Emerging solution of large-scale unit commitment problem by Stochastic Priority List

Electric Power Systems Research 76 (2006) 283–292

Emerging solution of large-scale unit commitment problemby Stochastic Priority List

Tomonobu Senjyua,∗, Tsukasa Miyagia, Ahmed Yousuf Sabera,Naomitsu Urasakia, Toshihisa Funabashib,1

a Faculty of Engineering, University of the Ryukyus, 1 Senbaru Nishihara-cho Nakagami, Okinawa 903-0213, Japanb Meidensha Corporation Riverside Building 36-2, Nihonbashi Hokozakicho, Chuo-ku, Tokyo 103-8515, Japan

Received 16 January 2005; received in revised form 25 May 2005; accepted 6 July 2005Available online 29 September 2005

Abstract

This paper presents a new approach for unit commitment problem using Stochastic Priority List method. In this method, rapidly some initial unitcommitment schedules are generated by Priority List method and priority based stochastic window system. Excess units are added with systemdependent probability distribution to avoid overlooking a desired solution during repeated search. Constraints are not considered in this stage. Thens only tos alculations.T r the systemsu o the resultso©

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rccoisoiiivwo

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s oneer

ari-ed toicles.vided

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chedules are modified gradually using the problem specific heuristics to fulfill constraints. To reduce calculations, heuristics are appliedtheolutions, which can be expected to improve. Besides, sign vector is introduced to reduce economic load dispatch (ELD) overhead rechis process is repeated for optimal solution. The proposed method is tested using the reported problem data set. Simulation results fop to 100-unit are compared to previous reported results. Numerical results show an improvement in solution cost and time compared tbtained from Genetic Algorithm and others.2005 Elsevier B.V. All rights reserved.

eywords: Gray zone; Heuristic; Probability distribution; Stochastic Priority; Unit commitment; Window system

. Introduction

In recent years, the further economical operation has beenequired to earn the maximum profit from the deregulation ofompetitive electric power market. Therefore, the optimal unitommitment (UC) solution is an essential factor in planning andperation of power systems. The basic goal of the UC problem

s to properly schedule the on/off state of all the units in theystem. In addition to fulfill a large number of constraints, theptimal UC should meet the predicted load demand, calculated

n advance plus the spinning reserve requirements at every timenterval such that the total cost is minimum. The UC problems formulated as a combinatorial optimization problem with 0–1ariables which represent off–on states and continuous variableshich represent unit power generations. However, the numberf combinations of 0–1 variables grows exponentially as being

∗ Corresponding author. Tel.: +81 98 895 8686; fax: +81 98 895 8708.E-mail addresses: [email protected] (T. Senjyu),

[email protected] (T. Funabashi).1 Tel.: +81 35 641 7509; fax: +81 35 641 9310.

a large-scale problem. Therefore, this problem is known aof the problems which is the most difficult to solve in powsystems.

A bibliographical survey on UC methods reveals that vous numerical optimization techniques have been employapproach the UC problem in more than 150 published artThe methods being used to solve the UC problem can be diinto the following categories[1–20]:

• Priority List (PL) [1–3];• Branch-and-Bound (BB)[4–6];• Lagrangian Relaxation (LR)[7–10];• Meta-heuristics[11–20].

For the PL method, units are committed in ascending oof the unit full-load cost so that the most economic unit is cmitted first and the most expensive unit at last in order to mthe load demand. The PL method is very fast but highly hetic and produces schedules with relatively higher operationThe BB method has the danger of a deficiency of storage city and increases the calculation time enormously as be

378-7796/$ – see front matter © 2005 Elsevier B.V. All rights reserved.oi:10.1016/j.epsr.2005.07.002

284 T. Senjyu et al. / Electric Power Systems Research 76 (2006) 283–292

large-scale problem. The Lagrangian Relaxation method con-centrates on finding an appropriate co-ordination technique forgenerating feasible primal solutions, while minimizing the dual-ity gap. The main problem with the LR method is the difficultyencountered in obtaining feasible solutions.

