Ahmed Yousuf Saber2008

8/22/2019 Ahmed Yousuf Saber2008

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Scalable unit commitment by memory-bounded ant colonyoptimization with A local search

Ahmed Yousuf Saber *, Abdulaziz Mohammed Alshareef

Department of Electrical and Computer Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia

Received 11 September 2006; received in revised form 22 December 2007; accepted 29 January 2008

Abstract

Ant colony optimization (ACO) is successfully applied in optimization problems. Performance of the basic ACO for small problemswith moderate dimension and searching space is satisfactory. As the searching space grows exponentially in the large-scale unit commit-ment problem, the basic ACO is not applicable for the vast size of pheromone matrix of ACO in practical time and physical computer-memory limit. However, memory-bounded methods prune the least-promising nodes to fit the system in computer memory. Therefore,the authors propose memory-bounded ant colony optimization (MACO) in this paper for the scalable (no restriction for system size) unitcommitment problem. This MACO intelligently solves the limitation of computer memory, and does not permit the system to growbeyond a bound on memory. In the memory-bounded ACO implementation, A heuristic is introduced to increase local searching abilityand probabilistic nearest neighbor method is applied to estimate pheromone intensity for the forgotten value. Finally, the benchmarkdata sets and existing methods are used to show the effectiveness of the proposed method. 2008 Elsevier Ltd. All rights reserved.

Keywords: Ant colony optimization; A local search; Economic load dispatch; Unit commitment

1. Introduction

Effective scheduling of available energy resources forsatisfying load demand has become an important task inmodern power systems. Unit commitment (UC) in powersystems involves to properly schedule on/off states of allgenerators in a system. In addition to fulfill a large numberof constraints, the optimal UC should meet forecasted loaddemand calculated in advance, plus spinning reserve

requirement at every time interval such that the total costis minimum. The UC problem is a combinatorial optimiza-tion problem with both binary and continuous variables.The number of combinations of 01 variables grows expo-nentially for a large-scale UC problem. Therefore, the UCis one of the most difficult problems in optimization area.

A bibliographical survey on UC methods reveals thatvarious numerical optimization techniques have beenemployed to approach the UC problem in more than 200published articles since the last 3-decade. Among thesemethods, the priority list (PL) [13] is very fast but highlyheuristic and gives schedules with relatively higher opera-tion cost. Branch-and-bound (BB) methods [46] have thedanger of a deficiency of storage capacity and calculationtime increases enormously for a large-scale UC problem.

Lagrangian relaxation (LR) methods [711] concentrateon finding an appropriate co-ordination technique for gen-erating feasible primal solutions, while minimizing theduality gap. The main problem with an LR method is thedifficulty encountered in obtaining feasible solutions.

Meta-heuristic methods [1237] are iterative techniquesthat can search not only local optimal solutions but alsoa global optimal solution depending on problem domainand execution time limit. In the meta-heuristic methods,the techniques frequently applied to the UC problem aregenetic algorithm (GA), tabu search (TS), evolutionary

0142-0615/$ - see front matter 2008 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijepes.2008.01.001

* Corresponding author. Tel.: +966 5422 69809.E-mail addresses: [email protected], [email protected], aysaber@

gmail.com (A.Y. Saber).

www.elsevier.com/locate/ijepes

Available online at www.sciencedirect.com

Electrical Power and Energy Systems 30 (2008) 403414
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programming (EP), simulated annealing (SA), particleswarm optimization (PSO), etc. They are general-purposesearching techniques based on principles inspired fromthe genetic and evolution mechanisms observed in naturalsystems and populations of living beings. These methodshave the advantage of searching the solution space more

thoroughly. The main difficulty is their sensitivity ofparameters.There are two popular swarm inspired methods in com-

putational intelligence areas: particle swarm optimization(PSO) and ant colony optimization (ACO). PSO is themost simple and promising, and it requires less computa-tion time, though it faces difficulties for solving discreteoptimization problems [1825]. Inspired by the food-seek-ing behavior of real ants, ant systems, attributable to Dor-igo et al., has demonstrated itself to be an efficient, effectivetool for combinatorial optimization problems. However,the typical version of the ACO in [2630] is near aboutimpossible even for a moderate size of the UC problem

in practical time limit and computer storage capacity.Fuzzy UC models are also available in [3843]. How-

ever, they are imprecise and need sufficient previous statis-tics to model the imprecision. Some general optimizationsoftware packages are also available for modeling and opti-mization [44,45]. Software companies urge state-of-the-arttechnologies to update their products.

The ACO for the large-scale UC problem is introducedfirst time in this paper and the rest of the paper is organizedas follows. In Section 2, problem formulation and con-straints of the UC are discussed. The proposed method,applied distributions and important operations are

explained in Section 3. Simulation results are reported inSection 4. Finally, conclusion is drawn in Section 5.

