Electrical Power and Energy Systems 32 (2010) 1099–1105

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Electrical Power and Energy Systems

journal homepage: www.elsevier .com/locate / i jepes

Enhanced Lagrangian relaxation solution to the generation scheduling problem

Farid Benhamida *, Bendaoud AbdelbarIRECOM Laboratory, Department of Electrical and Electronic Engineering, Djillali Liabes University, Sidi Bel Abbes 22000, Algeria

a r t i c l e i n f o

Article history:Received 2 April 2009Received in revised form 13 November 2009Accepted 7 June 2010

Keywords:Unit commitmentGeneration schedulingLagrangian relaxationUnit classification

0142-0615/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.ijepes.2010.06.007

* Corresponding author.E-mail address: farid.benhamida@yahoo.fr (F. Benh

a b s t r a c t

This paper proposes an enhanced Lagrangian relaxation (LR) solution to the generation scheduling prob-lem of thermal units, known as unit commitment (UCP). The proposed solution method is characterizedby a new Matlab function created to determine the optimal path of the dual problem, in addition, the ini-tialization of Lagrangian multipliers in our method is based on both unit and time interval classification.The proposed algorithm is distinguished by a flexible adjustment of Lagrangian multipliers, and dynamicsearch for uncertain stage scheduling, using a Lagrangian relaxation–dynamic programming method (LR–DP). After the LR best feasible solution is reached, a unit decommitment is used to enhance the solutionwhen identical or similar units exist in the same system. The proposed algorithm is tested and comparedto conventional Lagrangian relaxation (LR), genetic algorithm (GA), evolutionary programming (EP),Lagrangian relaxation and genetic algorithm (LRGA), and genetic algorithm based on unit characteristicclassification (GAUC) on systems with the number of generating units in the range of 10–100. The totalsystem production cost of the proposed algorithm is less than the others especially for the larger numberof generating units. Computational time was found to increase almost increases linearly with system size,which is favorable for large-scale implementation.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Unit commitment problem (UCP) is a nonlinear, mixed integercombinatorial optimization problem. It is defined as the problemof how to schedule generators economically in a power systemin order to meet the requirements of load and spinning reserve.Usually this problem is considered over some period of time, suchas the 24 h of a day or the 168 h of a week. It is a difficult problemto solve in which the solution procedures involve the economicdispatch problem as a sub-problem.

Since the problem was introduced, several solution methodshave been developed. However, they differ in the solution quality,computational efficiency and the size of the problem they cansolve. These methods or approaches have ranged from highly com-plex and theoretically complicated methods to simplified methods.

In the past, various approaches such as DP [1], branch-and-bound B&B [2] and Lagrangian relaxation (LR) [3] were proposedfor solving the UCP. However, not all of these methods are regardedas feasible and/or practical as the size of the system increases.

For moderately sized production systems, exact methods, suchas dynamic programming (DP) or (B&B) [2] can be used to solvethe UCP, successfully. For larger systems, exact methods fail be-cause the size of the solution space increases exponentially withthe number of time periods and units in the system. As a result,

ll rights reserved.

amida).

the computation time of exact methods becomes impractical. Inthese cases heuristic methods (evolutionary programming (EP),Tabu Search (TS), Simulated Annealing (SA), Genetic Algorithms(GA), etc.) can be used to produce near optimal solutions in a rea-sonable computation time. For heuristic methods optimality is notgiven such a high priority but the emphasis is on finding good solu-tions in a short time. This often results in the solution methodbeing more simple and transparent than exact solution methods[4].

The application of LR in the scheduling of power generationswas proposed in the late 1970s. These earlier methods used LR tosubstitute the common linear programming (LP) relaxation ap-proach as a lower bound in the B&B technique [5]. In this regard,great improvement of computational efficiency was achieved com-pared with previous B&B algorithms.

In recent years, methods based on LR, have become the mostdominant ones. This approach has shown some potential in dealingwith systems that consist of hundreds of generating units and ismotivated by the separable nature of the problem, and severalexamples have been reported in the literature.

