Programming of thermoelectric generation systems based on a heuristic composition of ant colonies

Electrical Power and Energy Systems 44 (2013) 134–145

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Electrical Power and Energy Systems

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Programming of thermoelectric generation systems based on a heuristiccomposition of ant colonies

Ivo C. Silva Jr., Flávia R. do Nascimento, Edimar J. de Oliveira, André L.M. Marcato,Leonardo W. de Oliveira ⇑, João A. Passos FilhoDepartment of Electrical Engineering, Federal University at Juiz de Fora (UFJF), Juiz de Fora, MG, Brazil

a r t i c l e i n f o a b s t r a c t

Article history:Received 9 March 2012Received in revised form 26 June 2012Accepted 5 July 2012Available online 9 August 2012

Keywords:Lagrange coefficientsSensibility matrixThermal unit commitmentAnt colony optimization (ACO)

0142-0615/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijepes.2012.07.036

⇑ Corresponding author. Tel.: +55 32 8861 1754/21E-mail address: [email protected] (L.W. d

Studies related to biologically inspired optimization techniques, which are used for daily operationalscheduling of thermoelectric generation systems, indicate that combinations of biologically inspiredcomputation methods together with other optimization techniques have an important role to play inobtaining the best solutions in the shortest amount of processing time.

Following this line of research, this article uses a methodology based on optimization by an ant colonyto minimize the daily scheduling cost of thermoelectric units. The proposed model uses a SensitivityMatrix (SM) based on the information provided by the Lagrange multipliers to improve the biologicallyinspired search process. Thus, a percentage of the individuals in the colony use this information in theevolutionary process of the colony. The results achieved through the simulations indicate that the useof the SM results in quality solutions with a reduced number of individuals.

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1. Introduction

The daily operations scheduling of thermoelectric generationsystems consists of determining a dispatch strategy of the generat-ing units to meet the demand for energy that satisfies the operat-ing and functional constraints of the electrical power system [1].

The problem can be divided into two sub-problems: (i) onereferring to the determination of the units that should be in oper-ation, given the requested demand, ‘‘Thermal Unit Commitment-TUC’’, and (ii) the other referring to the determination of the powergenerated by each of the units in service, ‘‘Economic Dispatch’’. Dueto the variation in the load over time, operational scheduling in-volves decisions of the generation system every hour, within thehorizon of 1 day–2 weeks [2].

The representation of costs and operating constraints increasesthe complexity of the problem through temporal coupling of thedown/up decisions for the generating units, resulting in a mixed-integer, non-linear programming problem with the following un-ique aspects [3]: (i) non-convex solution region, (ii) high computa-tional time due to the combinatory nature of the problem and (iii)dynamic decision process. Therefore, it can be seen that these as-pects require constant improvement to the existing algorithms.

The method proposed by [4] uses a priority operation list basedon the economic characteristics of the units. Other studies use this

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02 3484.e Oliveira).

technique to mitigate the unfeasibility of enumerating all the pos-sible solutions [5,6].

Dynamic scheduling was the first method based on optimiza-tion applied to the problem [7], offering advantages because it con-siders non-convex and non-linear problems. However, this methodrequires working in a discrete space, demanding a large memorycapacity and high computational time [8,9].

Lagrangian relaxation [10–12] separates the constraints of theproblem, allowing the solution to be found through sub-prob-lems solved by dynamic programming. However, due to thenon-convexity of the problem, there is no guarantee that theoptimality of the dual solution found will lead to a feasible pri-mal solution.

The study reported in [13] examined the biologically inspiredoptimization techniques used to solve the daily operatingscheduling of thermoelectric generation systems, with an emphasison optimization by ant colonies, artificial neural networks, geneticalgorithms and particle swarm optimization. Analyses indicate thatthe combination of biologically inspired computational methodsand other optimization techniques plays an important role inachieving better solutions in less processing time.

Reference [14] presents an investigation into the application ofthe genetic algorithm to solve the thermal unit commitmentproblem. A parallel structure was developed to handle the infeasi-bility problem in a structured and improved genetic algorithm.Typical constraints, such as the system power balance, minimumup and down times, start-up and shut-down ramps, wereconsidered

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I.C. Silva Jr. et al. / Electrical Power and Energy Systems 44 (2013) 134–145 135

Differential evolution based approaches have also been ap-plied to short-term electrical power generation scheduling [15].In this reference, a new way of applying a differential evolutionalgorithm, which comprises the search procedure involving bin-ary decision variables and the power dispatch calculation in aunique problem, is proposed. This approach eliminates the useof an iterative local search technique in all solution evaluations.

In [16], a hybrid Taguchi–Immune algorithm, which integratesthe Taguchi method and the traditional immune algorithm, is pre-sented to consider the unit commitment problem. The Taguchimethod is incorporated in the crossover operations to select thebetter gene for achieving crossover, consequently enhancing theimmune algorithm.

