IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 29, NO. 5, SEPTEMBER 2014 2347
Parallel Self-Adaptive Differential EvolutionAlgorithm for Solving Short-Term Hydro
Scheduling ProblemArnel Glotić, Student Member, IEEE, Adnan Glotić, Peter Kitak, Member, IEEE, Jože Pihler, Member, IEEE,
and Igor Tičar, Member, IEEE
Abstract—In order to optimize hydro power plants gener-ator scheduling according to 24-h system demand, a parallelself-adaptive differential evolution algorithm has been applied.The proposed algorithm presents a novel approach to consideringthe multi-population and utilization of the preselection step forthe improvements of the algorithm’s global search capabilities.A preselection step with the best, middle, and worst populations’individuals establishes the new trial vectors. This algorithm hasbeen verified on two different models. The first one consists ofeight power plants with real parameters, and the second oneconsists of four power plants, mostly used as a test model inscientific papers. The main goal of the optimization process is tosatisfy system demand for 24 h with a decreased usage of waterquantity per electrical energy unit. The initial and final states ofthe reservoirs must also be satisfied.
Index Terms—Algorithms, dispatching, hydroelectric powergeneration, optimization methods, parallel algorithms.
NOMENCLATURE
A. Hydro Power Plant Model
Parameters
Natural inflow to hydro plant in hour .
Required biological minimum flow of hydro plant.
Maximal reservoir volume of hydro plant .
Minimal reservoir volume of hydro plant .
Maximum allowed change of reservoir .
Maximal output power of hydro plant .
Minimal water discharge of hydro plant .
Maximal water discharge of hydro plant .
Manuscript received August 01, 2013; revised August 26, 2013, November06, 2013, December 13, 2013, December 23, 2013, and December 23, 2013; ac-cepted January 21, 2014. Date of publication February 07, 2014; date of currentversion August 15, 2014. Paper no. TPWRS-00992-2013.A. Glotić, P. Kitak, J. Pihler, and I. Tičar are with University of Maribor,
2348 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 29, NO. 5, SEPTEMBER 2014
the goal is to reduce thermal cost by hydroelectric genera-tion scheduling [1]–[7]. Power generation in today’s marketconditions requires continuous rationalization and profit max-imization. In this context, the motivation behind these workswas to optimize the operations of HPPs. Therefore, the maingoal is to satisfy the system demand with HPPs by optimalgenerator scheduling in order to decrease water quantity perproduced electrical energy unit (WQPEU) and decrease or eveneliminate water spillage. In order to create an optimizationalgorithm appropriate for real-world and practical applications,respectively, and for use in hydro generating companies, thealgorithm should have fast convergence time with the abilityto satisfy 24-h system demand in respect to all operationalconstrains and requirements.The optimization of cascade HPP reservoirs scheduling,
where downstream plant operation depends on upper plantsoperation in addition to a large solution space, is known as acomplex problem. Within the last 25 years, a wide range of op-timization methods has been applied for solving this problem.In order to achieve optimal production, several methods canbe implemented [7]. These can be generally classified into twomain groups: deterministic and heuristic methods. Determin-istic methods generally arrive at the same final solution throughthe same sequence of solutions, while heuristic methods can beconstructive (build a solution piece by piece) or improving (takea solution and alter it to find a better solution). Deterministicmethods include Lagrangian relaxation [8]–[10] and Bendersdecomposition-based methods [7], mixed-integer programming[11], [12], dynamic programming [13], [14] and sequentialquadratic programming (SQP) [15], [16]. In [15], by means ofthe Lagrangian relaxation, with the hydro scheduling problemsplit into a sequence of smaller and easier to solve subproblemsand afterwards, the nonlinear problem is successfully solvedby using the SQP method. The impact of combining the SQPmethod with an evolutionary algorithm in the process of deter-mining the optimal daily self-scheduling is shown in [16].Particle swarm optimization (PSO) [1], [17], Genetic al-
gorithms [18]–[20], evolutionary programming [21], anddifferential evolution algorithms [4], [5], [22]–[28] are clas-sified as heuristics methods. The differential evolution (DE)algorithm [22] is an efficient and robust global optimizationalgorithm, and therefore it has been selected in this paper as anappropriate optimization technique. Short-term optimizationusing DE [23] has been used on four cascade HPPs, where thebest objective value has been reached after 2000 generations.The authors in [23] highlighted the difficulty of setting the ade-quate control parameters and therefore applied the chaos theory[26] in order to improve the performance of the algorithm. Themodified DE presented in [5] includes a penalty factor duringthe objective function evaluation, which preserves the satisfiedfinal reservoirs levels of four cascade HPPs. In order to findoptimal system cost and emission, the algorithm [5] was firsttested on four cascade HPPs and one equivalent thermal plantand on the four cascade HPPs and three thermal plants. In[24], the authors combined the advantages of the two modifiedDE algorithms, where the grouping and shuffling operation iscarried out over the population periodically. The goal of thisapproach was to find the best reservoir scheduling of two HPPs,but without satisfying the final states of the reservoirs.