Based on the sharp bound provided by the Lagrangian dual opti-mum, it is expected that a sub-optimal feasible solution near thedual optimal point can be accepted as a proper solution for the pri-mal problem. A more direct and fairly efficient methodology whichhas used this idea was presented in [6] by Merlin, for UCP using LRmethod and validated at Electricite De France. Due to its reason-able performance, the successive improvement of the LR algorithm,

1100 F. Benhamida, B. Abdelbar / Electrical Power and Energy Systems 32 (2010) 1099–1105

in the last few years, has mainly followed the work in [6]. Theproblem which is supposed to be handled by this algorithm con-sists of thermal units only.

In [7], they combined LR, sequential UC based on the least re-serve cost index and unit decommitment (UD) based on the high-est average spinning reserve cost index. However, this methodcould not decommit some units that violate the minimum up timeconstraints even though the excessive reserve exists, leading to ahigher production cost.

In the advent of heuristic approaches, GA [8], EP [9], SA [10], andTS [11] have been proposed to solve the UC problems. Neverthe-less, the obtained results by GA, EP, and SA required a considerableamount of computational time especially for a large systemsize. There was an attempt to combine the LR and GA method(LRGA) to obtain a higher quality of UC solution in a shorter timeby using normalized Lagrange multipliers as the encoded parame-ter [12].

2. Unit commitment problem formulation

The objective of the UCP is to minimize the system operatingcosts, which is the sum of production and startup costs of all unitsover the entire study time span (e.g., 24 h), under the generatoroperational and spinning reserve constraints. Mathematically, theobjective function, or the total operating cost of the system canbe written as follows:

J ¼minPti ut

if ðPt

i ;uti Þ

¼minPti ut

i

XT

t¼1

XN

i¼1

uti FiðPt

i Þ þ Sti 1� ut�1

i

� �� � !ð1Þ

Subject to:

(1) The startup cost is modeled by the following function of theform:

Sti ¼

HSi; if Xtoff ;i 6 Tdown

i þ CHi

CSi; if Xtoff ;i > Tdown

i þ CHi

(ð2Þ

(2) Power balance

XN

i

uti P

ti ¼ Dt ð3Þ

(3) Spinning reserve requirements:

XN

i

uti P

maxi P Dt þ Rt ð4Þ

Generating limits:

uti P

mini 6 Pt

i 6 uti P

maxi ð5Þ

Minimum up time constraint:

Xt�1on;i � Tup

i

� �ðut�1

i � uti ÞP 0 ð6Þ

Xton;i ¼ ðX

t�1on;i þ 1Þut

i ð7Þ

Minimum down time constraint:

ðXt�1off ;i � Tdown

i Þðuti � ut�1

i ÞP 0 ð8Þ

Xtoff ;i ¼ ðX

t�1off ;i þ 1Þð1� ut

i Þ ð9Þ

Fuel cost function FiðPti Þ is frequently represented by the poly-

nomial function:

Fi Pti

� �¼ ai þ biP

ti þ ci Pt

i

� �2 ð10Þ

where Pti is the output power of unit i at period t (MW), FiðPt

i Þ isfuel cost of unit i when its output power is Pt

i ($), Sti is startup

price of unit i at period t ($), uti is commitment state of unit i

at period t (uti ¼ 1: unit is on-line and ut

i ¼ 0 unit is off-line), Nis total number of generating units, T is total number of schedul-ing periods, ai, bi, ci are coefficients for the quadratic cost curve ofgenerating unit i, Xt

off ;i;Xtoni are number of hours the unit has been

off-line/on-line (h), X0i is initial condition of a unit i at t = 0,

X0i > 0: on-line unit, X0

i < 0: off-line unit (h), Tupi is minimum

up time (h), Tdowni minimum down time (h), HSi, CSi are the unit’s

hot/cold startup cost ($), CHi is the cold start hour (h), Dt is cus-tomers’ demand in time interval t, Rt is the spinning reserverequirements;

3. An improved flexible Lagrangian relaxation technique

In the Lagrangian relaxation approach, the system operatingcost function of Eq. (1) of the unit commitment problem is relatedto the power balance and the spinning reserve constraints viatwo sets of Lagrangian multipliers to form a Lagrangian dualfunction.

LðP;u; k;lÞ ¼ f ðP;uÞ þXT

t¼1

kt Dt �XN

i¼1

uti P

ti

!

