constrained economic dispatch by deterministic tabu search approach

EUROPEAN TRANSACTIONS ON ELECTRICAL POWEREuro. Trans. Electr. Power 2004; 14:377–391Published online 2 August 2004 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/etep.31

Constrained economic dispatch by deterministictabu search approach

W. Ongsakul1,*,y, S. Dechanupaprittha2 and I. Ngamroo2

1Energy Field of Study, School of Environment, Resources and Development,Asian Institute of Technology, Pathumthani 12120, Thailand

2Electrical Power Engineering Program, Sirindhorn International Institute of Technology,Thammasat University, Pathumthani 12121, Thailand

SUMMARY

This paper presents a deterministic tabu search (DTS) algorithm for solving ramp rate constrained economicdispatch (ED) problems for generating units with non-monotonically and monotonically increasing incrementalcost (IC) functions. The proposed DTS uses proper tabu list, aspiration level, and reference unit rotation strategyto yield a high-quality solution. The developed DTS is tested on the systems with the number of generatingsystems in the range of 10 to 80 units over the entire dispatch periods. The obtained solutions are near the optimalsolutions of the unconstrained DTS (UDTS) and more economical than those obtained from the micro-geneticalgorithm based on migration and merit order loading solutions (MGAM-MOL), the genetic algorithm based onsimulated annealing solutions (GA-SA), simulated annealing (SA), genetic algorithm (GA), and merit orderloading (MOL) approaches, thereby leading to substantial generator fuel cost savings. Furthermore, DTS can beeasily parallelized to compromise the solution quality and computational speed-up for the best performance.Copyright # 2004 John Wiley & Sons, Ltd.

key words: tabu search; economic dispatch; micro-genetic algorithm; genetic algorithm; simulated annealing;merit order loading

1. INTRODUCTION

Economic dispatch (ED) is used to determine the optimal schedule of online generating units so as to

meet the load demand at the minimum operating cost under various system and generator operational

constraints. For generating units with non-monotonically and monotonically increasing incremental

cost (IC) functions, an ED solution based on the equal lambda principle could not result in the optimal

solution [1]. Therefore, dynamic programming [2], heuristic search techniques such as simulated

annealing (SA) [3], and genetic algorithm (GA) [4] were proposed to solve the ED problem to obtain

the near-optimal solutions. But the results were still far from the optimal solution. Furthermore, they

are expected to consume a large computational time on the large generating unit systems.

Copyright # 2004 John Wiley & Sons, Ltd.

*Correspondence to: W. Ongsakul, Energy Field of Study, School of Environment, Resources and Development, Asian Instituteof Technology, Pathumthani 12120, Thailand.yE-mail: [email protected]

Ongsakul et al. [5] proposed the micro-genetic algorithm based on migration and merit order

loading solutions (MGAM-MOL) for solving the constrained ED for generating units with linear

decreasing and decreasing staircase IC functions. However, the proposed MGAM-MOL was tested on

the generating units with monotonically decreasing IC functions only. Subsequently, they proposed a

genetic algorithm based on simulated annealing solutions (GA-SA) [6] to solve the ramp rate

constrained dynamic ED problems for the generating units with both non-monotonically and

monotonically increasing IC functions. It was shown that the solutions of the GA-SA were better

than GA and SA alone on the systems with the number of generating units in the range of 10 to 80.

Mantawy et al. [7] proposed the tabu search (TS) for solving unit commitment problems (UCP)

whereas the ED problem was solved via a quadratic programming routine. In the proposed TS, new

rules were introduced for randomly generating feasible trial solutions, in which all units would be

committed without violating the minimum up/down times and reserve constraints. Four different

criteria for constructing the tabu list restrictions were also investigated in order to find the appropriate

tabu list in terms of cost function, speed of solution, and allocated memory. Meanwhile, Mori et al. [8]

proposed the TS with restricted neighborhood of solution search for solving the UCP to speed-up the

computation. The ED solutions were determined by the equal IC method since the cost functions were

approximated by quadratic convex functions.

This paper proposes a deterministic TS (DTS) algorithm for solving constrained economic dispatch

(ED) problems for generating units with non-monotonically and monotonically increasing IC

functions. As the transmission losses are included, the proposed algorithm is tested and compared

to unconstrained DTS (UDTS), MGAM-MOL [5], GA-SA [6], SA, GA, and MOL on the 10, 20, 40,

and 80 generating unit systems.