The capability of finding the global optimum by using DEis closely connected with the proper selection of DE control pa-rameters [27] like the population size , the differential factorand crossover constant . To avoid this uncertainty, the
self-adaptive DE has been proposed in [25], [28]. The authorsin [29] applied dynamical differential factor combined withthe modified version of mutation strategy where population di-versity information was utilized in order to control the searchcapability and prevent the algorithm being stuck within a lo-cally optimal solution. As soon as the population diversity wasequal or lower than that prescribed, the algorithm automaticallyreplaced several worst individuals.The DE algorithm’s performance can be also improved by
parallelization [30]. The parallelization of the DE can be car-ried out on three levels [31]: the objective function evaluationlevel (master–slave model), the population level (island ormigrant models), and the elements level (cellular model). Theparallelization on objective evaluation level is generally usedin applications where the objective function evaluation is themost time consuming element within the optimization processand the elements level parallelization is generally used formassive parallel machines [31]. Therefore the parallelizationon the population level is used in this paper in similar way as in[31]–[33]. The proposed parallel differential evolution (PDE)uses self-adaptive parameters and consists of subpopulationswith an equal population size , where denotes the numberof CPUs. After all subpopulations have been executed withthe mutation, crossover, and evaluation step, the commonselection step is followed by using the novel best-middle-worst(BmW) strategy, where an optimal balance is achieved betweenconvergence time, population diversity, and global optimum.Parallel self-adaptive differential evolution (PSADE) has beenverified on two models, as shown in Fig. 1. It is compared withthe different parallelization of the DE algorithm and with theclassic DE algorithm by using different parameter sets. ThePSADE is also compared with the SQP method. The powersystem demand for both models must be satisfied by optimalgeneration schedule followed by the decreased usage of waterquantity per produced electrical energy unit m MWh anddecreased or eliminated water spillage.This paper is organized as follows. Mathematical models
are presented in Section II. The proposed PSADE with BmWstrategy are given in Section III. Simulation and results arepresented in Section IV, with conclusion given in Section V.
II. MATHEMATICAL MODELS OF HYDRO POWER PLANTS
This paper considers two different mathematical models.Model I [Fig. 1(a)] is a real parameters’ model consisting ofeight cascade HPPs located in Slovenia and Model II [Fig. 1(b)]as recently used in several papers [1]–[5]. The Model II consistsof four HPP with details presented in [2] and [5], and it usesfewer restrictions in comparison to Model I, which uses realrestrictions and requirements which must be fulfilled. The totalinflow for each HPP for Model I in the observed hour is thesum of the upstream plant flow through the turbine, the spillage,and the natural inflow:
(1)
GLOTIĆ et al.: PARALLEL SELF-ADAPTIVE DIFFERENTIAL EVOLUTION ALGORITHM FOR SOLVING SHORT-TERM HYDRO SCHEDULING PROBLEM 2349
Fig. 1. (a) Model I with eight HPPs. (b) Model II with four HPPs.
