
5/13/13)
20
30
40
50
60
Iteration Number
ARPAIE)Q4)Review)
70
80
90
100
20)
18 / 20
Conclusions and Ongoing Work
We are able to assess solution quality by means of using
information prices computed by progressive hedging. The implicit
bounds suggest that the unit commitment solutions generated by
PH are of high quality.
Lower bounding capability added to PySP
Further tuning required:
I
I
to manage the tradeoff between fast convergence and solution
quality,
to scale to very large instances (FERC/PJM unit commitment case)
19 / 20
Conclusions and Ongoing Work
We are able to assess solution quality by means of using
information prices computed by progressive hedging. The implicit
bounds suggest that the unit commitment solutions generated by
PH are of high quality.
Lower bounding capability added to PySP
Further tuning required:
I
I
to manage the tradeoff between fast convergence and solution
quality,
to scale to very large instances (FERC/PJM unit commitment case)
19 / 20
Conclusions and Ongoing Work
We are able to assess solution quality by means of using
information prices computed by progressive hedging. The implicit
bounds suggest that the unit commitment solutions generated by
PH are of high quality.
Lower bounding capability added to PySP
Further tuning required:
I
I
to manage the tradeoff between fast convergence and solution
quality,
to scale to very large instances (FERC/PJM unit commitment case)
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Conclusions and Ongoing Work
We are able to assess solution quality by means of using
information prices computed by progressive hedging. The implicit
bounds suggest that the unit commitment solutions generated by
PH are of high quality.
Lower bounding capability added to PySP
Further tuning required:
I
I
to manage the tradeoff between fast convergence and solution
quality,
to scale to very large instances (FERC/PJM unit commitment case)
19 / 20
Thank You!
Questions??
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