Four types of cost functions for thermal and combined cycle (CC) generating units obtained from

the Electricity Generating Authority of Thailand (EGAT) are used as the test data, as follows [9]:

* The second-order polynomial cost function of a thermal unit, Ci Pið Þ ¼ ai þ biPi þ ciP2i and its

linear increasing IC function, ICi Pið Þ ¼ bi þ 2ciPi, ci > 0.* The second-order polynomial cost function of a CC unit, Ci Pið Þ ¼ ai þ biPi þ ciP

2i and its linear

decreasing IC function, ICi Pið Þ ¼ bi þ 2ciPi, ci < 0.* The piecewise linear cost function of either the thermal unit or CC unit and its increasing

staircase IC functions are

Ci Pið Þ ¼ ai1 þ bi1Pi; Pi;min � Pi < Pi;int ð1aÞ¼ ai2 þ bi2Pi; Pi;int � Pi � Pi;max ð1bÞ

ICi Pið Þ ¼ bi1; Pi;min � Pi < Pi;int ð2aÞ¼ bi2; Pi;int � Pi � Pi;max ð2bÞ

where bi2 > bi1.* The piecewise linear cost function of a CC unit and its decreasing staircase IC function is the

same as Equations (1) and (2) except that bi1 > bi2.

The organization of this paper is as follows. The constrained ED problem formulation is introduced in

Section 2. Section 3 describes the DTS algorithm for the constrained ED problem. Subsequently, the

experimental results on the systems with the number of generating units in the range of 10 to 80 are

presented in Section 4. Parallel implementation issues are discussed in Section 5. Lastly, conclusions

are given.

378 W. ONGSAKUL, S. DECHANUPAPRITTHA AND I. NGAMROO

Copyright # 2004 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power 2004; 14:377–391

2. CONSTRAINED ECONOMIC DISPATCH (ED) PROBLEM FORMULATION

2.1. Constrained economic dispatch problem formulation

The objective of the constrained ED problem is to minimize the total generator fuel cost under various

system and generator operational constraints. The ramp rate of generating units is also included to

ensure the feasibility of outputs at the next time period. The constrained ED problem is formulated as:

Minimize CTðtÞ ¼XN

i¼1

CiðPiðtÞÞ ð3Þ

subject to a power balance constraint:

XN

i¼1

PiðtÞ ¼ PDðtÞ þ PLðtÞ ð4Þ

and inequality operational constraints of generating units including their ramp rate limits at time

period t:

Pi;lowðtÞ � PiðtÞ � Pi;highðtÞ; i ¼ 1; . . . ;N ð5Þ

2.2. Transmission losses

To achieve the true constrained ED, transmission losses must be taken into account. In this paper, the

traditional B matrix loss formula is used to calculate transmission losses as shown below [9]:

PLðtÞ ¼XN

i¼1

XN

j¼1

PiðtÞBijPjðtÞ þXN

i¼1

Bi0PiðtÞ þ B00 ð6Þ

3. DETERMINISTIC TABU SEARCH (DTS) ALGORITHM FOR CONSTRAINED ED

The deterministic tabu search (DTS) algorithm is a promising tool for combinatorial optimization

problems with any objective functions [7,10]. DTS is a meta-heuristic that is based on a deterministic

local search procedure with the ability to escape from trapping in local optima [11]. The algorithm can

start a search process from any given initial feasible solution to find a better solution. In this paper,

there are four operators used in the DTS algorithm for solving constrained ED: the trial solution

(move) generation, tabu list (TL) restriction, aspiration level (AL) criterion of the solution, and

reference unit rotation strategy.

3.1. Power balance constraint

The power output of the N generating units at a particular time period have to satisfy the power balance

constraint, unit operating limit, and ramp rate constraints. For arbitrary free unit power outputs Pi,

DETERMINISTIC TABU SEARCH APPROACH 379


Pi;lowðtÞ � PiðtÞ � Pi;high, i ¼ 1; . . . ;R� 1;Rþ 1; . . . ;N. The R-th dependent reference unit power

output is constrained by the power balance equation as:

PRðtÞ ¼ PDðtÞ þ PLðtÞ �XN

i¼1;i6¼R

PiðtÞ ð7Þ

In Equation (6), the transmission loss PLðtÞ can be written in terms of PRðtÞ as:

PLðtÞ ¼ A � P2RðtÞ þ B � PRðtÞ þ C ð8Þ

where,

A ¼ BRR

B ¼XN

j¼1;j 6¼R

BRjPjðtÞ þXN

i¼1;i 6¼R

PiðtÞBiR þ BR0

C ¼XN

i¼1;i 6¼R

XN

j¼1;j 6¼R

PiðtÞBijPjðtÞ þXN

i¼1;i 6¼R

Bi0PiðtÞ þ B00

Substituting PLðtÞ in Equation (7),

PDðtÞ þ A � P2RðtÞ þ B � PRðtÞ þ C �

XN

i¼1;i 6¼R

PiðtÞ � PRðtÞ ¼ 0

A � P2RðtÞ þ ðB� 1Þ � PRðtÞ þ C þ PDðtÞ �

XN

i¼1;i 6¼R

PiðtÞ ¼ 0

ð9Þ

The R-th reference unit power output, PRðtÞ, is obviously the solution of the quadratic Equation (9).

PRðtÞ is regarded as a feasible solution if it satisfies the ramp rate operational constraint in Equation (5).

3.2. Solution coding

The concatenated encoding method is employed in this paper, as shown in Figure 1. Each unit output

of N � 1 free units is encoded in a binary base string normalized over its operating range. This

encoding method stacks each unit’s normalized string in series with each other to construct the string

individual. Each unit string structure is assigned by the same number of n bits.

To obtain the actual generating power output of each unit for objective function evaluation, each

string individual is decoded to the decimal value by

PiðtÞ ¼ Pi;lowðtÞ þDi � Pi;highðtÞ � Pi;lowðtÞ

� �

2n � 1; i ¼ 1; . . . ;N ð10Þ



In this paper, 16 bits represent each free unit power output. The greater the number of bits per unit

power output, the higher the resolution will be.

3.3. Initialization

In the typical process of DTS, the initial solution of DTS has to be feasible [10]. Based on the

generator operating range ratio, Equations (11) and (12) are used to calculate the initial feasible power

outputs of N generating units as shown:

PG;leftðtÞ ¼ PDðtÞ þ PLðPiðtÞÞ �XN

i¼1

Pi;lowðtÞ ð11Þ

PiðtÞ ¼ Pi;lowðtÞ þðPi;highðtÞ � Pi;lowðtÞÞ � PG;leftðtÞPN

j¼1 ðPj;highðtÞ � Pj;lowðtÞÞ; i ¼ 1; . . . ;N ð12Þ

PiðtÞ in Equation (11) is initially set to Pi;lowðtÞ, i ¼ 1; . . . ;N. Equations (11) and (12) are then

calculated iteratively until an initial feasible solution, satisfying both the power balance equation and

generator operational constraints, is obtained.

3.4. Trial solution (move) generation

To generate a trial solution (move) or neighborhood of an initial solution, one bit of the binary string is

flipped one at a time. Starting from an initial feasible solution, DTS performs a deterministic advanced

local search [11]. The maximum number of trial solutions in each iteration is referred to a neighborhood

solution space (NS). By experiment, the NS is approximately set to 95% of the total number of bits

ðb0:95 � n� ðN � 1ÞcÞ to yield the best solution. As an example, the first, second, and third bits of the

initial solution are flipped one at a time to yield three trial solutions as shown in Figure 2.

Figure 1. 16 � N � 1ð Þ bit concatenated encoding scheme.

Figure 2. Example of generating trial solutions.



3.5. Tabu list restriction

TL is referred to as an adaptive memory. The mechanism of TL is to keep attributes (bit positions) of

the best solutions in the past iterations, in which each of the best solutions is used as an initial solution

to generate the trial solutions in each subsequent iteration. The attributes in the TL are temporarily

fixed and cannot be flipped to generate the new trial solution candidates unless the aspiration criterion

is met (to be discussed in Section 3.6). As the iteration proceeds, a new attribute enters into the TL as a

fixed attribute and the oldest attribute is released from the TL and becomes a free attribute, as shown in

Figure 3. In particular, TL affects the quality of the solution by controlling the search directions so that

the solution is not trapped in the local optima. The length of tabu list is also called the tabu length,

which is the control parameter of DTS [10]. Basically, the tabu length providing good solutions usually

grows with the size of the problems [7,11]. However, the appropriate size of the tabu length can be

identified by observing the quality of solutions. If the size of the tabu length is too small, the cycling of

solution occurs in the search process. On the other hand, if the size is too large, the search process will

be too restricted, which may deteriorate the solution. In our applications, bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin � ðN � 1Þ

pc is used to

determine the best size of TL [11].