A simulated annealing embedded evolutionary program-ming approach is presented in [17] to solve the hydro-thermalunit commitment problem. The objective is to determine thegeneration scheduling such that the total operating cost can beminimized when subjected to a variety of constraints.

Following this line of research, the objective of this paper is topresent an improvement in the ant colony optimization (ACO) evo-lution process for application to the daily operation planning prob-lem for thermoelectric generation units. The results obtained willbe evaluated through the simulation of systems that are widely de-scribed in the literature.

2. Formulation of the problem

The optimization problem referring to daily operational sched-uling of thermoelectric generator units can be formulated as fol-lows [18]:

MinOF ¼ Aþ B ð1Þ

where

A ¼XT

t¼1

XN

i¼1

½COiðPgiðtÞ�:FDOti ð1:aÞ

B ¼XT

t¼1

XN

i¼1

SCiðtÞ:FDOti :ð1� FDOt�1

i Þ ð1:bÞ

COi ¼ ai þ bi:PgiðtÞ þ ci:Pg2i ð1:cÞ

subject to

FDOti � PgiðtÞ � PlðtÞ ¼ 0 ð2Þ

XN

i¼1

FDOti � Pgmax

i P PlðtÞ þ rgðtÞ ð3Þ

Xoni ðtÞP Ton

i ð4Þ

Xoffi ðtÞP Toff

i ð5Þ

Pgmini 6 PgiðtÞ 6 Pgmax

i ð6Þ

where

N
total number of thermal units; T total operating period; i thermal unit index; t hour index;
FDOti

discrete variable [0, 1] of ‘‘ON/OFF’’ decision referringto unit i at hour t;

Pgi(t)
active power generated (MW) by thermal unit i athour t;
Pgmaxi
maximum limit of active power generated (MW) by
thermal unit i;

Pgmini

minimum limit of active power generated (MW) bythermal unit i;

Pl(t)
demand requested (MW) at hour t; rg(t) spinning reserve requested (MW) at hour t; Ton
i
minimum on time (h) of thermal unit i;
Toffi

minimum off time (h) of thermal unit i;

Xoni ðtÞ
time (h) during which thermal unit i is on;
Xoffi ðtÞ
time (h) during which thermal unit i is off;
SCi(t)
start-up cost (US$) of thermal unit i at hour t; ai, bi,
ci

coefficients related to fuel costs of thermal unit i ($/h,$/MWh and $/MW2h).

The Objective Function (OF), Eq. (1), consists of the minimizing thesum of the total operating cost (A) and the start-up costs (B) of thegenerating units during the operating period studied.

The active power balance constraint, Eq. (2), represents thestate of equilibrium of the load/generation of the system at alltimes. Eq. (3) is the spinning reserve of the system to meet unex-pected increases in load or deviations from the forecast.

The following constraints were applied to the generating units:(i) minimum up/down times, Eqs. (4) and (5), respectively; (ii)maximum and minimum production limits, Eq. (6).

To avoid the inherent difficulties associated with optimizationproblems of a discrete nature, the decision variables were allowedto assume continuous values [19] within the discrete interval [0–1]; this strategy aims to obtain the Lagrange coefficients associatedwith these variables. The sigmoid function (7) was adopted due toits similarity with the step function. Eq. (8) refers to the canaliza-tion of the argument of the sigmoid function.

FDOti ¼

exti � 1

exti þ 1

ð7Þ

xmini 6 xt

i 6 xmaxi :::pxi ð8Þ

where xti Argument of the sigmoid function of thermal unit i in hour

t, xmin, xmax Minimum and maximum limits of the sigmoid functionargument, pxi(t) Lagrange multiplier associated with the sigmoidfunction argument of unit i in hour t.

The objective of the formulation presented here is to enable acomparison of the proposed method with thermoelectric systemswidely described in the literature, which use the same modeling.However, it is important to mention that there are other signifi-cant constraints, such as ramp limits [20], prohibited operatingzones [20,21] and the ‘‘valve point’’ effect [22,23], inherent tothe generating units. These constraints increase the complexityof the sub-problem referring to the economic dispatch, makingit non-convex.

3. Proposed methodology

The proposed methodology is described below. Its objective isto solve the daily scheduling problem of thermoelectric generationsystems using biologically inspired optimization based on ant col-onies [24]. For this purpose, two extra stages will be used to helpthe colony in the search process: (i) relaxation of the discrete var-iable referring to the operation decision, enabling it to assume con-

Table 1Representation of an individual of the colony.

t TU1 TU2 TU3 TU4

1 1 1 1 02 0 1 1 03 1 1 0 1

136 I.C. Silva Jr. et al. / Electrical Power and Energy Systems 44 (2013) 134–145

tinuous values and allowing the possibility to obtain the LowerCost Limit (LCL) for the thermoelectric system under analysis and(ii) the use of a merit order based on the Lagrange multipliers,associated with the decision variables, to improve the criteria forthe colony search. The information resulting from these two stageswill be incorporated into the biologically inspired search criteria,as described below.