2. UC problem formulation

2.1. Nomenclature and acronyms

The following notations are used in this paper.N number of unitsH scheduling periodK number of buses with loadsL number of transmission linesIi

t

ith unit status at hour t (1/0 for on/off)Pit output power of ith unit at time tPmaxi maximum output limit of ith unitPmini minimum output limit of ith unitPmaxi t maximum output power of unit i at time t consid-

ering ramp ratePmini t minimum output power of unit i at time t consid-

ering ramp rateDt load demand at time tDKt load at bus K at time tRt system reserve requirement at hour tMUi=MDi minimum up/down time of unit iXoni

t

duration of continuously on of unit i at time t

Xoffi t duration of continuously off of unit i at time t

SCi start-up cost of unit iai integrated start-up cost and equipment mainte-

nance cost of unit ibi starting-up cost of unit i from cold conditionsdi time constant that characterizes unit i cooling

speed

FC fuel cost function,HCt; r;s hourly cost at hour t for the schedule s from theschedule r at hour t 1

Prt probability distribution at stage (hour) th-costi hot start cost of ith unitc-costi cold start cost of ith unitc-s-houri cold start hour of ith unitRURi ramp up rate of unit iRDRi ramp down rate of unit iFl real power flow limit on transmission line lCl the matrix relating generator output to power flow

on transmission line l

Mlength of the pheromone matrix

G number of trials at each hour (stage)sYt randomly generated Y6 Gth binary schedule at

hour ts0 initial pheromone intensityss pheromone intensity on the state of the hourly

schedule sgr;s visibility from state (hourly schedule) r to the next

state sELD economic load dispatchTC total cost

2.2. Objective function

Mathematically the UC is a minimization problem andthe objective of the UC problem is the minimization ofthe total cost which includes fuel cost and startup cost [46].

1. Fuel costFuel cost of a thermal unit is expressed as a second orderfunction of each unit output as below.

FCiPit ai biPit ciP2i t; 1where ai, bi, and ci are positive fuel cost coefficients.

2. Start-up costThe start-up cost for restarting a decommitted thermalunit is related to the temperature of the boiler. If theunit is cold, it is necessary to consume more fuel towarm up the boiler. If the unit has been decommittedfor a short while (which satisfies the minimum downtime), less energy will be needed to restart the unit.Usually the start-up cost is an exponential functionof off-time of a generating unit [47] as (2). In some sys-tems, a simplified step function of time-dependentstart-up cost is also applied using transition hourHoffi from hot to cold start, which is defined in [3].Start-up cost will be high cold cost

c-costi

when

down time duration Xoffi exceeds cold start hour

404 A.Y. Saber, A.M. Alshareef/ Electrical Power and Energy Systems 30 (2008) 403414


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c-s-houri in excess of minimum down time MDiand will be low hot cost h-costi when down timeduration does not exceed c-s-houri in excess of mini-mum down time as follows:

(a) Exponential function:

SCit ai bi 1 exp Xoff

i tdi

: 2

(b) Step function:

SCit h-costi : MDi 6 X

offi t 6 Hoffi

c-costi : Xoffi t > Hoffi

;

(3

Hoffi MDi c-s-houri: 4Depending on available data of the system, eitherexponential function or step function is used for thestart-up cost calculation.Therefore, the objective functionof the UC is

min TC XNi1

XHt1

FCiPitSCi1Iit1Iit

subject to 613 constraints: 5However, any new type of cost (e.g., maintenance cost,

emission cost, carrying cost, etc.) may be included in theobjective function according to the system operatorsdemand in the deregulated market.

2.3. Constraints

The constraints that must be satisfied during the optimi-zation process are as follows:

1. System power balanceThe generated power from all the committed units mustsatisfy the load demand plus network losses, which isdefined as below.

XNi1

IitPit Dt Losses: 6

2. Network lossesThe network losses are taken into account as functionsof respective generating units outputs using B coeffi-cients matrix.

Ploss XNi1

XNj1

PiBijPj XNi1

Bi0Pi B00: 7

Bij ijth element of loss coefficient symmetric matrix B.Bi0 ith element of the loss coefficient vector.B00 loss coefficient constant.

3. Spinning reserveTo maintain system reliability, adequate spinning reser-ves are required. Usually a fixed percentage of loaddemand or a constant amount is used as the spinning

reserve.

XNi1

IitPmaxi tP Dt Losses Rt: 8

4. Generation limitsEach unit has generation range, which is represented as-

Pmini 6 Pi

t

6 Pmaxi :

9

5. Minimum up/down timeOnce a unit is committed/decommitted, there is a prede-fined minimum time after it can be decommitted/com-mitted again.

1 Iit 1MUi 6 Xoni t; if Iit 1; 10Iit 1MDi 6 Xoffi t; if Iit 0: 11

6. Ramp rateFor each unit, output is limited by ramp up/down rateat each hour as follows:

Pmini

t

6 Pi

t

6 Pmaxi

t

;

12

where Pmini t maxPit 1 RDRi;Pmini andPmaxi t minPit 1 RURi;Pmaxi .

7. Prohibited operating zoneIn practical operation, the generation output Pi of unit imust avoid unit operation in the prohibited zones.Therefore, unit i must operate at one of the followingvalid operating zones.

Pi 2Pmini 6 Pi 6 P

li;1

Pui;j1 6 Pi 6 Pli;j; j 2; 3; . . . ;Zi

Pui;Zi 6 Pi 6 Pmaxi

;

8>:13

where Pli;j and Pui;j are lower and upper bounds of the jth

prohibited zone of unit i, and Zi is the number of prohib-ited zones of unit i.

8. Initial statusAt the beginning of schedule, the unit initial status mustbe taken into account.In the research, the above men-tioned constraints are considered. However, it can beextended to a more general model.

