a tabu search based hybrid optimization approach for a fuzzy modelled unit commitment problem

Electric Power Systems Research 76 (2006) 413–425

A tabu search based hybrid optimization approach for afuzzy modelled unit commitment problem

T. Aruldoss Albert Victoirea,∗, A. Ebenezer Jeyakumarb

a Department of Electrical and Electronics Engineering, Karunya Institute of Technology, Karunya Nagar,Coimbatore, Tamil Nadu 641114, India

b Department of Electrical and Electronics Engineering, Anna University, GCT campus, Coimbatore,Tamil Nadu 641013, India

Received 15 March 2005; received in revised form 26 July 2005; accepted 26 August 2005Available online 28 October 2005

Abstract

This article presents a solution model for the unit commitment problem (UCP) using fuzzy logic to address uncertainties in the problem. Hybridtabu search (TS), particle swarm optimization (PSO) and sequential quadratic programming (SQP) technique (hybrid TS–PSO–SQP) is used tos combiningt s of the loadd ved randomp e non-linearp mal PowerS olution withs©

K

1

stm(UsmUuefiu

are

ic tomandme.loadion isble itnablesnaler-stem.

ations

al, aopti-f itscou-blemiques

0d

chedule the generating units based on the fuzzy logic decisions. The fitness function for the hybrid TS–PSO–SQP is formulated byhe objective function of UCP and a penalty calculated from the fuzzy logic decisions. Fuzzy decisions are made based on the statisticemand error and spinning reserve maintained at each hour. TS are used to solve the combinatorial sub-problem of the UCP. An improerturbation scheme and a simple method for generating initial feasible commitment schedule are proposed for the TS method. Throgramming sub-problem of the UCP is solved using the hybrid PSO–SQP technique. Simulation results on a practical Neyveli Thertation system (NTPS) in India and several example systems validate, the presented UCP model is reasonable by ensuring quality sufficient level of spinning reserve throughout the scheduling horizon for secure operation of the system.2005 Elsevier B.V. All rights reserved.

eywords: Unit commitment; Fuzzy logic; Tabu search; Particle swarm optimization; Sequential quadratic programming

. Introduction

Unit commitment is the problem of determining the optimalet of generating units within a power system to be used duringhe next 1–7 days[1]. The unit commitment problem (UCP) toinimize production costs (mainly fuel cost) and transition costs

start-up/shut-down costs) is traditionally referred as cost-basedCP[2–12]. The problem is broken down into a combinatorialub-problem in unit status variables and non-linear program-ing sub-problem in unit power output variables[11]. In anyCP, the load demand to be met by the committed generatingnits will usually be a forecasted (crisp) one. Due to uncertainty,rrors will be always present in the forecasted one, which may

urther result in an over or underestimated solution[6]. Thus,t is advisable to formulate these imprecise concepts within thencertainty frame. Problems associated with the uncertainties in

∗ Corresponding author. Fax: +91 422 2615615.E-mail address: [email protected] (T.A.A. Victoire).

the UCP have been previously discussed and few of themlisted in Refs.[5–9].

The UCP model discussed in this article uses fuzzy logaddress the uncertainties. The model considers the load deerror and spinning reserve constraint in a fuzzy logic fraBased on the fuzzy membership functions of both thedemand error and spinning reserve constraint, a fuzzy decismade to penalize the objective function of the UCP and enato search for an acceptable schedule. This procedure also eflexibility in maintaining reserve at various levels. Thus, the fioptimal commitment of units will take into account the unctainties in the UCP and ensures a secure operation of the sy

To schedule the generating units and dispatch the generoptimally based on the decisions made by the fuzzy logic[7],the UCP solution procedure has to be involved. In generUCP is referred as a mixed combinatorial and non-linearmization problem. It is very complex to solve because oenormous dimension, non-linear objective function andpling constraints. Researchers studied this complex profor decades and many traditional and metaheuristic techn

378-7796/$ – see front matter © 2005 Elsevier B.V. All rights reserved.oi:10.1016/j.epsr.2005.08.004

414 T.A.A. Victoire, A.E. Jeyakumar / Electric Power Systems Research 76 (2006) 413–425

including artificial intelligence methods have been developed[5]. The traditional techniques include integer programming(IP), dynamic programming (DP), branch and bound, Ben-ders’ de-composition and Lagrangian relaxation (LR). Similarly,

metaheuristic approaches, such as simulated annealing (SA),tabu search (TS), genetic algorithms (GA) and evolutionary pro-gramming (EP) have been used to solve the UCP[5].

More recently hybrid techniques combining different opti-mization techniques were proposed to solve UCP. Few of theminclude Memetic algorithm, fuzzy-DP, ANN–DP, LR–GA andGA–TS–SA techniques[5]. These techniques accommodatemore constraints and produce better solutions in an acceptablecomputation time. This article emphasizes the effectiveness ofhybrid techniques by proposing another hybrid technique usingTS[14], PSO[15] and SQP[16] is proposed. The combinatorialpart of the UCP is solved using the TS method. The TS methodhas already proved itself as a powerful technique for solvingcombinatorial optimization problems[10,11,14]. A modifiedapproach for generating the trial schedule as the neighbourhoodof the current feasible schedule and a simple procedure to gen-e ente[

rob-l .P nedl pop-u mor enct aineo am-m tageo lemsS mpls singT

2

-t t ism

M )

lits.UC

assi

Min Ft =N∑

i=1

FCi(Pit) ($) (2)

where,

FCi(Pit) = aiP2it + biPit + ci (without valve-point effects)

FCi(Pit) = aiP2it + biPit + ci + |ei sin(fi(Pit min − Pit)| (with valve-point effects)

ai, bi andci are cost coefficients andei andfi are constantsfrom the valve point effect of theith generating unit.

Subject to

N∑i=1

Pit = Dt (3)

Pi min ≤ Pit ≤ Pi max, i = 1, 2, . . . , N (4)

(ii) Start-up cost

STi ={

Sci if Toff,i ≥ ticoldstart

Shi

(5)

Subject to the following constraints:(a) Load demand:

N∑Pit = Dt ∀ t = 1, 2, . . . , H (6)

t beg ilable,d tedi d asp shut-d

3

sett ob-l ndera sets[ con-s r the

rate the initial feasible schedule for the TS method is pres17].

