A Fuzzy Optimization-Based

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Electric Power Systems Research 72 (2004) 245252

A fuzzy optimization-based approach to largescale thermal unit commitment

M.M. El-Saadawia,, M.A. Tantawia, E. Tawfikb

a Elect. Power and Machines Department, Faculty of Engineering, Mansoura University, Mansoura 35516, Egyptb Engineering Laboratories, Mech. and Elect. Department, Ministry of Irrigation, Egypt

Received 11 August 2003; received in revised form 10 November 2003; accepted 24 April 2004

Available online 18 August 2004

Abstract

This paper presents a new fuzzy optimization based approach to solve the thermal unit commitment (UC) problem. In this approach load

demand, reserve requirements, and production cost are expressed by fuzzy set notations, while unit generation limits, ramp rate limits, and

minimum up/down limits arehandled as crisp constraints. A fuzzy optimization based algorithm is then, developed to find the optimal solution

by using fuzzy operations and if-then rules. Some heuristics such as dividing hourly load and generating units into levels are used to speed

the solution. The approach has been applied to a 38 units thermal power system. The results are compared with that obtained by the dynamic

programming (DP), the Lagrangianerelaxation (LR), constraint logic programming (CLP), and simulated annealing (SA) methods. The

achieved results prove the effectiveness, and validity of the proposed approach to solve the large-scale UC problem. The effects of unit ramp

rate limits and minimum up/down times are also, investigated.

2004 Published by Elsevier B.V.

Keywords: Optimization; Thermal unit; Dynamic programming; Fuzzy approach; Unit commitment

1. Introduction

Unit commitment (UC) is aimed at scheduling the gener-

ating units to serve the load demand at minimum operating

cost while meeting all unit and system constraints. The UC

problem often comprises thousands of 01 decision values as

well as continuous variables, and wide spectrum equality and

inequality constraints. Because of the large economic bene-

fits that could result from the improving unit scheduling, a

considerable attention has been devoted to develop problem

solution methods. Various mathematical programming and

heuristic based approaches such as dynamic programming

[1], neural networks [2], simulated annealing [3], evolution-

ary programming [4], constraint logic programming [5], ge-

netic algorithms [67], and Lagrangiane relaxation [810]

approaches have been devoted to solve the UC problem. It is

noticed that each of the previous methods has involved one

Corresponding author.

or more difficulties such as:

Reliance on heuristic, hence sub-optimal solutions.

High computational time for medium and large-scale sys-

tems.

Moreover, most of the previous methods dealt with the

problem as a crisp treatment (requiring to be satisfied exactly

or crisply all the time) although some of its parts are impre-

cise due to predicted demand variations (roughly values), this

may lead to uneconomic scheduling or non-feasible solutions[11]. Many researches aimed to apply the fuzzy systems as

effective alternatives for solving the UC problem [1115].

The use of fuzzy systems avoids the non-feasible solutions

by representing a membership function for each uncertainty

(objective or constraint). In Ref. [11], the problem was repre-

sented by mathematicalfuzzy optimization model, and then it

was converted to a crisp optimization one and solved by using

LR approach. The ramping rate and minimum up/down con-

straints have not been involved. The method may suffer the

0378-7796/$ see front matter 2004 Published by Elsevier B.V.

doi:10.1016/j.epsr.2004.04.009


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246 M.M. El-Saadawi et al. / Electric Power Systems Research 72 (2004) 245252

non-feasible solutions and complex computations due to the

dual nature of LR approach. In Refs. [12,14], fuzzy-neural

network approaches were used to solve the problem. The in-

puts to the fuzzy-neural network (i.e. load demands) were

considered as fuzzy variables. Then, a fuzzy neural mak-

ing decision has been developed to learn the neural network.