3. Proposed method

In nature, a real ant wandering in its surrounding envi-ronment will leave a biological trace, called pheromone,on its path. The intensity of left pheromone will possesshigher pheromone concentration and therefore, encouragesubsequent ants to follow it. As a result, an initially irreg-ular path from nest to food will eventually contract to ashorter path in ACO. ACO is successfully applied inTravelling Salesman Problem (TSP). Articles in [2630]only solve small size moderate-dimensional problemsusing basic ACO. A significant drawback of the ACO isthat the amount of memory required to store the phero-mone matrix is exponential of the number of units Nfor the UC problem. This pheromone matrix size will

A.Y. Saber, A.M. Alshareef / Electrical Power and Energy Systems 30 (2008) 403414 405


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be 2N H for an N-unit and H-hour scheduling periodUC problem, and it is not manageable in practical timeand computer storage even for the moderate size of aUC problem. This has led to the development of mem-ory-bounded extension of ACO for the scalable UC prob-lem. This idea comes from the fact that if there is no

space left, cull the least-promising node from the memoryand assign the best possible value in need of the forgottenone. With appropriate abstraction and modification, thisfact has led to develop a successful memory-boundedACO computational model for large combinatorial opti-mization problems, like the large-scale UC problem.Besides, A heuristic function and probabilistic nearestneighbor approach are introduced to improve localsearching ability and to estimate the forgotten value,respectively in the memory-bounded ACO.

3.1. The A heuristic function for the UC problem

A search is one of the best first search strategies, whichconsists of two functions: g is the exact cost up to currentstate and h is an estimated cost of minimum cost fromcurrent state to the next state [48].

fs; t gs; t hs; t 1: 14The core part of an intelligent searching algorithm is thedefinition of a proper heuristic function. For the UC prob-lem, fs; t is the A heuristic function where gs; t is theexact cost of hourly schedule s at time t and hs; t 1 isan estimate of the minimum cost of the next hour t 1from the schedule s. A good h

function is the most impor-

tant factor to obtain a good searching result. In this study,s is rescheduled as s0 to generate an approximate minimumcost schedule for the next hour t 1 as follows:

(i) I1tI2t . . .INt s.(ii) If Iit 1 and Xoni t < MUi then Iit 1 1.

(iii) If Iit 0 and Xoffi t < MDi then Iit 1 0.(iv) Rest of the generators are turned on in ascending order

of the best cost per produced unit (see Appendix I)until

PIit 1Pmaxi t 1P Dt 1 Losses

Rt 1.(v) s0

I1

t

1

I2

t

1

. . .IN

t

1

.

In this study, gs0; t 1 is considered as the near mini-mum cost of the next hour. Therefore, (14) is rewritten asbelow for simplicity.

fs; t gs; t gs0; t 1: 15There is no guarantee that s0 is the minimum cost scheduleat hour t 1 from s. A better heuristic function generatesbetter results. The effect ofh function will be discussed inAppendix II.

Instead of Greedy search in standard ACO, the above A

heuristics in the proposed MACO increase local searching

ability.

MACO therefore has better balance between local andglobal searching abilities.

3.2. Memory-bounded ACO algorithm

Memory-bounded ACO operations for the large-scale

UC problem are given below:Step 1: InitializationMatrix size M H is set for the pheromone of the

memory-bounded ACO considering the computer memorycapacity. Initially, pheromone intensities are set to s0 for allthe hourly schedules.

Step 2: Transition

It is not possible to check all possible combination ofstates (hourly schedules) at each stage (hour) even for amoderate size of the UC problem. A predefined numberG of hourly schedules are generated by random bits flip-ping, and select only one schedule of them stochastically(e.g., roulette wheel selection) based on the following

functions:

gs; t 1=HCt; r;s

1XNi1

,FCiPit ECiPit

MCiPit SCi1 Iit 1Iit; 16gs0; t 1 1=HCt 1;s;s0

1XNi1

,FCiPit 1 ECiPit 1

MCiP

it

1 SCi

1

Iit

Iit

1;17

gr;s fs; t gs; t gs0; t 1; 18

Prtr;s ssgr;sbP

u2Ssugr; ub;s 2 S; 19

where r is the selected hourly schedule at hour t 1, s isone of the randomly generated Ghourly schedules at hour tand s0 is the minimum cost schedule for the next hour gen-erated from s using Section 3.1. Values of Iit 1, Iitand Iit 1 come from r, s and s0, respectively. Hourlyschedule s is checked immediately in Section 3.5 and mod-

ified (if needed) to manage the constraints (8), (10), (11).gr;s is a heuristic function called visibility here. A heuris-tic is proposed for the g to increase local searching abil-ity. Thus, it is the reciprocal of the hourly cost as well asthe estimated minimum cost of the next hour (18). ss isthe deposited pheromone intensity for s. In this research,probabilistic nearest neighbor method is applied to esti-mate the pheromone intensity ss for the forgotten valuein the memory-bounded ACO (see Section 3.4). Power lev-els Pi are calculated for the active (ON) units of theschedules from economic load dispatch to fulfill other con-straints. (6), (9) and (11), (13). If any constraint is violated

after the economic load dispatch, a large penalty value is

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added. b is a flexible parameter used to control the relativeweight of two factors in (19). Typical value ofb is 2. There-fore, according to (16)(19), a better hourly schedule withlower cost or the schedule with higher pheromone intensity

will be selected with higher probability at each stage (hour).Eqs. (16)(19) are repeated for all the H stages (hours)and G hourly schedules (states) of each stage. Only onehourly schedule is selected from each stage as mentionedabove. All the selected hourly schedules are then mergedto generate a UC solution. At the next step, pheromonewill be updated only for the generated UC solution thatconsists of the selected hourly schedules.