The non-linear optimization part (economic dispatch pem (EDP)) is solved using hybrid PSO–SQP[18] techniqueSO[15,18] is one of the modern heuristic algorithms gai

ots of attention in various power system applications. It is alation based stochastic search technique with reduced meequirement, computationally effective and easier to implemompared to other evolutionary algorithms. SQP[16,18]seemso be the best non-linear programming methods for constrptimization. It outperforms every other non-linear progring method in terms of efficiency, accuracy and percenf successful solutions, over a large number of test probimulation results on a practical system and several exaystems show the applicability of the fuzzy UCP model uS–PSO–SQP method for real time practical system.

. Problem formulation

Classically the UCP is formulated as follows[1]: the objecive function of the UCP to minimize the production cosathematically formulated as,

in (Fuel cost+ Start-up cost+ Shut-down cost) ($) (1

(i) Fuel costFor a given set ofN committed units at hourt, the total fue

cost, is minimized by economically dispatching the unThis is a non-linear programming sub-problem of theproblem, commonly referred as EDP and it can be clcally stated as,

d

ryt

d

.e

-

i=1

(b) Capacity limits:

XitPi min ≤ Pit ≤ XitPi max,

t = 1, . . . , H, i = 1, . . . , N (7)

(c) Minimum uptime and minimum downtime

Toff,i ≥ Tdown,i (8)

Ton,i ≥ Tup,i (9)

(d) Spinning reserve:

N∑i=1

XitPi max ≥ Dt + SRt t = 1, . . . , H (10)

The total amount of power available at each hour musreater than the load demanded. The reserve power avaenoted by SRt, is used when a unit fails or an unexpec

ncrease in load occurs. The fuel cost is also often referreroduction cost. In this article it is assumed that there is noown cost for any generating unit.

. Fuzzy logic modelling of the UC problem

Fuzzy set theory is a generalization of traditional crispheory. As the underlying formulation of any optimization prem relies on the set structure, optimization problems un uncertain environment can be reformulated using fuzzy

6,7]. In this section, a fuzzy decision model is developedidering the uncertainties in the constraints. The input fo

T.A.A. Victoire, A.E. Jeyakumar / Electric Power Systems Research 76 (2006) 413–425 415

fuzzy logic system is the error in the load demand and spinningreserve. Based on the input, a penalty is determined and addedwith the objective function. This enables to ensure a reasonableschedule with sufficient reserve at any hour in the schedulinghorizon. Thus, the load balance and the reserve constraints aretreated as fuzzy constraints as they are related to the imprecisehourly loads. Although constraints like up-time, down-time, etc.,are still considered crisp. Following sub-sections discuss this indetail.

3.1. Uncertainties in the constraints

Since the load balance and spinning reserve requirementsconstraints are related to the imprecise hourly load demand, theyare treated as fuzzy constraints. To implement this, the uncer-tainties in load demand error are taken into consideration forload balance constraints, while the spinning reserve constraintsare considered as soft limits in a fuzzy logic frame. Based on theerror statistics and the fuzzy membership function of the loaddemand error, the fuzzy load demand is calculated as follows.

As mentioned previously, there will be always error inthe forecasted load demand. Thus, the actual load demand isexpressed as:

Dactual= Dforecsated+ �D (11)

i herf rs ned.

s ant isd , PLa um;S cribet r typMi isf ereaf ncet tivea malt s ont ribei rrord

rrori low

TT

L

VLMSV

and medium level, it is proposed to have a smaller membershipvalue assuming that the actual load will be closer to the fore-casted one. When the error statistics reports the error is largerand more, then it is proposed to have a larger membership valueto confirm that there is a considerable discrimination betweenthe actual load and the forecasted one.

µD =

1

1 + 2.333(

�D

M+)2

∀ �D ≥ 0 & M+ < 0.04

1

1 + 2.333(

∆D

M−)2

, ∀ �D ≤ 0 & M− > −0.03

1

1 +(

�D

M+)4

, ∀ �D > 0 & M+ > 0.04

1

1 +(

�D

M−)4

, ∀ �D < 0 & M− < −0.03

(12)

Where the percentage error in load demand�D is,

�D = Dactual− Dforecasted

Dforecasted× 100% (13)

In Eq. (12), the first two expressions have been chosen togive a membership value of 0.3 when the error in load demandis equal toM+ or M−. That means, the actual load would bec of thel ionsh n thee rei astedv ror isl

ande sen,w ,

µ

w errorM r),i 3 and0

rt ctiveoβ eset osedf od iss func-t int o thet ions.

ned,t bridT lized,t Basedo each

From the above equation, it is clear thatDactualand�D aremprecise, and are therefore characterized by fuzzy sets. Tore, if the membership functions for�D based on the errotatistics is known then the actual demand can be determi

Based on the considerations on the hourly error statistiche average error typeM+ or M−, the error in load demandescribed in seven linguistic variables, NL, NM, NS, PS, PMnd PVL. Where N, negative; P, positive; L, large; M, medi, small and V, very. The following procedure is used to des

he seven linguistic variables. The data for the average erro+ orM− is given inTable 1a. It may be noticed fromTable 1a,

n positive average error (M+), the difference in VS and Sound to be very small so they are combined as single S. Whor all other variables there is a considerable difference. Hehere are four variables in positive side. Similarly, in negaverage error, the difference in VS and S is found to be so combine as single S. Similarly, there are three variablehe negative side. Thus, the error in load demand is descn seven variables. This description will vary for different eata.

The fuzzy membership functions for the load demand es chosen such a way that, when the error found is within the

able 1ahe average error type

inguistic variable M+ M−L 0.054 −0.033

0.047 −0.0280.018 −0.0210.015 −0.017

S 0.013 −0.014

e-

d

e

s,

l

d

loser to the forecasted value and the membership valueoad demand error is small. Similarly, the last two expressave been chosen to give a membership value of 0.5 wherror in load demand is equal toM+ or M−. That means, the

s a large difference between the actual load and the forecalue and the membership value of the load demand erarge.

To determine a proper membership function for load demrror, the following general membership function is chohose shape is adjusted through a finite set of parameters

D = 1

1 + αξβ,

hereξ is the ratio of the load demand error to the average(positive for positive error and negative for negative erroα

s taken as 2.333 and 1.0 to get a membership value of 0..5, respectively.