þXT

t¼1

lt Dt þ Rt �XN

i¼1

uti P

maxi

!ð11Þ

The LR procedure solves the UCP through the dual problemoptimization procedure attempting to reach the constrainedoptimum.

The dual procedure attempts to maximize the Lagrangian withrespect to the Lagrangian multipliers kt and lt, while minimizing itwith respect to the other variables Pt

i ;uti subject to the unit con-

straints in Eq. (5) through Eq. (9). The dual problem is thus thesearch of the dual solution (Q) expressed as:

Q ¼maxkt ;lt

minPt

i ;uti

LðP;u; k;lÞ !

kt P 0 and lt P 0 ð12Þ

The Lagrangian function of Eq. (11) is rewritten as

LðP;u; k;lÞ ¼ f ðP;uÞ �XT

t¼1

ktXN

i¼1

uti P

ti �XT

t¼1

ltXN

i¼1

uti P

maxi

þXT

t¼1

ktDt þXT

t¼1

ltðDt þ RtÞ ð13Þ

When the Lagrangian multipliers kt(k) and lt(k) are fixed for iter-ation k, the last two terms of the Lagrangian in Eq. (13) are con-stant and can be dropped from the minimization problem. Hence,the system (coupling) constraints can be relaxed and the search forthe dual solution can be done through the minimization of theLagrangian as:

minPt

i ;uti

LðP;u; kðkÞ;lðkÞÞ ¼minPt

i ;uti

XT

t¼1

XN

i¼1

uti

�Fi Pt

i

� �þ St

i 1� ut�1i

� �

� ktðkÞPti � ltðkÞPmax

i

!ð14Þ

Then, the minimum of the Lagrangian function is solved foreach generating unit over the time horizon, that is

F. Benhamida, B. Abdelbar / Electrical Power and Energy Systems 32 (2010) 1099–1105 1101

minPt

i ;uti

LðP;u; kðkÞ;lðkÞÞ ¼XN

i¼1

minPt

i ;uti

XT

t¼1

uti

�Fi Pt

i

� �

þ Sti 1� ut�1

i

� �� � ktðkÞPt

i � ltðkÞPmaxi

!ð15Þ

Subject to constraints in Eq. (5) through Eq. (9).

3.1. The dual problem optimization

In the Lagrangian relaxation method, the dual solution is ob-tained for each unit separately.

When the state uti ¼ 0, the value of the function to be minimized

is equal zero (the unit is off-line).When the state ut

i ¼ 1, the value to be minimized is:

Fi Pti

� �� ktðkÞPt

i ð16Þ

The startup cost and the last term in Eq. (15) are dropped sincethe minimization is with respect to Pt

i .When the units’ fuel costfunctions are represented as polynomial functions as in Eq. (10),the minimum of Eq. (16) can be found by taking its first derivative.

d Fi Pti

� �� ktPt

i

� �=dPt

i ¼ dFi Pti

� �=dPt

i � kt ¼ 0 ð17Þ

Hence,

PtðkÞi ¼ ðktðkÞ � biÞ=2ci ð18Þ� If PtðkÞ

i < Pmini then PtðkÞ

i ¼ Pmini ð19Þ

� If PtðkÞi > Pmax

i then PtðkÞi ¼ Pmax

i ð20Þ

For known ktðkÞ, then PtðkÞi is obtained by Eq. (18) through Eq.

(20).

3.1.1. A new Matlab function to determine the optimal pathTo minimize the term in Eq. (15) for each unit, over the sched-

uled time T, subject to minimum up and down time constraints inEqs. (6)–(9), DP is often used to determine the optimal schedule.Dynamic programming CPU time increases at least linearly with Nand T (upper bounded by N[4(T � 1) + 2] additions and 2N(T � 1)comparisons) [3,13].

A reduction of the search domain, which is defined by 2T combi-nations, can be made by discarding the infeasible combinationsfrom the domain. The optimal combination which minimize Eq.(15) for a unit i can be determined by direct evaluation of all feasiblecombinations. A Matlab function is developed for this purpose.

This function gives all feasible combinations (mi) of a unit i overthe scheduling period T which satisfy the minimum up and downtime constraints given its initial state and condition.

Function input: X0i ; T; T

upi ; T

downi .