Step 3: Updating pheromone intensity

The experience accumulated in Step 2 is then used tomodify the pheromone intensity by the updating rule

s

s

1

e

s

s

e

1

TC;

20

where s is the selected hourly schedule at each hour of theUC solution and e is an evaporation factor, which lies be-tween 0 and 1. TCis calculated from (5) for the generatedUC solution in Step 2 and (20) is repeated for H hours.In this way, pheromone intensity of the selected hourlyschedules only will be updated and value of initial intensitys0 was chosen carefully so that 1=TCshould be greater thanss in (20) to increase the pheromone intensity of thepromising solutions gradually. It increases informationsharing and conveying mechanism of standard ACO.

As the pheromone matrix is memory-bounded, the leastpheromone intensity is pruned to make room for morepromising hourly schedule to be inserted. In this way, solu-tions of the lower pheromone intensity are eliminated grad-ually and the system will converge finally.

Step 4: Stopping conditionThe ACO loop is stopped running when there is no sig-

nificant improvement in the solution for a long time or themaximum number of iterations is reached. The best solu-tion is reported if the stopping condition of ACO isreached, otherwise Steps 2 and 3 are repeated.

Fig. 1 shows the general searching for the proposedmemory-bounded ACO. Text intensity of bold s in Fig. 1indicates the pheromone intensity level there. In this figure,

the last state sGHat stage H indicates randomly generated

Gth schedule at hour H. Fig. 2 shows the flowchart of theproposed memory-bounded ACO.

3.3. Data structure of the memory-bounded ACO

Predefined M H limited size matrix is saved for thepheromone, as the pheromone intensity values of all possi-ble states are not possible to store for the exponentiallygrowing UC problem. The value of

Mshould be chosen

based on the available physical computer memory.

Is11; ss11 Is12; ss12 Is1H; ss1HIs21; ss21 Is22; ss22 Is2H; ss2H

..

....

..

....

IsM1; ssM1 IsM2;ssM2 IsMH; ssMH

;

where an entry IsYt; ssYt indicates real-valued pheromoneintensity ssYt of randomly generated Yth binary scheduleat hour t and the UC schedule is converted to the equiva-lent single integer value IsYt to save the memory. IsYt isstored with its intensity, as all possible combinations are

not tried. For an example, Is23; ss23 29; 0:075

Ini.

s11

s21

sG1

(s11)

(s21)

(sG1)

Stage 1

(Hour 1)

(Ini,s11)

s12

s22

sG2

(s12)

(s22)

(sG2)

Stage 2

(Hour 2)

s1H

s2H

sGH

(s1H)

(s2H)

(sGH)

StageH

(HourH)

(Ini,s21)

(Ini,sG1)

(s21,s12)

(s21,s22)

(s21,

sG2)

(.,s1H)

(.,s2H)

(.,sGH)

Fig. 1. Searching in memory-bounded ACO.

Start

Visibility () is calculated with the help of

proposed A* heuristic function (14-18) for

better local search.

Stopping

criterion ?

End

No

Yes

Print result.

For each hour, select one hourly schedule

using roulette wheel selection based on(), () and (19).

Pheromone intensity () is calculated by

proposed probabilistic nearest neighbor

method for the forgotten value.

Initialize parameters, especially size of phero-

mone matrix to fit into the physical memory.

For each hour, generate G hourly schedules

by random n-bit flipping.

Insert and update pheromone for the sel-

ected hourly schedules using (20). Least

pheromone intensity will be pruned.

Large penalty value is added in (16-17) if

any UC constraint is violated after ELD.

Fig. 2. Algorithmic flowchart of the proposed memory-bounded ACO.

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indicates that the randomly generated 2nd schedule at 3rdhour is 11101b or 29d and its pheromone intensity is 0.075.At the current iteration, bold entries are inserted at eachcolumn of the matrix from Fig. 1 where the selected UCsolution is s21sM2 . . .s2H and the corresponding leastpheromone intensity entries are pruned.

3.4. Probabilistic nearest neighbor pheromone s inmemory-bounded ACO

For an hourly schedule s at hour t, its pheromone inten-sity ss is searched at the tth column of the stored M Hmemory matrix. If the entry Ist; sst exists, sst is thedesired pheromone intensity for (19), i.e., ss sst.However in many cases, ss is not available in the mem-ory, as matrix length M is not large enough to fit the whole(huge) matrix into the physical computer memory. This ishappened because either the s is a new schedule or the entry

Ist; s

st

was pruned for its least pheromone intensity

value and it is assumed that their possibilities are q and1 q, respectively.

q 11

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiIterationNo

W

q ; 21

where q is exponentially decreasing with respect to itera-tion numbers and higher value of denominational weight,W indicates higher probability of generating newschedules.

For the memory-bounded ACO, the authors proposeprobabilistic nearest neighbor approach to estimate thess if it is not found in the memory as follows:

(i) Set stemp1 s0, as the s may be new.(ii) Calculate minimum Hamming distance between s and

schedules of tth column sit, Min{Ham(s, sit),i 1; 2; . . . ;M}. Let, Ham(s, sqt) is the minimumfor the qth entry of tth column. If there is the sameminimum Hamming distance for more than oneentry, any one can be chosen randomly.Assign stemp2 ssqt, as the s may be pruned and thepheromone intensity of the nearest neighbor scheduleis considered (restored).

(iii) ss qstemp1 1 qstemp2.s0 is initial pheromone intensity and q is calculated at

the beginning of the iteration from (21).

3.5. Constraints management

Owing to the large set of physical and operational con-straints (6)(13) inherent in the UC problem, the randomlygenerated new solutions may not satisfy all the constraints.Constraints are handled in two ways direct repair andindirect penalty methods. Constraint (8) is fulfilled bydirect repair method to accelerate the system. Other con-

straints are managed by indirect penalty method to dis-courage the invalid solutions. Therefore, all types ofconstraint are tolerable in the proposed method.