A suitable and reasonable value forβ is to be chosen. Fohis,β is assumed 2 at one time and 4 at other time, irrespef the range of average error. In addition, as shown in Eq.(12),is assumed 2 for small error and 4 for large error. With th

hree membership functions for load demand error, the propuzzy modelled UCP solution using TS–PSO–SQP methimulated on example systems and found the membershipion shown in Eq.(12) is appropriate for the present studyerms of producing the load demand at each hour and alsotal production cost compared to other membership funct

Once the fuzzy or actual load demand is determihe scheduling of generating units is done using the hyS–PSO–SQP technique. Once a feasible schedule is fina

he spinning reserve maintained at each hour is calculated.n the forecasted requirement of the spinning reserve at


Fig. 1. Membership function of the spinning reserve requirement.

hour, the fuzzy frame of the spinning reserve is described infive linguistic variables PVS, PS, PM, PL and PVL. Spinningreserve is expected to be maintained irrespective of the actualload demand the system supplies. Thus, it is sufficient to describespinning reserve in just two variables, one for less than therequired and one for more than the required. When the spin-ning reserve is found less than the required, the fuzzy decisionis modelled to penalize the objective. In order to make the deci-sion precise and to have a wide range of possible solution, thespinning reserve is described in five variables. It is also acceptedthat the spinning reserve shall also be described in three or sevenvariables. If it is described in three variables, there need to becompromise in precision and the wide range of possibility ofsolution. If it is described in seven variables, better results maybe obtained at an extra computational expense.

The proposed membership function is shown inFig. 1. Here, atrapezoidal membership function is used as this, resembles sim-ilar to human thinking, which defines possibility as range ratherthan a point. InFig. 1, R is required reserve,R1 isR + R × 0.15,R2 is R + R × 0.25,R3 is R + R × 0.4, R4 is R + R × 0.5, R5 isR + R × 0.65,R6 is R + R × 0.75 andR7 is R + R × 0.9.

3.2. Imposing the uncertainties into the objective function

Once the spinning reserve is calculated for a schedule, apenalty factor based on the input membership functions, thel etemf im-u tpum M,P

rispv ddet y in

TF

R

PPPPP

Fig. 2. Membership function of the output penalty factor.

the problem is included with the objective function of the UCP.From the above discussion it is clear that, the uncertainties inthe UCP is addressed by imposing the imprecise data in boththe objective function and the constraints using the fuzzy logic.This way of solving the UCP ensures flexibility in maintainingreserve during uncertainties. Thus, the final schedule producedis more reliable and reasonable for practical cases.

4. Tabu search

Tabu search is a meta-heuristic local improvement methodthat uses the history of the search. It is a particularly successfulstrategy for solving many practical combinatorial optimizationproblems[10,11,14]. The central idea of the method is the use ofadaptive memory, which prevents convergence to local optima,by driving the search to different parts of the search space. Intabu search, a new solution is produced from the neighbourhoodof the current solution and accepted immediately as the newcurrent solution if, and only if, it satisfies the objective. Foreach current solution, if all new solutions have already beenexamined and discarded, the solution from the candidate listthat gave the smallest increase in cost is chosen. Thus, unlikerepeated descent, tabu search does not stop at local optima, butis able to climb out of them to explore new regions.

However, it is likely that as soon as the method has movedo calm ands ra ve isr n thel xed-l objecta cted,i if ar ed bya evel.

tus ift ctivef Con-s in thet inedw aspi-r ingt ng ist d the

oad demand error and spinning reserve requirement is dined using the fuzzy decision matrix shown inTable 1b. This

uzzy decision matrix is derived after performing number of slations and also by consulting with the field experts. The ouembership function with five linguistic variables Z, PS, PL and PVL for the penalty factor is shown inFig. 2.Once the fuzzy value of penalty factor is determined, its c

alue is calculated and a proportionate value of penalty is ao the objective function of the UCP. Thus, the uncertaint

able 1buzzy decision matrix

�D

NL NM NS PS PM PL PVL

VS PVL PVL PL PM Z Z ZS PL PM PM PS Z Z PSM PM PM PS Z PS PM PML PS Z Z PS PM PM PLVL Z Z Z PM PL PVL PVL

r-

t

d

ut of a local minimum, it will quickly return there, as that loinimum may be so attractive. To prevent such repetition

ubsequent cycling, a tabu list of size ‘L’ is used. Wheneven acceptance is made, certain information about the moecorded on the tabu list, with the oldest such information oist being discarded, so that the tabu list operates like a fiength pipe whenever an object is pushed in at one end, thet the other end falls out. A move is said to be tabu, or restri

f it and any of the information is on the tabu list. However,estricted move is advantageous, the tabu may be overruln aspiration criterion, expressed in the form of aspiration l

The aspiration criterion used here, override the tabu stahe restricted move yields a solution, which has better objeunction than the one obtained earlier for the same move.equently, the aspiration level associated with each moveabu list is equal to the value of the objective function obtahen performing that move. The effectiveness of using the

ation criterion is to add some flexibility in the TS by directhe search towards the attractive moves. Short-term cyclihus prevented, as the information held on the tabu list an


aspiration criterion are designed to prevent solutions from re-visiting the last ‘L’ solutions. IfL is too small, there is a dangerthat cycling is not prevented; if it is too large, time is wasted andthe tabu criterion may simply be too restrictive.

TS used to solve the combinatorial optimization problem con-tain three major steps:

Step (i): Generating randomly feasible trial schedules.Step (ii): Evaluating the objective function of the given sched-ule by solving the EDP.Step (iii): Applying the TS procedures to accept or reject theschedule and its solution in hand.

TS have been already applied to UC independently[10] aswell as integrated with other techniques[11].

5. Particle swarm optimization

PSO, a population-based search algorithm, is initialized witha population of random solutions within the feasible range,called particles. Each particle in PSO flies through the searchspace with a velocity that is dynamically adjusted according toits own and its companion’s historical behaviours. The particleshave a tendency to fly toward better search areas over the courseof a search process[15,18]. Two commonly used PSOs are theglobal version and the local version. This is achieved by dynam-i run.I ergef hilet ettes is af

ticle

icle.ticlet.ate

ticle.l bese. Ifo th.ise,

n thn ma-t husa d.