Function output: ½uti;j� which is a (T �mi) matrix containing all

feasible combination as

uti;j

h i¼

u1i;1 u1

i;2 . . . u1i;mi

u2i;1 u2

i;2 . . . u2i;mi

..

. ... ..

.

uTi;1 uT

i;2 . . . uTi;mi

0BBBBB@

1CCCCCA; i ¼ 1; . . . ;N; ½ui;j� ¼

u1i;j

u2i;j

..

.

uTi;j

2666664

3777775;

j ¼ 1; . . . ;mi; ð21Þ

Then the optimal solution PðkÞi and its corresponding path ui,j

(combination), given the Lagrangian multipliers kðkÞ;lðkÞ of itera-tion k, is obtained by the following procedure:

Step 1: Running the Matlab function to obtain all feasible com-binations (mi) of unit i over the scheduling period T which sat-isfies the minimum up and down time constraints, given itsinitial state and condition.

Step 2: For each ui,j, j = 1, . . . , mi, t = 1, . . . , T calculate the contri-bution term of unit i in a specific period t using the followingequation

uti;j Fi PtðkÞ

i

� �þ St

i 1� ut�1i;j

� �� ktðkÞPtðkÞ

i � ltðkÞPmaxi

n oj ¼ 1; . . . ;mi t ¼ 1; . . . ; T ð22Þ

Step 3: For each ui,j, j = 1, . . . , mi, calculate the contribution termwhich correspond to unit i over the total period T using the fol-lowing equation:

XT

t¼1

uti;j Fi PtðkÞ

i

� �þ St

i 1� ut�1i;j

� �� ktðkÞPtðkÞ

i � ltðkÞPmaxi

n oh ij ¼ 1; . . . ;mi ð23Þ

Step 4: Obtain optimal solution PðkÞi and its corresponding pathui,j (combination), by taking the least valued contribution termobtained in step 3. We have not to check the path vis-à-vis theminimum up and down time since it is a feasible one.Step 5: Repeat step 1–4 for all units to obtain PðkÞ; uðkÞ.

The values of the system variables PðkÞ;uðkÞ; kðkÞ;lðkÞ are substi-tuted back into the Lagrangian Eq. (11), LðPðkÞ;uðkÞ; kðkÞ;lðkÞÞ o deter-mine the dual solution Q(k):

L PðkÞ;uðkÞ; kðkÞ;lðkÞ� �

¼ f PtðkÞi ;utðkÞ

i

� �

þXT

t¼1

ktðkÞ Dt �XN

i¼1

utðkÞi PtðkÞ

i

!

þXT

t¼1

ltðkÞ Dt þ Rt �XN

i¼1

utðkÞi Pmax

i

!ð24Þ

Provided that the dual solution is feasible with respect to thespinning reserve constraint of Eq. (4) and the following constraintregarding the minimum output power of the scheduled units issatisfied:

XN

i¼1

uti P

mini 6 Dt 8 t ¼ 1; . . . ; T ð25Þ

The inequalities related to the spinning reserve constraints Eq.(4) do not impose an upper bound on the amount of reserve. How-ever, common sense for an economic schedule indicates that thereshould not be too much excess MW reserve because it would cer-tainly increase the cost associated with the corresponding dualsolution. Therefore, in the searching algorithm, a slack term (st) isincluded in the reserve constraint to assess the quality of the dualsolution. The upper-bound limit introduced by the slack term re-stricts the solution space and therefore may prevent the optimalsolution to be found. In addition, the value of the slack term mayaffect the convergence of the process. Unfortunately, there is nomathematical guideline for properly selecting the value of slackterm (st) [13].

Hence, in the searching algorithm, the following constraints areincluded implicitly to test the validity of the commitmentschedule.

Dt þ Rt6

XN

i

uti P

maxi 6 Dt þ Rt þ st ð26Þ

In this paper st is specified using a new heuristic algorithmbased on both unit and time interval classification.