If the spinning reserve (8) is violated at any hourly sche-dule, the system suffers from deficiency in units. Then,decommitted units of that hourly schedule are forced to

turn on randomly until (8) is satisfied.For an hourly schedule, if the minimum up/down timeconstraint (10) and (11) is not satisfied considering the pre-vious and the next possible hourly schedule(s), a very largevalue is directly assigned to HC in (16) and (17) to dis-courage the solution.

Other constraints, namely: system power balance (6),generation limit (9), ramp rate (12) and prohibited zone(13) need power levels, and are fulfilled in ELD. If any con-straint is violated after the ELD, a large penalty value isadded to HC in (16) and (17).

3.6. ELD calculation

ELD consists of finding optimal distribution of powerdemand among running units, satisfying (6), (7), (9), (12),(13) constraints. ELD is performed if an hourly scheduleis able to satisfy minimum up/down time and spinningreserve. Some articles [22,23] have already been publishedon economic load dispatch using another swarm inspiredmethod, PSO and the constraints are successfully managedthere. Modified PSO, which consists of 4-vectors instead of3-vectors of standard PSO, is applied here for the ELD cal-culation, as (i) ELD does not need any discrete optimiza-tion that is suitable for the PSO, (ii) PSO is also a swarm

inspired optimization method like the ACO, (iii) PSO isthe most simple and promising method, and it requires lesscomputation time.

Recall that, velocity changes of standard PSO consistof three parts: momentum part, cognitive part and socialpart. These three parts are not enough for complex poly-nomial optimization of constrained ELD. Therefore,modified PSO is applied for the ELD problem. The fol-lowing equations are used in the proposed modified PSOfor ELD.

vij w vij XI

r

1

cr rand Prmrij pij; 22

uij vijkVik vijffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPNj1v

2ij

q ; 23

pij pij Pf

i1jErrorij 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiIteration

p uij; 24

where i represents particle and j represents unit. Prmrij is

the jth component of rth promising value of particle i inthe swarm. In standard PSO, I 2, Prm1ij pbestij andPrm

2ij gbestj. In this study, authors use up to 3 promis-

ing values (4 vectors including momentum) for the calcula-tion of velocity vector, i.e. I 3, Prm1ij pbestij andPrm

2ij gbestj and

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Prm3ij

Prm1ij randomjPrm2ij Prm1ij j;

ifPrm2ij 6 Prm1ij

Pminj randomPmaxj Pminj ;otherwise

:

8>>>>>>>:

25

Researchers can choose number and values of Prmrij ; rP 3

depending on specific problem complexity and prior

knowledge. One of the choices is shown in (25) wherer 3. The better choice generates faster convergence.However, many Prmr parameters spend excess executiontime.

The normalized vector Ui ui1; ui2; . . . ; uiN of a non-zero vector Vi vi1; vi2; . . . ; viN is a unit vector co-direc-tional with Vi which indicates direction of next movement.Besides, length of next step (movement) should depend ontotal amount of errors for constraints violation and currentiteration number as in (24). Therefore, only swarm optimi-zation techniques are applied in the UC scheduling andeconomic load dispatch in this study.

4. Simulation results

Generator instances are proposed in [49] for simulation;however, it is in initial phase and reported results are notavailable for the instances using recent promising methodsfor comparison. In this paper, three standard data sets aretested for the constraints described in Section 2.3.

All calculations have been run on Intel(R) Celeron(TM)CPU, 256 MB RAM, Windows 2000 OS and C/C++ com-piler. Convergence mainly depends on the proper setting ofparameters. The proposed memory-bounded ACO param-eters values for the UC problem are as follows:

Pheromone matrix length, M 500.Number of trials (states) at each hour (stage), G 500.Evaporation factor, e 0:60.Initial pheromone intensity, s0 0:0000000340 N=10.b 2, denominational weight, W 10.Number of bits flipping, n dN=5e,Maximum number of MACO iterations = 100.

Case 1: Input data of 6-unit system, which consists of 26buses and 46 transmission lines, are shown in Table 1 andB coefficients (base capacity 100 MVA) for network losses

are shown below.

Economic load dispatch is the most important andtime-consuming part in the UC problem. This exampleconsists of most of the constraints of Section 2.3. Testresults using different methods are shown in Table 2 for1 h. The best, worst and average findings of the proposedmethod are reported together with their cost variation asa percentage of the best solution. It always converges andvariation is tolerable. For more than sufficient iterations,best, worst and average results are near about the sameand the variation is negligible. Average cost of 10 runs

is near to the best result and the solutions found presentsmall variation. These facts strongly demonstrate therobustness of the proposed method. Percentage of suc-cess, and maximum and minimum execution timewere not reported for GA and PSO (reported in [22])methods.

Table 3 shows the comparison of the proposed methodto the most recent methods (e.g. GA and PSO reportedin [22]) with respect to the generated best output. Accord-ing to Tables 2 and 3, the proposed method provides thelowest cost schedule and fulfills all the constraints for the6-unit system.

Case 2: The unit characteristics of a real system (calledTaipower 38-unit system) and load demand are collectedfrom [50]. In order to perform simulations on the samecondition in [5,33,49], the spinning reserve requirement isassumed to be 11% of the load demand, and total schedul-ing period is 24 h.

