using lagrangian relaxation in optimisation of unit ... · 3 lagrangian relaxation for optimisation...

Fakulteta za Elektrotehniko

Eva Thorin, Heike Brand, Christoph Weber

Using Lagrangian relaxation in

optimisation of unit commitment and planning

OSCOGEN Discussion Paper No. 3

February 2002

Contract No. ENK5-CT-2000-00094 Project co-funded by the European Community under the 5th Framework Programme (1998-2002)

Contract No. BBW 00.0627 Project co-funded by the Swiss Federal Agency for Education and Science

Contract No. CEU-BS-1/2001 Project co-funded by Termoelektrarna toplarna Ljubljana, d.o.o.

1 INTRODUCTION........................................................................................................................................ 1

2 BASIC THEORY......................................................................................................................................... 1

3 LAGRANGIAN RELAXATION FOR OPTIMISATION OF UNIT COMMITMENT AND DISPATCHING..................................................................................................................................................... 2

3.1 PROBLEMS DESCRIBED IN THE LITERATURE........................................................................................... 2 3.2 FORMULATION FOR A SIMPLIFIED CHP-PROBLEM ................................................................................. 5

4 ASPECTS ON SOLVING THE LAGRANGIAN RELAXED PROBLEM............................................ 8

5 CONCLUSIONS .......................................................................................................................................... 9

6 LITERATURE ............................................................................................................................................. 9

IER Stuttgart 1 v 20.02.02

1 Introduction

Lagrangian relaxation algorithms seem to be efficient in speeding up the solving of optimisation problems and there are several examples in the literature where they have been used for optimisation of unit commitment and dispatching /Aoki et al, 1989/, /Cheng et al, 2000/, /Dotzauer, 2001/, /Huonker, 2001/, /Stern et al, 2001/, /Virmani et al, 1989/. Within the OSCOGEN project we have problems to reach satisfactory solutions within reasonable time with the deterministic model for the long-term planning. Using the Lagrangian relaxation can be one way to enhance the computational performance.

2 Basic Theory

The basic concept of Lagrangian relaxation is illustrated with the minimisation problem below. The constraints are divided into two types, the equalities gj(x) (j= 1,…, m) and inequalities hj(x) (j=1,….p).

{ }

,.......,1 0 ,......,10)(.

)(min

Xx

pjhmjxgts

xf

j

j

X

∈

=≤

==

The idea is to decompose the problem, called the primal problem, to several smaller problems that are easier to solve. This can then be done through creation of the relaxed problem by including the constraints into the objective function. By multiplying the constraints with the Lagrange multipliers, λ and µ, respectively, and including them in the objective function, the primal problem is transformed into an unconstrained optimisation problem, called the relaxed problem.

{ }Xxts

xhxgxf TT

x

∈

++=Φ

.

)()()(min),( µλµλ

0. ≥µ

{ }),(max,

Φ µλµλ

Φ(λ,µ) is the dual objective function. By maximising Φ with respect to λ and µ we get a lower bound for the feasible solution of the primal problem. (a high value of the multipliers indicate that the constraints are considered to a higher degree).

ts


)()(),(),( *** xfxf ≤≤Φ≤Φ µλµλ

),()( µλδ Φ−= xf

* indicates the optimal solution The difference between the solution of the dual and primal problem is called the duality gap, δ

From the optimal dual variables we get the optimal solution of the relaxed problem and if the duality gap is zero this is also the solution of the primal problem. If a duality gap exists between the optimal solutions, the solution of the relaxed problem does not need to be primal feasible and then a primal feasible solution has to be constructed from the results of the relaxed problem.

3 Lagrangian Relaxation for optimisation of unit commitment and dispatching

3.1 Problems described in the literature

In /Dotzauer, 2001/ short-term planning of power and heating systems with algorithms based on Lagrangian relaxation is considered. Two basic models, one for the power production problem and one for the heat production problem, are used. The total system cost is minimised and the model includes start-up costs. For the heat production problem heat storage is included. Minimum operation and shut-down times are also considered.

