economic dispatch with prohibited operating zones using fast computation evolutionary programming...

Electric Power Systems Research 70 (2004) 245–252

Economic dispatch with prohibited operating zones using fastcomputation evolutionary programming algorithm

P. Somasundaram∗, K. Kuppusamy, R.P. Kumudini Devi

Power System Engineering Division, School of Electrical and Electronics Engineering,Anna University, Chennai 600025, Tamilnadu, India

Received 28 July 2003; received in revised form 6 November 2003; accepted 9 December 2003

Abstract

This paper presents an efficient and simple approach for solving the economic dispatch (ED) problem with units having prohibited operatingzones. The operating region of the units having prohibited zones is broken into isolated feasible sub-regions which results in multiple decisionspaces for the economic dispatch problem. The optimal solution will lie in one of the feasible decision spaces and can be found using theconventionalλ–δ iterative method in each of the feasible decision spaces. But, this elaborate search procedure is time consuming and notacceptable for on-line application. In this paper, a simple and novel approach is proposed. In this approach, the optimal solution and thecorresponding optimum system lambda are determined using an efficient fast computation evolutionary programming algorithm (FCEPA)without considering the prohibited operating zones. Then, a small set of advantageous decision spaces is formed by combining the feasiblesub-regions of the fuel cost curve intervening the prohibited zones in the neighbourhood of the optimal system lambda. A penalty cost foreach advantageous decision space is judiciously computed using participation factor. The most advantageous decision space is found out bycomparing the penalty cost of the decision spaces. The optimal solution in the most advantageous decision space is obtained using the FCEPA.The proposed algorithm is tested on a number of sample systems with units possessing prohibited zones. The study results reveal that theproposed approach is computationally efficient and would be a competent method for solving economic dispatch problem with units havingprohibited operating zones.© 2004 Elsevier B.V. All rights reserved.

Keywords: Prohibited operating zones; Decision space; Economic dispatch; Evolutionary programming

1. Introduction

Many methods have been proposed in the literature[1–5]to solve ED problem with units having prohibited operatingzones. This paper presents a new algorithm based on evolu-tionary programming (EP) for determining the optimal load-ing of generators having prohibited operating zones. Thefuel cost curve of units with prohibited zones are broken intoseveral isolated feasible sub-regions. These isolated feasiblesub-regions form multiple decision spaces. A decision spacemay be feasible or infeasible with respect to system demandand the optimal solution of the dispatch problem will residein one of the feasible decision spaces[1,2].

Fan and McDonald[2] presented a method that determinesa small set of advantageous decision spaces and selects the

∗ Corresponding author.E-mail address: [email protected] (P. Somasundaram).

most advantageous decision space among them. The optimalsolution in the most advantageous decision space is obtainedby performing the conventionalλ–δ iterative search. In[5],ED problem with prohibited operating zones is solved usingstandard evolutionary programming algorithm (SEPA) tak-ing the generator outputs as decision variables and fuel costas fitness function. If the optimal generation schedule lie inthe prohibited zone, then they are re-dispatched to nearestboundary of the prohibited zone. Major drawbacks of thisalgorithm are very slow and inconsistent convergence, largenumber of iterations, large number of decision variables andindeterministic stopping criteria, etc. To overcome the abovedifficulties, in this paper, a new approach based on EP andparticipation factor is proposed.

In this proposed approach, the ED problem is solved us-ing fast computation evolutionary programming algorithm(FCEPA) with system lambda as decision variable and powermismatch as fitness function without considering prohibitedzones. A small set of advantageous decision spaces is formed

0378-7796/$ – see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.epsr.2003.12.013

246 P. Somasundaram et al. / Electric Power Systems Research 70 (2004) 245–252

by combining the feasible sub-regions of the fuel cost curveintervening the prohibited zones in the neighbourhood ofoptimum system lambda. A simple and novel method whichis based on the well-known participation factor[6] is usedto estimate a penalty cost for each selected feasible advan-tageous decision space and the most advantageous decisionspace is chosen by penalty cost comparison. The optimalsolution of the dispatch problem which lie in the most ad-vantageous decision space is computed using the FCEPA.This method is simple and faster and requires only two ex-ecutions of EP based ED to get the solution.