3.1. Stage 1 – Data entry

In this stage, necessary entry parameters are defined for theperformance of the proposed algorithm, such as (i) the numberof individuals in the colony, (ii) evaporation rates, (iii) maximumnumber of iterations, (iv) number of generating units, (v) hourlyload demand, (vi) minimum up/down times and (vii) merit order(sensitivity matrix). In addition, other entry information is theinitial condition of the first hour of operation of each of the gen-erating units in the system under analysis.

3.2. Stage 2 – Calculation of lower cost limit

In the second stage, the problem is completely relaxed. Toachieve this relaxation, the discrete variables of the continuousoperation decision (FDO) will be used. In other words, the decisionvariables can assume any continuous value within the discreteoperation interval [0, 1]. In addition, the constraints relative tothe start and stop times are ignored, and the lower generation lim-its are considered to be equal to zero. The objective of this stage isto obtain, through the calculation of the operating cost (LCL), a sen-sitivity for the colony search process to evaluate the distance be-tween solutions obtained by the colony (discrete solution) inrelation to the LCL (continuous solution). It should be noted thatthe discrete solution sought will have a cost equal to or greaterthan the LCL.

3.3. Stage 3 – Sensitivity Matrix (SM)

The proposed algorithm uses Lagrange multipliers to identifythe most technically and economically relevant generating unitssuch that this information can be used in ant colony optimization.The multipliers can be obtained as follows:

(i) Rigid interval of canalization for FDO argument,0 6 xt

i 6 0:0001, of all generating units existing in the elec-tric system. The intention is to evaluate the sensitivity ofthe objective function in relation to the activation trend ofeach one of the thermoelectric units according the requesteddemand, that is, FDOðxt

i Þ ! 1.(ii) the inclusion of a new, fictitious generating unit, called the

convergence unit (PgC). The purpose of this new unit is toenable the convergence of the optimization problem due tothe narrowing of the canalization constraint of the FDO. Fur-thermore, this unit is characterized by high operating costwhen compared with all other existing units and by beingable to meet the demand reserve of the electric systemalone.

After the inclusion of this new unit, an additional share (C), asdescribed by Eq. (9), is included in the traditional OF, as describedby Eq. (1), with the purpose of enabling the convergence and theachievement of Lagrange multipliers associated with the operatingdecision function arguments.

C ¼XT

t¼1

l:PgCðtÞ ð9Þ

where l Operating cost of the thermoelectric convergence unit ($/MW h), PgC(t) Active power (MW) generated by the convergencethermoelectric power plant at hour t.

Based on the aforementioned considerations, the process ofoptimization will initially present the trend of placing the mosteconomic generating units in service; however, with the imposi-tion that 0 6 xt

i 6 0:0001, the power balance constraints and spin-ning reserve (Eqs. (2) and (3)) will only be met through theconvergence unit. Therefore, this unit must meet the demandand the spinning reserve requested by the system alone, despitethe high cost associated with the convergence unit. Thus, the solu-tion of the optimization problem is represented by the values ofthe Lagrange multipliers associated with the operating decisionfunction arguments of each one of the existing generating units.

With the known multipliers, it is possible to obtain a SensitivityMatrix (SM), Eq. (10). This matrix is composed of Lagrange multi-pliers associated with the FDO’s and indicates the sensitivity ofthe objective function in relation to the activation trend of eachthermoelectric power plant in terms of the hourly demandthroughout the operating period.

SM ¼

px1ðtÞ px2ðtÞ � � � pxNðtÞpx1ðt þ 1Þ px2ðt þ 1Þ � � � pxNðt þ 1Þ

: : � � � :

px1ðTÞ px2ðTÞ � � � pxNðTÞ

26664

37775 ð10Þ

As the values of the multipliers are negative because they re-flect a reduction of the objective function in relation to the trendof placing units in service, an hourly merit order is obtainedthrough the lowest to highest classification of the Lagrange multi-plier values. This information is included in the optimization pro-cess by the ant colony with the purpose of increasing theefficiency of the search in the solution region. Further details ofthe formation of the SM can be obtained in [18,25].

3.4. Stage 4 – Initial colony

The search process begins with a population of individuals(ants) that represent information referring to the discrete solutionof the problem. This set of individuals, which represents a set ofsolutions, is called a colony.

Each solution is represented by a matrix, whose size is given by(i) matrix lines, which represent the number of hours of operationand (ii) matrix columns, which represent the number of existingThermoelectric Units (TUs). The elements of this matrix are associ-ated with the hourly operational decisions (0–1). Table 1 uses as anexample an individual representing an operating solution for fourgenerating units and 3 h of operation.