1102 F. Benhamida, B. Abdelbar / Electrical Power and Energy Systems 32 (2010) 1099–1105

3.2. A new initial scheduling of UC

The initial values of Lagrangian multipliers are very critical tothe LR solution since they may prevent LR from reaching the opti-mal solution or require a longer computational time to reach one[14]. Different initial values may also lead LR to different solutions.In [15], the initial multiplier kt was set to the hourly system mar-ginal cost of the schedule to satisfy the power balance constraintand the initial multiplier lt was set to zero, leading to an infeasibleinitial solution. Alternatively, the initial multiplier kt was set to thehourly system marginal cost of the schedule to satisfy both thepower balance and spinning reserve constraint, whereas the initialmultiplier lt was set to zero which was generally lower than theoptimal value [16].

An initialization procedure which intends to create a high qual-ity feasible schedule in the first iteration is described here, basedon unit and time interval classification.

3.2.1. Unit classificationIn general, generation units can be classified into three types:

base load units with low operation cost Fi, high startup cost Si,and long minimum up/down times Tup

i ; Tdowni ; intermediate load

units with medium operating cost, medium startup cost and med-ium minimum up/down time, and peak load units with high oper-ation cost, low startup cost and short minimum up/down time.Base load units should not be shut down. In other words they con-stitute the must run constraint. Intermediate load units could becommitted during on-peak and decommitted during off-peak peri-ods. Finally, peak load units could be frequently turned on and off.

Following this classification, the N units of an N-unit system canbe classified into a set Nb of base load units, a set NI of intermediateload units and a set Np of peak load units according to unit full loadaverage production costs (flac) and unit operational constraintswhere:

flac ¼ F Pmaxi

� �=Pmax

i in $=MWh ð27Þ

Table 1Initial scheduling of UCP based on unit and time interval classification.

Time set Unit sets

Nb NI Np

t 2 Tbd uti ¼ 1 ut

i ¼ 0 uti ¼ 0

t e TId uti ¼ 1 ut

i ¼ 1 uti ¼ 0

t e TI � TId uti ¼ 1 ut

i ¼ 1 uti Initialized

based on flact e TI0 � TId � Tbd ut

i ¼ 1 uti Initialized

based on flacut

i ¼ 0

3.2.2. Time interval classificationThe overall study period is decomposed into several interval

classes as follow:

(1) Tbd presents the set of scheduling intervals t where t 2 Tb

and the upper-bound limit of the spinning reserve is satis-fied:

Pi2Nb

Pmaxi � Dt � Rt

6 st ; Tb being the set of schedulingintervals t where base units can produce enough power tosatisfy the inequality

Pi2Nb

Pmaxi P Dt þ Rt . Hence, during

the intervals (Tbd) only base units are committed.(2) TId presents the set of scheduling intervals t where

t 2 TI \ TI0 , and in whichP

i2Nb[NIPmax

i � Dt � Rt6 st . Here TI

presents the set of scheduling intervals t where the groupof base and intermediate units cannot produce enoughpower to satisfy the constraint:

Pi2Nb[NI

Pmaxi 6 Dt þ Rtþ

mini2NI

ðPmaxi Þ, while TI0 presents the set of scheduling intervals

t where the base and intermediate units grouped can pro-duce enough power to satisfy the spinning reserve con-straint

Pi2Nb[NI

Pmaxi P Dt þ Rt . At these intervals (TId) both

base and intermediate units are committed. Note thatTb � TI0.

(3) TI – TId give the set of scheduling hour’s t, where peak unitsmust be committed, for these scheduling periods the peakunits are selected one by one, based on the flac, until enoughcapacity is reached to fulfill the spinning reserve constraints.

(4) TI0 � TId � Tbd give the set of scheduling hour’s t, where inter-mediate units must be committed, for these scheduling peri-ods intermediate units are selected one by one, based on the

flac, until enough capacity is reached to fulfill the spinningreserve constraints.

The slack term (st) is defined as

If t 2 Tb; st ¼maxi2Nb

Pmaxi

� �ð28Þ

If t 2 TI; st ¼maxi2Np

Pmaxi

� �ð29Þ

If t 2 TI0 Þ and t R Tb; st ¼maxi2NI

Pmaxi

� �ð30Þ

Table 1 gives the initial commitment states of different sets Nb,NI and Np during different time interval classes.