Test results are shown in Table 4. The best, worst, andthe average findings of the proposed method are reportedtogether both considering and neglecting the ramp rateconstraint. It also always converges and operating cost var-iation is tolerable. Operating cost considering ramp rate ishigher than the cost neglecting it, as the ramp rate con-straint sometimes prohibits to run low (high) cost units at

the maximum (minimum) generation limit. In case of

Bij

0:0017 0:0012 0:0007 0:0001 0:0005 0:00020:0012 0:0014 0:0009 0:0001 0:0006 0:00010:0007 0:0009 0:0031 0:0000 0:0010 0:0006

0:0001 0:0001 0:0000 0:0024 0:0006 0:00080:0005 0:0006 0:0010 0:0006 0:0129 0:00020:0002 0:0001 0:0006 0:0008 0:0002 0:0150

;

Bi0 1:0e30:3908 0:12970:70470:05910:2161 0:6635; B00 0:0056:

Table 1Units characteristics of 6-unit system

No. of units ai ($) bi ($/MW) ci ($/MW2) Pmaxi (MW) P

mini (MW) P

0i (MW) RURi (MW/h) RDRi (MW/h) Prohibited zones (MW)

1 240 7.0 0.0070 500 100 440 80 120 [210 240] [350 380]2 200 10.0 0.0095 200 50 170 50 90 [90 110] [140 160]3 220 8.5 0.0090 300 80 200 65 100 [150 170] [210 240]4 200 11.0 0.0090 150 50 150 50 90 [80 90] [110 120]

5 220 10.5 0.0080 200 50 190 50 90 [90 110] [140 150]6 190 12.0 0.0075 120 50 110 50 90 [75 85] [100 105]

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neglecting ramp rate, time will also be saved, as the con-straints (see Section 2.3) are fulfilled easily.

Table 5 shows the comparison of the proposed methodto the popular methods, e.g. DP reported in [5], LRreported in [11], SA reported in [33], and CLP reportedin [49] with respect to the total cost and execution time.From the results, it is clear that the proposed method isrobust and it is applicable to real systems.

Case 3: Base 10-unit system is also used to compare withother popular methods. The load demand and unit charac-teristics of the base 10-unit system are collected from [21].In order to perform simulations on the same condition of[12,14,15,17,21], the spinning reserve requirement isassumed to be 10% of the load demand, and total schedul-ing period is 24 h. The simulations include test runs up to100-unit systems. For the 20(40)-unit system, the base 10units are duplicated (copied four times) and the loaddemand is multiplied by two(four).

Test results are shown in Table 6. The best, worst, and

the average findings of the proposed method are reportedtogether with standard deviation of cost variation. Italways converges. The variation is little bit high, as somestates are pruned to fit the memory, though it is tolerable.Average cost and execution time of 10 runs are near to themiddle position between their maximum and minimum val-

ues. So, it is clear that solutions are not biased and they areequally distributed between the best and the worst solu-tions. These facts strongly demonstrate the robustness ofthe proposed memory-bounded ACO for the UC problem.Besides, the execution time is not exponentially increasedwith respect to the number of units, as the length of thepheromone matrix and the number of generated hourlystates at each stage (hour) are fixed.

Table 7 shows the comparison of the proposed methodto the most recent methods, e.g., integer-coded GA (ICGA)reported in [12], Lagrangian relaxation and genetic algo-rithm (LRGA) reported in [14], genetic algorithm (GA),dynamic programming (DP) and Lagrangian relaxation(LR) reported in [15], evolutionary programming (EP)reported in [17], and hybrid particle swarm optimization(HPSO) reported in [21] with respect to the total cost. -indicates that no result is reported in the correspondingarticle for the large-scale UC problem. Average results ofthe proposed memory-bounded ACO is comparable withother methods, though it was impossible to model thelarge-scale UC problem for the typical ACO. From Table7, it is obvious that the memory-bounded ACO is accept-able for the large-scale UC problem, though it is mentioned

that the cost variation is little bit high, as all possible com-binations are not checked at each stage.

Scaling ability is one of the major features of optimiza-tion methods. Execution time complexity of each optimiza-tion method is very important for its application to realsystems. Table 8 shows time comparison. Execution time

Table 2Comparison of test results of 6-unit system using different methods

Method Success (%) Total cost Execution time

Best ($) Worst ($) Average ($) Variation (%) Maximum (s) Minimum (s) Average (s)

Proposed method 100 15,443.20 15,485.97 15,448.66 0.27 1.63 1.31 1.46PSO[22] 15,450.00 15,492.00 15,454.00 0.27 14.89

GA 15,459.00 15,524.00 15,469.00 0.42 41.58

Table 3Comparison of the best output of 6-unit system using different methods

Power outputs GAmethod

PSOmethod

Proposedmethod

Unit 1 (MW) 474.807 447.497 445.832Unit 2 (MW) 178.636 173.322 173.865Unit 3 (MW) 262.208 263.474 263.682Unit 4 (MW) 134.282 139.059 139.274Unit 5 (MW) 151.903 165.476 167.157Unit 6 (MW) 74.181 87.128 85.654

Network loss (MW) 13.0217 12.9584 12.4639Total generated power (MW) 1276.0300 1276.0100 1275.4636Load demand (MW) 1263.00 1263.00 1263.00Error (MW) 0.0083 0.0516 0.0005Cost ($) 15,459.00 15,450.00 15,443.20

Table 4Test results of the proposed method for cost and time after 10 runs considering/neglecting ramp rate constraint of 38-unit system