The fuzzy-neural networks have not involve any optimiza-tion technique, hence they suffer sub-optimal solutions. The

authors have developed a heuristic approach to solve the UC

problem [15], but that approach has ignored some crisp con-

straints such as unit ramp rate limits and minimum up/down

time limits. In this paper, the UC problem is expressed by

mathematical fuzzy model to include the fuzzy variables, and

a fuzzy optimization based approach is proposed to solve the

problem. The solution approach divides the generating units

into groupsto minimize thesize of calculationsand thesearch

space. A crisp economic dispatch is incorporated with the ap-

proach to handle crisp constraints. The effects of unit ramp

rate limits and minimum up/down time limits on the fuzzy

optimization based approach have been investigated.

2. A background about the used fuzzy concepts

Fuzzy set theory provides a natural platform to model

fuzzy relationships such as essentially or roughly vari-

ables, and adds the dimension of fuzziness or uncertainty

to the conventional set theory. This section introduces defini-

tions for some fuzzy concepts used in the proposed approach.

A fuzzy setA(x) is defined by a grade of membership func-

tion, A(X), where A(X) [0,1]. For two fuzzy sets A and B:

AB(x) = Max(A(x), B(x)) (1)

AB(x) = Min(A(x), B(x)) (2)

The union () and intersection () operations represent the

OR, and AND operators, respectively.

Define a universe ofn alternatives,A = (al, a2, . . ., an), and

a set ofrobjectives and constraints, B = (B1, B2, . . ., Br),

let Bi (a) is the membership degree of an alternative a in

an objective or constraint Bi, and O is a decision function

to satisfy all objectives and constraints simultaneously, i.e.

O = B1 B2 . . . Br (3)

With assumption that wi(x) [0,1], is the weight of the

objective or constraint Bi, then the degree of membership

function is expressed by:

Bwi = Max(wi, Bi ) (4)

where wi is the complement ofwi.

Then, the overall membership degree of the alternative a

for performing the objectives and constraints simultaneously

is:

o(a) = Min(w1(a), w2(a), . . . , wr(a)) (5)

The optimal decision a can be obtained by verifying the

following relation:

o(a

) = Max(o(a)) (6)Notes:

if a constraint Bi is neglected, i.e. wi = 0, then wi = 1,

and Bwi = 1,

if the weight of a constraint Bi is assumed to be equal to

unity thus, Bwi = Bi .

3. Problem formulation

To clearly present the problem formulation, the crisp

model will be introduced, and followed by the fuzzy version.

3.1. Crisp problem formulation

The objective function and associated constraints of the

thermal UC problem are given as follows:

(A) Objective function

Min C = Min

Tt=1

Ni=1

[Ut(i)OCi[Pt(i)]

+SCi (1 Ut1(i))] (7)

(B) System constraints

Load balance :

Ni=1

Ut(i) Pi(i) = Pd(t) (8)

Spinning reserve :

Ni=1

Ut(i)Rct(i) R(t) (9)

(C) Unit constraints: The thermal unit constraints include

generation output limits, ramp rate limits, and minimum

up/down time limits. These constraints are handled as

crisp constraints either during executing the economic

dispatch or at refinement stage as will be discussed later.

3.2. Fuzzy problem formulation

3.2.1. Definition of the fuzzy variables and quantities in

UC

The load demand depends on weather variables, social be-

havior of customers, etc. Theforecasted demandis imprecise,

thus it can be described as a fuzzy quantity. Any variable as-

sociated with the system load will be considered as a fuzzy

variable. Thus, unit generation production cost and spinning

reserve are fuzzy quantities.

To obtain an optimal commitment scheduling under the

fuzzy environments: production cost, load demand equal-


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M.M. El-Saadawi et al. / Electric Power Systems Research 72 (2004) 245252 247

ity constraint, and spinning reserve inequality relation con-

straint are all expressed in fuzzy set notations. On other hand,

the crisp quantities include: limits on unit outputs, minimum

up/down times, and ramp rate constraints.

3.2.2. Definition of thresholds and membership

functions for fuzzy variablesThe values of thresholds are obtained by last experiences

for the operator. These values can be defined as follows [11].

3.2.2.1. Load balance membership function. The predicted

system load deviation is usually from 2 to 5% [10]. The

load balance constraint can be represented by the following

fuzzy equality relation:

Ni=1

Ut(i) Pi(i) = Pd(t) (10)

The thresholds for system demand can be defined as follows:Nominal demand: having the maximum degree of grade

in membership function i.e. degree one. The nominal value

for demand equals the mean value of the predicted demand.