3.2.3. Initial value of Lagrangian multipliersThe initial value of Lagrangian multipliers kt(0) are set as follow:

(1) For each hour t e TI0 � TId � Tbd and t e TI � TId, the group ofidentical units with the least (flac) will be committed onegroup by one group until the spinning reserves constraintis satisfied as shown in Table 1. Subsequently, economic dis-patch in each hour is carried out to obtain the hourly equallambda which is initially set to Lagrangian multipliers kt(0).

(2) For each t 2 Tbd [ TId, as at these periods a predefined UC isestablished as shown in Table 1. Lagrangian multiplierskt(0) are set to the hourly equal lambda, after running an eco-nomic dispatch program for these periods.

The initial value of each non-negative Lagrangian multiplierslt(0) is set as follow:

ltð0Þ ¼max maxi¼1;...;Mt

1Pmax

i

Fi Pti

� �þ St

i

Tupi

� ktð0ÞPti

� �;0

�;

t ¼ 1; . . . T

ð31Þ

Mt: is the marginal unit with the highest (flac), giving the suffi-cient spinning reserve at hour t.

3.3. Updating of the lagrangian multiplier

In general, adjusting Lagrangian multiplier by sub-gradientmethod is not efficient in the presence of the spinning reserve con-straint [6]; one of the shortcomings of this method is the slow con-vergence. The LR performance is heavily dependent on the methodused to update the multipliers. In this paper, a flexible sub-gradi-ent rule is proposed to update the Lagrangian multiplier and de-signed such that the step size is large at the beginning ofiterations and smaller as the iteration grows. Each non-negativekt and lt are adaptively updated by,

ktðkÞ ¼max 0; ktðk�1Þ þ PtM

ðqþ h� kÞ � normðPMÞ

�t ¼ 1; . . . ; T ð32Þ

ltðkÞ ¼max ltðk�1Þ þ SRtM

ðqþ h� kÞ � normðSRMÞ;0

�t ¼ 1; . . . ; T ð33Þ

F. Benhamida, B. Abdelbar / Electrical Power and Energy Systems 32 (2010) 1099–1105 1103

Where

PtM ¼ Dt �

XN

i¼1

uti P

ti ð34Þ

SRtM ¼ Dt þ Rt �

XN

i¼1

uti P

maxi ð35Þ

normðPMÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðP1

MÞ2 þ ðP2

MÞ2 þ . . . . . . . . . . . .þ ðPT

MÞ2

qð36Þ

normðSRMÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðSR1

MÞ2 þ ðSR2

MÞ2 þ . . . . . . . . . . . .þ ðSRT

MÞ2

qð37Þ

The values of q and h are divided into four cases depending onthe signs of PM and SRM.

Case (1) PtM P 0 and SRt

M P 0: updating both kt and lt byusing q = 0.03 and h = 0.06.Case (2) Pt

M < 0 and SRtM < 0: updating both kt and lt by

using q = 0.5 and h = 0.3.Case (3) Pt

M < 0 and SRtM > 0: updating only lt by using q = 0.03

and h = 0.06.Case (4) Pt

M > 0 and SRtM < 0: updating only kt by using q = 0.5

and h = 0.3.

The general guidelines for selecting their values are explained in[17].

In fact, updating the two multipliers kt and lt in hour t mustmove them in the same direction. In hour t, if Pt

M and SRtM have

the same signs, either positive or negative, kt and lt will be up-dated (increase or decrease) by Eqs. (32) and (33), respectively.

When the total dual generation output is larger than the loadin that hour (Pt

M < 0) but the spinning reserve is insufficient(SRt

M > 0), more committed unit(s) are required to satisfy the spin-ning reserve constraints. However, updating kt by Eq. (32) will de-crease its value, resulting in committing less units. Therefore, when(Pt

M < 0) and (SRtM > 0), only lt will be updated. On the contrary,

when the spinning reserve is sufficient (SRtM < 0), but the total dual

generation output is less than the load in that hour (PtM > 0),

updating lt by Eq. (33) will decrease its value, resulting in commit-ting less units. Therefore, when (Pt

M > 0) and (SRtM < 0), only kt will

be updated.Note that the sub-gradient method generally needs a large

number of iterations to converge to near the dual optimum [17].The proposed flexible sub-gradient method using high-quality ini-tial feasible multipliers proved to require much lower number ofiterations to converge, leading to much less computational time.