2. Problem formulation

The ED problem with some units possessing prohibitedzones can be stated as:

minFT =∑j∈Ω

Fj(Pj) (1)

Subject to:

(i) Power-balance constraint (referred as power mismatch)

f =∑j∈Ω

Pj − PD − PL = 0 (2)

(ii) Spinning reserve (SR) constraints∑j

Sj ≥ SR (3)

where

Sj = min(Pj,max − Pj), Sj,max, ∀j ∈ (Ω − ω) (4)

Sj = 0, ∀j ∈ ω (5)

(iii) Generation limit constraint

Pj,min ≤ Pj ≤ Pj,max, ∀j ∈ (Ω − ω) (6)

The additional constraints for units with prohibited op-erating zone are

Pj,min ≤ Pj ≤ Plj,1

or

Puj,k−1 ≤ Pj ≤ Pl

j,k, k = 2,3, . . . , nj

or

Puj,nj

≤ Pj ≤ Pj,max, ∀j ∈ ω

(7)

An effective upper generation limitPeff1j,max for each

unit j ∈ (Ω − ω) is first calculated to satisfy the spin-ning reserve constraints for the units without prohibitedzones as explained in[1] andEq. (6)becomes

Pj,min ≤ Pj ≤ Peff1j,max, ∀j ∈ (Ω − ω) (8)

(iv) Ramp-rate constraint

The ramp-rate constraint restricts lower and upperlimit to the effective lower limit and upper limit givenby

Peff2j,min = maxPj,min, Pj0 − DRj, ∀j ∈ Ω (9)

Peff2j,max =

minPeff1

j,max, Pj0 + URj, ∀j ∈ (Ω − ω)

minPj,max, Pj0 + URj, ∀j ∈ ω

(10)

Now, we have

Peff2j,min ≤ Pj ≤ Peff2

j,max, ∀j ∈ Ω (11)

The constraints inEq. (7) implies that if an unit hasnjprohibited zones its operating region will be broken intonj + 1 isolated feasible sub-regions, resulting in multipledecision spaces for the ED problem. The number of totaldisjoint decision spaces is given by

N =∏j∈ω

(nj + 1) (12)

The optimal solution will reside in one of the feasible deci-sion spaces.

3. Proposed approach

The proposed approach is based on the basis that the op-timal solution with prohibited zones is most likely to lie inone of the feasible decision spaces, which are in the neigh-bourhood of the optimal solution obtained without consid-ering the prohibited zones.

An overview of the various steps of the proposed approachare outlined below.

Step I. Solve the economic dispatch (ED) problem withoutconsidering the prohibited zones using FCEPA and obtainthe optimum system lambda and generation schedule.

Step II. Assemble the disjoint advantageous decisionspaces by combining the feasible sub-regions of the fuelcost curves in the neighbourhood of the optimum systemlambda. Retain only the feasible advantageous decisionspaces.

Step III. Find out the most advantageous decision spacefrom the feasible advantageous decision spaces by compar-ing the penalty cost of the decision spaces.

Step IV. Obtain the optimal solution in the most advanta-geous decision space using FCEPA.

The steps I–IV are explained in detail in the followingsections.

P. Somasundaram et al. / Electric Power Systems Research 70 (2004) 245–252 247

Step I (solution of ED problem using FCEPA). Theco-ordination equations for the ED problem without con-sidering the prohibited operating zones are

dFjdPj

+ λ∂PL

∂Pj= λ, ∀j ∈ Ω (13)

f =∑j∈Ω

Pj − PD − PL = 0 (14)

and the inequality constraint:

Peff2j,min ≤ Pj ≤ Peff2

j,max, ∀j ∈ Ω (15)

the transmission loss is expressed as:

PL =∑i∈Ω

∑j∈Ω

PiBijPj (16)

The solution for the ED problem can be obtained by solvingthe set ofEqs. (13)–(15)which may be stated as follows.

