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8/13/2019 16- Lagrangian Relaxation Solution of Unit Commitment
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ECE573 2001-2013 George Gross, University of Illinois at Urbana-Champaign; All Rights Reserved 1
ECE573
Power System Operations andControl
16. Lagrangian Relaxation Solution of UnitCommitment
George Gross
Department of Electrical and Computer Engineering
University of Illinois at Urbana-Champaign
ECE573 2001-2013 George Gross, University of Illinois at Urbana-Champaign; All Rights Reserved 2
Problem statement
Problem formulation
Application of the Lagrangian relaxation approach
Summary
OUTLINE
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ECE573 2001-2013 George Gross, University of Illinois at Urbana-Champaign; All Rights Reserved 3
UC PROBLEM STATEMENT
1 1
1 1
1
1
1 2
K N
i i i
k i
K NF S
i i i i i
k i
M
i i i
N
i
i=N
i i i
i
min f u ,p f u k ,p k
f u k ,p k + f u k
f u k ,p k
s.t.
p k d k
k = , ,...K
r u k ,p k k
u , p S
ECE573 2001-2013 George Gross, University of Illinois at Urbana-Champaign; All Rights Reserved 4
Large-scale optimization
Mixed-integer nonlinear programming problem
commitment decision variables are integer
variables
costs are nonlinear and not continuous
reserves are nonlinear functions
the feasible solution set forms a highly
constrained region
Additively separablecost function
UC PROBLEM CHARACTERISTICS
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ECE573 2001-2013 George Gross, University of Illinois at Urbana-Champaign; All Rights Reserved 5
LAGRANGIAN RELAXATION (LR)
Lagrangian relaxation has been applied to solve
the unit commitment problem since the 1970s
Lagrangian relaxation is a technique that makes
extensive use of duality theory in nonlinear
programming
There are commercial packages that solve the UC
problem for very large-scale power systems
ECE573 2001-2013 George Gross, University of Illinois at Urbana-Champaign; All Rights Reserved 6
REFERENCES
John A. Muckstadt and Sherri A. Koenig, An
Application of Lagrangian Relaxation to
Scheduling in Power-Generation Systems,
Operations Research, vol. 25, pp. 387-403, 1977.
A. Merlin and P. Sandrin, "A New Method for Unit
Commitment at EdF,IEEE Transactions on Power
Apparatus and Systems, vol. PAS-102, no.5, pp. 1218-
1225, May 1983.
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REFERENCES
A. I. Cohen and V. R. Sherkat, Optimization-
Based Methods for OperationsScheduling,IEEEProceedings, vol. 57, no.12, pp. 1574-1592,November1987.
B.F. Hobbs, M.H. Rothkopf, R.P. ONeill and H.-P.
Chao, eds., The Next Generation of Electric
Power Unit Commitment Modules,Kluwer
Academic Publishers, Boston, 2001.
ECE573 2001-2013 George Gross, University of Illinois at Urbana-Champaign; All Rights Reserved 8
GENERAL DUALITY THEORY
with and continuously
differentiable functions
n
f :n m
g :
( )
n
m i n f x
s.t .
Pg x 0
x S
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KUHN-TUCKER OPTIMALITYCONDITIONS
We construct the Lagrangian
Assume for (P)and that (P)satisfies
the constraint qualification
Let be optimal for (P); then, there exists
such that and
Tx , f x g x
nS =
*
x
* m * 0 * * Tx* T *
x , 0
g x 0
stationaritycomplementary-slackness
ECE573 2001-2013 George Gross, University of Illinois at Urbana-Champaign; All Rights Reserved 10
THE DUAL PROBLEM
The dual function is defined to be
The dual problem is
:h min x , x S
( )
{ : }
max h
s.t D
h exi sts, 0
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GLOBAL OPTIMALITY CONDITIONS
Let and . Suppose that andsatisfy the following three conditions:
(i i) feasibility:
(i i i) complementary
slackness:
Then, is a saddle point of and
is optimal for (P) and is optimal for (D) .