3.4. Dynamic economic dispatch (DED) [18]

To replace conventional economic dispatch algorithm, a moreaccurate and flexible problem formulation of DED is developed tofacilitate the interaction with UC schedule, The DED solver usethe Hopfield Neural Network, which make it a very fast solverand suitable to UCP.

If the 24-h schedule is feasible at iteration k, a DED is carried outto determine the optimal generation power outputs for each of the24 h, and the total production cost J(k).

3.5. Checking for convergence

The convergence of the proposed LR-UC algorithm can be mea-sured by the relative duality gap between the primal and dualsolutions.

Relative duality gap ¼ ððJðkÞ � Q ðkÞÞ=Q ðkÞÞ � 100 ð38Þ

The process stops when the relative duality gap is smaller thana pre-specified tolerance e, or when a pre-specified maximumnumber of iterations is reached.

The sensitivity of the integer variables corresponding to thegenerating unit statuses (ut

i ) to small adjustments in the Lagrang-ian multipliers may cause the algorithm to oscillate around theoptimal solution. As such, there is no guarantee that the solutionachieved in the last iteration of the iterative process will be feasibleor optimal. Hence, in the computational model developed in thepaper, a running record of the feasible solutions is kept so thatthe final solution is the one corresponding to the most economicalschedule, i.e., the one with the minimum primal solution (J).

4. Identical unit decommitment

When identical or similar units exist the LR could find onlysub-optimal solutions [14]. These units have the identical costparameters ai, bi, ci, and startup cost which will be simultaneouslycommitted or decommitted. This will not lead to the optimal solu-tion because committing one unit at a time will be less expensivethan committing a whole group of units, which may lead to overcommitment. Thus, after committing a group of identical units, aunit of which is decommitted one at a time if it does not violatethe minimum up time constraint until either the spinning reserverequirement is not satisfied or there is only one unit left. The iden-tical unit decommitment procedure is as follows:

Step (1): Get the initial feasible solution ½uti �, i = 1, . . . , N, t =

1, . . . , T.Step (2): Calculate the excess spinning reserve of every hours,

Rtex ¼

XN

i

uti P

maxi � Dt þ Rt ð39Þ

Step (3): Initialize t = 1Step (4): Initialize i = 1Step (5): If the excess spinning reserve Rt

ex is greater than themaximum generation of unit i, and this unit is already commit-ted, check if decommitting the unit would violate its minimumup time constraints.

Decommit the unit i,

if Xton;i ¼ 1; and X

tþTupi�1

on;i ¼ Tupi : ð40Þ

or if Xton;i > Tup

i ; and Xtþ1off ;i ¼ 1: ð41Þ

or if Xton;i ¼ 1; and

XT

k¼t

uki ¼ T � kþ 1: ð42Þ

or if Tupi ¼ 1 ð43Þ

Otherwise, let the unit committed.

Step (6): If t = T stop, else go to step 7Step (7): Update ½ut

i � and Rtex, replace i by i+1,

Step (8): If i = N, replace t by t+1, and go to step 4 .Otherwise, goto step 5.

5. Numerical results

A 10-unit system [8] is selected as a test system. System dataand load demand are given in Tables 2 and 3. The spinning reserveis assumed to be 10% of the demand. The 20, 40, 60, 80, and 100unit systems are obtained by duplicating the 10-unit base case,whereas the load demand are adjusted in proportion to the systemsize. The proposed LRUC uses the developed Matlab function todetermine the optimal path. A maximum allowable number of 50iterations was set as a stopping criteria.

Table 3Demand of the 10-unit 24 h test system.

Hour Load (MW) Hour Load (MW) Hour Load (MW)

1 700 9 1300 17 10002 750 10 1400 18 11003 850 11 1450 19 12004 950 12 1500 20 14005 1000 13 1400 21 13006 1100 14 1300 22 11007 1150 15 1200 23 9008 1200 16 1050 24 800

0 0 1 1 1

…… 1 1 1 1 1 0 1 0 1 1

1 0 0 1 1

t …… 21 22

Intermediate unit coding at t = 21

All possible intermediate coding a t = 22

Fig. 1. Dynamic programming search

Table 4Solution of 10 – unit 24-h using the proposed LRUC method.