Determine the decision variableλ, to minimise the powermismatch (Eq. (14)) to zero satisfyingEqs. (13) and (15).Alternatively, the problem may be viewed as an optimisationproblem with an objective to minimize the power mismatchto zero subject to the equality and inequality constraints inEqs. (13) and (15), respectively, withλ as decision variable.The problem stated above is solved using EP taking systemlambda as decision variable and power mismatch as fitnessfunction.

• Algorithmic steps

Step 1 (initialisation of single parentλp). A single parentis deterministically generated as:

λp = λ1 + λ2 + · · · + λng

ng(17)

where

λj = dFj(Pj)

dPj× Lj, ∀j ∈ Ω

Lj = 1

1 − dPL/dPj

Pj = Pj,max∑Pj,max

× PD

(18)

The system demand is distributed among the units in pro-portion to their capacity as stated inEq. (18)for fixing initialgeneration schedule.

The generation schedule and power mismatchfp (fitnessvalue) corresponding toλp are obtained fromEqs. (13) and(14).

Step 2 (generation of offsprings). A set ofNp effective off-springs are generated as stated follows.

The ith offspring is generated as:

λi = λp + Ni(0, σ2p)

whereNi(0, σ2p) is the normal random number generated

from the normal distribution curve with standard deviationσp.

Retainλi in the set only if:

(i) fp is positive andNi(0, σ2p) is negative;

(ii) fp is negative andNi(0, σ2p) is positive and abort the

offsprings in all other cases.

An offspring is effective and useful only if its fitness func-tion converges with respect to its parent’s fitness functionvalue or else the offspring is ineffective. In the above twocases, the fitness function, namely the power mismatch cor-responding to the offspringfi converges with respect to itsparent’s fitness function valuefp while in all other casesits fitness function diverges. The above selective process isused to generateNp offspringsλi, i = 1,2, . . . , Np from thesingle parent.

3.1. Selection of normal distribution curve

It may be recalled that in conventional gradient optimi-sation methods, the step size to move along the negativegradient direction is fixed arbitrarily to start with and re-duced progressively during subsequent iterations to achievefaster and non-oscillatory convergence. On similar groundsthe normal distribution curve may be generated by fixing ar-bitrary width (maximum range ofλ) as displayed inFig. 1and the width may be reduced successively during subse-quent offspring generations to get faster and non-oscillatoryconvergence.

3.2. Selection of σp for normal distribution curve

The maximum permissible range of initial variation in thedecision variableλp is fixed judiciously based on experience.The maximum range of the random variation inλ may betaken asλ = ±t, wheret = α(λmax − λmin), α being anarbitrary constant in the range 0–1. The normal distributioncurve is constructed such that the probability of occurrenceof λ ≥ t is zero.

Mathematically

1√2π × σp

∫ ∞

t

e− 1

2

(λ−µσp

)2

= 0 (19)

Fig. 1. Search range.


The above equation is solved to computeσp by settingthe RHS value to a small value as:

1√2π × σp

∫ ∞

t

e− 1

2

(λ−µσp

)2

= 0.00003 (20)

The solution of the above equation is obtained by using thestandard normal tables[7]. The width ofλ is progressivelydecreased during successive generation of offsprings.

Step 3. The generation schedule and power mismatch cor-responding to offspringsλi, i = 1,2, . . . , Np are computedby solvingEqs. (13) and (14).

Step 4. The best offspring having the minimum fitness func-tion value is selected from the populationNp and the con-vergence is checked. If the convergence criterion is satisfied,then the optimum is reached. Otherwise the steps 2 and 3are repeated by taking this offspring as the parent.

Step II (formation of advantageous decision spaces). Theprohibited operating zones of units leads to multiple decisionspaces for the economic dispatch problem. The operatingrange of a decision space can be defined as:∑j=Ω

Pl,Rij ,

∑j=Ω

Pu,Rij − SR

(21)

A decision space is feasible with respect to system de-mand,PD and spinning reserve,SR if Eq. (22)is satisfied.∑j=Ω

Pl,Rij ≤ PD ≤

∑j=Ω

Pu,Rij − SR (22)

The decision spaces formed by considering feasiblesub-regions in the neighbourhood of the optimal solutionobtained without considering the prohibited zones are calledthe advantageous decision spaces. Formation of advanta-geous decision spaces are explained by considering theincremental cost curves of the three-unit system as shownin Fig. 2.