The last two HPPs in Model I are canal-based types where theflow merges with the riverbed flow at the end of the canal. Forthe old riverbed flow, time delay to the downstream plant mustbe considered. These last two HPPs also have the required bio-logical minimum flow which must always be provided to theriverbed and it is different for winter and summer time. There-fore, the total inflow of the last HPPs in Model I is
(2)where is the time delay from entry of old riverbed to thedownstream HPP and is the water flow through the oldriverbed, which is usually used at a time when the reservoir 6is on the maximum storage level and the water discharge fromHPP 6 has already reached the maximum allowed discharge.The total inflow for last two HPPs in Model II consist of thesum of upstream HPPs discharges as
(3)
(4)
The inflow water in both models can be used for charging reser-voirs up to the maximal reservoirs’ volume or used incombination with the flow gained from discharging reservoirs,
by considering that the reservoirs values in the observed hourof -th plant must be between the minimal and the maximallevel
(5)of each individual reservoir . The water discharge on each HPPmust also be within the minimal and maximal allowed waterdischarge of th HPP as
(6)Model I also includes maximum allowed change of the thHPP reservoir level in hour compared with as:
(7)
In cases where the inflow exceeds the maximal allowed flowthrough the turbines of the th HPP in hour and the reservoirvolume reaches the maximal allowed level, then the spillageis unavoidable:
(8)
The hydro generator output power is expressed as
(9)
where represents the hydropower generationcoefficients. The input data and coefficients for Model I wereobtained from the company Dravske elektrarne Maribor andHSE Group, respectively. The input data and coefficients forModel II can be found in [2] and [5].
III. APPLICATION OF THE MODIFIED DE ALGORITHM
The PSADE algorithm has been used as parallel implemen-tation of the multipopulation algorithm. The flow chart of theproposed algorithm is shown in Fig. 2.
A. Initialization and Implementation of the ConsideredProblem
The initialized population for the presented mathematicalmodels consists of the values for the hydro discharges of eachindividual model’s HPP and the individual hours during theobserved day (24 h). For instance, the Model I with eight HPPsand within observed intervals over a 24-h period with 1-htime steps represents a problem dimension of 192 (96 forModel II). Therefore, the initial population is composed of-dimensional vectors as follows:
(10)
where is the set of hydro discharges of the th HPP whichcontains randomly chosen values . Theinitial population is distributed on CPU cores as shown in Fig. 3.The mutation and crossover step is run on each core, and the
new trial members are created by using self-adaptive parame-ters and . Each th place in population contains water
2350 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 29, NO. 5, SEPTEMBER 2014
Fig. 2. Flow chart of parallel implementation of multipopulation DE.
Fig. 3. HPP model integration into PSADE algorithm.
discharges for all HPPs. The discharges for each -th HPP arecollected and used as input data of th HPP.The generator output of the th HPP in hour is calculated as
in (9); afterwards, the evaluation step is followed. The step-by-step procedure of the evaluation step is shown in Fig. 4.When the evaluation step is finished by all cores, the trial pop-
ulation is collected and sorted according to the members’ objec-tive values, from minimum to maximum. The new trial popula-tion is formed by the proposed BmW strategywhich is describedbelow. Afterwards, each trial member from trial population is
Fig. 4. Step-by-step procedure of PSADE algorithm.
compared with a target member. Members with better objectivefunction values obtain positions in the target population.
B. Mutation and Crossover With Self-Adaptive Parameters
DE algorithm control parameters selection requires certaineffort in order to find proper values for individual optimizationproblems. In order to avoid this, the authors of [25] proposed aDE algorithm with self-adaptive parameters and . In thispaper, the idea of self-adaptive parameters for DE from [25] isexpanded as
if
elseif
otherwise(11)
elseif
otherwise(12)
where , ,, , , , and are randomly generated
GLOTIĆ et al.: PARALLEL SELF-ADAPTIVE DIFFERENTIAL EVOLUTION ALGORITHM FOR SOLVING SHORT-TERM HYDRO SCHEDULING PROBLEM 2351
numbers from interval [0, 1], and is thedifference factor for each th vector (10) in generation .