{ } down times-shut andoperation minimum

0,1)( )()()()()(

)()(

)()()(.

)()())1(1()())()()())()(((min

u

,,,,

,,u EL,

,,,

1

0,

1

1

2,

2

,

∈≤≤

=

≥

−−+++

∑

∑

∑∑∑∑==

tOtOtPtPtOtP

tPtP

tPtOtPts

ttOtOtOttPttPt

uuELMAXuuELMIN

uuDEMANDEL

uRESELuuELMAX

T

t uuutuuuuELu

T

t uuELuOPEL

γααα


{ } 0,1)( )()()(

)()()( )()1()(

)()()()()(

)()()(.

)()())1(1()())()()())()(((min

u

,,,,,

,,,,,,,

,,,

,,,,

,,,,,

1

0,

1

1

2,

2

,,

∈≤≤

≤≤−−=

≤≤

=+

−−+++

∑

∑∑∑∑==

tOtEtEtE

tPtPtPtPtEtE

tOtPtPtOtP

tPtPtPts

ttOtOtOttPttPt

sTHMAXsTHsTHMIN

soutTHMAXsoutTHsTH,outMIN

soutTHsTHTH,s

uuTHMAXuTHuTH,uMIN

uuDEMANDTHsoutTHuTH

T

t uuuuuuuTHu

T

t uuTHuEOPTH

γααα

down times-shut andoperation minimum

)(, tP uEL

)(, tP uTH

)(tuα

)(tuγ

)(tOu

)(, tP RESEL

)(, tP DEMANDEL

)(, tP DEMNADTH

= the electric power produced by unit u at time t

)(, tE sTH

= the heat produced by unit u at time t

soutTHP ,,

= constant in the production cost equation for unit u at time t

= start-up cost for unit u at time t

= variable indicating the on-off status for the unit k at time i

= the spinning reserve at time t

= the electric power demand at time t

= the heat demand at time t

min),(,

=ΦPEL

µλ

= energy content of storage s at time t

= heat taken from storage s at time t

. Pts MIN

For the power production problem the relaxed problem is constructed by including the restrictions for the electric demand and the reserve requirements into the objective function of the primal problem.


0,1)( )()()()()(

)()()()()()()(

)()())1(1()())()()())()(((

,,,,,

1,,,

1,,

1

0,

1

1

2,

2

∈≤≤

−+

−+

−−+++

∑ ∑∑ ∑

∑∑∑∑

==

==

tOtOtPtPtOt

tOtPtPttPtPt

ttOtOtOttPttPt

u

uuELMAXuELuuEL

T

t uuuELMAXRESEL

T

t uuELDEMANDEL

T

t uuuuuuuELu

T

t uuELu

Oµλ

γααα

For given Lagrange multipliers the relaxed problem then decomposes into one sub-problem for each production unit.


{ }down times-shut andoperation minimum

0,1)( )()()()( .

)()())1(1(

)())()()()())()(())()(((min

u

,,,,

1

1,,

0,

12,

2

,

∈≤≤

−−

+−+−+

∑

∑

=

=

tOtOPtPtOtPts

ttOtO

tOtPtttPtttPt

uuELMAXuELuEL,uMIN

I

iuuu

T

tuuELMAXuuELuuELu

OPEL

γ

µαλαα

( ))()()()()(.

)()1()()(

()()()()())1(1(

)())()()())()(((

min),(

,,,,,

1,,,,

1,,

1

0,

1

1

2,

2

,,

≤≤

+−−+

−+−−+

++

=Φ

∑

∑ ∑∑∑

∑∑

=

==

=

tOtPtPtOt Pts

tPtEtEt

PtPtttOtO

tOttPttPt

uuTHMAXuTHuuTHMIN

T

tsoutTHsTHsTH

T

t uuTHDEMANDTH

T

t uuuu

uuuTHu

T

t uuTHu

EOPTH

µ

λγ

ααα

µλ

For the heat production problem the restrictions for the heat demand and the heat storage are included in the objective function of the primal problem.