6

osta f th

constrained optimization techniques called SQP is chosen asa fine-tuning algorithm for the PSO. Classically, these con-strained optimization techniques transform the main probleminto simple sub-problems, which are then solved iteratively.The transformation process, transforms the main constrainedproblem to a simple unconstrained problem by using penaltyfunction for constraints, which are closer or far from the con-straint boundary. In this way the constrained problem is solvedsequentially by parameterized unconstrained optimization tech-niques. These methods are now considered relatively inefficient,and are replaced by techniques, which focus on the solution ofKhun–Tucker conditions. The solution of Khun–Tucker equa-tions forms the basis to many non-linear programming methodsincluding SQP. SQP[16] proves as a best non-linear program-ming methods for constrained optimization. It outperforms everyother non-linear programming method in terms of efficiency,accuracy and percentage of successful solutions, over a largenumber of test problems. The method resembles closely to New-ton’s method for constrained optimization just as is done forunconstrained optimization. At each iteration an approxima-tion is made of the Hessian of the Lagrangian function usinga Broyden–Fletcher–Goldfarb–Shanno quasi-Newton updatingmethod. This is then used to generate a Quadratic Programmingsub-problem whose solution is used to form a search directionfor a line search procedure. The SQP subroutine for the EDP isadopted as it is in[18].

7h

singT

angeand.

load

(1),e TSybrid

o thefuzzy

em-and

cost

wiseent

ande Thisc inede n the

cally adjusting an inertia weight over the course of PSOt has been reported that the global version of PSO convast, but with potential to converge to a local minimum. Whe local version of PSO might have more chances to find bolutions slowly. The pseudo-code of the PSO techniqueollows [15]:

Step (i): Initial searching points and velocities of each parare generated randomly within the allowable range.Step (ii): Evaluate the fitness function value for each partThe current searching point is set to local best for each parThe best-evaluated value of local best is set to global besStep (iii): Modify the searching point of each particle, updthe inertia weight and increment the iteration count.Step (iv): Evaluate the fitness function value for each parIf the current value is better compared to the previous locaof the particle, the previous is replaced by the current valuthe best value of all current local best is better compared tprevious global best, the former is replaced with the laterStep (v): If the termination criterion is satisfied exit. Otherwgo to Step (iii).

From the above discussion, PSO does not depend oature of the function it minimizes rather it uses the infor

ion of the evaluation function to explore the search space. Tpproximations made in traditional methods can be avoide

. Sequential quadratic programming

Fine-tuning is one of the main drawback faced by almll heuristic based approaches. To overcome this, one o

s

rs

.

t

e

e

,

e

. Pseudo-code for the solution of FMUCP based onybrid TS–PSO–SQP

The pseudo-code for the fuzzy modelled UCP solution uS–PSO–SQP is as follows:

Step (1): Select randomly a percentage error (within the rof average error) in load demand to calculate the fuzzy demStep (2): Calculate the fuzzy membership value of all thedemand errors taking into account the type of error.Step (3): For the fuzzy load demand obtained in Stepdetermine the feasible commitment schedule using thmethod also determine the total operating cost using the hPSO–SQP method.Step (4): Calculate the spinning reserve corresponding tfeasible schedule in Step (3). Calculate the correspondingmembership value of spinning reserve.Step (5): Calculate the penalty factor based on the fuzzy mbership values of both the load demand error (Step (1))spinning reserve requirements (Step (4)).Step (6): Add the calculated penalty (Step (5)) to the totalobtained in Step (3).Step (7): Is termination criterion reached, terminate. Othergo to Step (3) by perturbing the current feasible commitmschedule of the TS method.

In Step (1) above, it may be noticed that the load demrror is randomly selected within the average error range.onstitutes slight variations in the fuzzy load demand obtavery time. In order to have a compromise and to convince i


demand obtained, the following reasonable assumption is made.Irrespective of the load demanded per hour in the schedulinghorizon, the reserve maintained must be equal or greater thanthe required reserve of the system. To realize this, the fuzzydecisions are carefully made to ensure that sufficient reserve isalways maintained in a reduced operating cost. This flexibilityin the reserve maintenance using the presented model, assures asecure and reliable operation of the system.

7.1. Proposed TS based UC solution algorithm

The following steps enumerate the algorithm for the TS basedUC [10]. As discussed previously, the EDP sub-problem will besolved using the hybrid PSO–SQP[18] technique.

Step (1): Get the system data.Step (2): Randomly generate an initial feasible commitmentschedule, and evaluate its cost function. Set this schedule ascurrent schedule UCC, with current cost function value FSC.Step (3): Set the current schedule and solution as global best,thereby UCGB = UCC and FSGB = FSC.Step (4): Find a set of trial schedules S(UCt

l) that are neigh-bours to the current schedule UCC, and their corresponding costfunction values. Sort the cost function values in the ascendingorder S(FStl). Identify the best trial schedule UCtP in that set,with cost function value FS.

,tiontep

o toon

er-p

set

thec

7s

siblec theO nts ran-d ule itT ism

asf ct a

generating uniti, and an hourt. The current schedule is examinedto find the state of generating uniti, at hourt. If the selectedunit is OFF during t and if the OFF interval (the length of thisinterval is found by moving forward and backward aroundt),Toff,i is equal toTdown,i, then change the status of that unit toON over the entireToff,i interval. If Toff,i > Tdown,i, then there isa flexibility in determining the number of hours,Tch for whichthe status shall be changed. Purushothama et al.[11] suggestedan option to select, changing the status at either the beginningor at the end of the intervalToff,i.

In this article, the modification proposed is as follows.

Let, Tch = Tr + Tdown,i (14)

where,

Tr = UD(0, Toff,i − Tdown,i) (15)

UD(0,Toff,i − Tdown,i) denotes a uniform distribution of discretenumbers ranging over the interval [0,Toff,i − Tdown,i].

According to Purushothama et al.[11], a singleTch using(14) will be selected by randomly selecting a singleTr using(15). Here, in the proposed method, it is suggested to selectall possibleTch for all Tr = 0, 1, 2,. . . Toff,i − Tdown,i, insteadof a singleTr as did by Mantawy et al. From these differentTchvalues, various feasible schedules are generated and among themt en ast ings le.

hed-

thed in

S -s in

en

7

dulefm r tos

S rwise

S

S ll

P

Step (5): If FSP≥ FSGB go to Step (6), else set UCGB = UCPand FSGB = FSP and go to Step (6).Step (6): If the trial schedule UCtl is NOT in the tabu listupdate the tabu list, aspiration criterion and current soluUCt

l = UCtP and FStl = FSP and go to Step (9), else go to S

(7).Step (7): If the aspiration criterion test is NOT satisfied gStep (8) else, set UCtl = UCt

P, update the aspiration criteriand go to Step (9).Step (8): If end of the S(FStl) is reached, go to Step (9), othwise let UCt

P be the next solution in the S(FStl) and go to Ste

(6).Step (9): Stop if the termination criterion is satisfied, elset = t + 1 and go to Step (4).