3.6. Aspiration level criterion

Another important component of DTS is the aspiration level (AL) criterion. The major role of the AL

is to allow the fixed (tabued) attributes included in the TL to move if that move yields a more

economical solution. There are many different aspiration level criteria used [11]. The AL used in this

paper is to override the tabu status if the move of the tabued attribute yields a solution that is cheaper

than the best solution reached. After the AL is satisfied, updating the TL is carried out by moving the

tabued attribute back to the first position of the TL.

3.7. Reference unit rotation strategy

The reference unit rotation strategy is also used to reduce the search effort towards the optimal

solution. Regardless of fixed attributes in the TL, the R-th reference unit is freely selected. More

specifically, the R-th reference unit rotates from the first unit at the first iteration to the second unit at

the second iteration and so on as shown in Figure 4.

3.8. Procedure of DTS for solving constrained ED

To apply the DTS for solving the constrained ED problem, an initial feasible solution is generated. A

move to a neighborhood solution is performed if either a move is not tabued or, when a move is tabued,

it passes the AL test. The best solution reached is updated during the search process until the

maximum allowable number of iterations is reached.

The DTS procedure can be described as follows:

Figure 3. Mechanism of tabu list.



Step 1: Read the unit operating limits, heat input–output characteristics, ramp rate constraint limits,

fuel cost of each unit, a forecast load demand at t ¼ 1, transmission loss B-matrix, and initial

power outputs at t ¼ 0.

Step 2: Specify the length of TL, kmax, and size of NS.

Step 3: Initialize the time period t to one.

Step 4: Initialize the iteration counter k to one and empty the TL.

Step 5: Generate Xðk;0Þ by Equations (11) and (12).

Step 6: Initialize AL by setting Xb ¼ Xðk;0Þ.Step 7: Initialize the R-th reference unit to one.

Step 8: Execute DTS process:

Step 8.1: Initialize the trial counter m to one.

Step 8.2: Generate a trial solution Xðk;mÞ from Xðk;0Þ.Step 8.3: If Xðk;mÞ is not feasible, go to Step 8.9.

Step 8.4: If Xðk;mÞ is the first feasible solution, set Xkcb ¼ Xðk;mÞ.

Step 8.5: Perform the tabu test. If Xðk;mÞ is tabued, then go to Step 8.8.

Step 8.6: If C Xðk;mÞ� �< C Xk

cb

� �, set Xk

cb ¼ Xðk;mÞ. Otherwise, go to Step 8.9.

Step 8.7: If C Xðk;mÞ� �< C Xbð Þ, then update AL by setting Xb ¼ Xðk;mÞ. Go to Step 8.9.

Step 8.8: Perform the AL test. If C Xðk;mÞ� �< C Xbð Þ, set Xk

cb ¼ Xðk;mÞ, and update AL by

setting Xb ¼ Xðk;mÞ.Step 8.9: If m is less than NS, m ¼ mþ 1 and return to Step 8.2.

Step 8.10: If there is a feasible solution, set Xðkþ1;0Þ ¼ Xkcb and update TL. Otherwise, set

Xðkþ1;0Þ ¼ Xb.

Step 9: If k < kmax, set k ¼ k þ 1, update the R-th reference unit and return to Step 8.

Step 10: The Xb is the solution at time period t.

Step 11: If t < T , let t ¼ t þ 1, read a forecast load demand at time period t, and return to Step 4.

Otherwise, DTS is terminated.

4. EXPERIMENTAL RESULTS

The proposed DTS algorithm is developed by using the C-language on a Pentium II 266 MHz installed

with the Linux operating system. DTS is tested on the systems consisting of generating units with

Figure 4. Reference unit rotation strategy.



linear increasing IC, linear decreasing IC, increasing staircase IC, and decreasing staircase IC

functions. The fuel cost, operating limit, ramp rate, and coefficients of input–output functions are

shown in Table I. An input–output function is given as Ai þ BiPi þ CiP2i . The generator cost function

is the product of the input–output function and its fuel cost.

In this paper, two types of load demand are used: the monotonically increasing load demand and the

daily load demand. Table II gives the ranges of 10, 20, 40, and 80 generating unit systems for

monotonically increasing load demand. The load demands increase from the minimum loads to the

maximum loads with the given step sizes at every 15 minutes.