The meta-heuristic methods are iterative search techniquesthat can search not only local optimal solution but also globaloptimal solution and can treat various constraints. In the meta-heuristic methods, the techniques frequently applied to the UCproblem are Genetic Algorithm (GA), Evolution Program (EP),Tabu Search (TS), Simulated Annealing (SA), etc. The GA andEP are general-purpose search techniques based on principlesinspired from the genetic and evolution mechanisms observedin natural systems and populations of living beings. The featureof the GA is to search retaining candidates for the solutions.These methods have the advantage of searching the solutionspace thoroughly, and avoiding premature convergence to localminima. The main difficulty is their sensitivity to the choiceof parameters. However, in case of large-scale problem theyconsume a lot of time and space due to their random and iterativenature.

The TS is an approach, which avoids the cycle of solutionsand searches the whole space. Tabu List is used to restrict cer-tain areas of the search space that have already been searched.SA methods maintain a probability distribution for higher costsolutions on the track upwards to the hills. In the original simu-l timi that itho lemT alcul se ol uet

tion,w irst,t mec has-t nals ionsa duco utionT lutioi

tion2 ned.T cs ae rep nsa

2

2

c-costi cold start cost of unitic-s-houri cold start hour of unitiCF total costD (t) load demand at hourtFi(Pi(t)) fuel cost of uniti at hourth-costi hot start cost of unitiHRi heat rate of unitiHoff

i transition hour from hot to cold start of unitiIi (t) off/on state of uniti at hourtN number of generatorsPi (t) generation of uniti at hourtPmax

i maximum generation limit of unitiPmin

i minimum generation limit of unitiRt spinning reserve at hourtSCi start-up cost of unitiSDi shut-down cost of unitiS number of solutions applying heuristicsT scheduling periodT on

i minimum up time of unitiT off

i minimum down time of unitiW number of solutions included by window systemXon

i (t) time duration for which uniti has been continuouslyon at timet

Xoffi (t) time duration for which uniti has been continuously

off at timetX number of units inside the window

2

UCp

2tal

c s thes ally,o

M

order(

F

w nonn

hichi thism eu e tosI singt inEto

ated annealing algorithm, a large share of the computations spent in randomly generating and evaluating solutionsurn out to be infeasible. Authors always try to merge SA wther methods where SA solves one of the parts of UC probhese meta-heuristic methods can solve within suitable c

ation time for reasonably sized problems. However, in caarge-scale problem they consume a lot of time and space dheir iterative nature.

In this paper, we propose a new unit commitment soluhich applies a fast Stochastic Priority List (SPL) method. F

he PL method yields rapidly an initial solution with linear tiomplexity. Then, plurality of solutions are produced by stocic window system based on derived one for multidirectioearch. Finally, among all solutions only potential solutre selected for some appropriate heuristics, which can reperation cost and this process is repeated for optimal solherefore, the proposed method provides a satisfactory so

n terms of both cost and execution time.The rest of the paper is organized as follows. In Sec

, problem formulation and constraints of UC are mentiohe proposed method, important terms, and used heuristixplained in Section3. Simulation conditions and results aresented in Sections4 and 5, respectively. At last, conclusiore drawn in Section6.

. Problem formulation

.1. Notation

The following notations are used throughout the paper:

et

.-fto

e.n

re

.2. Formulation

The objective function and associated constraints of theroblem are as follows.

.2.1. Objective functionThe objective of UC problem is the minimization of the to

ost. The total cost, CF over the entire scheduling period ium of the fuel and start-up cost for all the units. Mathematicverall objective function of the UC problem is as follows[12]:

in CF =N∑

i=1

T∑t=1

[Fi(Pi(t)) + SCi(1 − Ii(t − 1))]Ii(t). (1)

Fuel cost of a thermal unit is expressed as a secondparabolic) function of each unit output as follows:

i(Pi(t)) = ai + bi · Pi(t) + ci · P2i (t), (2)

hereai, bi, andci represent the unit cost coefficients andegative constants.

