Introduction
Context
Unit commitment under uncertainty
Solution strategy
Summary
Unit-commitment with joint probability constraints
W. van Ackooij1
1
OSIRIS Department
EDF R&D
7 Boulevard Gaspard Monge; 9120 Palaiseau ; France
MINO/COST, 2016
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Introduction
Context
Unit commitment under uncertainty
Solution strategy
Summary
Outline
1
Introduction
Introduction
2
Context
A Genco’s view
Generation
System wide
3
Unit commitment under uncertainty
appetizer: unit commitment without uncertainty
unit commitment with uncertainty
problem summary
4
Solution strategy
solving this problem
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Introduction
Context
Unit commitment under uncertainty
1
Introduction
Introduction
2
Context
A Genco’s view
Generation
System wide
3
Unit commitment under uncertainty
appetizer: unit commitment without uncertainty
unit commitment with uncertainty
problem summary
4
Solution strategy
solving this problem
Solution strategy
Summary
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Introduction
Context
Unit commitment under uncertainty
Solution strategy
Summary
Introduction
Introduction
In this talk we will sketch how MINLP problems can arise in practice in
energy management
This will be illustrated through the unit-commitment problem
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Introduction
Context
Unit commitment under uncertainty
1
Introduction
Introduction
2
Context
A Genco’s view
Generation
System wide
3
Unit commitment under uncertainty
appetizer: unit commitment without uncertainty
unit commitment with uncertainty
problem summary
4
Solution strategy
solving this problem
Solution strategy
Summary
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Introduction
Context
Unit commitment under uncertainty
Solution strategy
Summary
A Genco’s view
General objectives
We will consider the viewpoint of a generation company
It has to satisfy the load from its customers
It has to deal with the technical constraints of its assets
This has to be done in a cost-effective way
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Introduction
Context
Unit commitment under uncertainty
Solution strategy
Summary
Generation
Assets
Generation assets consist of
Thermal plants (conventional and nuclear)
Hydro valleys
Renewable generation
Various contracts
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Introduction
Context
Unit commitment under uncertainty
Solution strategy
Summary
Generation
Thermal plants
Thermal plants are subject to various
restrictions on their way of functioning
Copyright: EDF brandcenter : EDF/MERAT PIERRE
Limit on variation of power (gradient)
minimum up/down times
maximum number of starts per day
limitation on power modulations
Copyright: EDF brandcenter : EDF/CONTY Bruno
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Introduction
Context
Unit commitment under uncertainty
Solution strategy
Summary
Generation
Thermal plants II
The costs of thermal plants decompose into
start up cost: a cost paid each time the plant starts up. This cost depends
on the number of hours the plant has been offline (when the plant is cold,
more energy is needed to heat up)
fixed cost: paid whenever the plant is running
proportional cost: depends on the amount of energy produced and consumed (e.g. for instance on the coal used)
CO2 allowance cost: Conventional thermal plants emit CO2 , for which
allowances need to be bought, which induce a cost.
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Introduction
Context
Unit commitment under uncertainty
Solution strategy
Summary
Generation
Hydro valleys I
Hydro valleys consist of
Reservoirs of varying capacity retaining water.
Turbines : when water is released downward, energy is generated
Copyright: EDF brandcenter : EDF/PATRICE
DHUMES
Pumps : water can be pumped into a reservoir
for being released later when production is more
economic (partial storage of electricity)
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Introduction
Context
Unit commitment under uncertainty
Solution strategy
Summary
Generation
Hydro valleys II
A hydro valley is
a set of connected reservoirs, turbines and pumps
Constraints imply:
Production level bounds
Reservoir bounds have to be satisfied
Production has to be relatively
stable
Pumping and Turbining can’t be
done simultaneously.
Efficiency depends on water
height and is non-linear
Inflows come from melting of snow
/ rain
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Introduction
Context
Unit commitment under uncertainty
Solution strategy
Summary
Generation
Hydro valleys III
The efficiency curve
depicts the amount of MW generated for a given flow rate
It depends on the height of the top
reservoir (water head effect).
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Introduction
Context
Unit commitment under uncertainty
Solution strategy
Summary
Generation
Renewable generation
Renewable generation is mostly intermittent
Wind generation: Wind farms produce around
25% of installed capacity on average with highly
varying output levels. In Germany over the course
of a few hours one can loose up 8 GW (the equivalent of 8 nuclear plants).