C. Subpopulations Evaluation Step
The subpopulation on each CPU core consists of trialvectors which are evaluated by using the objective function. Inthis paper, the objective function is composed of three separateobjective functions which are merged into a singleobjective function (22) by using the weighted sum method[34]. This method transforms a set of three objectives of a givenproblem into a single objective by using weights , , andwhich are defined by the user in proportion to the individualobjective’s importance. Each weight can take any number frominterval [0, 1], but the sum of the weights should be equal to 1.When the value of 1/3 is set as the weight for each individual ob-jective function, the chances of functions being minimized areequal for all. A set of different weights leads to a set of differentsolutions. Therefore, a different set of weights has been testedand the set that obtained the best result by all objectives, hasbeen chosen as author’s appropriate selection for weights ,, and , which were 0.65, 0.25, and 0.1. Objective evalu-
ates agreement between the system demand and productionobtained by PSADE as follows:
(13)
(14)
(15)
(16)
where is the sensitivity of the equation, and is the hydroscheduling time step and the basic time interval, respectively.The sensitivity of the equation has an impact on the functionvalue in (14), as shown in Fig. 5(a), where the system demandthrough 100-h period has been set to 10 MWh and the pro-duction set to increase from 0 to 10 MWh with 0.1 MWh stepthrough a 100-h period as shown in Fig. 5(b). It is obvious thatbigger equation sensitivities have higher and faster impactson those parts, where the differences between system demandand production are smaller. The authors set the sensitivity ofequation and time-step to value 1. The mismatch betweensystem demand and production for each hour is calculated byusing (14). The larger mismatch on time interval [1, ] is ob-tained by using (15) and then the obtained value is used in (16).This helps the objective function not just to follow the averagemismatch value, but also to follow the higher mismatch. Theand are weights and both have been set as 0.5 to ensure bal-ance between the largest mismatch of system demand foundedin hour and the averagemismatch value for time interval, withthe exception of hour .The impact of weights set and (example: set 1 in
Fig. 6(a) is and ) on the final agree-ment between system demand and production obtained by thePSADE and applied to Model II is shown in Fig. 6(b). When
Fig. 5. Impact of (a) the sensitivity of equation depends on (b) demand andproduction.
the weight is set low (sets 1–7 in Fig. 6(b)), the priority tothe largest mismatch of the system demand is decreased andtherefore the average mismatch is adequately larger. The bestagreement is achieved when both weights are set to 0.5 [set 10in Fig. 6(b)].The second objective in
(17)
evaluates the water quantity used per produced electrical en-ergy, which means that the total water discharge obtained byHPPs units is divided by the total produced energy. Because thesecond objective is normalized between 0 and 1, the total max-imal energy of th HPP unit is divided by the total maximal en-ergy of th HPP unit and used in (17). The third objective shownin
(18)
is complementary to the second objective and it evaluates thewater spillage. The restriction (6) can be integrated directlywithin the mathematical model by cutting the parameters withvalues outside the allowed boundaries or it can be consideredthrough objective function (22) by using correction factorfor the objective function, as calculated in
(19)
(20)
(21)
2352 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 29, NO. 5, SEPTEMBER 2014
Fig. 6. Impact of (a) 19 different weights set and to (b) final agreement between the system demand and the production, averaged over ten independentruns.
Fig. 7. Correction matrix for objective function.
The separate correction factors obtained from (19) actuallypresent a correction matrix, as shown in Fig. 7.The influence of the correction factor on the objective func-
tion value is greater when the individual is closer to the lowerallowable parameter’s bounds and the impact is smaller whenthe individual is closer to zero. and are weights and bothhave been set as 0.5 to ensure the balance between the largestdeviation from minimal output power across all HPPs obtainedin (20) and averaged deviation for the whole time interval ,with the exception of hour .Correction factor obtained from (21) is applied onto a
single objective function (22). The has a significant impacton the process of keeping or moving exceeding parameters intothe allowable interval. When the output power on each th HPPin each hour during time-interval satisfies the output powerrestrictions, the correction factor will be equal to 1 and thereforewill not have any impact on objective function:
(22)
D. Forming New Trial Vector by BmW Strategy
After all of the subpopulations have been processed by themutation, crossover, and evaluation steps on each CPU core,the subpopulations are sent to the master CPU core. The masterCPU core collects sub-populations into a temporary populationwhich is -times larger in comparison to the original initial pop-ulation, and where denotes the number of CPU core. After-wards, the master CPU core sorts the temporary population ac-cording to their objective function values, from best to worst, asshown in Fig. 8.