The intersection of theλopt line obtained without consid-ering prohibited zones and the incremental cost curves ofthe units gives the optimum schedule. The optimum sched-ule of units 1 and 2 falls in the prohibited zone while thatof unit 3 lies in feasible sub-region,R3,2.

Fig. 2. Incremental cost curves.

A search range for the systemλ is defined in the vicinity ofλopt. If the optimal schedule of a unit falls in the prohibitedzone then its generation schedule can be re-dispatched to itsadjacent feasible sub-region such that the incremental costremains in the search range. This can be mathematicallydescribed as:

λs,min ≤ λs ≤ λs,max (23)

with

λs,min = minj∈ω′

Lj

dFjdPj

∣∣∣∣Pj=P l

j,k

(24)

and

λs,max = maxj∈ω′

Lj

dFjdPj

∣∣∣∣Pj=Pu

j,k

(25)

wherek is the active prohibited zone ofjth unit.For units 1 and 2 the generation can be re-dispatched to

the feasible sub-regionR1,1 or R1,2 andR2,1 or R2,2, respec-tively. While for unit 3 its generation may be re-dispatchedto any one of the two neighbouring feasible sub-regionsR3,1 andR3,3 or may remain in the feasible sub-regionR3,2itself.

Thus, the number of advantageous decision spacesM willhave the range

2ω′ ≤ M ≤ 2ω

′ · 3(ω−ω′) (26)

Among theN number of decision spaces, only theM ad-vantageous decision spaces are selected. There are 12 advan-tageous decision spaces for the three-unit system as given inTable 1. The units without prohibited zones will have onlyone region between its minimum and maximum limits andwill be included in all advantageous decision spaces. Theadvantageous decision spaces which are satisfyingEq. (22)are the feasible advantageous decision spaces.

Step III (selection of most advantageous decision space).Cost penalties of advantageous decision spaces: a penaltycost is computed for each feasible advantageous decisionspace using a simple procedure based on participation fac-tor and the most advantageous decision space is selectedby penalty cost comparison. The penalty cost of a feasible


Table 1Advantageous decision spaces

Sub-region Advantageous decision spaces

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12

Unit 1 R1,1 R1,1 R1,1 R1,1 R1,1 R1,1 R1,2 R1,2 R1,2 R1,2 R1,2 R1,2



advantageous decision space is the increment of generationcost from that of the optimal dispatch obtained without con-sidering the prohibited zones.

If a unit say, thejth unit has its optimal generation dispatchin a prohibited zonek, having its upper boundary generationPuj,k and lower boundary generationP l

j,k, then the feasiblesub-region ofjth unit in an advantageous decision spacewill be either the feasible sub-region abovePu

j,k or below

P lj,k. Since thejth unit optimum schedule is in prohibited

zonek, it is to be re-dispatched either toPuj,k or P l

j,k There-dispatch involves a change in generation. The net changein generation in a decision spaceRi due to re-dispatch iscomputed as:

PRinet =∑j∈ω′

(Pj,opt − ρRij P lj,k − (1 − ρRij )Pu

j,k) (27)

where

ρRij =

1 if generation is redispatched toP lj,k

0 if generation is redispatched toPuj,k

The net change in the generation in a decision space maybe positive or negative. The net change in generation,PRinetis re-dispatched to all the units according toEq. (28)subjectto the operating restrictions due to prohibited zones as wellas upper and lower limits of the units.

Pj = PFj × PRinet, ∀j ∈ Ω (28)

This re-dispatched feasible generation schedule in the de-cision spaceRi differs from the optimum schedule obtainedwithout considering the prohibited zones. The difference be-tween the fuel cost of optimal solution and the re-dispatchedfeasible solution is defined as penalty cost.

The penalty cost for theRith feasible advantageous de-cision space is computed as:

PCRi =∑j∈Ω

F(Pj,opt + Pj) −∑j∈Ω

F(Pj,opt) (29)

The most advantageous decision space is the one thatgives the minimum penalty cost.