Start-up cost for restarting a decommitted thermal unit, ws related to the temperature of the boiler, is included in

odel. The generator start-up cost, SCi depends on the time thnit has been off prior to start-up. Feasible solutions havatisfy minimum down time constraint of all units in Eq.(8).n this paper, time-dependent start-up cost is simplified uransition hour (Hoff

i ) from hot to cold start, which is definedq.(4). Start-up cost will be high cold cost (c-costi) when down

ime duration (Xoffi ) exceeds cold start hour (c-s-houri) in excess

f minimum down time (T offi ) and will be low hot cost (h-costi)

T. Senjyu et al. / Electric Power Systems Research 76 (2006) 283–292 285

when down time duration does not exceed c-s-houri in excess ofminimum down time as follows[3,13]:

SCi ={

h-costi : T offi ≤ Xoff

i ≤ Hoffi

c-costi : Xoffi > Hoff

i

, (3)

Hoffi = T off

i + c-s-houri. (4)

Shut down cost, SD is usually a constant value for each unit.In this paper, the shut down cost has been taken equal to 0 forall units and is excluded from the objective function.

2.2.2. ConstraintsThe constraints that must be satisfied during the optimization

process are as follows:

(a) System power balanceThe generated power from all the committed units must

satisfy the load demand, which is defined as

D(t) =N∑

i=1

Pi(t). (5)

(b) Spinning reserveTo maintain system reliability, adequate spinning reserves

are required.

(d as

(ede-ted.

(ust

3

uesw ought pact thoi ovedb SPLm byP singP e ofS unit

for generation cost are off states, whereas cheaper units are set-tled on states. In this time, constraints are not considered. Then,plurality of solutions, generated by priority based stochastic win-dow system, are included in this system for multi-directionalsearch. Excess units are added with predefined probability dis-tribution to avoid overlooking a desired solution during repeatedsearch. Then we apply several heuristics repeatedly on pluralityof promising initial solutions. Applying heuristics to all solu-tions is a time consuming process. Hence, using a proper pruningcondition, the proposed method carries out the reduction ofnumber of solutions for applying heuristics. Then the solutions,which are not potentially moderate, are reduced. A step-by-stepproposed SPL method for the UC problem is outlined as follows:

Proposed SPL algorithm for UCStep 0: Initialize the parameters andi = 1.Step 1: Make priority list of the units.Step 2: Appoint the highest priority units for base load.Step 3: Generate an initial solutionV0 by PL method.Step 4: IncludeW plurality of solutionsV[W] by priority basedwindow system.Step 5: Turn excess units on with system dependent probabilitydistribution Prexcess.Step 6: Prune (W–S) unpromising solutions.Step 7: Set current solution,Vc = V[i].

nd

i-

t

PL

I , andh

3

axi-m vels.S wer.A ef

H

N∑i=1

Ii(t) · Pmaxi ≥ D(t) + Rt. (6)

c) Generation limitsEach unit has generation range, which is represente

Pmini ≤ Pi(t) ≤ Pmax

i . (7)

d) Minimum up/down timeOnce unit is committed/decommitted, there is a pr

fined minimum time after it can be decommitted/commit

T oni ≤ Xon

i

T offi ≤ Xoff

i .

}(8)

e) Initial statusAt the beginning of schedule, the unit initial status m

be taken into account.

. Explanation of the SPL method

GA, EP, etc. are general-purpose optimization techniqhich are being used very frequently these days for UC th

hey are slow. PL is very fast but cannot search the solution shoroughly (e.g. Greedy Search). So, Stochastic PL mes introduced here and slowness of iterative SPL is impry introduced sign vector. Now, we explain the proposedethod. In SPL method, we get rapidly an initial solutionL method with necessary improvement. Advantage of uL method is to obtain a very fast solution, which is the basPL method. PL method decides that the more expensive

,

ed

s

Step 8: Set new solution,Vn = Vc.Step 9: Manage peak loads ofVc considering priority.Step 10: Handle minimum up/down time constraint ofVc.Step 11: Apply gray zone modification ofVc.Step 12: Eliminate excess and deficient in units ofVc graduallybased on priority.Step 13: Solve economic load dispatch (ELD) intelligently aevaluate cost CF(Vc).Step 14: If CF(Vc) > CF(Vn) thenVc = Vn.Step 15: If not terminating condition then go to Step 8 (termnating condition of each solution).Step 16: StoreV[i] = Vc and increasei by 1 for the trial of nexsolution.Step 17: If i ≤ S then go to Step 7 (stopping criterion of Sloop).Step 18: Print out the best solution, Min CF(V[i]) for i = 1, 2,. . ., S.

n the following subsections, all the important steps, termseuristics of the proposed method will be described.