Fig. 8. Forming the new trial vector by BmW strategy.
After the population is collected and sorted, the new trial pop-ulation is created by the proposed BmW strategy. The trialpopulation has the original size of the initial population and itconsists of best, middle, and worstvectors from the temporary population. The , and areuser-selected weights. Their selection influences the algorithm’sglobal search capability, as shown in Section IV.
E. Selection
In the selection step, vectors from the new trial populationare compared with the target (parent) vectors. When the trial
vector has a lower objective value compared to its parent it re-places one, otherwise the parent retains its position in the pop-ulation. After the selection step is finished and the stopping cri-terion has not been met, the obtained population is forwardedto the CPU cores. The process repeats all steps shown in Fig. 2and Fig. 8 until the stopping criteria is met.
IV. SIMULATION AND RESULTS
The proposed PSADE was implemented in MATLAB envi-ronment using an Intel Core i7 3.5-GHz and 32-GB RAM
GLOTIĆ et al.: PARALLEL SELF-ADAPTIVE DIFFERENTIAL EVOLUTION ALGORITHM FOR SOLVING SHORT-TERM HYDRO SCHEDULING PROBLEM 2353
TABLE IRESULTS OBTAINED BY DIFFERENT METHODS FOR MODEL I IN COMPARISON WITH REAL DATA FROM SCADA
Number of function evaluations. Water quantity per produced electrical energy unit
Fig. 9. Final agreement between the system demand and the production obtained by different optimization methods and applied to Model I.
TABLE IIOPTIMAL HYDRO SCHEDULING OBTAINED WITH PSADE FOR MODEL I
Deviation between total hydro generation obtained by PSADE and SCADA
computer. The performance of the proposed algorithm has beentested on two test models.
A. Test Model I
For the optimization process, the real system demand datafrom SCADA [35] have been used as the reference for theModel I. The effectiveness of the PSADE algorithm is evidentfrom Table I, where the results obtained by different opti-mization methods are shown in comparison with data fromSCADA. The classic DE and SQP method dissatisfied the
system demand, but the PSADE algorithm has successfullysatisfied it in regard to the total water discharge in comparisonwith SCADA. The final agreement between the system demandand the production obtained by those algorithms is shown inFig. 9. The total production of all HPPs in each individual hourobtained with the PSADE algorithm is compared in Table IIwith the system demand from SCADA.The stopping criterion for the optimization algorithm was the
maximal number of generations (1500). The weightsfor forming the new trial vectors by BmW strategy have an
2354 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 29, NO. 5, SEPTEMBER 2014
TABLE IIIEXPERIMENTAL RESULTS, AVERAGED OVER TEN INDEPENDENT RUNS OF PROPOSED PSADE
WITH DIFFERENT SETS OF WEIGHTS FOR THE BMW SELECTION STEP FOR MODEL I
Number of function evaluations. Water quantity per produced electrical energy unit
Fig. 10. Objective function values through generations obtained with the pro-posed PSADE algorithm and different control parameters (Model I) (, , , ).
impact on the algorithms capabilities of reaching global solu-tions. Therefore the different sets of weights have been testedand the results are shown in Table III. The best objective func-tion value during the tests were obtained by weight sets, and , and population size .