Ramp rate Success (%) Total cost Execution time

Best (M$) Worst (M$) Average (M$) Variation (%) Maximum (s) Minimum (s) Average (s)

Yes 100 203.32 209.61 205.49 3.1 135.82 110.63 124.71

No 100 200.46 205.58 201.84 2.55 126.36 102.85 111.92

Table 5Comparison of the total cost and timeDP [5], LR [11], SA [33], CLP [49],and the average finding of the proposed method

Algorithm DP LR SA CLP Proposedmethod

Cost (M$) with ramp rate 215.2 214.5 215.6 213.9 205.49Time (s) 199 29 2,589 17 124.71

Cost (M$) with out ramprate

201.5 209.0 207.8 208.1 201.84

Time (s) 24 7 1690 10 111.92



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is acceptable, as it is in second. Execution time is not expo-nentially growing with respect to the scaling of the UCproblem, as the number of trials at each stage is fixed forall the systems. The proposed method needs moderateCPU time, as it has better internal information sharingand conveying mechanisms that help it to converge quicklyand the results are collected from 100 iterations only.According to Table 8, only ICGA is faster than the pro-posed MACO method. However, it is important to men-

tion that in ICGA, the running cost is considerablyhigher than the proposed MACO because the ICGA usesonly 300 generations and 50 chromosomes which are notsufficient (see notes of Table 7) for the ICGA.

Therefore, considering the cases, it is better than non-swarm based optimization (deterministic) methods, e.g.,DP, LR, etc., as they do not have any information gather-ing mechanism like the pheromone in ACO, and they can-not solve large-scale UC problem in suitable time andphysical computer memory limit. The proposed MACOshares many common parts of GA and EP. However, the

MACO has better information sharing and conveyingmechanisms, and thus may possess less randomness thanGA and EP. It is better than PSO for the UC scheduling,as the PSO is not suitable for solving the discrete optimiza-tion problem, which is one of the parts of the UC problem,and the PSO suffers from the lack of balance between glo-bal and local searching abilities [21]. It is also better thanthe hybrid methods, as they have the above difficulties oftheir member methods. Only LRGA produced better result

than MACO for the 40-unit system, as it consumed unac-ceptably long execution time (2165 s) and it might there-fore, reach very near to the global best solution. In theproposed method, cost per unit power is little bit higherfor larger system, as the allocated memory is fixed for allthe systems and larger systems suffer more than that ofsmaller systems.

Fig. 3 shows the convergence of the memory-boundedACO. In the beginning, it converges quickly, then con-verges slowly at the middle of the iteration and then veryslowly or steady from the near final iterations. Therefore,

Table 7Comparison of total cost - ICGA [12], LRGA [14], GA [15], DP [15], LR [15], EP [17], AG [31], HPSO [21] and the average finding of the proposedmemory-bounded ACO

No. of units Total cost ($)

ICGA [12] LRGA [14] GA [15] DP [15] LR [15]

Best Worst Average Best Worst Average Best Worst Average Best Worst Average Best Worst Average

10 566,404 564,800 565,825 570,032 565,825 N/A N/A 565,825 N/A N/A20 1,127,244 1,122,622 1,126,243 1,132,059 1,130,660 N/A N/A40 2,254,123 2,242,178 2,251,911 2,259,706 2,258,503 N/A N/A

No. of units Total cost ($)

EP [17] AG [31] HPSO [21] MACO

Best Worst Average Best Worst Average Best Worst Average Best Worst Average

10 564,551 566,231 565,352 564,005 563,942 565,785 564,772 561,453 563,212 562,34120 1,125,494 1,129,793 1,127,257 1,124,651 1,115,347 1,125,245 1,119,18740 2,249,093 2,256,085 2,252,612 2,242,012 2,256,802 2,248,090

Notes: indicates that result is not reported in the corresponding reference.ICGA: Generations = 300, chromosomes = 50, crossover and mutation prob. = not constant.LRGA: Generations = 500, population size for 10, 20, 40-units = 60, 80, 80, respectively, duality gap = 0.02, crossover and mutation prob. = 0.8, 0.0333,respectively.GA: Generations for 10, 20, 40-units = 500, 1000, 2000, respectively, chromosomes = 50, adaptive crossover prob. range = 0.40.9 and adaptive mutationprob. range = 0.0040.024.DP: Complete state enumerations for only 10-unit system.LR: Multiplier increasing factor = 0.01 and decreasing factor = 0.97.EP: Generations for 10, 20, 40-units = 500, 1000, 2000, respectively, population size = 50.AG: Population size = 70, ini. temperature = 900, crossover prob. = 0.8 and mutation prob. = 0.1.HPSO: Maximum iterations = 1000, population size = 20, inertia weight, w = 1.0, q1 = 2.8, q2 = 1.2.

MACO: Iteration no. = 100, pheromone matrix length = 500, no. of trials at each stage = 500, = 0.60.

Table 6Test results of the proposed memory-bounded ACO for cost and execution time after 10 runs (base 10-unit system)

No. of units Success (%) Total cost Execution time

Best ($) Worst ($) Average ($) Standard deviation (K$) Maximum (s) Minimum (s) Average (s)

10 100 561,453.72 563,212.60 562,341.38 0.58 38.86 34.91 37.9920 100 1,115,347.51 1,125,245.34 1,119,187.48 2.54 94.53 85.76 90.14

40 100 2,242,012.92 2,256,802.58 2,248,090.84 3.96 288.48 275.62 283.05100 100 5,607,532.93 5,622,378.38 5,613,814.03 5.13 1174.74 1121.52 1146.26



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the memory-bounded ACO holds the above importantcharacteristic of a good optimization method.