.1. Priority list

Heat rate (HR), cost per produced unit, of a unit at its mum output power is less than that at other output power leo, it is expected to run a unit at its maximum output pot maximum output power, HR of uniti is calculated by th

ollowing equation:

Ri = Fi(Pmaxi )

Pmaxi

. (9)


Table 1Heat rate at maximum power

Unit HR ($/MW) Unit HR ($/MW)

1 18.61 6 27.452 19.53 7 33.453 22.24 8 38.154 22.01 9 39.485 23.12 10 40.07

Heat rate at maximum output power of 10-unit system is givenin Table 1and corresponding priority list is shown inTable 5.According toTables 1 and 5, priority of Unit 1 is the highestand it stands at the bottom of the priority list as its heat rate isminimum and the goal of UC is the minimization of cost of Eq.(1). On the other hand, priority list is created according to thedescending order of maximum output limit and in case of thesame maximum limit, heat rate is considered in[3]. In this paper,maximum output limit criterion is ignored to create priority listas the generator, which has the highest maximum output limitin a system, may not produce the least cost power.

3.2. Solution string coding

We have introduced(N + 1)× T binary string matrix (Table 2)where firstN × T binary string matrix represents UC solution andthe last row vector indicates any change of the schedule. Eachbit of the last row vector is a sign whether any change is occurredor not at each hour for SPL operations. This introduced vectorwill prohibit to recalculate the cost of unchanged hour schedule(overhead).

3.3. Units for base load

Load demand can be divided into two parts. One is DC (base)p dul-i y bec di ayse fa on byP wer,w eett ulingp nits1 d att nits

Table 3Load demand for 24 h

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

Fig. 1. State of the must-run units (scheduling periodT hours).

because base load demand is 700 MW for standard 10-unit sys-tems and 24-h scheduling period inTable 3. In Fig. 1, these twounits are fixed as on state. In iterative process, this techniquereduces the total search space and helps to get better solutionsquickly. However, must-run units for base load are not effectivein deterministic single-run process like[3].

3.4. Generating an initial solution

The generation of initial solution is important, particularly,for the UC problem. Usually, the initial solution is generated atrandom. However, by this technique, it is difficult to get optimalsolution for the UC problem with many constraints and result isdecline in solution quality as well as long execution time[3].

Priority list as shown inFig. 2 is created based on heat rate.A unit, which can run with the least heat rate at its maximumoutput power, is located at the bottom of the list and other unitsare located in ascending order of heat rate at maximum outputtowards the top of the list. The most costly unit, which is on thetop of the list, has the lowest priority. Then, units are committeduntil the load demand and the spinning reserve requirements rep-resented by Eq.(6)are fulfilled according to the priority list orderat each hour. In addition, units of equal heat rate at maximumoutput power are selected randomly or start-up cost, minimumup time, minimum down time, etc. may be considered. All signbits are set 1 for an initial solution as the schedule is new.

TB

U

1 12 1...N 0S 0

art, which is the minimum load demand of the entire scheng period and other part is AC (fluctuating) load, which mahanged randomly at each hour. InTable 3, base load demans 700 MW. In power system the units, which must run, alwxist for the above base load demand. In[3], 20% (constant) oll units are considered as must-run units. Since the solutiL method is started from the unit of the cheapest output poe desire the highest priority (lowest HR) units, which m

he base load demand, will remain on during whole schederiod rather than altering on/off. In the proposed method Uand 2 (maximum output 455 MW each), which are locate

he bottom of the priority list, are detected as must-run u

able 2inary solution string

nit Period (1–24)

1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1

. . .

0 0 0 0 0 0 0 0 1 1 1ign 0 0 1 0 0 0 0 0 0 1 0

1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 0 0 0 0 1 1 1 01 0 0 0 0 0 0 0 1 0 0 1


Fig. 2. Initial solution by Priority List method.

In this paper, an initial solutionV0 is generated using the PLmethod. However, the initial solution obtained by PL methodcannot fulfill all constraints. Merely characteristics that is costlyunits are off status and cheap units are on status, can be imple-mented to obtain an approximate solution.