By using these settings for Model I the result of the optimizationprocess satisfied system demand with more than 4.2of the water in reservoirs saved in comparison with the manualoperation of the dispatching personnel. According to the av-erage water quantity used per produced electrical energy unit,the saved water quantity is equivalent to 176.2 MWh of energy.The disagreement between production according to PSADE al-gorithm and demand from SCADA was negligible. The perfor-mance of the PSADE algorithm with the self-adaptive parame-ters (11) and (12) in comparison with the self-adaptive param-eters from [25] and the set of various fixed control parametersare shown in Fig. 10.In order to verify the proposed method for parallelizing the
DE algorithm, the comparison between the proposed algorithmPSADE and the other three methods from [30]–[32] are shownin Fig. 11. The constant denotes migration rate, which deter-mines how many of the best individuals from one sub-popula-tion will replace the same number of the worst individuals fromother subpopulations.The constant defines the migration interval which denotes
that trough each generation, migration will be set up. Other two
Fig. 11. Objective function values through generations obtained with the pro-posed PSADE algorithm and different parallelization types (Model I).
TABLE IVRESULTS OBTAINED BY DIFFERENT METHODS FOR MODEL II IN
COMPARISON WITH DATA FROM [5]
number of function evaluations. water quantity per produced electricalenergy unit
techniques of parallelization DE algorithm are used from [31]and [32]. The best results are obtained by using the proposedalgorithm.
B. Test Model II
The results obtained by different optimization methods incomparison with the system demand from [5] are shown inTable IV. The optimal solution is obtained by the PSADEalgorithm, where the optimal hydro schedule shown in Table Vsaved more than 228.000 water in reservoirs, compared to[5] as shown in Table IV. This saving is equal to 200.5 MWhof electrical energy. To evaluate performance of proposedalgorithm the standard test model combined of four HPPs,has also been used. The reference system demand obtained
GLOTIĆ et al.: PARALLEL SELF-ADAPTIVE DIFFERENTIAL EVOLUTION ALGORITHM FOR SOLVING SHORT-TERM HYDRO SCHEDULING PROBLEM 2355
TABLE VOPTIMAL SCHEDULING OBTAINED WITH PSADE FOR MODEL II
TABLE VIHYDRO STORAGE VOLUMES FOR MODEL II OBTAINED WITH OPTIMAL SCHEDULING,
TOTAL HYDRO GENERATION, SYSTEM DEMAND FROM [5] AND DEVIATION IN %
Deviation between total hydro generation obtained by PSADE and system demand [5]
Fig. 12. Objective function values through generations with different controlparameter of the proposed PSADE implemented on Model II ( ,
, , ).
from [5], where authors show water discharges and reservoirsvolumes, are used for comparison purposes in Table VI. The
final reservoir levels from [5] are also satisfied and shown inTable VI. To optimize the hydro scheduling for Model II thePSADE algorithm has also been applied. The comparison ofresults obtained for different methods for setting the algorithm’scontrol parameters is shown in Fig. 12. The comparisons ofdifferent methods for parallelization [30]–[32] are shown inFig. 14. The maximum generation number for Model II was setat 1200. The best objective function value was obtained withweights set , and , and .The weight selection impact for the novel BmW strategy isshown in Table VII.The classic DE control parameters for both models were set
to: , , , and strategy to DE/rand/1.The SQP method for both models has been used with MATLABOptimization Toolkit by default settings, where the differentrandom initial points have been used, and the best obtained op-timization result for Model II is shown in Table IV. Comparisonof the final agreement between each the system demand and theproduction obtained by PSADE, classic DE and SQP is shownin Fig. 13, separately for each individual hour. The associated
2356 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 29, NO. 5, SEPTEMBER 2014
Fig. 13. Final agreement between the system demand and the production obtained by different optimization methods and applied to Model II.
TABLE VIIEXPERIMENTAL RESULTS, AVERAGED OVER TEN INDEPENDENT RUNS OF THE PROPOSED PSADE ALGORITHM
WITH DIFFERENT SETS OF WEIGHTS FOR THE BMW SELECTION STEP FOR MODEL II
Number of function evaluations. Water quantity per produced electrical energy unit
Fig. 14. Objective function values through generations for different paral-lelization of PSADE and DE algorithm implemented on Model II.
reservoirs’ volumes at optimal hydro scheduling and the dis-agreements between total hydro energy production and systemdemand in [5] are shown in Table VI.