The termination is done when there is no improvement inurrent solution for a pre-specified number of iterations.

.2. Improved random perturbation (IRP) of the currentchedule

The algorithm begins with a randomly generated feaommitment schedule UCC (a commitment schedule denotesN/OFF status of theN units over theH hours) as the curre

chedule[17]. Then a trial feasible schedule is generated byom perturbation of the current schedule. A feasible sched

he one, which satisfies the constraints given by Eqs.(8)–(10).he random perturbation scheme proposed in this articleodification of the one proposed by Mantawy et al.[10].Purushothama et al.[11] perturbed the current schedule

ollows. Given a current feasible schedule, randomly sele

;

s

a

he best feasible schedule (commitment schedule) is choshe next feasible neighbourhood trial schedule. The followteps enumerate this procedure for finalizing a trial schedu

Step I: Identify all the feasible schedules (commitment scules) from allTr = 0, 1, 2,. . . Toff,i − Tdown,i.

Step II: Find the corresponding cost value by evaluatingcost function of all the feasible schedules obtaineStep I.

tep III: Find the best commitment schedule UCP corresponding to the best-cost value from all the cost valueStep II.

The best commitment schedule UCP of Step III above is theighbourhood trial schedule of the current schedule UCC.

.3. Generating the initial feasible commitment schedule

A simple procedure for generating initial feasible scheor the TS method is proposed as follows[17]: identify theust-not-run units from the initial status of the units, prio

cheduling.

tep a: If there are no must-not-run units, go to Step b, othego to Step c:

tep b: If PCmax≥ Dmax and PCmin ≤ Dmin, switch ON all theN units for allH hours and exit.

tep c: If PCmmax ≥ Dmax and PCmmin ≤ Dmin, switch ON altheN units (excluding the must-not-run units) for alHhours and exit.


where,

PCmax =N∑

i=1

Pi max, PCmin =N∑

i=1

Pi min,

PCmmax =

N∑i = 1i�=NRU

Pi max, PCmmin =

N∑i = 1i�=NRU

Pi min

Dmax/Dmin is the maximum/minimum load demand in the wholescheduling horizon. NRU is the index of must-not-run units athour t, so that at hourt, the maximum number of the units arekept must-not-run.

This approach is validated from the experiments conducted togenerate initial feasible schedule on several UC problems avail-able in Ref.[5]. The example systems used in this article alsovalidates this. If this way of generating initial feasible scheduleis not possible for any UC problem, then the technique proposedby Mantawy et al.[10], is suggested.

8. Numerical results

The proposed technique has been implemented in MATLABon a 933 MHz Pentium PC. The performance of the algorithmhas been evaluated through simulation. Fuzzy logic decisionsare made using the fuzzy logic toolbox. Simulation studies haveb dulit ticalN on-s temh ersv ion.A DP)i thirde , onlt rrori n inT

sedt ingc thodT onc cur-r ectiv

function than the one obtained before for the same move. Forthe hybrid PSO–SQP technique the simulation parameters sug-gested in Ref.[18] is adopted. Since the prime emphasis ofthis article is to include uncertainty in the UCP and observehow the imprecise data affect the final solution, the presentedUCP formulation does not include ramp rate limit and securityconstraints. The final production cost reported for each exam-ple system is obtained after an average of 30 trial runs of theTS–PSO–SQP technique. Also the actual load demand shownin this section is determined based on the magnitude of the typeof average error at each hour in the scheduling horizon. Therebyfrom Table 1a, if the error is mentioned large it has more possi-bility to have a positive error and similarly if the error mentionedis small it has more possibility to have a negative error.

Before illustrating the UCP systems with uncertainty, a stan-dard widely used example system without considering uncer-tainty is taken and solved using TS–PSO–SQP technique tovalidate the effectiveness of the method compared to the tradi-tional and heuristic methods. This test case had been previouslysolved using several conventional and meta-heuristic techniques[19,20]. The system data[21] is given in Appendix. The finalcommitment schedule of the TS–PSO–SQP technique is shownin Table 3. The final results show that the production cost hasbeen reduced to US$ 563,937 when solved using the proposedtechnique, compared to the cost of US$ 563,977 of the Enhancedadaptive LR method of Ref.[19], which seems to have producedt risticm casesh hod iss

oneI dix.B di SQPm pre-s red tot zy)d rizonb fore-c odela orig-i byt 8,057o

TL

Hand

12345678

een carried out on three example test cases, over a scheime horizon of 24 h. The first example system is a praceyveli Thermal Power Station system (NTPS) in India cisting of 7 unit[12]. The second and the third example sysave 10 units each[2,13]. The third example system considalve-point effects into account in its units fuel cost functs said earlier, the non-linear programming sub-problem (E

s solved using hybrid PSO–SQP technique only for thexample system. For the remaining two example systemshe SQP routine is used. The linguistic description of en the hourly demand of the example systems is showable 2.

After performing several trial experiments using the propoechnique for the UC problems of this article, the followontrol parameters are found to be most fit for the TS meabu list size,L = 7 and termination condition = 30. Aspiratiriterion used in this article overrides the tabu status if theent solution associated with tabu status has a better obj

able 2inguistic description of error in the hourly demand

Examplesystem 1 and 3

Examplesystem 2

H Examplesystem 1

M VL 9 SM L 10 MS L 11 MS L 12 ML L 13 MVL M 14 MVL M 15 LL M 16 S

ng

y

.

e

he best solution compared to all other traditional and heuethods for this system. In the same way, several testave been studied and found that the TS–PSO–SQP metuperior in producing quality solution for UCP.

The first example system is a practical NTPS (India) ZI 7-unit system. The data for this system is given in Appenased on the average error typeTable 1a, the fuzzy deman

s calculated and scheduled using the hybrid TS–PSO–ethod. Following observations were derived from the

ented hybrid TS–PSO–SQP based UCP model compahe classical UCP formulation. The total power actually (fuzemanded by the system over the 24 h of the scheduling hoased on the error statistics is 15,126 MW compared to theasted demand of 15,120 MW. In addition the presented mlso maintained a reserve of 754 MW compared to 430 MW

nally maintained. Finally, the total production cost obtainedhe presented model is Rs. 1,547,033 compared to Rs. 1,54riginally invested.