The search process begins in a completely random manner, andthe initial solutions are chosen randomly. However, to ensure thefeasibility of the solutions generated, the process of constructingthe solution is performed hour by hour such that the hourly sumof the maximum generation capacities of the thermoelectric unitsin operation is greater than the hourly demand to be met and theforecast hourly reserve, according to Eq. (11).

XN

i¼1

FDOiðxti Þ � Pgmax

i P PlðtÞ þ rgðtÞ ð11Þ

Fig. 1. Roulette referring to solutions generated by builder ants.

Table 2Sensitivity matrix – example.

t TU1 TU2

1 �3.72 �3.622 �3.88 �3.913 �3.91 �4.024 �6.80 �4.005 �7.60 �4.106 �3.56 �3.28


3.5. Stage 5 – Validation of the colony

The objective of this stage is to ensure the feasibility of eachsolution generated. Therefore, the feasibility analysis process isperformed every hour such that the hourly sum of the maximumgeneration capacities of the thermoelectric units placed into oper-ation is greater than the demand and the reserve forecast for eachhour. In addition, it should also be verified whether there are anyviolations of down time and start time of the units. These viola-tions, if they exist, are treated using the heuristic procedures de-scribed in [18]. These correction procedures ensure the feasibilityof the solutions generated throughout the entire iterative process.

3.6. Stage 6 – Economic dispatch

Once the feasible operation schedule has been obtained by eachindividual, the economic dispatch is performed to obtain thehourly active power for each thermal unit and also to obtain theOF associated with each individual. The dispatch is performedusing the primal–dual interior-point method [26–27].

3.7. Stage 7 – Obtaining the Initial Pheromone Matrix (PM)

The initial pheromone matrix presents the same structure usedin the representation of the individual; in other words, it is repre-sented by a matrix whose size is a function of the number of hoursof operation and the number of existing thermoelectric units. Thepheromone deposits of each individual are stored in the PM. Thisdeposit is proportional to the quality of the solution. Thus, solu-tions (OF) with operating costs that are close to the LCL contributemore decisively to the PM matrix, while solutions that have oper-ating costs farther from the LCL contribute in a more superficialmanner. The pheromone deposited by the individual m is given by:

DPMm ¼ 1jOFm�LCLj if FDOt

i ¼ 1

DPMm ¼ 0 if FDOti ¼ 0

(ð12Þ

where DPMm Quantity of pheromone deposited by ant m, OFm Valueof the objective function obtained by ant m.

Thus, the total quantity of initial pheromone deposited in thePM is given by Eq. (13).

PM0 ¼XNA

m¼1

ðb� DPMmÞ ð13Þ

where PM0 Initial pheromone matrix, NA Number of ants in thecolony.

In Eq. (13), the parameter b is a constant used to emphasize theproximity of the solution to the LCL. This multiplicative factor isempirical and equal to 1000.

3.8. Stage 8 – New colony

Using the proposed methodology, the new colony is basicallycomposed of three types of individuals:

(i) Soldier ants – These individuals walk through the solutionregion totally at random and represent 10% of the colony.These individuals play an important role in diversificationof the colony search process.

(ii) Builder ants – These ants walk through the solution regionaccording to the order of merit based on schedule informa-tion from the Lagrange multipliers, obtained through theprocess described in Stage 3. They represent 10% of the col-ony and bring relevant information to the colony from aprior optimization process.

(iii) Worker ants – These ants walk through the solution regionbased on the pheromone matrix (PM) and represent 80% ofthe colony. It is important to emphasize that the solutionsobtained with ants type (i) and (ii) must been consideredin the pheromone matrix.

For the solutions generated by the builder ants, some criteriawere adopted to allow greater diversification of the solutions gen-erated, as a result of the merit order established. Thus, a drawing isheld, represented through a roulette wheel, as shown in Fig. 1,where the merit order to be used at hour t will be given accordingto the region drawn. The regions are described as follows:

� Zone A (40%) – the merit order of the previous hour is used, SM(t);� Zone B (20%) – the merit order of the following hour is used, SM

(t + 1);� Zone C (20%) – the merit order of the current hour is used, SM

(t � 1);� Zone D (20%) – the merit order of a random hour is used, SM (?).

The mechanism described above can be exemplified as follows.Let us assume that (a) the SM is in Table 2, (b) the system has twothermal units and 6 h of operation and (c) a builder ant is buildingthe solution via the SM. For the second hour of operation, there arefour probabilities: (i) Zone A – Probability 40% – It will be chosenthe order of merit determined by the SM at t = 1, (TU1–TU2); (ii)Zone B – Probability 20% – It will be chosen the merit-order estab-lished by the SM at t = 3, (TU2–TU1); (iii) Zone C – Probability 20%– It will adopted the merit-order of the SM in t = 2, (TU2–TU1); and(iv) Zone D – Probability 20% – It will be adopted a merit-order ran-domly choose by the SM (TU1–TU2) or (TU2–TU1). The percent-ages for each slice were those that had the best results in ourexperiments.