V. CONCLUSION
The main goal of this research was to modify the standardDE algorithm in order to improve its performance when ap-plied to solve short-term hydropower scheduling optimization
problem. This means that the algorithm should be capable ofsatisfying system demand and the fulfillment of a given sched-uling plan, respectively within the context of the optimal pro-duction allocation by minimizing the used water quantity perproduced electrical energy unit. Therefore, the presented par-allel self-adaptive differential evolution algorithm has been de-veloped and tested on two different models, one real-parameterHPPmodel and one standard test model mostly used in scientificpublications. The experimental results for both models showedthat the novel approach for DE algorithm parallelization alongwith the proposed BmW strategy and the introduced correctionmatrix, significantly improved the performance of the standardDE algorithm. The BmW strategy is used in preselection step,and it controls the balance between global search capability andpopulation diversity, while the correction matrix has a signifi-cant impact on the process of keeping or moving exceeding pa-rameters into the allowable intervals. The proposed algorithmalso uses a novel approach for the self-adaptations of DE con-trol parameters, which improves the algorithms’ performance.The PSADE algorithm, therefore, outperformed the original al-gorithm as well as the other modified algorithms used in thisstudy. According to the obtained results it can be concludedthat the proposed algorithm enables significant savings whenapplied to hydro scheduling problem. It is also evident that thePSADE convergence time is appropriate for the usage of the al-gorithm for real-world and practical applications, respectively.
GLOTIĆ et al.: PARALLEL SELF-ADAPTIVE DIFFERENTIAL EVOLUTION ALGORITHM FOR SOLVING SHORT-TERM HYDRO SCHEDULING PROBLEM 2357
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Arnel Glotić (S’12) was born in Slovenj Gradec,Slovenia, in 1987. He received the B.S. degreein electrical engineering from the University ofMaribor, Maribor, Slovenia, in 2011, where he iscurrently working toward the Ph.D. degree at theInstitute for Power Engineering.His current research interest includes multi-objec-
tive optimization in the application of evolutionarytechniques for electrical power system.
Adnan Glotićwas born in Slovenj Gradec, Slovenia,in 1980. He received the B.S. and Ph.D. degreesin electrical engineering from the University ofMaribor, Maribor, Slovenia, in 2006 and 2011,respectively.Since 2011, he has been with the Holding
Slovenske elektrarne d.o.o., Ljubljana, Slovenia,within the R&D Department. His special fields of in-terest are electrical power network and optimizationmethods.
Peter Kitak (M’09) was born in Ptuj, Slovenia,in 1974. He received the B.S. and Ph.D. degreesin electrical engineering from the University ofMaribor, Maribor, in 1999 and 2006, respectively.Since 2002, he has been a Faculty Member with
the University of Maribor, Maribor, Slovenia. Hisspecial fields of interest are modeling insulators andswitchgears, numerical calculation of electromag-netic fields, and optimization methods.
2358 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 29, NO. 5, SEPTEMBER 2014
Jože Pihler (M’09) was born in Ptuj, Slovenia, in1955. He received the B.S., M.S., and Ph.D. degreesin electrical engineering from the University ofMaribor, Maribor, Slovenia, in 1978, 1991, and1995, respectively.Currently he is a Full Professor with the Univer-
sity of Maribor, Maribor, Slovenia, where his specialfields of interest are switching devices and optimiza-tion methods.
Igor Tičar (M’00) was born in Maribor, Slovenia, in1949. He received the B.S. degree in electrical engi-neering from the University of Ljubljana, Ljubljana,Slovenia, in 1975, and the M.S. and Ph.D. degreesfrom the University of Maribor, Maribor, Slovenia,in 1981 and 1993, respectively.In 1977, he joined the University of Maribor,
Maribor, Slovenia, as an Assistant, where he iscurrently a Full Professor. His special fields ofinterest are also optimization methods.