Solar generation
Copyright: EDF brandcenter : EDF/Marc DIDIER
Geothermal generation
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Introduction
Context
Unit commitment under uncertainty
Solution strategy
Summary
System wide
Uncertainties
Load is impacted by
Temperature
Cloud cover
Economic context
client portfolio
Generation is impacted by
Available hydro inflows (water levels in the reservoirs)
Random failures of plants
Market depth (availability of a products)
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Introduction
Context
Unit commitment under uncertainty
Solution strategy
Summary
System wide
Meta-uncertainty
The system is moreover subject to
Economical changes
Legislatorial changes
New technologies and use for electrical energy
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Introduction
Context
Unit commitment under uncertainty
1
Introduction
Introduction
2
Context
A Genco’s view
Generation
System wide
3
Unit commitment under uncertainty
appetizer: unit commitment without uncertainty
unit commitment with uncertainty
problem summary
4
Solution strategy
solving this problem
Solution strategy
Summary
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Introduction
Context
Unit commitment under uncertainty
Solution strategy
Summary
appetizer: unit commitment without uncertainty
introduction
Goal:
Find the most cost effective production schedule (today for tomorrow) that
satisfies the operational constraints and balances with customer load.
This is a large scale problem, since Edf’s portfolio consists of
58 Nuclear plants
150 Thermal plants
50 Hydro valleys totalling some 500 hydro plants.
There are moreover 96 half hourly time steps to account for
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Introduction
Context
appetizer: unit commitment without uncertainty
Map of french power plants:
Unit commitment under uncertainty
Solution strategy
Summary
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Introduction
Context
Unit commitment under uncertainty
Solution strategy
Summary
appetizer: unit commitment without uncertainty
A bird’s view of the structure
Let I be the set of units (valleys, thermal) and xi ∈ Rni the related decision
vector (containing at least the production levels for the 96 time steps.
Let x = (x1 , ..., xm ) ∈ Rn be the total decision vector, A a T × n matrix
summing up power and D ∈ RT customer load.
Then the unit-commitment problem can be written as:
X
minn
fi (xi )
x∈R
i∈I
s.t. xi ∈ Xi
Ax ≥ D,
where Xi ⊆ Rni is a set of technically feasible schedules.
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Introduction
Context
Unit commitment under uncertainty
Solution strategy
Summary
appetizer: unit commitment without uncertainty
Difficulties of this problem
This innocent looking problem is actually rather difficult, because
Both m and n are large
The set Xi is typically non-convex and (partially) discrete
The linking constraint Ax ≥ D couples the units together (system wide
constraint)
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Introduction
Context
Unit commitment under uncertainty
Solution strategy
Summary
appetizer: unit commitment without uncertainty
Basic thermal model: MINLP
λ, some market price (or Lagrangian dual signal)
u commitment variable, fixed running cost cf
p production variable with startup cost cs , proportional cost c and bounds
pmin , pmax
ramping rates g+ , g− > 0 expressed in MW /h
minimum up and down times τ+ , τ− expressed in a number of time steps
T , total set of time steps, ∆t, time step duration
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Introduction
Context
Unit commitment under uncertainty
Solution strategy
Summary
appetizer: unit commitment without uncertainty
Basic thermal model: equations
minp,u,z∈RT ×{0,1}2T
+
s.t.
(c − λ)T p∆t + cfT u∆t + csT z
pmin (t)u(t) ≤ p(t) ≤ pmax (t)u(t)
p(t) ≤ p(t − 1) + u(t − 1)g+ ∆t + (1 − u(t − 1))s+ ∆t
p(t − 1) ≤ p(t) + u(t)g− ∆t + (1 − u(t))s− ∆t
u(t) ≥ u(r ) − u(r − 1), for t, r = t − τ+ + 1, ..., t − 1
u(t) ≤ 1 − u(r − 1) + u(r ), for t, r = t − τ− + 1, ..., t − 1
u(t) − u(t − 1) ≤ z(t)
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Introduction
Context
Unit commitment under uncertainty
Solution strategy
Summary
appetizer: unit commitment without uncertainty
Basic cascaded reservoir model: MINLP
In very basic models, cascaded reservoir management is a linear programming problem.
However, when taking into account water head effect, the actual production as a function of flow rate can be a high order polynomial (typical
modelling in Brazil)
As an alternative path the set of feasible flow rates may belong to a discrete set giving an integer flavour to the problem.
