16- lagrangian relaxation solution of unit commitment

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


Problem statement

Problem formulation

Application of the Lagrangian relaxation approach

Summary

OUTLINE


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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


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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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


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.


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


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:


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


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


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


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


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



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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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


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


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


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


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


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

- - - - - - - - - - - - - - - - - - - - -- - - -


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 ,


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* *

,


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


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


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

16- lagrangian relaxation solution of unit commitment

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