3Examplesystem 2

H Examplesystem 1 and 3

Examplesystem 2

S 17 L SS 18 VL SS 19 VL SVS 20 S SM 21 S SS 22 S SS 23 S SS 24 S M


Table 3Power demand and final commitment schedule generated using the proposedTS–PSO–SQP technique

Hour(MW)

Power Unit status

1 2 3 4 5 6 7 8 9 10

1 700 1 1 0 0 0 0 0 0 0 02 750 1 1 0 0 0 0 0 0 0 03 850 1 1 0 0 1 0 0 0 0 04 950 1 1 0 0 1 0 0 0 0 05 1000 1 1 0 1 1 0 0 0 0 06 1100 1 1 1 1 1 0 0 0 0 07 1150 1 1 1 1 1 0 0 0 0 08 1200 1 1 1 1 1 0 0 0 0 09 1300 1 1 1 1 1 1 1 0 0 0

10 1400 1 1 1 1 1 1 1 1 0 011 1450 1 1 1 1 1 1 1 1 1 012 1500 1 1 1 1 1 1 1 1 1 113 1400 1 1 1 1 1 1 1 1 0 014 1300 1 1 1 1 1 1 1 0 0 015 1200 1 1 1 1 1 0 0 0 0 016 1050 1 1 1 1 1 0 0 0 0 017 1000 1 1 1 1 1 0 0 0 0 018 1100 1 1 1 1 1 0 0 0 0 019 1200 1 1 1 1 1 0 0 0 0 020 1400 1 1 1 1 1 1 1 1 0 021 1300 1 1 1 1 1 1 1 0 0 022 1100 1 1 0 0 1 1 1 0 0 023 900 1 1 0 0 0 1 0 0 0 024 800 1 1 0 0 0 0 0 0 0 0

Thus, the inclusion of uncertainties in the UCP formula-tion increases the reliability of secure operation of the sys-tem by maintaining sufficiently large reserve. Also the hybridTS–PSO–SQP technique produced a better cost saving even foran increased actual demand compared to the forecasted demand.The load demand and the final schedule obtained by the hybridTS–PSO–SQP technique is shown inTable 4. Table 5summa-rizes the various results considering the maximum and minimumpossible load demands determined using the presented model.

The second example system is adopted as it had been pre-viously discussed considering uncertainties. The system data isavailable in Ref.[2]. The average error type for this system istaken from Ref.[6]. This system is solved in two different cases.Firstly with the same fuzzy load demand proposed in Ref.[9]. InRef. [9], the total power actually (fuzzy) demanded by the sys-tem over the 24 h of the scheduling horizon based on the errorstatistics is 51,601 MW compared to the forecasted demand of53,095 MW. For this fuzzy load demand, the presented modelmaintained a reserve of 23,007 MW compared to 22,781 MWmaintained in[9]. The total production cost obtained by the pre-sented model is US$ 529,993 compared to US$ 530,480 by Ref.[9]. The final schedule obtained by the hybrid TS–PSO–SQPtechnique is shown inTable 6.

Secondly the fuzzy load demand is derived using the pre-sented model of this article. Based on the error statistics thefuzzy demand is calculated is given inTable 7along with variouso y thes erros nd o

Table 4Actual load demand and the final commitment schedule for example system 1

Hour Power demand(MW)

Unit status

1 2 3 4 5 6 7

1 451 0 0 0 1 1 0 12 453 0 0 0 1 1 0 13 455 0 0 0 1 1 0 14 427 0 0 0 1 1 0 15 521 1 0 0 1 1 0 16 773 1 0 1 1 1 1 17 779 1 0 1 1 1 1 18 733 1 0 1 1 1 1 19 767 1 0 1 1 1 1 1

10 759 1 0 1 1 1 1 111 753 1 0 1 1 1 1 112 755 1 0 1 1 1 1 113 779 1 0 1 1 1 1 114 773 1 0 1 1 1 1 115 749 1 0 1 1 1 1 116 753 1 0 1 1 1 1 117 577 1 0 1 1 1 1 118 743 1 0 1 1 1 1 119 737 1 0 1 1 1 1 120 531 0 0 1 1 1 0 121 527 0 0 1 1 1 0 122 457 0 0 0 1 1 0 123 441 0 0 0 1 1 0 124 433 0 0 0 1 1 0 1

53,095 MW. In addition the presented model also maintaineda reserve of 21947.7 MW compared to 13,025 MW originallymaintained. Finally, the total production cost obtained by thepresented model is US$ 535,773 compared to US$ 537,372originally invested. In this case it may be noted that the loadsupplied by the presented model is less compared to the fore-casted. Although the reserve maintained is sufficiently large forsecure operation of the system. This is ensured fromTable 7as itsummarizes the results considering the maximum and minimumpossible load demands determined using the presented model.

The third example system is adopted as it has its fuelcost function includes the valve-point effects. These valvepoint effects contribute non-convexity in the cost function.Hence, this example is more appropriate to be solved using theproposed TS–PSO–SQP hybrid approach. The average errortype and linguistic description of error in the hourly demand isassumed as same as in the case of second example system, asthe load demand pattern of this system is same as the patternof the second example system. The system data (partial) isavailable in[13] and the remaining data is given inTable 8.A reserve of 6% of the load demand is assumed. Based on theerror statistics the fuzzy demand is calculated and shown inTable 9along with other results.