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Introduction
Context
Unit commitment under uncertainty
Solution strategy
Summary
unit commitment with uncertainty
Uncertainty I
At least the following sources of uncertainty need to be accounted for:
uncertainty on net load
uncertainty on inflows in each reservoir
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Introduction
Context
Unit commitment under uncertainty
Solution strategy
Summary
unit commitment with uncertainty
Uncertainty II
In unit-commitment the decision (the schedule) has to be decided upon prior
to observing uncertainty. As a consequence, the equations:
Ax ≥ D,
(1)
r
r
vmin
≤ v0 + Ar xi + I ≤ vmax
,
(2)
wherein net load D is uncertain
wherein the inflows I are uncertain
need to be given an appropriate meaning.
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Introduction
Context
Unit commitment under uncertainty
Solution strategy
Summary
unit commitment with uncertainty
Probability constraints
One way of doing so is through the use of probability constraints:
P[Ax ≥ D] ≥ p,
(3)
r
r
P[vmin
≤ v0 + Ar xi + I ≤ vmax
] ≥ p,
(4)
note that these are joint constraints.
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Introduction
Context
Unit commitment under uncertainty
Solution strategy
Summary
unit commitment with uncertainty
Probability constraints: more NLP
The feasible set of the probability constraint:
P[g(x, ξ) ≥ 0] ≥ p,
(5)
is convex if g is jointly quasi-concave and ξ has a density with respect
to the Lebesgue measure with generalized concavity properties (for instance, ξ is multivariate-Gaussian, Student, Dirichlet, Uniform on a convex support). (e.g., [Prékopa(1995)])
The feasible set of a joint probability constraint is non-polyhedral in general, e.g., level sets of x 7→ P[ξ ≤ x], ξ ∼ N (0, I):
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Introduction
Context
Unit commitment under uncertainty
Solution strategy
Summary
problem summary
The structure: a large scale MINLP
The full unit-commitment problem under uncertainty is a large scale MINLP:
min
(x1 ,...,xm )∈Rn
X
fi (xi )
i∈I
s.t. xi ∈ Xi
P[Ax ≥ D] ≥ p,
where Xi ⊆ Rni is a set of technically feasible schedules.
some of the “sub-problems”: minxi ∈Xi fi (xi ) are NLPs too. Others are
MILPs.
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Introduction
Context
Unit commitment under uncertainty
Solution strategy
Summary
problem summary
Discussion of the structure
Note that the unit-commitment problem is
a nice MINLP if cascaded reservoir management problems have convex
continuous relaxations. Then depending on the law of underlying uncertainty the continuous relaxation of the problem has a convex feasible set.
a not so nice MINLP if cascaded reservoir management involves detailled
modelling of the water-head effect. Still there is some structure that can
be exploited.
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Introduction
Context
Unit commitment under uncertainty
1
Introduction
Introduction
2
Context
A Genco’s view
Generation
System wide
3
Unit commitment under uncertainty
appetizer: unit commitment without uncertainty
unit commitment with uncertainty
problem summary
4
Solution strategy
solving this problem
Solution strategy
Summary
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Introduction
Context
Unit commitment under uncertainty
Solution strategy
Summary
solving this problem
An equivalent primal
Let us consider the equivalent model:
min
(x1 ,...,xm ,z)∈Rn
X
fi (xi )
i∈I
s.t. xi ∈ Xi
Ax = z
P[z ≥ D] ≥ p,
where Xi ⊆ Rni is a set of technically feasible schedules.
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Introduction
Context
Unit commitment under uncertainty
Solution strategy
Summary
solving this problem
And its dual
And its Lagrangian dual mapping Θ : RT → R defined as
Θ(λ) :=
min
(x1 ,...,xm ,z)
(
X
)
T
fi (xi ) + λ (Ax − z) : xi ∈ Xi , P[z ≥ D] ≥ p
i∈I
(6)
Note that λ 7→ Θ(λ) is concave and can be maximized by a bundle
method.
The subproblems are easier: some are MILPs, others NLPs
In the harder situation some subproblems are MINLPs, but smaller ones
(maybe even polynomial optimization problems).