The solutions generated by the worker ants are obtainedthrough a decision rule, which indicates the probability ðPt

NÞ thateach ant switches on the thermal unit N at time t. Eq. (14) describeshow this probability is calculated. Thus, this probability is associ-ated with the pheromone concentration, and it will correct theant movements toward the best solution. This mechanism wasoriginally proposed in [24].


PtN ¼

ðPMtNÞ

/

XN

i¼1

ðPMti;Þ

/

ð14Þ

where / is the control parameter that determines the relative weightof the influence of pheromone concentration. Thus, u ¼ 1 wasadopted.

Table 3Operating cost – 4TU’s system.

Simulations Operating cost ($)

Case A 74,480.00Case B 74,476.00

Fig. 2. Convergence – 4TU’s system: Case A

Fig. 3. Convergence – 4TU’s system: Ca

To hold the drawing, a roulette wheel is mounted for each hourof the planning horizon, where each generator represents a sectionof the roulette wheel. The size of each section is proportional to thequantity of pheromone of the generating unit in relation to thesum of the total pheromone accumulated at instant t. Thus, solu-tions with a larger quantity of pheromone have higher chances ofbeing chosen. However, there may be generating units that presenta null amount of pheromone at a certain time. Thus, to give all gen-erating units a chance to be chosen, these units are included in theroulette wheel with a very small probability.

It is worth noting the existence of an ant called the Queen in thecolony, which is unique, and its purpose is to strengthen thepheromone trail of the best solution found during the searchprocess.

– without using the sensitivity matrix.

se B – using the sensitivity matrix.


3.9. Stage 9 – Validation of the colony

After the new colony is defined, it is validated, as in Stage 5.

3.10. Stage 10 – Economic dispatch

The hourly power generated by each unit is defined, as in Stage6.

3.11. Stage 11 – Pheromone matrix update

The deposit in the pheromone matrix (PM) has the same struc-ture as Eq. (12). These deposits have the purpose of registering theinformation of the solutions found by all colony individuals, whichis the so-called collective intelligence. The deposit is cumulative,which makes the quantity of pheromone stronger in the best solu-tions the colony finds.

The evaporation phenomenon is represented by the evaporationcoefficient, q, which may range from zero to the unit value and isgiven by:

PMs ¼ ð1� qÞ � PMðs�1Þ þXNA

m¼1

b� DPMsm ð15Þ

where PMs is the total quantity of pheromone deposited in PM by allants in iteration s, PM(s�1) is the total quantity of pheromone depos-ited in PM by all ants in iteration s – 1, DPMs

m is the quantity ofpheromone deposited by ant m in iteration s.

The proposed algorithm uses a differentiated evaporation rate,where the solutions that are closer to the LCL present an evapora-tion rate lower than the most distant solutions. Therefore, the fol-lowing values for the evaporation rate were adopted: (i) anevaporation rate of 25% for the best solution found, (ii) an evapora-tion rate of 75% for the worst solution found, and (iii) an evapora-tion rate of 50% for the remaining solutions. Thus, the search is notstagnant in places with the worst quality.

Fig. 4. Pheromone matrix – 4TU’s system: Case A – without using the sensitivitymatrix.

3.12. Stage 12 – Stop criteria and convergence

Until convergence is obtained, the iterative process keepsreturning to Stage 8. In this paper, two parameters were adoptedas stop criteria: (i) Stagnation: The iterative process is terminatedwhen the best solution found by the colony is the same for a cer-tain number of consecutive iterations previously established and(ii) Number of iterations: The iterative process is terminated whenthe maximum number of iterations, previously determined, isreached. Thus, the process of bio-inspired search is terminatedwhen one of the established criteria is reached.

The initial solutions created at the beginning of the algorithmhave a strong impact on the convergence process because thesolution region is not convex. Thus, if the initial solutions are gen-erated near the optimum, the convergence process will become

Fig. 5. Pheromone matrix – 4TU’s system: Case B – using the sensitivity matrix.

Table 4Economic dispatch – 4TU’s system.

t TU1 (MW) TU2 (MW) TU3 (MW) TU4 (MW)

1 300 150 0 02 300 205 25 03 300 250 30 204 300 215 25 05 300 0 80 206 255 0 25 07 265 0 25 08 300 200 0 0

Table 5Performance of operating cost - 4TU’s system.

System Best Average Worst Average time (s)

4TU’s $74,476.00 $74,476.00 $74,476.00 27.03


more efficient. Moreover, the ant number parameter is associatedwith the convergence process efficiency. Increasing the numberof ants improves the convergence but worsens the convergencetime. The main contribution of this work, which will be describedbelow, is that the size of ant colony can be small with a high effi-ciency in the convergence process when the SM is taken intoaccount.