Step IV. Optimal solution in the most advantageous deci-sion space is obtained using FCEPA as explained in step I.

4. Examples and discussion

The effectiveness of the proposed approach is demon-strated by solving a few example dispatch problems.

4.1. Example 1

A system with five on-line units represented by the fol-lowing input–output cost function is considered.

Fj(Pj) = 1 × 10−6P3j + 0.001P2

j + 8Pj + 350 $/h,

j = 1–5

and the operating limits:

120 MW ≤ Pj ≤ 450 MW, j = 1–5

Units 1, 2 and 3 have prohibited zones and units 4 and5 do not. The prohibited zones are given inTable 2. Thetransmission loss penalty factor is assumed to be unity forall units. The system demand is 1175 MW.

As per Eq. (12), there are 27 decision spaces for thedispatch problem. The optimal generation schedule with-out considering prohibited zones is determined by using theFCEPA and is found to be

Pj,opt = 235 MW, j = 1–5

and the corresponding optimum system lambda is

λopt = 8.63567 $/MWh

Apparently, unit 1 is in the first feasible sub-region,while units 2 and 3 are in prohibited zones. Unit 1 hasone adjacent feasible sub-region [275, 315 MW]. Basedon Eqs. (23)–(25), the search range of systemλ isfound to be [8.52, 8.7587 $/MWh]. The incremental cost,8.7769 $/MWh corresponding to the upper bound of the firstprohibited zone of unit 1 does not fall in the search range.Thus, a set of four advantageous decision spaces,R1,R2,R3 andR4 are defined, as shown inTable 3. All the four

Table 2Prohibited zones of the units—five-unit system

Unit Zone 1 (MW) Zone 2 (MW)

1 [240, 275] [315, 375]2 [210, 270] [300, 390]3 [200, 250] [290, 370]


Table 3Advantageous decision spaces

Sub-region Advantageous decision spaces

R1 R2 R3 R4

Unit 1 [120, 240] [120, 240] [120, 240] [120, 240]Unit 2 [120, 210] [120, 210] [270, 300] [270, 300]Unit 3 [120, 200] [250, 290] [120, 200] [250, 290]Unit 4 [120, 450] [120, 450] [120, 450] [120, 450]Unit 5 [120, 450] [120, 450] [120, 450] [120, 450]Operating range of the decision space [600, 1550] [730, 1640] [750, 1640] [880, 1730]

advantageous decision spaces are feasible with respect tosystem demand.

The penalty cost forR1 is calculated as follows.The net change in generation due to prohibited operating

zones of units 2 and 3 is obtained as follows.

PR1net = (P2,opt − P1

2,1) + (P3,opt − P13,1) = (235.0 − 210)

+ (235− 200) = 60 MW

This net change in generation is re-dispatched to all theunits based on participation factors as:

P1 = 5 MW, P2 = 0 MW, P3 = 0 MW,

P4 = 27.5 MW, P5 = 27.5 MW

According toEq. (29)penalty cost

PCR1 = 11496.77905− 11491.01953= 5.75952($/h)

The optimal schedule and penalty cost for each feasibleadvantageous decision space is given inTable 4. SpaceR2

Table 4Optimal schedule and penalty cost

Unit generation (MW) Optimal schedulewithout prohibited zones

Optimal schedule for decision spaces

R1 R2 R3 R4

P1 235.0 240.0 238.33 235.0 218.33P2 235.0 210.0 210.00 270.0 270.0P3 235.0 200.0 250.00 200.0 250.0P4 235.0 262.5 238.33 235.0 218.33P5 235.0 262.5 238.33 235.0 218.33λopt ($/MWh) 8.63567 8.73172 8.64707 8.63567 8.57968Fuel cost ($/h) 11491.02 11496.77 11492.50 11495.19 11494.95Penalty cost ($/h) – 5.75952 1.49463 4.17749 3.92627

Table 5Comparison of generation schedule for five-unit system

Unit generation (MW) λ–δ iterative method[1] λ–δ iterative method[2] SEPA with unit generationas variable[5]

Proposed method

P1 238.33 238.33 240.0 238.33P2 210.0 210.0 210.0 210.00P3 250.0 250.0 250.0 250.00P4 238.33 238.33 223.07 238.33P5 238.33 238.33 251.93 238.33Fuel cost ($/h) 11492.51 11492.51 11493.23 11492.5Computation time (ms) 12.1 4.6 2750 5.4

is the most advantageous decision space as its penalty costis minimum. The optimal solution in this most advantageousdecision space is obtained using the FCEPA.Table 5presentsthe results obtained using theλ–δ iterative method[1], λ–δiterative method[2], SEPA with unit generations as decisionvariables[5] and the proposed method.