For 20, 40, and 80 generating unit systems, the multiples of the RY_CC, KN_CC, IPT_CC, and

BPK_T input–output data are used. The scaled EGAT daily load demands for different generating unit

systems are also used. Figure 5 shows an example of the scaled EGAT daily load curve for the 40

generating unit system. The B-matrices for any given system size are randomly generated so that the

transmission losses are in the range of 1 to 2.1% of the total load demands.

The tabu lengths of 12, 17, 24, and 35 are used for the 10, 20, 40, and 80 generating unit system,

respectively. These parameters yield the best results. The DTS algorithms stop after 500 iterations.

The results are compared to unconstrained DTS (UDTS), MGAM-MOL [5], GA-SA [6], SA, GA, and

MOL based on Ci Pi;max

� �=Pi;max for both monotonically increasing load demands and daily load

demands as shown in Tables III and IV. Obviously, the obtained solutions are near the optimal

Table I. Input–output characteristics of Rayong CC (RY_CC), Bangpakong thermal (BPK_T), Khanom CC(KN_CC), and Independent Power IPT_CC units.

Unit Pmin Pint Pmax Fuel cost Ramp rate(MW) (MW) (MW) (Baht/Gcal) Input–output coefficients (Gcal/h) (MW/min)

Ai Bi Ci URi DRi

RY_CC#1 100 — 300 �123.26930 3.1770742 �0.002752183RY_CC#2 100 — 300 273.80 �127.79030 3.2410401 �0.003104120

8 8RY_CC#3 100 — 150 �30.27881 2.2650570 �0.000923850RY_CC#4 100 — 300 �127.79030 3.2410401 �0.003104120BPK_T#1 280 — 525.5 89.212409 2.158260 0.000111BPK_T#2 280 — 526.5 372.381 106.132003 2.101667 0.000201

5 5BPK_T#3 280 — 576 134.028279 1.890687 0.000425BPK_T#4 280 — 576 80.666760 2.163603 0.000107KN_CC 376 495 315.143 113.5361104 1.856133141 0

8 8495 678 206.6947723 1.667493427 0

IPT_CC 320 510 — 180.4651584 1.45977807 0— 510–650 — 315.143 79.4196883 1.65809420 0 15 15— 650 700 �141.8048526 1.99828107 0

Table II. Ranges of monotonically increasing load demands for 10, 20, 40, and 80 generating unit systems.

No. of Minimum load Maximum load Increasing Number ofunits demand (MW) demand (MW) step size (MW) time periods (T)

10 2220 4500 10 22820 4400 9070 10 46740 8780 18 160 20 46980 17 560 36 320 40 469



Figure 5. EGAT daily load curve scaled for the 40 unit system.

Table III. Comparison of total generator fuel cost of various methods for monotonically increasing load demand.

No. of units Method Total generator %Total cost CPU timefuel cost (Baht) difference per load demand (sec)

UDTS 137 571 020 0.0000 N/ADTS 137 604 060 0.0240 2.1220MGAM-MOL 137 629 199 0.0423 6.9650

10 GA-SA 137 641 399 0.0512 10.8760SA 137 702 765 0.0958 2.9220GA 137 852 686 0.2047 7.5340MOL 138 026 055 0.3308 0.0002


20 GA-SA 563 132 029 0.0793 26.0500SA 563 586 592 0.1601 7.6420GA 564 279 779 0.2733 17.1500MOL 565 001 762 0.4116 0.0007

UDTS 1 130 503 496 0.0000 N/ADTS 1 130 537 803 0.0030 111.0810MGAM-MOL 1 131 220 011 0.0634 37.0390

40 GA-SA 1 133 006 773 0.2214 72.5550SA 1 134 667 957 0.3684 25.2050MOL 1 135 185 542 0.4142 0.0025GA 1 136 092 285 0.4944 43.3350

UDTS 2 260 860 347 0.0000 N/ADTS 2 260 875 096 0.0007 855.6370MGAM-MOL 2 262 561 463 0.0752 102.7994

80 GA-SA 2 268 904 652 0.3558 228.3640MOL 2 269 852 713 0.3977 0.0101SA 2 270 572 225 0.4296 89.1720GA 2 278 352 718 0.7737 121.6090



solutions of UDTS and more economical than those obtained from the other algorithms. Even though

UDTS solutions are less expensive than the others, they may not be feasible since UDTS does not take

the ramp rate constraints into account. Note the CPU times per load demand are obtained from the runs

on a Pentium II 266 MHz installed with the Linux operating system.