{ }down timesshut andoperation minimum

0,1)( )()()(

)()(

)()

u

,.,,,

,,,,,,,,

,,

∈≤≤

≤≤

−

tOtEtEtE

tPPtP

tPt

sTHMAXsTHsTHMIN

soutTHMAXsoutTHsoutTHMIN

soutTH

)()()( )()()(.

)()())1()()(()())()((min

,,,,,

,,,,,,,,

1,,,,,,

tEtEtEtPtPt Pts

tPttEtEttPtt

sTHMAXsTHsTHMIN

soutTHMAXsoutTHsoutTHMIN

T

tDEMANDTHsTHsTHsoutTHEPTH

≤≤≤≤

+−−+−∑

=

λµλµ

( )


0,1)( )()()()()(.

)()())1(1()())()())()(())()(((min

,,,,,

1

0,

12,

2

,

∈≤≤

=−−++−+∑

=

tOtOtPtPtOt Pts

ttOtOtOttPtttPt

u

uuTHMAXuTHuuTHMIN

T

tuuuuuuTHuuTHuOPTH

γαλαα

For given Lagrange multipliers the relaxed problem decomposes into one sub-problem for each production unit and one problem for the heat storage.


The heat storage problem can then be further separated into one problem for each time interval. The power production problem described above has also been studied in /Cheng et al, 2000/. /Virmani et al, 1989/ describe a similar power production problem. They have included a more complicated reserve restriction. In [Aoki et al, 1989] also a similar power production problem is described. They, however, consider long-term optimisation (one week up to one month). A pump-storage hydro unit as well as an assigned fuel restriction are also included in their problem.

3.2 Formulation for a simplified CHP-problem

In our long-term model we consider both the power production and the heat production problem simultaneously. Further we maximise the profit instead of minimising the costs since we also include the possibility to sell electric power at the spot market. Take-or-pay contracts for buying electric power and fuels are also considered. Here we consider a system of CHP turbines, with given heat and electric power demand and the possibility to buy electric power from a take-or-pay contract and to buy and sell electric power from the spot market. The problem can then be described as follows:

{ } 01

0 0

0

01

0101

1 1

110

11

,O)t(O)t(P)t(P

)t(O)t(P)t(P)t(O)t(P

)t(OP)t( P)t( OP)t(OP)t(P)t(OP

)t(P)t(P

)t(PP)t(P

)t(P

)t(P)t(P)t(P)t(P)t(Ps. t

tcos)t(P)t(tcos)t(P

)))t(P)t(P((tcos)t(price)t(Pmax

u

uuuu,THu,EL

uuuu,THu,ELuuuu,TH

uu,TH,MAXu,THuu,TH,MIN

uu,EL,MAXu,ELuu,EL,MIN

DEMAND,THu

u,TH

SPOT,SELL,EL

TOP,BUY,EL,MAXTOP,BUY,EL

SPOT,BUY,EL

SPOT,SELL,ELu

DEMAND,ELTOP,BUY,ELSPOT,BUY,ELu,EL

T

t

T

tTOPTOP,BUY,ELSPOTSPOTx,BUY,EL

u,THuu,ELuu

T

t ur

T

tSPOTSPOT,SELL,EL

O,P,P THEL

∈+≥

+≤≤+

≤≤≤≤

≥

≥≤≤

≥

+=++

−−

−+−

∑

∑

∑ ∑

∑∑∑

= =

==

γγββεε

βαα

)(,, tP SPOTSELLEL

= electric power sold at the spot market at time t


)(tpriceSPOT

rtcos

)t(P u,EL

)t(P u,TH

uα

uuu ,, εγβ

)t(Ou

)t(P SPOT,BUY,EL

)t(tcos SPOT

)t(P TOP,BUY,EL

TOPtcos

)t(P DEMAND,EL

)t(P DEMAND,TH

= the price for the electric power when you sell it at the spot market at time t

= the cost for fuel r

= the electric power produced by unit u at time t

= the heat produced by unit u at time t

= constant for unit u in fuel consumption equation

= constants for unit u in equations describing the PQ-chart

= variable indicating the on-off status for the unit u at time t

= electric power bought at the spot market at time t

= the cost for buying electric power at the spot market at time t

= electric power bought from take-or-pay contract at time t

= the cost for buying electric power from the take-or-pay contract

= the electric power demand at time t

= the heat demand at time t

The fuel consumption is described with a linear equation with the constants α1 and α0. The relation between the power and heat production is described with plant-characteristics maps, so called PQ-charts. They consist of three straight lines with the constants β1and β0, ε1and ε0 and γ1 and γ0, respectively. By including the first two restrictions we get the following relaxed problem:


{ } 01

0 0

0

u

01

0101

1

1

1 1

,)t(O)t(O)t(P)t(P


)t(PP)t(P

)t(P)t(OP)t( P )t(OP

)t(OP)t(P)t(OPs. t

)t(P)t(P)t(

)t(P)t(P)t(P

)t(P)t(P)t(

tcos)t(P)t(tcos)t(P

max),(

uuuu,THu,EL

uuuu,THu,ELuuuu,TH

SPOT,SELL,EL


SPOT,BUY,EL



T

t uu,THDEMAND,TH

T

t TOP,BUY,ELSPOT,BUY,ELu

u,EL

SPOT,SELL,ELDEMAND,EL

t tTOPTOP,BUY,ELSPOTSPOT,BUY,EL

O,P,P THEL

∈+≥

+≤≤+

≥≤≤

≥≤≤

≤≤

−−

−−−

+−

−−

=Φ

∑ ∑

∑ ∑

∑ ∑

=

=

= =

γγββεε

µ

λµλ

110

11)))t(P)t(P((tcos)t(price)t(P

T T

u,THuu,ELuu

T

t ur

T

tSPOTSPOT,SELL,EL

−+−∑∑∑==

βαα

{ }),(min,

The dual problem then becomes:

µλµλ

Φ

( )

1

110

)t(OP)t(P)t(OPs. t

)t(P)t()t(P)t()))t(P)t(P((tcosmax


T

tu,THu,ELu,THuu,ELuurO,P,P THEL

≤≤

++−+−∑

=

µλβαα

For given λ and µ the relaxed problem can be decomposed to one problem for each CHP turbine and one problem for the terms that are not turbine-dependent.

{ } 01

01

0101

,)t(O)t(O)t(P)t(P


)t(OP)t( P)t( OP

u

uuuu,THu,EL

uuuu,THEL,uuuuu,TH


∈+≥

+≤≤+

≤≤

γγββεε

´


)t(PP)t(P

P)t(P)t(P)t(Pt.s

)t(P)t()t(P)t()tcos)t((P))t(tcos)t()(t(P))t()t(price)(t(P

max

TOP,BUY,ELu

u,EL,MAXSPOT,SELL,EL


DEMAND,ELSPOT,BUY,EL

T

t DEMAND,THDEMAND,ELTOPTOP,BUY,EL

SPOTSPOT,BUY,ELSPOTSPOT,SELL,EL

PEL

+≤≤

≤≤≤≤

−−−+

−+−

∑

∑=

0

0 0

1 µλλλλ

The amount of bought electricity from the spot market can never be higher than the electricity demand. The amount of electricity sold to the spot market can never be higher than what can be produced by the turbines and what can be bought from contracts. The cost for buying electric power at the spot market, , is always higher than the

price for selling, . Therefore, the optimally bought amount of electric power

from the spot market will always be zero when the sold amount is non-zero. The opposite is also true.

(t) tcos SPOT

)t(priceSPOT

Since we do not have any time-dependent restrictions the relaxed problem (as well as the primal problem) can be further decomposed to one problem for each time interval. If minimal operation and shut-down times are included we will get a dependency between the time intervals for the part of the relaxed problem related to the CHP turbines. If a limit for the amount bought from the take-or pay contract is included we will also get a time dependency for the sub-problem including the contract terms.