FromTable 5, it can be seen that the optimal schedule ob-tained with the proposed method is exactly same as that ofresults obtained with[1,2]. It can be observed that the opti-mum schedule obtained with the proposed method is differ-ent from that of results obtained with[5]. But the optimalfuel cost is almost the same. The computation time (IBMPC, 1 GHz) for the method[1,2], SEPA with unit generationas variable[5] and the proposed method are given in the lastrow of Table 5. With the proposed method there is 55% re-duction in computation time when compared to method[1]and 99% reduction in computation time when compared tomethod[5]. The computation time for the proposed methodis closer to the time obtained with[2]. This shows the com-putational efficiency of the proposed method.


Table 6Prohibited zones of the units—15-unit system

Unit Zone 1 (MW) Zone 2 (MW) Zone 3 (MW)

2 [185, 225] [305, 335] [420, 450]5 [180, 200] [260, 335] [390, 420]6 [230, 255] [365, 395] [430, 455]

12 [30, 55] [65, 75] –

Table 7Optimal schedule and penalty cost

Unitgeneration(MW)

Optimal schedulewithoutprohibited zones

Optimal schedulein the advantageousdecision spaces

R1 R2

P1 455.00 455.0 450.0P2 455.00 455.0 450.0P3 130.00 130.0 130.0P4 130.00 130.0 130.0P5 317.83 260.0 335.0P6 460.00 460.0 455.0P7 465.00 465.0 465.0P8 60.00 60.0 60.0P9 25.00 25.0 25.0P10 20.00 70.0 20.0P11 20.00 20.0 20.0P12 57.16 65.0 55.0P13 25.00 25.0 25.0P14 15.00 15.0 15.0P15 15.00 15.0 15.0λopt ($/MWh) 10.5303 10.8884 10.3391Fuel cost ($/h) 32542.41 32558.35 32544.97Penalty cost ($/h) – 15.9350 2.5670

4.2. Example 2

A practical example used by Lee and Breipohl in[1] isconsidered. The system has 15 on-line units that supply a

Table 9Comparison of generation schedule for 15-unit system

Unit generation (MW) λ–δ iterative method[1] λ–δ iterative method[2] SEPA with unitgeneration as variable[5]

Proposed method

P1 450.0 450.0 446.98 450.0P2 450.0 450.0 451.50 450.0P3 130.0 130.0 130.00 130.0P4 130.0 130.0 130.00 130.0P5 335.0 335.0 335.02 335.0P6 455.0 455.0 456.11 455.0P7 465.0 465.0 464.91 465.0P8 60.0 60.0 60.00 60.0P9 25.0 25.0 25.00 25.0P10 20.0 20.0 20.00 20.0P11 20.0 20.0 20.01 20.0P12 55.0 55.0 55.46 55.0P13 25.0 25.0 25.00 25.0P14 15.0 15.0 15.01 15.0P15 15.0 15.0 15.00 15.0Fuel cost ($/h) 32544.99 32544.99 32545.20 32544.97Computation time (ms) 34.7 11.5 3000 9.4

Table 8Adjacent feasible sub-regions

Unit Lower sub-region Upper sub-region

MW λi,max ($/MWh) MW λi,min ($/MWh)

2 [355, 420] 10.3737 – –6 [395, 430] 10.3589 – –

12 [20, 30] 10.2308 [75, 80] 10.7269

system demand of 2650 MW with a system spinning reserverequirement of 200 MW. No regulating margin is required.The prohibited zones are given inTable 6. Among the on-lineunits, four of them (units 2, 5, 6 and 12) have prohibited op-erating zones that form 192 decision spaces for the dispatchproblem.