Range of predictedload variation (d(t)): themaximum

range of variation of the hourly predicted demand. It has the

least value of membership degree, i.e. degree zero. In this

study, d(t) is taken equal to5%. The membershipfunction

of the above fuzzy equality (=) can be described as:

d() =

1 if = Pd

1(Pd )

difPd d < < Pd

1 ( Pd)

difPd < < Pd + d

0 else where

(11)

3.2.2.2. Spinning reserve membership function. The spin-

ning reserve constraint can be described as fuzzy inequality

relation as follows:

Ni=1

Ut(i) Rct(i)R2(t) (12)

There are two thresholds for the reserve variable:

Nominal reserve: this value verifies the greatest degree ofsatisfaction, i.e. degree one.

Minimum acceptable reserve (least value): this value ver-

ifies a completely unacceptable degree of satisfaction, i.e.

degree zero.

The membership function of the fuzzy inequality () is

described by:

R() =

1 if R(t)

1 (R(t) )

R(t)ifR(t) R(t) < < R(t)

0 else where(13)

3.2.2.3. Cost membership function. The cost function

should be essentially smaller than or equal to some aspi-

ration level ideal cost, C0. Thus the cost function can be

expressed in fuzzy inequality relation as follows:

C(t)C0(t) (14)

The thresholds of the cost are:The ideal cost level (C0): it has the maximum grade of

membership i.e. degree one. Selecting this level may be

subjective and dependent on specific practice; one good can-

didate for the ideal cost (C0) is the cost of the crisp problem

with nominal system demand and reserve requirements.

The highest acceptable cost level (C0 + C): it has the

least degree of membership, i.e. degree zero. It can be

determined by choosing C as a certain percentage of C0based on the operators experience. In this study, Cis taken

equal to 20% ofC0 based on authors experience. The fuzzy

inequality () can be described by the cost membership

function as follows:

c() =

1 if0 C0(C0 + C )

CifC0 < C0 + C

0 else where

(15)

4. A proposed solution strategy

The proposed solution strategy consists of three stages.

In the first stage, both hourly load and generating units are

divided into divisions. This technique allows searching the

hourly schedule of units in only one division and thus speeds

the solution. In the second stage, a fuzzy optimization based

algorithm is proposed to find an optimal solution for the UC

problem by using fuzzy operations and if-then rules. In the

third stage, a refinement of the obtained solution is achieved

by adjusting unit minimum up/down time constraint.

4.1. Stage 1: dividing the load and generating units

In this stage, both the hourly load and generating units are

divided into base, medium, and peak divisions. This tech-

nique is executed as follows:

1. Dividing the hourly load into:

Base load(Base): equal to the minimum value of the given

hourly load.

Peak load: any load which is larger than 90% of the max-

imum hourly load is considered as peak load.

Medium load: is the load between base and peak values.

2. All generating units are arranged in a priority list on the

basis of their full load average costs. The lowest produc-

tions cost units are put at the top of the list, whereas the

highest ones are on the bottom.

3. From the priority list a number of generating units are

chosen to supply the required load plus the reserve values.

The units are divided to:


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Base units: these units usually have the highest starting

costs, longest minimum up/down times and lowest pro-

duction costs. These units are always committed during

the studied period.

Peak units: these units have the highest production costs

and the shortest minimum up/down times. A list of all

possible states for these units is made (a state means: aunit combination or unit schedule).

Medium units: these units have characteristics in between

the last two types. A list of all possible states for these

units is made.

The first group of units has to supply the base load, while

the second and third groups have to supply the medium and

peak loads respectively.