*
x S * *x ** *
x x , x Sminimizes overall
*g x 0* T *g x 0
* *x ,*
x *x ,
(i) minimality:
ECE573 2001-2013 George Gross, University of Illinois at Urbana-Champaign; All Rights Reserved 12
STRONG DUALITY
Let and . Then, satisfy the
global optimality conditions if and only if
(i) is feasible for (P)
(i i) is feasible for (D)
(i i i)
*
x S* * *x ,
*
*
x
* *f x h m ax h :
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WEAK DUALITY AND CONVEXITY
For any feasible for (P
), and any
The weak duality theorem implies that the value
of the dual at a feasible point is a lower bound for
the primal problem
x
:
T
h m i n x , x S
x ,
f x g x
f x
ECE573 2001-2013 George Gross, University of Illinois at Urbana-Champaign; All Rights Reserved 14
WEAK DUALITY AND CONVEXITY
A nonlinear program is convex if and
are differentiable and convex
At the optimum of the convex nonlinear
programming problem (P), the K-Tconditions and
the global optimality condition hold
g f
*x
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THE UC PROBLEM
The UCproblem is a mixed-integer nonlinear
programming problem: the K-T conditions and
the global optimality condition need not hold
In the UCproblem, the objective and constraints
are non-convex; moreover, they are non-
differentiable due to the presence of discrete
variables
Weak duality condition holds since , the dual
function, is concave in the dual variables
h
ECE573 2001-2013 George Gross, University of Illinois at Urbana-Champaign; All Rights Reserved 16
APPLICATION OF LR TO UC
The basic idea in LRis to get as tight a lower
bound as possible to the optimal solution of (P)
Let be the solution of
then, from weak duality we have
constitutes a tight lower bound for
m ax h 0 ,:
* *
h f x *h
*
*
f x
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THE DUALITY GAP
f x
h
1
2
3
the true dual
optimumb
the true primal
optimumc
a computable
soluti on of (P)
d
a computable
soluti on of (D)a
ECE573 2001-2013 George Gross, University of Illinois at Urbana-Champaign; All Rights Reserved 18
1is the optimization defect of the primal
problem (P)
3is the optimization defect of the dual
problem (D)
2is the difference between the true optimum
solution of the primal problem and the true
optimum solution of the dual problem, given that
only weak duality holds and is called the duality
gap; it can be shown that 2is a decreasing
function of the dimensionality of the problem
THE DUALITY GAP
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Let be the computable solution of
and so is an approximation of
The duali ty gap indicates the
difference between the approximate minimum and
the lower bound; the gap is a decreasing function
of the dimension of the primal problem (P)
RELATIVE DUALITY GAP
x
* * *
x Sh m i n x , x ,
*
f x h
*
xx
ECE573 2001-2013 George Gross, University of Illinois at Urbana-Champaign; All Rights Reserved 20
The relative duali ty gapis expressed by the ratio
Typically, the relative duali ty gapis small: for
practical UCproblems it is of the order of 0.5%
The relative duali ty gapis often used as the
stopping criterion in computational schemes
RELATIVE DUALITY GAP
*
f x h
f x
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PRACTICAL CONSIDERATIONS
While the minimization of the primal problem may
be easy, the maximization of the dual problem
may be difficult
The primal solution obtained may not generally
satisfy the system-wide coupling constraints
(primal infeasible) due to the non-convexity of the
constraints
ECE573 2001-2013 George Gross, University of Illinois at Urbana-Champaign; All Rights Reserved 22
PRACTICAL CONSIDERATIONS
In actual computations, is not computed;
rather, an approximation of is obtained,
with computed by some numerical scheme;
the goodness of the approximation is very
much a function of the form of the dual function
Then, is computed to be the point at which
*
x Sx , min x ,
*
x
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PRACTICAL CONSIDERATIONS
The feasibility of is tested: for infeasible, a
feasible approximation is computed and used
The estimate of the computable duality gap is then
given by the difference
f x h
x
x
x
ECE573 2001-2013 George Gross, University of Illinois at Urbana-Champaign; All Rights Reserved 24
THE UC LAGRANGIAN FUNCTION
1 1
1
1
K N
i i i
k i
N
k i
i
N
k i i i
i
u , p , , f u k ,p k
d k p k
k r u k ,p k
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THE UC DUAL PROBLEM
The Lagrangian dual function is given by
The Lagrangian dual problem is to determine
that mazimizes , i.e,
u , p S
h , m in u , p , , 0
:
0
m ax h ,
,h
, ,h
ECE573 2001-2013 George Gross, University of Illinois at Urbana-Champaign; All Rights Reserved 26
UC IN THE LAGRANGIANRELAXATION FRAMEWORK
1 1
1
u , p S
N K
i i i k i u , p S
i k
k i i i
K
k k
k
min u , p , , =
min f u k ,p k p k
r u k ,p k
d k k
constant for
given and
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UC IN THE LR FRAMEWORK
1
1ii i
N
i
i =
K
i i i i k i u , p S k
k i i i
h , h ,
h , min f u k ,p k p k
r u k ,p k
1 1
1
ii i
N K
i i i k i u , p S i = k
k i i i
N
i
i=
= min f u k ,p k p k
r u k ,p k constan t
= h , constant
ECE573 2001-2013 George Gross, University of Illinois at Urbana-Champaign; All Rights Reserved 28
UNIT i SUBPROBLEM
Therefore, the dual problem is decomposable into
Nseparable subproblems
For fixed values of the Lagrangian multipliers
, the subproblem may be solved using a
dynamic programming based approach; the
curse of dimensionality may thereby be reduced,
at least to some extent
th
i
,
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THE DUAL PROBLEM SOLUTION
The dual function is nonlinear and concave, and is
generally nondifferentiable
The computation of a near-optimal dual solution
which is feasible for the primal problem is the
most challenging aspect of the LRapproach
The subgradienttechnique is a general approach
which is useful for this purpose; there are many
heuristics-based computational schemes that use
subgradient information to solve the dual problem
ECE573 2001-2013 George Gross, University of Illinois at Urbana-Champaign; All Rights Reserved 31
EXAMPLE
f x = x
x
f xx
at x = 0, a subgradient can
be any number in[- 1 , + 1]
1
-1
0x
f xa subgradient of
f x
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AN IMPORTANT FACT
Consider the general optimization problem
( )
min f x
s.t.