As shown in Tables III and IV, the required average CPU time per load step of DTS increases as the

generating unit system size increases due to the longer neighborhood solutions. Figure 6 shows the

convergence comparison of all methods on the 40 generating unit system with 10 000 MW load

demand. Obviously, the proposed DTS converges to the less-expensive total generator fuel cost

solution at a faster rate than the other methods. It is clear that even with the same CPU times given, the

other methods cannot determine solutions as well as DTS.

5. PARALLEL IMPLEMENTATION ISSUES

To speed-up the computation of DTS by compromising the solution quality, the proposed DTS can be

easily parallelized and implemented on parallel machines. In particular, DTS is parallelized by

Table IV. Comparison of total generator fuel cost of various methods for daily load demand.

No. of units Method Total generator %Total cost CPU timefuel cost (Baht) difference per load demand (sec)


10 GA-SA 58 095 199 0.0818 10.8760SA 58 132 810 0.1466 2.9220GA 58 193 757 0.2516 7.5340MOL 58 269 171 0.3815 0.0002


20 GA-SA 113 949 813 0.1378 26.0500SA 114 047 687 0.2238 7.6420GA 114 156 979 0.3198 14.1500MOL 114 332 935 0.4745 0.0007


40 GA-SA 228 036 931 0.2238 72.5550SA 228 414 559 0.4560 25.2050MOL 228 450 426 0.4718 0.0025GA 228 608 830 0.5414 43.3350


80 GA-SA 456 940 652 0.4148 228.3640MOL 457 204 202 0.4727 0.0101SA 457 303 849 0.4946 89.1720GA 459 146 978 0.8996 121.6090



decomposing the entire neighborhood solution into NSN equal-size sub-neighborhoods (species),

where NSN is set to the given number of processors (p) as shown in Figure 7. For load balancing

purposes, NS in each species is set to bn� N=NSNc. Each species is mapped to each processor for

parallel implementation. For the reference unit rotation strategy of parallel DTS, the R-th reference

units of each species are initially set to the first units in the fixed regions. For the second iteration, the

R-th reference units are moved to the second units in the fixed regions of each species, and so on. For

every specified epoch (G) of 20 iterations, the best solution reached among species is selected as an

initial solution for all species through solution exchange among processors.

The parallel DTS algorithm with NSN species can be described as follows:

Step 1: Each processor reads the input data (Step 1 in Section 3.8).

Step 2: Each processor specifies the length of TL, kmax, size of NS, and epoch generation (G).

Step 3: Each processor initializes the time period t to one.

Step 4: Each processor initializes the iteration counter (k) to one and empties the TL.

Step 5: Each processor generates Xðk;0Þ by Equations (11) and (12).

Step 6: Each processor initializes AL by setting Xb equal to Xðk;0Þ.Step 7: Each processor initializes the R-th reference unit to the first unit in its fixed region.

Figure 6. Convergence comparison on the 40 unit system with 10 000 MW load demand.

Figure 7. Example of sub-neighborhood decomposition.



Step 8: Each processor executes the DTS process on its own species (Step 8 in Section 3.8).

Step 9: Each processor checks whether k is indivisible by G and k < kmax. If yes, then go to Step 15.

Step 10: Each client processor sends its best solution reached (Xb) to the host processor.

Step 11: The host processor receives solution candidates from all client processors and determines the

best solution among all received solutions.

Step 12: The host processor broadcasts the best solution reached to all client processors.

Step 13: Each client processor sets the solution received as its initial solution and update AL.

Step 14: Each processor checks whether k ¼ kmax. If yes, then go to Step 16.

Step 15: Each processor moves the R-th reference unit to the next unit in its fixed region, sets

k ¼ k þ 1, and returns to Step 8.

Step 16: The Xb of all species is the solution at time period t.

Step 17: Each processor checks whether t < T. If yes, then let t ¼ t þ 1, read a forecast load at the

time period t, and return to Step 4. Otherwise, parallel DTS is terminated.

With the best species solution exchange and load balancing strategies, the experimental speed-up is

expected to be high [12]. As shown in Table V, the entire neighborhood solution is equally divided into

4, 8, 16, and 32 species for different parallel DTS. As the entire neighborhood is decomposed into

species, the ratio of the total number of feasible solutions to the total number of trial solutions in each

species is reduced, resulting in fewer required CPU times of sequential runs of parallel DTS. The

solutions of parallel DTS with different species size slightly deteriorate compared to those of DTS (or

parallel DTS with one species) due to a slightly lower total number of feasible solutions in search

space. With a different number of species, the trade-off between the experimental speed-ups and

quality of solutions has to be compromised for the best performance on real parallel machines. Note

Table V. Comparison of total generator fuel costs of sequential runs of parallel DTS for daily load demand.