The optimal solution without considering the prohib-ited zones is given inTable 7. It can be seen that unit 5alone fall in prohibited zone [260, 335 MW]. Accordingto Eqs. (23)–(25), the search range of systemλ is foundto be [10.5066, 10.5374 $/MWh]. The adjacent feasiblesub-regions of the other three units are given inTable 8.

Table 8shows the incremental cost of the adjacent feasi-ble sub-regions do not fall in the search range ofλ. Thus,there are only two advantageous decision spaces. The fea-sible sub-region [200, 260 MW], below the active prohib-ited zone of unit 5 is in decision spaceR1 and the feasiblesub-region [335, 390 MW], above the prohibited zone is inR2. Both spacesR1 andR2 are feasible with respect tothe system demand and the system spinning reserve require-ment. The penalty cost and optimal solution for each of thefeasible advantageous decision spaces are given inTable 7from which it can be noted that spaceR2 is the most advan-tageous decision space. The optimal solution in this mostadvantageous decision space is obtained using the FCEPA.Table 9presents the results obtained using the method[1,2],


SEPA with unit generations as decision variables[5] and theproposed method.

FromTable 9, it can be seen that the optimal schedule ob-tained with the proposed method is exactly same as that ofresults obtained with[1,2]. The computation time (IBM PC,1 GHz) for the method[1], method[2], SEPA with unit gen-eration as variable[5] and the proposed method are given inthe last row ofTable 9. With the proposed method there is72, 22 and 99% reduction in computation time when com-pared to methods[1,2,5], respectively.

The analysis shows that the proposed method is reliableand uses a simple procedure to select the most advanta-geous decision space. It requires only two executions of theFCEPA, one for determining the optimal solution withoutconsidering the prohibited zones and the other for determin-ing the optimal solution in the most advantageous decisionspace. The comparison of computation time shows that theproposed method is competitive to method[2] and betterthan the methods[1,5].

5. Conclusion

This paper presents an efficient and simple approachfor solving the ED problem with units having prohibitedoperating zones. Initially, the optimal solution without con-sidering the prohibited zones is obtained by using FCEPA.A small set of advantageous decision spaces is formedby combining the feasible sub-regions of the fuel costcurve intervening the prohibited zones in the neighbour-hood of optimum system lambda. A simple and novelmethod which is based on the well-known participationfactor is used to estimate a penalty cost for each selectedfeasible advantageous decision space and the most advan-tageous decision space is chosen by penalty cost com-parison. The optimal solution in the most advantageousdecision space is obtained by using FCEPA. The effec-tiveness of the proposed approach has been tested on anumber of sample systems. The proposed approach is rela-tively simple, reliable and efficient and suitable for on-lineapplications.

Appendix A. List of symbols

Bij loss coefficientDRj ramp-rate limit ofjth unit as generation

decreasesFj(Pj) input/output cost function ofjth unitk index of prohibited zones of a unitLj transmission loss penalty factor ofjth unitnj number of prohibited zones ofjth unitng number of dispatchable on-line unitsPD system demand

Pj power output ofjth generatorPj change in generation atjth unitPj0 power generation ofjth generator at previous

hourPj,opt optimum schedule ofjth unit without

considering prohibited zonePj,max jth unit upper generation limitPj,min jth unit lower generation limitP lj,k lower bound of thekth prohibited zone of

jth unitP

l,Rij lower limit of the feasible sub-region ofjth unit

with respect to decision spaceRiPuj,k upper bound of thekth prohibited zone of

jth unitP

u,Rij upper limit of the feasible sub-region of

jth unit with respect to decision spaceRiPL transmission lossPCRi penalty cost for decision spaceRi.PFj participation factor ofjth unitSj jth unit SR contributionSj,max jth unit maximum SR contributionSR total SR requirementURj ramp-rate limit ofjth unit as generation increases

Greek lettersλ system lambdaλmax maximum value of system lambdaλmin minimum value of system lambdaµ meanΩ set of all dispatchable on-line unitsω set of units which have prohibited

operating zone(s)ω′ set of units in active prohibited zones

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