P
g x 0
x S
ECE573 2001-2013 George Gross, University of Illinois at Urbana-Champaign; All Rights Reserved 33
AN IMPORTANT FACT
Given a ,
then, is a subgradient of at since
*
0
T* *ar gmi n ar gmi n x x , = f x g x x S x S
*g x*
*T* * *h h g x 0 h
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A UC PROBLEM SUBGRADIENT
h
=
h
1
1
1
1
1 1
1 1 1
N
ii =
N
i
i =
N
i i i
i =
N
i i i
i =
d p
d K p K
r u , p
K r u K ,p K
- - - - - - - - - - - - - - - - - - - - -- - - -
ECE573 2001-2013 George Gross, University of Illinois at Urbana-Champaign; All Rights Reserved 35
SUBGRADIENT APPROACH
The subgradient approach consists of iterations
of the form :
where, is the positive scalar stepsize
1
1
Tv v
v v
v
v v Tv v
h ,
h ,
v
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A SUBGRADIENT
Is a physically meaningful quantity
Measures how binding the primal constraints are
Provides a basis for a stopping criterion for the
maximization in the dual problem
Is useful in maximizing h ,
ECE573 2001-2013 George Gross, University of Illinois at Urbana-Champaign; All Rights Reserved 37
ECONOMIC INTERPRETATION
establishes prices for system
requirements (energy and capacity)
central coordinator
monitors responses of the
unit to the requirement
optimizes its performance given
the value of its contribution and
its operating costs and subject to
its constraints
uniti unitN
generation, reserves
i i,
unit1
iip ,r
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For a differentiable function , the vector
is the gradient of at
The vector is a subgradient of
at if f is not differentiable
f u , p * *u , p
f u , p * *u , p
f
SENSITIVITY ANALYSIS
TT T* *
,
TT T* *
,
ECE573 2001-2013 George Gross, University of Illinois at Urbana-Champaign; All Rights Reserved 39
THE LR APPROACH
Advantages
detailed representation of the complex
characteristics of the units
highly flexiblemodular and expandable
duality stopping criterion is a function of the
problem dimension
useful for evaluation of marginal costs
Disadvantages
provides only suboptimal solutions
computationally intensive approach
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In the competitive electricity market environment,
the unit commitment function, when solved by a
central decision-maker to coordinate resource
scheduling and operations, may lead to equity
problems since not all the units are owned by a
single entity
For the LR-
based solution only near-optimality
is possible and since there may be many near-
optimal solutions, problems of discrimination
UC IN COMPETITIVE ELECTRICITYMARKETS
ECE573 2001-2013 George Gross, University of Illinois at Urbana-Champaign; All Rights Reserved 41
may arise when ownership is vested among many
different entities
Two solutions, which provide approximately
equal values of the objective function, may yield
very different schedules of individual resources
which, in turn, vary significantly in terms of costs,
profits, and commitments
UC IN COMPETITIVE ELECTRICITYMARKETS
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REVIEW OF KEY POINTS OF LR
UCis a large-scale problem
integer nature of the commitment variables
global constraints
numerous local constraints
very complex problem
The role of heuristics in any optimization based
approach is critically important to:
obtain reasonably good results
determine feasible solutions
reduce overall computation
ECE573 2001-2013 George Gross, University of Illinois at Urbana-Champaign; All Rights Reserved 43
REVIEW OF KEY POINTS LR
LRis one of the most important optimization
methods in practice; it works by substituting the
original problem by a sequence of a set of
simpler decomposed subproblems
Currently, LRis the most efficient method, but its
equity and efficiency are severely challenged in
the competitive environment
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REVIEW OF KEY POINTS LR
Recently, a lot of interest has arisen in usingmixed integer programming MIPapproaches to
solve UC
The MIPapplication to UCis very challenging
and requires effective use of heuristics to solve
large-scale problems
Unless the MIPsolves the problem exactly, the
inequity issues persist