3.13. Stage 13 – Final solution

After the convergence of the iterative process, the final solutionis obtained through the queen ant. This solution provides the dailyprogramming of the operation, which corresponds to the lowestoperating cost found by the search process.

Fig. 6. Comparison of operating costs – 4TU’s system.

Table 6Operating cost – 10TU’s system.

Simulations Operating cost ($)

Case A 581,432.00Case B 563,937.00

Fig. 7. Convergence – 10TU’s system: Case A

4. Results

To verify the efficiency of the proposed methodology, the TUCprocedure was applied to a system with four thermal units [28]and to systems with 10, 20, 40, and 100 thermal units, as proposedby Kazarlis et al. [29]. The results obtained by the proposed meth-odology will be compared with other results present in theliterature.

The algorithm used to implement the methodology was devel-oped in the MatLab� environment, version 2008b, and executedwith a Pentium Dual Core processor, 1.86 Hz, 2 Gb RAM. For thealgorithm based on ant colony optimization, the following param-eters were considered: (i) a colony composed of 50 ants (10%builder ants, 10% soldier ants and 80% worker ants), (ii) a maxi-mum stagnation allowed of 10 iterations, (iii) a maximum of a hun-dred iterations, (iv) an evaporation rate for the best solution of 25%,(v) an evaporation rate of the worst solution of 75%, and (vi) anevaporation rate of the other solutions of 50%.

4.1. System with four thermal units

This system is composed of four thermal units with a planninghorizon of 8 h. The demand and data for the thermoelectric unitsfor this system can be found in [28].

The following simulation conditions were considered: (i) CaseA: without using the sensitivity matrix, that is, without the infor-mation from the Lagrange multipliers via builder ants and (ii) CaseB: using the sensitivity matrix, that is, with the information fromLagrange multipliers via builder ants, 10% of the colony.

In Table 3, the operating costs are presented for each one of thecases mentioned above. Comparing the results, it can be observedthat the use of the sensitivity matrix improved the algorithm effi-ciency, resulting in lower operating cost. Although the percentageof economic difference is small, in real systems, this differencetends to be more significant.

Figs. 2 and 3 present the evolution of the operating cost duringthe iterative process for the simulations, considering Cases A and B.Because it is a small-size system, both simulated cases reach stag-

– without using the sensitivity matrix.


nation with nearly the same number of iterations: five iterationsfor Case A and six iterations for Case B.

In Figs. 4 and 5, it is possible to observe the distribution of pher-omones deposited by the ants during the whole iterative process,where the white dots refer to the operational thermal units thatare ‘‘ON’’ during the eight-four hours. The red region representshigh pheromone concentration, and the blue region has a lowerpheromone level. Given that the Sensitivity Matrix takes into ac-count the Lagrange multipliers, the efficiency of the search processincreases significantly. Fig. 5 shows that the best solution has ahigh pheromone concentration. This conclusion cannot be ob-served in Case A, which does not use the SM. Fig. 4 correspondsto Case A, where there are several final solutions with low phero-mone concentration. Therefore, the efficiency of the convergenceprocess decreases without the SM.

The economic dispatch is shown in Table 4. The total operationcost, given by Eq. (1), is $ 74,476.00. Table 5 presents the values ofoperating costs referring to a cycle of 10 simulations of the pro-posed algorithm in which the best solution, the worst solution,the average value of the solutions and the average time of simula-tion were recorded.

The result achieved with the proposed methodology for the 4TU’s system, shown in Fig. 6, was compared to other methodolo-gies found in the literature, including (i) the Memetic Algorithm(MA) [28] and (ii) Particle Swarm Optimization–Lagrangian Relax-ation (PSO–LR) [30].

The operating cost obtained by the proposed methodology islower than that of the other methods. As previously mentioned,although the percentage economic difference is minimal, in largersystems, the difference tends to be more significant.

Fig. 9. Pheromone matrix – 10 TU’s system: Case A – without using the sensitivitymatrix.

4.2. Results on the systems in Kazarlis et al.

The data for the system with 10 generating units are presentedin [29]. The data for the systems having 20, 40 and 100 generat-ing units can be obtained by replicating the base system (10units) and the demand according to the number of units. Thespinning reserve is assumed to be 10% of the total demand, asproposed in [29]. The following simulation conditions were con-

Fig. 8. Convergence – 10TU’s system: Ca

sidered: (i) Case A: without using the sensitivity matrix, that is,without the information from Lagrange multipliers via builderants and (ii) Case B: using the sensitivity matrix, that is, withthe information from Lagrange multipliers via builder ants, whichcomposed 10% of the colony.