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Introduction
Context
Unit commitment under uncertainty
Solution strategy
Summary
solving this problem
Getting a primal solution
Over the course of solving the dual
We generate a set of primal feasible iterates x ` := (x1` , ..., xm` , z ` )
At convergence we obtain a set of simplex weights α1 , ..., αL ,
The pseudo-solution: x̂ :=
PL
`=1
α` x ` is optimal for the bi-dual problem
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Introduction
Context
Unit commitment under uncertainty
Solution strategy
Summary
solving this problem
Getting a primal solution
The idea is to pick ẑ that satisfies P[ẑ ≥ D] ≥ p and interpret it as the
load that needs to be produced.
Under appropriate conditions, the cascaded reservoir subproblems have
convex feasible sets too (but non-polyhedral), so then too (x̂)i can be
taken as the schedule
Then, we can employ well-established primary recovery heuristics with
ẑ − Aret x ret as target load. Here Aret x ret stands for the generation of
the retained (sub-)schedules.
Doing so we can obtain a solution within 0.5 % of optimality (on a problem
involving some 105 thermal units, 40 hydro valleys and 96 time steps).
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Introduction
Context
Unit commitment under uncertainty
Solution strategy
Summary
Summary
This talk illustrates how MINLP problems arise in practice.
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Introduction
Context
Unit commitment under uncertainty
Solution strategy
Summary
Some references I
On decomposition: C. Sagastizábal. Divide to conquer: Decomposition
methods for energy optimization.
Mathematical Programming, 134(1):187–222, 2012
On bundle methods: J.B. Hiriart-Urruty and C. Lemaréchal. Convex Analysis and Minimization Algorithms I.
Number 305 in Grundlehren der mathematischen Wissenschaften. SpringerVerlag, 2nd edition, 1996
On unit commitment: M. Tahanan, W. van Ackooij, A. Frangioni, and F. Lacalandra. Large-scale unit commitment under uncertainty: a literature survey.
4OR, 13(2):115–171, 2015.
doi: 10.1007/s10288-014-0279-y
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Introduction
Context
Unit commitment under uncertainty
Solution strategy
Summary
Some references II
On the results and principles illustrated here W. van Ackooij. Decomposition
approaches for block-structured chance-constrained programs with application to hydro-thermal unit commitment.
Mathematical Methods of Operations Research, 80(3):227–253, 2014
W. van Ackooij. A comparison of four approaches from stochastic programming for large-scale unit-commitment.
To Appear in EURO Journal on Computational Optimization, pages 1–19,
2015.
doi: 10.1007/s13675-015-0051-x
On some recent theoretical advances on probability constraints : W. van
Ackooij and R. Henrion. (sub-) gradient formulae for probability functions
of random inequality systems under gaussian distribution.
Submitted, WIAS preprint 2230, pages 1–24, 2016
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Appendix
Bibliography
Bibliography I
[[Hiriart-Urruty and Lemaréchal(1996)]
]J.B. Hiriart-Urruty and
C. Lemaréchal.
Convex Analysis and Minimization Algorithms I.
Number 305 in Grundlehren der mathematischen Wissenschaften.
Springer-Verlag, 2nd edition, 1996.
[[Prékopa(1995)]
Stochastic Programming.
Kluwer, Dordrecht, 1995.
]A. Prékopa.
[[Sagastizábal(2012)]
]C. Sagastizábal.
Divide to conquer: Decomposition methods for energy optimization.
Mathematical Programming, 134(1):187–222, 2012.
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Appendix
Bibliography
Bibliography II
[[Tahanan et al.(2015)Tahanan, van Ackooij, Frangioni, and Lacalandra]
]M. Tahanan, W. van Ackooij, A. Frangioni, and F. Lacalandra.
Large-scale unit commitment under uncertainty: a literature survey.
4OR, 13(2):115–171, 2015.
doi: 10.1007/s10288-014-0279-y.
[[van Ackooij(2014)]
]W. van Ackooij.
Decomposition approaches for block-structured chance-constrained programs with application to hydro-thermal unit commitment.
Mathematical Methods of Operations Research, 80(3):227–253, 2014.
[[van Ackooij(2015)]
]W. van Ackooij.
A comparison of four approaches from stochastic programming for largescale unit-commitment.
To Appear in EURO Journal on Computational Optimization, pages 1–19,
2015.
doi: 10.1007/s13675-015-0051-x.
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Appendix
Bibliography
Bibliography III
[[van Ackooij and Henrion(2016)]
]W. van Ackooij and R. Henrion.
(sub-) gradient formulae for probability functions of random inequality systems under gaussian distribution.
Submitted, WIAS preprint 2230, pages 1–24, 2016.
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