3.5. Plurality of solutions

It is already mentioned that the PL method produces sched-ules with relatively higher operation cost. So, plurality of solu-tions by window system are included here based on obtainedsolution by PL method at this point to search in multiple direc-tions. An X-window consists of units priority numbered from(n − X + 1) ton at hourt if there aren number of on units in theschedule at the same hour of a schedule by PL method. Must-rununits are not included in window system. The technique of gen-erating multiple solutions is by making stochastic interchangebetween decommitted units of priority list andX units locatedinside the window (thick lines filled area) of a schedule as shownin Fig. 3for each hour. Here,X is window size. Each decommit-ted unit has been assigned a probability according to inverse ofHR as below:

Prj(t) = 1/HRj∑Ni=1

1−Ii(t)HRi

(1 − Ij(t)) (10)

a

q

N uniti uletw ge[ d ifr hat

Fig. 3. Units located inside the window (case ofX = 2).

qi−1(t) < r ≤ qi(t). Decommitted units of lower HR have betterchance to participate for this window exchange process. Onesolution is achieved when a setup of the last period is completed.The same procedure is repeated andW number of solutionsV[W]are generated whereW is a predefined number. This stochasticand global window system is modified version of random andlocal window system in[3]. This is introduced here for multi-directional search.

3.6. Stochastic excess units

Prexcess= K log10(N). (12)

To satisfy Eq.(8) for all the units, more number of unitsneeds to be on than the actual load demand. For a larger scaleproblem, condition(8) will be tighter and search space willincrease exponentially. So, after satisfying Eq.(6), rest of theunits will be turned on with a predefined logarithmic prob-ability distribution of number of units as in Eq.(12). Thebest parameter value ofK up to 100-unit systems and 24 hscheduling period is 0.35 from simulation. Graphical distri-bution of excess units is shown inFig. 4. According to thegraph, each decommitted unit has a 45% probability to beturned on in a 20-unit system. These excess units are addedso that the optimal solution is not overlooked during repeatedsearching.

3

olu-t uris-t

Y

nd cumulative probabilityqj(t) for each decommitted unit is

j(t) =j∑

k=1

Prk(t). (11)

ow for each one unit inside the window, one decommitteds selected for the interchange of their states based on roheel selection. A random numberr is generated from the ran

0. . .1] and the first (lowest HR) decommitted unit is selecte< q1(t), otherwise theith decommitted unit is selected such t

te

.7. Solution pruning

In order to reduce computational efforts, unpromising sions are reduced by adding following condition and then heics are applied only on remaining solutions[3].

=T∑

t=1

∣∣∣∣∣N∑

i=1

Ii(t) · Pmaxi − (D(t) + Rt)

∣∣∣∣∣ . (13)


Fig. 4. Probability distribution of excess units.

Actually, Y measures the excess and deficient amount of powerover the entire scheduling period. So, Eq.(13) is calculatedfor each solution and solutions are sorted in ascending orderof Y. After rearrangement, each heuristic explained below isapplied to the firstS promising solutions and the rest solutionsare pruned.

3.8. Incorporated heuristics

3.8.1. Peak load managementIn the period of peak load, how the constraint represented by

Eq. (8) for relatively large units of minimum up/down time issatisfied poses an important problem. For example, assume thatthe minimum up/down time of Unit 5 shown inFig. 5is relativelylarge (6 h) and Unit 5 does not need to generate electricity at hour(H + 2). Unit 5 is shut down usually but since it must restart inthe next hour (H + 3), Unit 5 remains on in the previous hour(H + 2). On the contrary, it is also possible to satisfy minimumdown time by shutting down the Unit 5. However, other costlyunits are started on and also included start-up costs, as a resultthe total cost will be higher. Hence, this heuristic decides tocontinue starting on higher priority large units according to thepriority list order between peak load hours. From prior simula-tion, we assume that an inserted unit is selected from bottom ofthe priority list so that the total cost becomes the cheapest[3].

3b-

l enc thish ivid-u imei low.A ei

is a2 d 1’ssn ertedb rs or

Fig. 5. Consideration of the minimum up/down time of the period inserted intothe load peak.

Fig. 6. Minimum up/down time repair (Ton = Toff = 4 h).

one of the neighbor streams is split and after inversion near splitpart is merged to itself. Between the two options, better one ischosen for which less number of bitwise operations is needed.Here, neighbor may be left or right and they are equally likely.