For this system, the total power actually (fuzzy) demandedby the system over the 24 h of the scheduling horizon basedon the error statistics is 44,922 MW compared to 44,580 MW,t l alsom rig-i byt 2,046

ther results. The total power actually (fuzzy) demanded bystem over the 24 h of the scheduling horizon based on thetatistics is 52980.3 MW compared to the forecasted dema

rf

he demand forecasted. In addition the presented modeaintained a reserve of 4814 MW compared to 4472 MW o

nally maintained. Finally, the total production cost obtainedhe presented model is US$ 981,921 compared to US$ 98


Table 5Summary of results obtained for example system 1

Hour Maximum possibleactual demand (MW)

Actual (fuzzy) demand obtainedusing the presented model (MW)

Minimum possibleactual demand (MW)

Crisp forecasteddemand (MW)

1 491.2 451.0 450.8 466.02 484.6 453.0 450.1 463.03 481.5 455.0 447.2 460.04 454.3 427.0 421.9 434.05 554.7 521.0 515.2 530.06 856.6 773.0 822.2 840.07 856.6 779.0 822.2 840.08 772.0 733.0 741.0 757.09 786.5 767.0 762.0 775.010 784.5 759.0 760.0 773.011 781.5 753.0 757.1 770.012 787.9 755.0 766.9 778.013 772.0 779.0 741.0 757.014 789.6 773.0 764.9 778.015 781.5 749.0 757.1 770.016 775.4 753.0 751.2 764.017 606.9 577.0 588.0 598.018 603.9 743.0 585.0 595.019 553.1 737.0 535.9 545.020 546.0 531.0 529.0 538.021 543.0 527.0 526.0 535.022 472.9 457.0 458.2 466.023 455.7 441.0 441.5 449.024 447.7 433.0 429.7 439.0

Total demand supplied (MW) 15439.4 15126.0 14824.0 15120.0

Total capacity committed (MW) 16000 15880 15880 15550

Reserve maintained (MW) 560 754 1056 430

Production cost (Rs.) 1573321 1547033 1543179 1548057

Table 6Power demand and the final commitment schedule for example system 2

Hour Power demand(MW)

Unit status

1 2 3 4 5 6 7 8 9 10

1 871 0 1 0 1 0 1 0 0 1 12 1030 0 1 0 1 0 1 0 0 1 13 927 0 1 0 1 0 1 0 0 1 14 876 0 1 0 1 0 1 0 0 1 15 1035 0 1 0 1 0 1 0 0 1 16 1190 0 1 1 1 0 1 1 0 1 17 2029 0 1 1 1 0 1 1 0 1 18 2472 0 1 1 1 0 1 1 0 1 19 2936 0 1 1 1 0 1 1 0 1 1

10 3245 0 1 1 1 0 1 1 1 1 111 2805 0 1 1 1 0 1 1 1 1 112 3502 0 1 1 1 0 1 1 1 1 113 3373 0 1 1 1 0 1 1 1 1 114 2508 0 1 1 1 0 1 1 1 1 015 2781 0 1 1 1 0 1 1 1 1 016 2627 0 1 1 1 0 1 1 1 1 017 2316 0 1 1 1 0 1 1 1 1 018 2720 0 1 1 1 0 1 1 1 1 019 3399 0 1 1 1 0 1 1 1 1 020 2987 0 1 1 1 0 1 1 1 1 021 2189 0 1 1 1 0 1 0 1 1 022 1700 0 1 0 1 0 1 0 1 1 023 1105 0 1 0 1 0 1 0 1 1 024 978 0 1 0 1 0 1 0 1 1 0

originally invested. Thus, the inclusion of uncertainties in theUCP formulation increases the reliability of secure operation ofthe system by maintaining sufficiently large reserve. Also thehybrid TS–PSO–SQP technique produced a better cost savingeven for an increased actual demand compared to the forecasteddemand.Table 9show various results considering the maxi-mum and minimum possible load demands determined usingpresented model.

To ensure the reliability and robustness of the presentedmodel, 100 trail runs were carried out using the third exam-ple system. In each of the trial run the simulation parameters ofthe hybrid TS–PSO–SQP method was unaltered. The total loaddemanded and total reserve maintained by the scheduled unitsin the whole scheduling horizon is plotted against each trial run.The plot is shown inFig. 3. FromTables 5, 7 and 9andFig. 3, thepresented model always maintains adequate reserve even whenthe load demand supplied by the committed units is high. And itmaintains sufficiently large reserve when the load demand sup-plied by the committed units is low. In all the trial runs a secureoperation of the system is guaranteed by the presented modelby allowing a flexibility in reserve maintenance as it considersuncertainties in the UCP.

Utilities evaluate a reasonable amount of spinning reserve,as it is essential to satisfy system security constraints when thepower system encounters a contingency. Most utilities operat-ing rules require spinning reserve to be greater than the capacity



Hour Maximumpossible actualdemand (MW)

Actual (fuzzy) demandobtained using thepresented model (MW)

Minimumpossible actualdemand (MW)

Actual (fuzzy) demandobtained using the modelpresented in Ref.[9] (MW)


1 1080.5 1035 991.5 871.3 10252 1046.7 1007.7 972.1 1030 10003 942 907.1 874.9 927 9004 889.7 856.6 826.3 875.5 8505 1072.8 1033.1 996.4 1035.3 10256 1427.7 1393.9 1370.4 1190 14007 2009 1961.3 1928.3 2029.2 19708 2447.5 2389.4 2349.2 2472 24009 2892.4 2840 2802.2 2935.4 285010 3196.9 3139 3097.1 3244.6 315011 3349.1 3288.5 3244.6 2805.1 330012 3443.1 3389.9 3351.6 3502 340013 3339.8 3279 3205.7 3373.3 327514 2993.9 2938.1 2900.5 2507.5 295015 2740.2 2689.1 2654.7 2781 270016 2587.9 2539.7 2507.2 2626.4 255017 2765.5 2714 2679.3 2316.3 272518 3247.6 3187.1 3146.3 2720 320019 3349.1 3286.5 3244.6 3399 330020 2943.2 2888.2 2851.3 2987 290021 2156.6 2116.2 2089.3 2188.8 212522 1674.6 1643.2 1622.3 1699.5 165023 1319.3 1294.7 1278.2 1105 130024 1172.8 1163 1125.7 977.5 1150

Total demand supplied (MW) 54087.9 52980.3 52109.7 51601 53095

Total capacity committed (MW) 75320 74928 74528 74608 66120

Reserve maintained (MW) 21232 21947.7 22418.3 23007 13025

Production cost ($) 539721 535773 533217 529993 537372

Table 8System data for example system 3

Unit 1 2 3 4 5 6 7 8 9 10

Minimum up-time (h) 8 8 5 5 6 3 3 1 1 1Minimum down-time (h) 8 8 5 5 6 3 3 1 1 1Cold start hour (h) 5 5 4 4 4 2 2 0 0 0Initial status (h) 8 8 −5 −5 −6 −3 −3 −1 −1 −1Hot start-up cost ($) 4500 5000 550 560 900 170 260 30 30 30Cold start-up cost ($) 9000 10000 1100 1120 1800 340 520 60 60 60