4.2. Stage 2: a proposed fuzzy algorithm (FA)

This algorithm classifies the load at each hour, i.e. base,

medium or peak load, then calls the states of the correspond-

ing units, candidates the feasible states, and searches among

them about the best state which has the maximum overall

membership degree to be thesolutionat thepresent hour. Dur-

ingsearching, the operating cost for each state is computed by

executing an economic dispatch (ED), thus the fuzzy mem-

bership function degrees for theproduction cost, load demand

balance, and spinning reserve constraints are calculated. The

fuzzy overall membership function degree for each state is

the minimum degree among the last three degrees. The unit

output and ramping rate crisp constraints are handled during

the ED. In the following subsections the used economic dis-

patch, the crisp constraints handling, and the fuzzy solution

procedure are addressed.

4.2.1. Economic dispatch

The ED is carried out as a crisp process to compute the

output of units and thus the crisp operating cost, the process

is executed in this paper before and after UC. Before UC,

the ED is achieved to compute the operating cost for each

candidate state at each time step; this cost is taken later as

fuzzy quantity during solving the UC. For base loads, the ED

is only executed over base units. For medium loads, the pro-

cess is executed over base units and the committed medium

units in each of medium states. While, for peak loads, it is

achieved over base, medium, and the committed peak units

in each of peak states. The ED is executed by committing

the units at their minimum outputs, if the generation is lesser

than the load, then the most efficient unit (according to the

priority list) has to increase its generation until either its max-

imum output or its ramp up rate is reached, then the next unit

in the priority list is considered. The process continues until

the load demand is supplied. Once the outputs of units are

computed, the operating cost of the state can be calculated

by using the unit cost functions. After achieving UC, an ED

is resorted again to compute the exact economic cost. In case

using quadratic fuel functions, the ED is achieved by Lambda

() criteria technique.

4.2.2. Crisp unit constraints handling

As mentioned before, the unit output limits and ramping

rate constraints are handled during executing the ED, using

the if-then relations as follows:

ifPmax(i) Pt1(i) RUR(i) then Pmax(i,t) = Pmax(i)

otherwise, Pmax(i, t) = Pt1(i)+ RUR(i).

Similarly,

ifPt1(i) Pmin(i) RDR(i) then Pmin(i,t) = Pmin(i)

otherwise, Pmin(i,t) = Pt1(i) RDR(i).

Thus, while realizing the ED, the unit output limits are taken

as:

Pmin(i, t) Pt(i) Pmax(i, t)

4.2.3. The fuzzy solution procedureThe procedure consists of the following steps:

Step 1:

h = 1. All base units must be run, i.e. status ON.

Step 2:

Checkif load (h) Base then, the state is only the base

units are ON.

Calculate the economic dispatch (ED) and go to step 6.

Else ifload (h) > Med go to step 7.

Else, check ifNb

i=1

Pmax(i) [load(h)+ reserve], then the

state is only the base units are ON,

Calculate the economic dispatch and go to step 6.Else: call the states for medium units and go to step 3.

Step 3:

Thebase units are ON,the medium units are scheduled and

the peak units are OFF. For each state, only the candidated

state (k) is the state that verifies the following condition:Nb

i=1

Pmax(i)+

Nmj=1

Pmax(j) Uh(j) 0.95 load(h)

The value (0.95) is assumed on basis of the deviation in

forecasting load (5%) to make a fuzzy relation.

Step 4:

For each candidate (k) compute the following:

load demand PG, thus calculate d(k) using Eq. (11);

reserve amount, Res(k), then calculate R(k) using Eq.

(2);

production cost by an ED, then calculate c(k) using

Eq. (15) and

overall membership degree o(k), using Eq. (5):

o(k) = min(R(k), d(k), . . . , c(k))

Step 5:

Choose the best state, which has the maximum overall

membership degree st(k) to be the solution at hour (h)

by using Eq. (6):

st(h) = max(o(1), o(2), . . . , o(km))


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Table 1

38-units system data

Unit no. Pmax (MW) Pmin (MW) a ($) b ($/MW) c ($/MW2) Start up cost ($) MUT (h) MDT (h) RUR (MW/h) RDR (MW/h)

1 550 220 64782 0796.9 0.3133 805000 18 8 92 138

2 550 220 64782 796.9 0.3133 805000 18 8 92 138

3 500 200 64670 795.5 0.3127 805000 18 8 84 120

4 500 200 64670 795.5 0.3127 805000 18 8 84 120

5 500 200 64670 795.5 0.3127 805000 18 8 84 1206 500 200 64670 795.5 0.3127 805000 18 8 84 120