Let, schedule (bit patterns) of uniti for T hours is

and bit stream does not satisfyminimum up/down time constraint.

Fig. 7. Gray zone for start-up cost.

.8.2. Handle minimum up/down timeMinimum up/down time is the vital constraint of UC pro

em. Till now, minimum up/down time of all units has not beonsidered. In order to satisfy the minimum up/down time,euristic looks for the hours when schedule changes for indal unit and corrects to on or off until the minimum up/down t

s satisfied. A pattern matching algorithm is introduced ben example is shown inFig. 6 where minimum up/down tim

s 4 h.For each unit, bit patterns of entire scheduling period

-class (0 and 1) system. In a pattern, bit streams of 0’s antand for off and on states of a unit, respectively. If Eq.(8) isot satisfied for a stream, either this stream vector is invy high speed X-OR operation and merged to its neighbo


Option 1:

Option 2 (left):

Or (right):

Here, is the opposite bit stream ofP andPnm indicates the

stream, which consists of bit-positionm to n. After bitwise oper-

ations, length of bit streamsPT1 , and must be

greater than or equal to minimum up/down time of uniti.

3.8.3. Gray zone modificationTo reduce the cost, the proposed gray zone modification meth-

ods, as shown inFig. 7, is applied. The gray zone for start-upcost indicates the state of (Toff + c-s-hour)-th off hour when a unitjust enters into cold start-up from hot start-up. As cold start-upcost is higher than hot start-up cost, it is desirable that the unitstarts with hot start-up cost if possible[3]. The method inFig. 7searches the gray zone for start-up cost and modifies scheduleby turning the unit on just 1 h (period) before.

3nted,

e t anyp mit-t willb thee nd ane da

cted ifE ord-i onu esto orarc its ad heatr

P

Ntr gesw pr twoe tiont entirs nistic

.,d d o

according to the ascending order of heat rate. This heuristicjudges whether the schedule can satisfy Eq.(8) at every startingup unit, and keeps up remaining starting up if the unit schedulecan satisfy the constraint or shuts down the unit otherwise. Theabove operation is carried out until Eqs.(6) and(8) are fulfilled.Elimination of excess and deficient in units assists more than arepair algorithm.

Any modification of the schedule for any heuristic, sign vec-tor will be simultaneously changed accordingly. In repeatediterations, the unchanged hour schedule will be ignored for ELDrecalculation.

3.9. ELD calculation

The economic load dispatch (ELD) is a computational inten-sive part in unit commitment problem to evaluate solutions.Therefore, to save the computational efforts, ELD is performedvery carefully and intelligently using the following criteria:

Criterion 1: ELD is performed if the schedule is able to satisfythe spinning reserve and minimum up/down time constraintsafter applying heuristics.Criterion 2: ELD is performed only for the hours when theintroduced sign bits are 1’s that indicate the schedule has beenchanged due to SPL operations to reduce overhead.Criterion 3: In iteration method such as SPL, relaxed ELD

final

A foren

3

fors is-i s nos um-b ngers olu-t

4

and1 d on

.8.4. Elimination of excess and deficient in unitsWhen the above heuristics are successfully impleme

xcess started or deficiency in units may be generated aeriod. In order to reduce total cost, units should not be com

ed more without violating the constraints. So, excess unitse turned off gradually. The procedure of commitment ofxcess started and deficient in units are explained below axample of operation is shown inFig. 8where units are indicates the excess started or deficient in units.

Excess units are tested at each hour and they are deteq.(6) is satisfied after shutting down one or more units. Acc

ng to the priority, the minimum number of higher prioritynits, that can satisfy Eq.(6), are determined at each hour. Rf the on units are considered as excess units that are tempopied to an excess array, EA. Elements of the excess unecided to turn off gradually according to the normalizedate of the excess units as follows:

roff (u) = HRu∑i ∈ EAHRi

. (14)

ow for each excess unit, a random numberr is generated fromhe range [0. . .1] and an excess unit,u will be turned off if≤ Proff (u). When shutting down one unit, a heuristic judhether the schedule can satisfy Eqs.(6) and(8) and keeps u

emaining shut down if the schedule can satisfy mentionedquations, and starts the unit up again if not. At each itera

he above operation is carried out for each excess unit andcheduling period. No excess units are turned off determially.