Fig. 3. Total power supplied and reserve maintained for example system 3 for100 trial runs.

of the largest online generator or a fraction of the load or somecombination of both, but there is no hard and fast rule to maintaincertain level of spinning reserve. Thus, this constraint is identi-fied as soft limits and ultimately provides an option to reduce therunning cost. But at the same time it is expected to maintain a rea-sonable level of security in the system generation by maintainingsufficient reserve at each hour. To enable this, a UCP model ispresented to determine the load demand at each hour based onthe error statistics. Further this demand is scheduled and fuzzydecisions direct the search to produce a reasonable solution tothe problem. The error statistics varies based on several factorslike weather, seasons, etc., thus the actual load demand for asystem also vary. Keeping this in mind simulation results con-sidering the two extreme possibilities of load demand is providedin Tables 5, 7 and 9. Finally, the UCP formulation including the



Hour Maximum possibleactual demand (MW)

Actual (fuzzy) demand obtainedusing the presented model (MW)

Minimum possibleactual demand (MW)


1 1092.1 1072.9 1002.1 10362 1161.8 1126.8 1079.1 11103 1316.7 1271.3 1223 12584 1471.6 1427.2 1366.8 14065 1549.1 1496.9 1438.8 14806 1660.2 1606.9 1593.6 16287 1735.7 1693.1 1666 17028 1811.1 1751.1 1738.4 17769 1952.6 1902.8 1891.7 192410 2102.8 2067.3 2037.2 207211 2177.9 2138.9 2110 214612 2248.1 2216.9 2188.4 222013 2113 2063 2028.2 207214 1952.6 1920.7 1891.7 192415 1802.4 1770.1 1746.2 177616 1577.1 1549.2 1527.9 155417 1502 1477.3 1455.2 148018 1652.2 1622.9 1600.7 162819 1802.4 1770.1 1746.2 177620 2102.8 2068.9 2037.2 207221 1952.6 1921 1891.7 192422 1652.2 1626.7 1600.7 162823 1351.8 1329.2 1309.6 133224 1207.4 1179.1 1159 1184

Total demand supplied (MW) 40948.2 40071 39329.4 40108

Total capacity committed (MW) 45508 44922 44922 44580

Reserve maintained (MW) 4559.8 4851 5592.6 4472

Production cost ($) 983791 981921 979543 982046

ramping rate limits, security constraints, etc. makes the prob-lem more realistic. The extended research considering the aboveissues is underway and will be presented in a future paper.

9. Conclusion

A UCP solution model using fuzzy logic is presented toaddress the uncertainties in the problem. A hybrid TS–PSO–SQPtechnique guided by the fuzzy decisions is proposed to sched-ule the generating units. The fuzzy decisions are made basedon the input membership functions, the load demand error andthe spinning reserve. Based on the fuzzy decisions, a penaltyis added to the production cost (objective function) to formu-late the fitness function so as to guide the search to find an

optimal and more practical solution. A simple method to gen-erate initial solution and an improved random perturbation forthe TS method is proposed to effectively solve the combina-torial sub-problem. Hybrid PSO–SQP method is used to solvethe non-linear programming sub-problem. Extensive numericalsimulations show, when uncertainties are considered, the pre-sented model improves the secure operation of the system bymaintaining sufficient reserve at each hour irrespective of theload it supplies. Thus, the presented model incorporates prac-tical aspects regarding flexibility in maintaining the spinningreserve to effectively schedule the generating units and therebysecurely operate the system.

Appendix A

Unit data for the example system 1

Unit Pmin Pmax c b a Sc Sh Tup Tdown tcold start

1 15 60 750 70 0.255 4250 2125 1 1 02 20 80 1250 75 0.198 5050 2525 3 3 23 30 100 2000 70 0.198 5700 2850 4 4 34 25 120 1600 70 0.191 4700 2350 5 5 45 50 150 1450 75 0.106 5650 2825 6 6 56 50 150 4950 65 0.0675 14100 7050 6 6 57 75 200 4100 60 0.074 11350 5675 7 7 5


Load demand for the example system 1

H 1 2 3 4 5 6 7 8 9 10 11 12Load (MW) 840 757 775 773 770 778 757 778 770 764 598 595

H 13 14 15 16 17 18 19 20 21 22 23 24Load (MW) 545 538 535 466 449 439 466 463 460 434 530 840

Unit data for the example system taken from[21]

Unit Pmin Pmax c b a Sc Sh Tup Tdown tcold start

1 150 455 1000 16.19 0.00048 9000 4500 8 8 52 150 455 970 17.26 0.00031 10000 5000 8 8 53 20 130 700 16.6 0.002 1100 550 5 5 44 20 130 680 16.5 0.00211 1120 560 5 5 45 25 162 450 19.7 0.00398 1800 900 6 6 46 20 80 370 22.26 0.00712 340 170 3 3 27 25 85 480 27.74 0.00079 520 260 3 3 28 10 55 660 25.92 0.00413 60 30 1 1 09 10 55 665 27.27 0.00222 60 30 1 1 0

10 10 55 670 27.79 0.00173 60 30 1 1 0

Load demand for the example system taken from[21]

H 1 2 3 4 5 6 7 8 9 10 11 12Load (MW) 700 750 850 950 1000 1100 1150 1200 1300 1400 1450 1500

H 13 14 15 16 17 18 19 20 21 22 23 24Load (MW) 1400 1300 1200 1050 1000 1100 1200 1400 1300 1100 900 800

List of symbolsFCi(Pit) fuel cost function of uniti ($)STi start-up of cost of uniti ($)Dt load demand at timet (MW)SRt power reserve at timet (MW)Pit power produced by uniti at timet (MW)Rit reserve maintained by uniti at timet (MW)Xit ON/OFF status of uniti at timetN number of generating unitsH number of hours in the scheduling horizonPi min/Pi max minimum/maximum generation capability of unit

i (MW)Toff,i/Ton,i OFF/ON time of uniti (h)Tup,i/Tdown,i minimum up/down-time of uniti (h)Shi/Sci start-up costs incurred for a hot/cold start for uniti ($)ticold start number of hours that it takes for the boiler of uniti to

cool down (h)rand(0,1) random number between 0 and 1

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a tabu search based hybrid optimization approach for a fuzzy modelled unit commitment problem

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