Tables 911 show the sensitivity of the parameters.Results depend on the allocated memory for the memory-bounded ACO. Mainly vast memory is needed to storethe trials (states) of each stage (hour) and for the largepheromone matrix. According to Table 9, results are notsatisfactory when the system suffers from the lack of mem-ory. Initial pheromone intensity and evaporation factor

depend on systems. Initial pheromone intensity should besuch that pheromone intensity will increase for all theselected hourly schedules in (20). From Table 10, the bestevaporation factor value is 0.6 for the UC problem. Forother values of , either the past experience or the currentknowledge is given more precedence than required andthus proper coordination of them is not maintained. FromTable 11, the best denominational weight is 10 for the UCproblem. For other values ofW, either the new solution orthe pruned solution is assumed to be generated more fre-quently than the actual case.

5. Conclusion

This paper introduces the memory-bounded ACO forthe scalable UC problem. Between the swarm inspired opti-mization methods, ACO is suitable for the combinatorialoptimization problem. Therefore, the authors try to modelthe memory-bounded ACO for the scalable UC problem.In this study, our contributions are (i) the proper executionof the memory-bounded version of ACO for the exponen-tially growing scalable UC problem that was impossible tosolve by the basic ACO, (ii) the implementation of A heu-ristic in the MACO method to increase the solution qual-ity, (iii) the introduction of probabilistic nearest neighbor

approach to estimate pheromone intensity for the forgottenvalue, and (iv) an appropriate introduction of the modified

PSO by incorporating more promising values in velocityvector for the ELD in UC problem.

Advantages of the proposed method are discussedbelow.

It is an improved version of standard ACO. It has better information sharing and conveying

mechanisms.

It does not permit the system to grow beyond a boundon memory. It has better balance between local and global searching

abilities. It can handle more constraints and higher order cost

polynomials without extra concentration/effort. Only swarm optimization techniques are applied in the

UC scheduling and ELD.

From this study, it is an important notice that its perfor-mance is moderate and the result variation is little bit high.Finally, this study is a first look at the memory-boundedACO for the scalable UC problem. In future, there is

enough scope to study on developing more appropriateheuristic function for the UC problem and preservinginformation of the pruned nodes from better heuristicestimation.

Appendix I. Best cost per produced unit

The cost function for a thermal generator is parabolicaccording to (1). Cost per produced unit (CPU) is definedin (A1).

CPUi

FCiPiPi

ai

Pi bi

ciPi

$=MW

:

A1

Table 8Comparison of execution time

No. of units Execution time (s)

ICGA [12] LRGA [14] GA [15] EP [17] MACO

10 7.40 518 221 100 37.9920 22.40 1147 733 340 90.14

40 58.30 2165 2,697 1176 283.05100 242.50 1146.26

1.12e+06

1.13e+06

1.14e+06

1.15e+06

1.16e+06

0 10 20 30 40 50 60 70 80 90 100

Cost($)

Generations

Fig. 3. Convergence of the memory-bounded ACO.

Table 9Effect of allocated memory for 20-unit system

No. of trials, G M 300 M 400 M 500300 1,149,358 1,138,098 1,129,437400 1,143,575 1,131,184 1,123,656500 1,138,516 1,126,744 1,119,187

Table 11Sensitivity of denominational weight, Wfor 20-unit system where iterationno. = 100, M 500, G 500, 0:6W 5 W 10 W 15 W 201,124,244 1,119,187 1,123,019 1,124,234

Table 10Sensitivity of evaporation factor, for 20-unit system where iterationno. = 100, M 500, G 500, W 10 0:2 0:4 0:6 0:81,132,857 1,124,218 1,119,187 1,127,149

412 A.Y. Saber, A.M. Alshareef / Electrical Power and Energy Systems 30 (2008) 403414


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It is expected to run a generator at its minimum (best)CPU. For the minimum CPU, first derivative of (A1) willbe zero.

dCPUidPi

0; i:e: Pi ffiffiffiffiai

ci

rMW:

Here, Pi must be in the valid range. IfPi exceeds the validoutput range, the nearest limit (Pmaxi or Pmini ) is considered.

Then, the best CPU of unit i is calculated from (A1) asfollow:

min CPUi ffiffiffiffiffiffiffiaicip bi ffiffiffiffiffiffiffiaicip 2 ffiffiffiffiffiffiffiaicip bi$=MW:Finally, units are sorted according to the minimum (best)CPU to assign priority for the heuristic function (15).

Appendix II. The effect of h function in memory-boundedACO

A heuristic function, (14) consists of two cost func-tions g is the exact deterministic cost function for theselected hourly schedule and h is the estimated cost effectat the next hour(s) for the selected hourly schedule [48]. Forthe UC problem, we can exactly calculate g function, as itdepends on the present state. On the other hand, h is atmost nearly predictable, as it depends on future state(s).Hence, the proper prediction or construction of the hfunction in (14) is the most important task. Better h func-tion increases the local searching ability in the current iter-ation. It is essential to keep in mind the following factors todesign the h function for the UC problem.

(i) Ifh is always lower than the actual cost of the nexthour, then the system will converge. However, thelower h() needs more possible trials and makes itslower convergence.

(ii) Ifh is exactly equal to the actual minimum cost ofthe next hour in minimization problem, then the sys-tem will follow the best path and makes it very fast.

(iii) Ifh is greater than the exact cost of the next hour inminimization problem, then the convergence and per-formance of the system are not guaranteed.

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