7 500 200 64670 795.5 0.3127 805000 18 8 84 120

8 500 200 64670 795.5 0.3127 805000 18 8 84 120

9 500 200 172832 915.7 0.7075 402500 7 7 128 256

10 500 114 172832 915.7 0.7075 402500 7 7 128 256

11 500 114 176003 884.2 0.7515 402500 7 7 128 256

12 500 114 173028 884.2 0.7083 402500 7 7 128 256

13 500 110 91340 1250.1 0.4211 575000 9 8 110 170

14 365 90 63440 1298.6 0.5145 575000 12 8 92 125

15 365 82 65468 1298.6 0.5691 575000 12 8 92 125

16 325 120 72282 1290.8 0.5691 575000 10 8 82 125

17 315 65 190928 238.1 2.5881 23000 1 1 320 70

18 315 65 285372 1149.5 3.8734 023000 1 1 320 70

19 315 65 271376 1269.1 3.6842 023000 1 1 320 70

20 272 120 39197 696.1 0.4921 575000 9 8 55 91

21 272 120 45576 660.2 0.5728 575000 9 8 55 91

22 260 110 28770 803.2 0.3572 460000 11 8 53 132

23 190 80 36902 818.2 0.9415 092000 14 7 48 98

24 150 10 105510 33.5 52.123 023000 1 1 460 20

25 125 60 22233 805.4 1.1421 115000 8 8 42 60

26 110 55 30953 707.1 2.0275 287500 14 7 28 56

27 75 35 17044 833.6 3.0744 253000 14 7 20 38

28 70 20 81079 2188.7 16.765 005750 1 1 70 30

29 70 20 124767 1024.4 26.355 005750 1 1 70 30

30 70 20 121915 837.1 30.575 005750 1 1 70 30

31 70 20 120780 1305.2 25.098 005750 1 1 75 30

32 60 20 104441 716.6 33.722 007670 1 1 70 30

33 60 25 83224 1633.9 23.915 007670 1 1 70 30

34 60 18 111281 969.5 32.562 007670 1 1 70 20

35 60 8 64142 2625.8 18.362 007670 1 1 70 20

36 60 25 103519 1633.9 23.915 007670 1 1 75 30

37 38 20 13547 694.7 8.482 069000 11 8 10 20

38 38 20 13518 655.9 9.693 069000 11 8 10 20

Step 6:

Put h = h + 1

if(h > 24) go to step 8.

Else, go to step 2.

Step 7:

The base and medium units are ON,

Check ifNb

i=1Pmax(i)+Nm

j=1Pmax(j) load(h)+

reservethen, the state is only the base and medium units are

ON, thus go back to step 3.

Else, call all states for peak units, for each state candidate

only the state k, which verifies the condition:

Npj=1

Pmax(j) Uh(j)+

Nbi=1

Pmax(i)+

Nmi=1

Pmax(i)

0.95 load(h)

Assume the number of candidates (km), and go to step 4.

Step 8:

Print the chosenstate at eachhour(h) to be thesub-optimal

schedule.

4.3. Stage 3: refinement of the schedule

According to the basis on which the units are selected, the

minimum up/down time constraints are always performed

for base and peak units. In more details, the base units are al-

ways committed during all study periods, while the minimum

up/down times for peak units are often equal to or less than

the time step (1 h). They may be violated for some medium

units. To insure complete verification for these constraints, a

filtering process is handled. This process is executed taking

into consideration the up/down times of the units schedule

for each load value and the units schedule for the next load

by checking the up/down times of units through the obtained

schedule. If the constraint is violated, next states are tested,

until the constraint is performed.

After this stage, a lambda-criteria ED is executed over the

resultant UC solution to compute the final total economic


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cost. The following remarks can be observed after applying

this stage:

The checked states (to perform the up/down constraint)

are chosen from the candidate states, i.e. they are feasible

states to other constrains.