No. of units NSN Total generator Total generator fuel %Total cost CPU timefuel cost (Baht) cost difference (Baht) difference (sec)

1 58 067 292 0 0.0000 0.67298332 58 072 982 5690 0.0098 0.734154

10 16 58 074 746 7454 0.0128 0.7295434 58 074 952 7660 0.0132 0.6894008 58 086 559 19 267 0.0332 0.717761

16 113 839 323 0 0.0000 4.1466251 113 840 601 1278 0.0011 5.341601

20 8 113 846 388 7065 0.0062 4.0972914 113 855 983 16 660 0.0146 4.075340

32 113 869 334 30 011 0.0264 4.187341

1 227 472 893 0 0.0000 40.0727328 227 515 015 42 122 0.0185 27.614079

40 16 227 533 911 61 018 0.0268 27.7969704 227 545 238 72 345 0.0318 27.532461

32 227 574 554 101 661 0.0447 27.882971

1 455 274 003 0 0.0000 438.0300008 455 543 579 269 576 0.0592 199.027500

80 16 455 670 279 396 276 0.0870 199.47030032 455 800 398 526 395 0.1156 199.8847004 456 319 960 1 045 957 0.2297 198.635200



that CPU times shown in Table Vare obtained from the sequential runs of parallel DTS algorithms on a

Pentium III 733 MHz installed with the Linux operating system.

6. CONCLUSION

In this paper, a new DTS algorithm is proposed to solve ramp rate constrained ED problems for

generating units with non-monotonically and monotonically increasing IC functions. The proposed

DTS employs the tabu list, aspiration level, and reference unit rotation strategy to yield a high-quality

solution. The proposed DTS algorithm is superior to MGAM-MOL, GA-SA, SA, GA, and MOL in

terms of the quality of solutions, thereby leading to substantial generator fuel cost savings. The DTS

can easily facilitate parallel implementation on real parallel machines to compromise the solution

quality and the experimental speed-ups for the best performance.

7. LIST OF SYMBOLS AND ABBREVIATIONS

Ai no-load coefficient constant of input–output function of the i-th unit

AL aspiration level criterion

Bi first coefficient of input–output function of the i-th unit

B00 loss coefficient constant

Bi0 i-th element of the loss coefficient vector

Bij ij-th element of the loss coefficient square matrix

BPK_T Bangpakong thermal unit

Ci second coefficient of input–output function of the i-th unit

CðXÞ objective function of solution X

CTðtÞ total generator fuel cost at time period t (bath/time period)

CiðPiðtÞÞ generator fuel cost of the i-th generating unit at time period t (baht/time period)

CC combined cycle unit

Di decimal integer value of binary string of the i-th unit

DRi down ramp limit of the i-th generating unit (MW/time period)

DTS deterministic tabu search

ED economic dispatch

EGAT Electricity Generating Authority of Thailand

G epoch iteration

GA genetic algorithm

GA-SA genetic algorithm based on simulated annealing solutions

IC incremental cost

IPT_CC Independent Power combined cycle unit

KN_CC Khanom combined cycle unit

MGAM-MOL micro-genetic algorithm based on migration and merit order loading solutions

MOL merit order loading

N total number of online generating units to be dispatched

NS neighborhood solution space

NSN number of equal-size species

Pi;min minimum power output of the i-th generating unit (MW)



Pi;max maximum power output of the i-th generating unit (MW)

PiðtÞ real power output of the i-th generating unit at time period t (MW)

PDðtÞ total real power load demand at time period t (MW)

PLðtÞ total transmission loss at time period t (MW)

Pi;lowðtÞ lowest possible power output of the i-th generating unit at time period t

(MW),Max Pi;min; Piðt � 1Þ � DRið Þ� �

Pi;highðtÞ highest possible power output of the i-th generating unit at time period t (MW),

Min Pi;max; Piðt � 1Þ þ URið Þ� �

R dependent reference unit power output

RY_CC Rayong combined cycle unit

SA simulated annealing

T total number of time periods in the time horizon

TL tabu list

TS tabu search

UDTS unconstrained deterministic tabu search

URi up ramp limit of the i-th generating unit (MW/time period)

Xb best solution reached

Xkcb current best trial solution at iteration k

Xðk;0Þ initial feasible solution at iteration k

Xðk;mÞ trial m solution at iteration k

ai no-load coefficient constant of cost function of the i-th unit

bi first coefficient of cost function of the i-th unit

ci second coefficient of cost function of the i-th unit

kmax maximum allowable number of iterations

n number of bits representing each unit power output

p number of processors

ACKNOWLEDGEMENT

The authors would like to thank the Electricity Generating Authority of Thailand (EGAT) for

providing the test data.