Inversely, in case of violating Eq.(6) at any period, i.eeficiency in units exists and decommitted units are starte

ilyre

,e-

n

may be performed at the beginning and exact ELD at neariterations.

ll sign bits are reset(0) after successful ELD calculations beext iteration starts.

.10. Stopping criterion and terminating condition

SPL loop is stopped running when trials are completedelectedS number of promising solutions and for each prom

ng solution, terminating condition is reached when there iignificant improvement of the solution or the maximum ner (50N in this paper) of iterations is reached. Relax and lotopping criterion and terminating condition produce better sions with longer execution time.

. Simulation conditions

The simulations include test runs for 10, 20, 40, 60, 8000-unit systems. Expansion of the number of units is base


Fig. 8. Commitment of excess started or deficient in units.

Table 4Unit characteristics and cost coefficients

Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit 7 Unit 8 Unit 9 Unit 10

Pmaxi (MW) 455 455 130 130 162 80 85 55 55 55

Pmini (MW) 150 150 20 20 25 20 25 10 10 10

ai ($) 1000 970 700 680 450 370 480 660 665 670bi ($/MW) 16.19 17.26 16.60 16.50 19.70 22.26 27.74 25.92 27.27 27.79ci ($/MW2) 0.00048 0.00031 0.002 0.00211 0.00398 0.00712 0.00079 0.00413 0.00222 0.00173T on

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

i (h) 8 8 5 5 6 3 3 1 1 1h-costi ($) 4500 5000 550 560 900 170 260 30 30 30c-costi ($) 9000 10000 1100 1120 1800 340 520 60 60 60c-s-houri (h) 5 5 4 4 4 2 2 0 0 0ini state (h) +8 +8 −5 −5 −6 −3 −3 −1 −1 −1

Table 5Priority list

Priority order 1 2 3 4 5Unit 1 2 4 3 5

Priority order 6 7 8 9 10Unit 6 7 8 9 10

10-unit system. For the 20-unit system, the base 10 units areduplicated and the load demand is multiplied by 2. Similarly,other unit systems are expanded too. The load demand and theunit characteristics, which are used in this paper, are given inTables 3 and 4, respectively[13,15]. In order to perform a sim-ulation on the same conditions of[13,15], the spinning reserverequirement is assumed to be 10% of the load demand and totalscheduling period is 24 h. All calculations have been run onIntel(R) Pentium-4 CPU (1.5 GHz), 128 MB RAM, Linux ver-sion 2.2.18 and gcc version 2.91.66. No extra temporary storageis needed exceptS number of promising solutions and some localvariables whereS is small.

In Table 4, ‘ini state’ represents the initial state of units onscheduling period and the positive sign indicates that the unitis on, whereas the negative sign indicates that the unit is off.The priority list created fromTable 4is shown inTable 5. Thesimulation parameters used in the SPL, such asX, W and S,are given inTable 6. The number of units inside the window isone, which was obtained the best result from prior simulation.Furthermore, for 40-unit systems we examine the effects on totalcost and execution time by varying window size,X.

Table 6Simulation parameters

Number of units inside the window X 1Number of solutions included by window system W 20Number of solutions applying several heuristics S 5

5. Simulation results

All the cases, average values of 10 runs are reported as pro-posed SPL results. It always converges and operating cost vari-ation is negligible. Solutions are not biased and they are equallydistributed between the best and worst solutions. These factsstrongly demonstrate the robustness of proposed SPL method.

Table 7shows the comparison of proposed SPL method toGA reported in[13] and EP reported in[15] with respect to totalcost and our method provides the lowest total cost schedule forall unit systems.Table 8shows execution time of SPL method

Table 7Comparison of total cost

No. of units Total cost ($)

Proposed SPL GA[13] EP[15]

10 564950 565825 56535220 1123938 1126243 112725740 2248645 2251911 225261260 3371178 3376625 337625580 4492909 4504933 4505536

100 5615530 5627437 5633800

W = 20,S = 5, X = 1.