Table 2Unit data of the 10-unit 24 h test system.

Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit 7 Unit 8 Unit 9 Unit10

Pmax (MW) 455 455 130 130 162 80 85 55 55 55Pmin (MW) 150 150 20 20 25 20 25 10 10 10a ($/h) 1000 970 700 680 450 370 480 660 665 670b ($/MWh) 16.19 17.26 16.60 16.50 19.70 22.26 27.74 25.92 27.27 27.79c ($/MW2h) 0.00048 0.00031 0.00200 0.00211 0.00398 0.00712 0.00079 0.00413 0.00222 0.00173Tup

i (h) 8 8 5 5 6 3 3 1 1 1

Tdowni (h) 8 8 5 5 6 3 3 1 1 1

HS 4500 5000 550 560 900 170 260 30 30 30CS 9000 10000 1100 1120 1800 340 520 60 60 60CH 5 5 4 4 4 2 2 0 0 0

Xi0

8 8 �5 �5 �6 �3 �3 �1 �1 �1

flac 18.61 19.53 22.24 22.01 23.12 27.45 33.45 38.14 39.48 40.06

1104 F. Benhamida, B. Abdelbar / Electrical Power and Energy Systems 32 (2010) 1099–1105

5.1. An improvement to the method

The behavior of the units during the iterative search of the LRbased solution and the preliminary schedule itself is assessed todefine the uncertain intervals, in which commitment states ofsome units are not certain. In this example these stages are 22and 23. Then, a dynamic search is performed at these stages, usinga DP solution to UC combined with LR (LR–DP), as shown in Fig. 1,where all possible and feasible paths with respect to minimum upand down time constraints are shown. The optimum path is distin-guished by bold lines. The UC solution schedule using the proposed

1 0 0 0 00 1 0 0 00 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1

23 24

units All possible intermediateunits coding a t = 23

for uncertain stages 22 and 23.

Table 5Comparison of total production costs.

Method COST($)

No. of units

10 20 40 60 80 100

LR [8] 565,825 1130,660 2258,503 3394,066 4526,022 5657,277GA [8] 565,825 1126,243 2251,911 3376,625 4504,933 5627,437EP [19] 564,551 1125,494 2249,093 3371,611 4498,479 5623,885LRGA [20] 564,800 1.122,622 2242,178 3371,079 4501,844 5613,127DPLR [21] 564,049 1.128,098 2256,195 3384,293 4512,391 5640,488GAUC [21] 563,977 1.125,516 2249,715 3375,063 4505,614 5640,488Proposed LR 563,937.69 1122,637 2243,245 3363,376 4484,915 5604,470

Table 6CPU time.

CPU time

No. of units

Method 10 20 40 60 80 100Proposed LR 10 14 25 39 64 80

F. Benhamida, B. Abdelbar / Electrical Power and Energy Systems 32 (2010) 1099–1105 1105

Lagrangian Relaxation combined to DP is shown in Table 4, dy-namic search for uncertain stages are highlighted.

Table 5 shows simulation results (production costs) obtained bythe proposed LR method compared with results obtained by LR [8],GA [8], EP [19], and the combined LRGA [20] and DPLR [21] meth-ods for 10, 20, 30, 40, 60, 80 and 100 units. Table 6 shows the sim-ulation time obtained by the proposed LR method which is carriedout on Pentium M 1.73 GHz processor. Because simulations werecarried out on different types of computers, simulation times arenot compared. It can be seen that the results of the proposed meth-od is better than other methods in term of total production cost. Itcan be seen that computational time increases almost linearly withsystem size.

6. Conclusion

This paper presents a Lagrangian relaxation solution to the gen-eration scheduling problem of thermal units. An initialization pro-cedure intends to create a high quality feasible schedule in the firstiteration is proposed, based on unit and time interval classification.The proposed LR is efficiently and effectively implemented to solvethe UC problem. The proposed LR total production costs over thescheduled time horizon are less than conventional LR, GA, EP,LRGA, and GAUC especially for the larger number of generatingunits. Moreover, the proposed LR CPU times increase almost line-arly with the system size as shown in Table 6, which is favorablefor large-scale implementation.

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