REFERENCES

1. Ongsakul W. Real-time economic dispatch using merit order loading for linear decreasing and staircase incremental costfunctions. Electric Power Systems Research 1999; 15:167–173.

2. Liang ZX, Glover JD. A zoom feature for a dynamic programming solution to economic dispatch including transmissionlosses. IEEE Transactions on Power Systems 1992; 7:544–550.

3. Wong KP, Fung CC. Simulated annealing based economic dispatch algorithm. IEE Proceedings: Generation, Transmissionand Distribution 1993; 140:509–515.

4. Sheble GB, Brittig K. Refined genetic algorithm—economic dispatch example. IEEE Transactions on Power Systems1995; 10:117–124.

5. Ongsakul W, Tippayachai J. Constrained economic dispatch by micro genetic algorithm based on migration and merit orderloading solutions. International Conference on Electric Utility Deregulation and Restructuring, and Power Technologies2000; pp. 510–517.

6. Ongsakul W, Ruangpayoongsak N. Constrained dynamic economic dispatch by simulated annealing/genetic algorithms.22nd International Conference on Power Industry Computer Applications (PICA) 2001; pp. 201–212.



7. Mantawy AH, Abdel-Magid YL, Selim SZ. Unit commitment by tabu search. IEE Proceedings: Generation, Transmissionand Distribution 1998; 145:56–64.

8. Mori H, Usami T. Unit commitment using tabu search with restricted neighborhood. International Conference on Intelli-gent Systems Applications to Power Systems 1996; pp. 422–427.

9. Wood AJ, Wollenberg BF. Power Generation, Operation and Control, 2nd edn. Wiley: New York, 1996.10. Glover F, Laguna M. Tabu Search. Kluwer: London, 2001.11. Rayward-Smith VJ, Osman IH, Reeves CR, Smith GD. Modern Heuristic Search Methods. Wiley: Chichester, 1996.12. Huang G, Ongsakul W. Managing the bottlenecks in parallel gauss–seidel type algorithms for power flow analysis. IEEE

Transactions on Power Systems 1994; 9:677–684.

AUTHORS’ BIOGRAPHIES

Weerakorn Ongsakul was born in 1967. He received his B.Eng. degree in ElectricalEngineering from Chulalongkorn University, Bangkok, Thailand in 1988, and received hisM.S. and Ph.D. degrees in Electrical Engineering from Texas A&M University, CollegeStation, in 1991 and 1994, respectively. During 1995–2000, he was teaching at the ElectricalEngineering Program, Sirindhorn International Institute of Technology (SIIT), ThammasatUniversity, Thailand. He joined Energy Field of Study, Asian Institute of Technology,Thailand in 2001. He is currently an Associate Professor and Coordinator there. His currentinterest is in power system operation and control, computer applications to power systems,AI applications to power systems, and power system deregulation.

Sanchai Dechanupaprittha was born in 1978. He received the B.Eng. and M.Sc. degrees inElectrical Engineering from SIIT, Thammasat University, Thailand in 2000 and 2003,respectively. His research interest is in applications of robust control to power systemdynamic and stability, and applications of heuristic methods to power system optimization.

Issarachai Ngamroo was born in 1970. He received his B.Eng. degree in ElectricalEngineering from King Mongkut’s Institute of Technology Ladkrabang (KMITL), Bangkok,Thailand in 1992. He was the recipient of a Monbusho scholarship from the JapaneseGovernment to continue his graduate study in Japan during 1993–2000. He received hisM.Eng. and Ph.D. degrees in Electrical Engineering in 1997 and 2000, respectively, fromOsaka University, Osaka, Japan. He is currently an Associate Professor and Chairperson ofthe Electrical Power Engineering Program, Sirindhorn International Institute of Technology(SIIT), Thammasat University. His research interests are in the areas of power system controland stabilization, flexible AC transmission system devices (FACTS) applications to powersystems, and robust control design of power system controllers.



constrained economic dispatch by deterministic tabu search approach

Documents