The changes in the status of units are usually in a nar-

row scale due to the basis on which the units are divided(0.53%) of unit status.

Dealing with the overall membership degree, o, leads to

taking simultaneously the fuzzy membership into account.

5. Case study

A computer program was written using the FORTRAN

language so that the proposed approach can be applied to

large-scale thermal power system. The approach is applied

to the 38 thermal unit system mentioned in [5] to compare

the results with that obtained by other methods. The systemdata are shown in Table 1, while the load data are given in

Table 2.

The application is executed under the same conditions

taken by Ref. [5], i.e. the start up costs are constants, shut

down costs are neglected, and with taking 11% of hourly load

as spinning reserve requirements into account (the thresh-

olds of the reserve constraint are 11 and 7%). Also, the ramp

up/down constraints for each of the units are taken into con-

sideration. The thresholds of load are 4800 MW for Base,

and 7450MW for Med. The units 17 through 19, 24 and 28

through 36 are peak units; the units 1 through 8, 20 through

Table 2

Load data of the studied system

Time (h) Load (MW)

1 5700

2 5400

3 5150

4 4850

5 4950

6 4800

7 4850

8 5400

9 6700

10 7850

11 8000

12 8100

13 6900

14 8150

15 8250

16 8000

17 7800

18 7100

19 6800

20 7300

21 7100

22 6800

23 6550

24 6450

23, and 25 through 27 are baseunits,while the units 9 through

16, 37, and 38 are medium units. After executing the solution

scenario as discussed before, the final results were obtained

as shown in Table 3, with total crisp cost of M$ 213.9 through

about 5 s.

This case was studied by the four methods DP, LR, SA,

and CLP [5]. The total crisp costs and the execution times ob-tained by the five methods are compared as shown in Table 4.

The comparison shows that the total cost obtained by the pro-

posed method has the second degree in respect of the lowest

cost (behind the CLP method), but its execution time is the

shortest one among the five methods. The comparison con-

firms the closeness of the overall results and proves that it

is an effective tool for solving the UC problem. Also, the

computing time of the proposed method is short due to less

computational process.

The crisp methods limit the spinning reserve to its nom-

inal value, while the FA deals with this constraint as a

fuzzy variable. Thus more expensive units may be selected

in the solutions obtained by crisp methods. Therefore, thecosts may increase in case of solutions obtained by crisp

methods. In fact, the comparison is made in this paper to

prove the validity of the proposed method (Ref. [13] com-

pared between a fuzzy logic approach and a crisp dynamic

programming approach for the same task). The compari-

son shows also that the proposed method is a fast tool.

The use of the method may save in the cost, where it is

not necessary to deal with the spinning reserve, as a crisp

variable.

On other hand, when applying the proposed method to

the same system with ignoring the ramp rate constraints, the

cost of the obtained solution was M$ 207.9, compared toM$ 213.9, when considering the ramp rate as illustrated in

Table 4. The results show that ramp rate constraints restrict

the outputs of some units to certain capacities during rising

or lowering the load, and thus some expensive units have to

be added. This demonstrates that the ramp rate constraints

cannot be neglected.

6. Conclusions

A new heuristic fuzzy method has been proposed for solv-

ing the thermal UC problem. The load demand, spinning

reserve, and operating cost are represented by fuzzy mem-

bership functions.

The proposed method handles the unit generation limits,

ramp rate limits, and minimum up/down time limits as

crisp constraints.

The method depends on using the fuzzy operations and the

if-then rules for searching about the best unit schedule.

A computer program was written and applied to a large

scale power system consists of 38-units.