4 Aspects on solving the Lagrangian relaxed problem

The dual problem of a Lagrangian relaxed problem is solved iteratively where the Lagrange multipliers are updated between each iteration. In /Dotzauer, 2001/ the procedure is described as follows:

1. Choose starting values for the multipliers. 2. Solve the relaxed problem for the current values of the multipliers. 3. Construct a primal feasible solution from the solution of the relaxed problem. 4. Stop if the convergence criterion is satisfied (the duality gap is sufficiently small). 5. Update the multipliers and go to 2.

The difficulty is then to find good starting values for the multipliers and a suitable algorithm to update the multipliers. In both /Dotzauer, 2001/ and /Huonker, 2000/ several previously used so-called sub-gradient methods are mentioned. In /Cheng et al, 2000/ genetic algorithms are used to update the multipliers. In /Aoki et al, 1989/ an algorithm based on the variable metric method for dual maximisation is presented.


For the sub-gradient method , where n is the number of the iteration step, λ is the Lagrange multiplier, α is the step length and s is the sub gradient. The sub-gradient equals the value of the restriction included in the relaxed problem. There are several methods to determine α, and in /Dotzauer, 2001/ some different methods are compared for the heat production problem described under 3.1.

nnnn sαλλ +=+1

Aoki et al /Aoki et al, 1989/ say that the advantage of the variable metric method they use for updating Lagrange multipliers over the traditional sub-gradient method is that their method prevents the solution from oscillating near the dual maximum. Also in this method the sub-gradients, s, are used . The equation for updating the multipliers is, however, different from the one shown above for the sub-gradient method. A method to solve the dual and primal problem must also be chosen. Dynamic programming seems to be common to use in connection with Lagrangian relaxation. For example in /Cheng et al, 2000/, /Dotzauer, 2001/, /Sterner et al, 2001/ and /Virmani et al, 1989/ this method is used. However, in /Dotzauer, 2001/ combinations with branch-and-bound methods are also mentioned and presented. In one example he uses a branch-and-bound framework for solving the primal problem.

5 Conclusions

To use the Lagrangian relaxation could be one way to improve the computational performance of the long-term OSCOGEN optimisation model. To start with we have to look more into how the algorithms for updating the Lagrange multipliers are best formulated for our problem and also how to best choose the starting values of the multipliers. Then we can try to solve a simplified problem in the form of the problem presented in Chapter 3.2 . Further we have to consider the best way to solve the primal and dual problem. We will look more into the possibilities to use dynamic programming.

6 Literature

/Aoki et al, 1989/ Aoki, K; Itoh, M.; Satoh, T.; Nara, K.; Kanezashi, M.: Optimal Long-Term Unit Commitment in Large Scale Systems Including Fuel Constrained Thermal and Pumped-Storage Hydro, In: IEEE Transactions on Power Systems, Vol. 4, No. 3, 1989 /Cheng et al, 2000/


Cheng, Chuan-Ping; Liu, Chih-Wen; Liu, Chun-Chang: Unit Commitment by Lagrangian Relaxation and Genetic Algorithms, In: IEEE Transactions on Power Systems, Vol. 15, No 2, 2000 /Dotzauer, 2001/ Dotzauer, Erik: Energy System Operation by Lagrangian Relaxation, Linköping Studies in Science and Technology, Dissertations No. 665, Division of Optimization, Departments of Mathematics, Linköpings universitet, Sweden, 2001 /Huonker, 2001/ Huonker, U.: Kraftwerkseinsatzplanung mit dem Lagrange Optimization System, In: Optimerung in der Energiversorgung, VDI-Berichte Nr 1627, 2001 /Stern et al, 2001/ Stern, B.; Haubrich, H.-J.; Ewert, A.: Stochastiche Optimerung von Kraftwerkeinsatz und Stromhandel zur Berücksichtung von Planungssicherheiten, In: Optimerung in der Energiversorgung, VDI-Berichte Nr. 1627, 2001 /Virmani et al, 1989/ Virmani, S.; Imhof, K.; Mukherjee, S.: Implementation of a Lagrangian Relaxation Based Unit Commitment Problem, In: IEEE Transactions on Power Systems, Vol. 4, No. 4, 1989

using lagrangian relaxation in optimisation of unit ... · 3 lagrangian relaxation for optimisation...

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