In Table 6, the operating costs are presented for each one of thecases mentioned above. A comparison of the results shows that theuse of the sensitivity matrix improved the algorithm efficiencyonce more, resulting in a lower operating cost. Because this systemis larger than the one previously reviewed, it can be observed thatthe percentage economic difference already becomes more signif-

se B – using the sensitivity matrix.


icant because, as the number of generating units to be consideredin the system increases, the level of freedom to choose the gener-ator that should become operational also increases.

Both simulations used the same initial conditions, and there-fore, the importance of the SM can be verified. Figs. 7 and 8

Table 7Performance of operating cost – 10TU’s system.

System Best Average Worst Average time (s)

10TU’s $563,937.00 $563,977.00 $563,945.00 60.70

Fig. 10. Pheromone matrix – 10TU’s system: Case B – using the sensitivity matrix.

Table 8Economic dispatch – 10TU’s system.

t TU1 (MW) TU2 (MW) TU3 (MW) TU4 (MW) TU5 (MW)

1 455 245 0 0 02 455 295 0 0 03 454 370 0 0 254 455 455 0 0 405 453 396 0 125 256 455 361 130 130 257 455 411 130 130 258 455 455 130 130 309 455 455 130 130 85

10 455 455 130 130 16211 455 455 130 130 16212 455 455 130 130 16213 455 455 130 130 16214 455 455 130 130 8515 455 455 130 130 3016 455 311 130 130 2517 455 261 130 130 2518 455 361 130 130 2519 455 455 130 130 3020 455 455 130 130 16221 455 455 130 130 8522 455 455 0 0 14523 455 425 0 0 024 455 345 0 0 0

show the evolution of the operating cost during the iterativeprocess for the simulation of Cases A and B. After theinsertion of information from the Lagrange multipliers, the algo-rithm was able to find a higher quality solution with feweriterations.

The deposition of pheromone during the search process for thesimulations of Cases A and B can be observed in Figs. 9 and 10,where the white dots refer to the operational thermal units thatare ‘‘ON’’ during the twenty-four hours.

In Fig. 9, a case in which the information from the Lagrangemultipliers is not added to the algorithm, the pheromone is spreaddiffusely throughout the region of the solution, indicating that theinformation of the colony is not converging. In Fig. 10, with theinclusion of the sensitivity matrix, the region with the largest con-centration of pheromone becomes well defined by the algorithm,indicating that the colony is more determined in relation to thesolution found.

Table 7 presents the values of the operating costs referring tothe best solution and the worst solution, the average value of thecost obtained and the computational time for the 10TU’s systemafter 10 simulations of the proposed methodology.

The economic dispatch is shown in Table 8, where the totaloperation cost, given by Eq. (1), is $563,937.00. The results ob-tained with the proposed methodology and with some recent ap-proaches in the technical literature are compared in Fig. 11.These methods are:

� Hybrid Ant System Priority List (HASP) [31]� Enhanced Lagrangian Relaxation (ELR) [32]� Selective Self Ant Colony Optimization (SSACO) [33]� Lagrangian Relaxation-Genetic Algorithm (LRGA) [34]� Modified Hybrid Particle Swarm Optimization (MHPSO) [35]� Evolutionary Programming (EP) [36]� Dynamic Programming Hopfield Neural Network (DPHNN)

[37]� Fuzzy Simulated Annealing Dynamic Programming (FSADP)

[38]� Dynamic Programming (DP) [39]� Genetic Algorithm (GA) [29]� Improved Binary Particle Swarm Optimization (IBPSO) [40]

TU6 (MW) TU7 (MW) TU8 (MW) TU9 (MW) TU10 (MW)

0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0

21 25 0 0 033 25 10 0 073 25 10 10 080 25 43 10 1033 25 10 0 021 25 0 0 0

0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0

33 25 10 0 021 25 0 0 020 25 0 0 020 0 0 0 0

0 0 0 0 0

Fig. 11. Comparison of results – 10TU’s system.

Table 9Analysis of sensitivity – percentage of builder ants.

Builders Systems10 TU’s 20 TU’s 40 TU’s 100 TU’s

>40% $563,999.00 $1,125,219.00 $2,246,928.00 $5,612,953.0020% $563,937.00 $1,121,088.00 $2,240,839.00 $5,594,263.0015% $563,937.00 $1,121,088.00 $2,240,839.00 $5,594,263.0010% $563,937.00 $1,121,088.00 $2,240,839.00 $5,594,263.00

0 $581,432.00 $1,184,284.00 $2,408,727.00 $6,072,421.00


Fig. 11 shows the comparison between the main results foundin the literature and the result generated by the proposed method-ology in which it can be confirmed that this methodology corre-sponds to the second best solution, $563,937.00, among all ofthem. However, it must be mentioned that the best solution, foundin [39], does not take into account the minimum operation timeconstraints of the thermal units.