A Comparison between the results obtained with consid-

ering and neglecting the ramp rates shows that: neglecting


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Table 3

Final results of the studied system

Hour Load (MW) Chosen state Production cost ($) Min total cost ($)

R d c o

l 5700 l 1 0.927 0.927 16234780 16234780

2 5400 0.86 1 0.948 0.86 6162851 22397630

3 5150 0.488 1 0.787 0.488 5734151 28131780

4 4850 0.94 1 0.72 0.72 5390191 335219705 4950 0.55 1 0.93 0.55 6487791 40009760

6 4800 0.67 1 0.73 0.67 5371404 45381160

7 4850 0.22 1 0.68 0.22 5422277 50803440

8 5400 0.35 1 0.66 0.35 9085918 59889360

9 6700 1 1 1 1 9281980 69171340

10 7850 0.226 1 0.912 0.226 10684470 79855820

11 8000 1 1 1 1 10595860 90451670

12 8100 0.877 1 0.833 0.833 10849670 101301300

13 6900 1 1 0.897 0.897 8367491 109668800

14 8150 0.46 1 0.63 0.46 12355280 122024100

15 8250 0.74 1 0.94 0.74 11242420 133266500

16 8000 1 1 0.67 0.67 10651340 143917900

17 7800 0.98 1 0.83 0.83 10345590 154263500

18 7100 1 1 0.84 0.84 8770456 163033900

19 6800 1 1 0.77 0.77 8366886 171400800

20 7300 0.87 1 0.73 0.73 9590061 180990900

21 7100 0.02 1 0.90 0.02 8770457 189761300

22 6800 1 1 0.77 0.77 8366886 198128200

23 6550 0.63 1 0.70 0.63 7957839 206086000

24 6450 0.55 1 0.84 0.55 7840516 213926600

the ramp rate constraints affects the accuracy of the applied

methodology. The ramp rate constraints restrict the outputs

of some units to certain capacities duringrisingor lowering

the load, and thus some expensive units have to be added.

Taking theramp rate constraints into account, thecrisp cost

was M$ 213.9 compared with M$ 215.2, 214.5, 215.6, and

213.8 for the DP, LR, SA and CLP methods respectively.

Also, the execution time taken by the proposed method

was only 5 s, while it was 199, 29, 2589 and 17 s for the

other four methods respectively.

The achieved results prove the validity and effectiveness

of the suggested method to solve large-scale UC problems.

The advantages of this method are:

Avoiding the complex calculations used in crisp opti-

mization based methods.

Minimizing the search space to find the solution and

thus its computational time is short.

Its principles can be expanded to consider complicated

cases (larger systemsmore constraints). Avoiding the non-feasible solutions or uneconomic

schedulesthat may be produced by using the crisp meth-

ods.

Table 4

Comparison between the proposed method and other methods

Algorithm DP LR SA CLP Proposed FA

Cost (M$) considering ramp rate 215.2 214.5 215.6 213.8 213.9

Cost (M$) neglecting ramp rate 201.5 209.0 207.8 208.1 207.9

Execution time (s) 199 29 2589 17 5

All constraints are involved, on conflict with other fuzzy

methods.

Moreover, its a new tool that can be used to solve the

UC problem.

List of symbols

Base a floating variable represents the base load level

C cost function of the UC problem

C0 ideal cost ($)

d maximum range of variation from the nominal de-

mand

Med a floating variable represents the medium load level

MUT minimum up time for unit i (h)

MDT minimum down time for unit i (h)

N total no of generating units

Nb no. of base unitsNm no. of medium units

Np no. of peak units

OCi fuel cost of unit i


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Pd(t) nominal demand (predicted load) at hour t(MW)

P(i, t) generation output of unit i at time t(MW)

Pmax(i) maximum generation capacity of unit i (MW)

Pmax(i, t) maximum available capacity of unit i at hour t

(MW)

Pmin(i, t) minimum available capacity of unit i at hour t

(MW)Rc(i, t) reserve contribution of unit i at time t

RDR(i) ramp down rate limit for unit i (MW/h)

Res(k) reserve value for state k

R(t) the nominal reserve at time t(MW)

RUR(i) ramp up rate limit for unit i (MW/h)

SCi start up cost of unit i ($)

U(i, t) status value of unit i at time t(for on status, Ut(i)

= 1 and for off status, Ut(i) = 0)

Greek letters

c(k) cost membership degree for state k

d(k) load demand membership degree for state k

o(i) overall membership degree for state k

R(x) membership degree for the reverse value x

Res(k) reserve membership degree for state k

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