Table 8Comparison of execution time

No. of units Execution time (s)

Proposed SPL GA[13] EP[15]

10 7.24 221 10020 16.32 733 34040 46.32 2697 117660 113.85 5840 226780 215.77 10036 3584

100 374.03 16733 6120

W = 20,S = 5, X = 1.

Fig. 9. Average time complexity curve of proposed SPL.

with GA[13] and EP[15]. According toTable 8, the SPL methodcan obtain better solutions rapidly even though the problem islarge because the generation of the feasible solutions is limitedby pruning and the number of ELD calculations is reduced aswell by sign vector.

Fig. 9 shows the graphical presentation on the average timecomplexity of proposed method as it is very difficult to derivea mathematical equation of average execution time. From thisfigure, its nature is quadratic of number of units with small coef-ficient for small size UC problems and it tends to be linear forlarge-scale problems. The approximate execution time of more

Fig. 11. Effect of window size (40-unit system:W = 20,S = 5, varyingX).

than 100 units systems or other important systems which are notreported here (e.g. 50-unit system) may also be predicted fromthis curve.Fig. 10 illustrates the simulation results to confirmeffectiveness of the SPL method visually.

In the proposed method, the generation of plurality of solu-tions is performed by stochastic change ofX units at each hour.Therefore, we are interested to display the effect on solutionquality and execution time by varyingX. Fig. 11(a) and (b) showthe simulation results of the total cost and execution time for 40-unit system in the same conditions asTable 6except varyingX.

Table 9Gray zone modification effect

No. of units Average total cost ($) Reduced cost ($)

Case 1 Case 2

10 564950 565156 20620 1123938 1124110 17240 2248645 2249044 39960 3371178 3372718 74880 4492909 4493343 434

100 5615530 5616142 612
Fig. 10. Comparison of proposed with others.


In order to verify effectiveness of the gray zone modification,we compare Case 1 with Case 2 in the same condition asTable 6and average value of simulation results is reported inTable 9.In Case 1, gray zone is modified for start-up cost but in Case 2,gray zone modification concept is not applied.

6. Conclusions

This paper presents a fast Stochastic Priority List methodfor the UC problem. Genetic algorithm is an absolutely ran-dom algorithm and tries to reach the goal randomly. It is a slowprocess as it needs long generation. Our contribution is by intro-ducing a nice combination of priority list method and problemspecific appropriate heuristics with necessary improvement toovercome the drawbacks of other existing methods. To reducehuge time complexity, PL method is applied as a techniquegetting initial solution rapidly. Variety of solutions have beenincluded by introducing stochastic window system to searchthe domain thoroughly from multiple directions. Schedules aremodified using suitable heuristics to avoid local optima. Con-dition for solution pruning can expedite the process. Besides,overhead ELD calculations are protected by introduced sign vec-tor. These attributes make SPL method more powerful, general,and absolutely robust. Advantages of our proposed method overother published methods are discussed below:

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simulation results. Recall that, it is, however, easy to implementand is not memory intensive. In future, further examinations onconstraints and intelligent heuristics are being expected.

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(i) Sometimes PL[3] is used to solve UC problem, howevePL searches a very small part within the whole UC dom(universe). On the other hand, stochastic PL method ittively searches large domain in this paper.

(ii) Optimization is not guaranteed in pure PL method. Hennecessary improvements, e.g. excess units, stochasticstitution in window system, suitable number of trials eare included in this paper to make it general, powerfulrobust.

(iii) General optimization methods[11–20] are fully randomprocess for UC and thus very time consuming. On the ohand in this paper, problem dependent heuristics makea semi-random and fast iterative UC search compareother iterative searches.

(iv) Complex calculations are technically avoided in this paAll calculations are simple here. There is neither comppenalty calculations (e.g. GA, EP), complex acceptaprobability calculations (e.g. SA) nor any complex mebership degree calculations (e.g. Fuzzy Algorithm) in S

(v) SPL method does not need huge memory like Tabu Se(vi) Overhead ELD is protected by introduced sign vecto

this paper.

Finally, the simulation results show a big improvement ofproposed method up to 100-unit systems, even though powmethods are set as benchmark. Hence, even for a large prothe proposed method can produce solutions within practicalcution time. The effectiveness of the SPL method is clear f

Emerging solution of large-scale unit commitment problem by Stochastic Priority List

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