Fig. 12 presents a chart that compares the results found in theliterature and the result obtained through the proposed methodol-ogy for the system, formed by a hundred generating units, whichconfirms that the daily programming of the cheapest operationcorresponds to the solution obtained through the proposed meth-odology, i.e., $5,594,263.00.

Another analysis performed concerns the influence of the per-centage of builder ants in the colony. Thus, Table 9 shows sensitiv-ity of the solutions obtained in relation to the number of builderants that exist in the colony.

From the table above, it can be confirmed that a colony com-posed of 10% builder ants is sufficient to ensure the efficiency ofthe proposed search process. For a number above this percentage

Fig. 12. Comparison of operatin

of builder ants in the colony, the process begins to be biased, andthe search process tends to be compromised.

The proposed methodology was simulated using the MatLabplatform on a 1.86-GHz personal computer. Table 10 shows thecomparisons of CPU times for the proposed and existing methodol-ogies considering 100 TUs. The processing time in per unit (pu)(second column of Table 10) was calculated with the CPU time re-quired for the machine above to run the proposed algorithm as abase time, based on the reference [41].

g costs – 100TU’s system.

Table 10Comparison of CPU times – 100 thermal units.

Methodology CPU time(pu)

CPU time(s)

CPU-frequency(GHz)

Operatingcost ($)

Proposed 1 418 1.86 5,594,263.00HASP 0.03 73 1.50 5,608,649.00GA 30.3 15733 1.50 5,610,293.00IBSO 0.57 295 1.50 5,623,885.00EP 12.6 6120 1.60 5,627,437.00

Table 11Summary of results.

Systems Proposed Literature

04 TU’s $74,476.00 $74,675.00 [28]10 TU’s $563,937.00 $553,507.00 [39]20 TU’s $1,121,088.00 $1,121,630.00 [41]40 TU’s $2,240,839.00 $2,243,648.00 [42]100 TU’s $5,594,263.00 $5,602,411.00 [33]


Table 10 confirms that, for the largest system being studied, theproposed methodology is the third most efficient in computationalterms, with the best solution found. However, it is worth notingthat the proposed methodology can be conveniently arranged forimplementation in an architecture with parallel processing, whichwould make it possible to reduce the computational time spent onlarger systems [42,43].

Table 11 summarizes the results obtained by the proposedmethodology for each case analyzed. We present the solutions ob-tained by the proposed methodology and the best solution found inthe literature.

5. Future developments

Future developments include (i) the inclusion of constraints onthe ramp rate load of the generating units, (ii) the consideration ofthe transmission system in daily operating scheduling of thermo-electric generation systems, (iii) the incorporation of constraintsinherent to the natural gas network (gas pipeline system) to modelthe possibility of gas shortage due to network operational limits,(iv) the modeling of prohibited operating zones and the ‘‘valvepoint’’ effect inherent to the generating units, and (v) an optionfor computational time reduction that uses parallel processing.Parallel processing makes it possible for an algorithm, which re-quires much computational time, to have its execution distributedto several processors and memories, allowing for a considerablereduction in simulation time.

6. Conclusions

The results reached through the simulations performed demon-strate the satisfactory efficiency of the methodology proposed forthe problem of the operational programming of thermoelectricgeneration systems. For the cases analyzed, the proposed method-ology required an acceptable computational effort, which providedoptimal and suboptimal solutions. The results obtained can becredited to the following factors: (i) the economic data comingfrom the Lagrange multipliers, associated with operation decisionfunctions in the identification of the generating units to be madeoperational and incorporated into the bio-inspired search processand (ii) the acquisition of the lower limit of cost, which tends todrive the solutions to a more promising solution region whenintroduced to determine the amount of pheromone to be depositedby the colony.

Regarding the analysis of the results: (i) The proposed methodis shown to be competitive in relation to the main optimizationtechniques used to solve the problem at hand, combining qualitysolutions and acceptable computational effort. (ii) The majorityof time, the comparison of the results shows very small financialpercentage differences between the methodologies, and there areeven alternations of performance (quality of the solutions) for dif-ferent systems. (iii) Despite the small economic differences be-tween the methodologies, it should be kept in mind that, for realand/or large systems, the amount of money involved is large, andtherefore, any savings that are obtained are welcome. (iv) Theguarantee of obtaining the global minimum point can only be at-tained by enumerating all possible combinations of operation,which is not always possible due to the high processing time. (v)Despite the good performance obtained by the proposed sensitivityindex, it should be kept in mind that there is no guarantee of opti-mality for the solutions; that is, similar to other methodologies, theproposed index may produce suboptimal solutions.

Acknowledgments

The authors would like to thank CAPES, CNPq and FAPEMIG forfinancial support.

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Documents

Programming of thermoelectric generation systems based on a heuristic composition of ant colonies