research article hybrid de-sqp method for solving combined...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 982305 7 pageshttpdxdoiorg1011552013982305

Research ArticleHybrid DE-SQP Method for Solving Combined Heat andPower Dynamic Economic Dispatch Problem

A M Elaiw12 X Xia3 and A M Shehata2

1 Department of Mathematics Faculty of Science King Abdulaziz University PO Box 80203 Jeddah 21589 Saudi Arabia2Department of Mathematics Faculty of Science Al-Azhar University Assiut 71511 Egypt3 Centre of New Energy Systems Department of Electrical Electronic and Computer Engineering University of PretoriaPretoria 0002 South Africa

Correspondence should be addressed to A M Elaiw a m elaiwyahoocom

Received 19 June 2013 Accepted 25 August 2013

Academic Editor Carla Roque

Copyright copy 2013 A M Elaiw et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Combined heat and power dynamic economic dispatch (CHPDED) plays a key role in economic operation of power systemsCHPDED determines the optimal heat and power schedule of committed generating units by minimizing the fuel cost under ramprate constraints and other constraints Due to complex characteristics heuristic and evolutionary based optimization approacheshave became effective tools to solve the CHPDED problem This paper proposes hybrid differential evolution (DE) and sequentialquadratic programming (SQP) to solve the CHPDED problem with nonsmooth and nonconvex cost function due to valve pointeffects DE is used as a global optimizer and SQP is used as a fine tuning to determine the optimal solution at the finalThe proposedhybrid DE-SQP method has been tested and compared to demonstrate its effectiveness

1 Introduction

In the past decades increasing demand for power and heatresulted in the existence of combined heat and power (CHP)units known as cogeneration or distributed generation Itproduces electricity and useful heat simultaneously Whilethe efficiency of the normal power generation is between50 and 60 the power and heat cogeneration increases theefficiency around 90 [1] Utilization of CHP units besidesconventional thermal power generating units and heat-onlyunits to satisfy heat and power load demands in an economi-cal manner emphasizes the need to combined heat and powereconomic dispatch (CHPED) The objective of the CHPEDproblem is to determine both power generation and heatproduction from units by minimizing the fuel cost such thatboth heat and power demands are met while the combinedheat and power units are operated in a bounded heat versuspower plane For most CHP units the heat productioncapacities depend on the power generation This mutualdependency of the CHP units introduce a complication tothe problem [2] In addition considering valve point effectsin the CHPED problem makes the problem nonsmooth with

multiple local optimal point which makes finding the globaloptimal challenging

Over the past few years a number of approaches havebeen developed for solving the CHPED problem with com-plex objective functions or constraints such as LagrangianRelaxation (LR) [3 4] Semidefinite Programming (SDP)[5] augmented Lagrange combined with Hopfield neuralnetwork [6] Harmony Search (HS) algorithm [1 7] GeneticAlgorithm (GA) [8] Ant Colony Search Algorithm (ACSA)[9] Mesh Adaptive Direct Search (MADS) algorithm [10]Self Adaptive Real-Coded Genetic Algorithm (SARGA)[2] Particle Swarm Optimization (PSO) [11 12] ArtificialImmune System (AIS) [13] and Evolutionary Programming(EP) [14] In [11 13] the valve point effects and the transmis-sion line losses are incorporated into the CHPED problem

The main drawbacks of the CHPED is that it may failto deal with the large variations of the heat and power loaddemands due to the ramp rate limits of the units moreover itdoes not have the look-ahead capability To overcome thesedrawbacks combined heat and power dynamic economicdispatch (CHPDED) problem is formulated with the objec-tive to determine the optimal heat and power schedule of

2 Mathematical Problems in Engineering

the committed units so as to meet the predicted heat andpower load demands over a time horizon at minimumoperating cost under ramp rate constraints and other con-straints [15] CHPDED has a look-ahead capability whichis necessary to schedule the load beforehand so that thesystem can anticipate sudden changes in power and heatdemands in the near future The ramp rate constraint isa dynamic constraint which is important to maintain thelife of the generators [16] Since the ramp rate constraintcouples the time intervals the CHPDEDproblem is a difficultoptimization problem If the ramp rate constraint is notincluded in the optimization problem theCHPDEDproblemis reduced to a set of uncoupled CHPED problems that caneasily be solved The traditional dynamic economic dispatch(DED) problem which considers only conventional thermalunits that provide only electric power has been studied byseveral authors (see eg [17 18] and the review paper [16])However the CHPDED problem has only been consideredin [15 19]

Differential Evolution (DE) algorithm which was pro-posed by Storn andPrice [20] is a population-based stochasticparallel search technique DE uses a rather greedy and lessstochastic approach to problem solving compared to otherevolutionary algorithms DE has the ability to handle opti-mization problems with nonsmoothnonconvex objectivefunctions [20] Moreover it has a simple structure and a goodconvergence property and it requires a few robust controlparameters [20] DE has been applied to the CHPEDproblemwith nonsmooth and nonconvex cost functions in [21]

The DE shares many similarities with evolutionary com-putation techniques such as Genetic Algorithms (GA) tech-niquesThe system is initialized with a population of randomsolutions and searches for optima by updating generationsDE has evolution operators such as crossover and mutationAlthough DE seems to be a good method to solve theCHPDED problem with nonsmooth and nonconvex costfunctions solutions obtained are just near global optimumwith long computation timeTherefore hybrid methods suchas DE-SQP can be effective in solving the CHPDED problemwith valve-point effects Hybrid DE-SQP method has beenused for solving the DED problem in [22 23]

The aim of this paper is to propose a hybrid DE-SQPmethod for solving the CHPDED problem with nonsmoothand nonconvex objective function DE is used as a base levelsearch for global exploration and SQP is used as a local searchto fine-tune the solution obtained fromDEThe effectivenessof the proposed method is shown for test system

2 Problem Formulation

In this section we formulate the CHPDED problem Thesystem under consideration has three types of generatingunits conventional thermal units (TU) CHP units andheat-only units (119867) The power is generated by conventionalthermal units and CHP units while the heat is generatedby CHP units and heat-only units The objective of theCHPDED problem is to minimize the systemrsquos productioncost so as tomeet the predicted heat and power load demands

over a time horizon under ramp rate and other constraintsThe following objectives and constraints are taken intoaccount in the formulation of the CHPDED problem

21 Objective Functions In this section we introduce thecost function of three types of generating units conventionalthermal units CHP units and heat-only units

ConventionalThermalUnitsThe cost function curve of a con-ventional thermal unit can be approximated by a quadraticfunction [24 25] Power plants commonly have multiplevalves which are used to control the power output of theunit When steam admission valves in conventional thermalunits are first open a sudden increase in losses is registeredwhich results in ripples in the cost function [16 26] Thisphenomenon is called as valve-point effects The generatorwith valve-point effects has very different input-output curvecompared with smooth cost function Taking the valve-pointeffects into consideration the fuel cost is expressed as the sumof a quadratic and sinusoidal functions [17 19 27]Thereforethe fuel cost function of the conventional thermal units isgiven by

119862TU119894(119875

TU119894119905)

= 119886119894+ 119887119894119875TU119894119905+ 119888119894(119875

TU119894119905)2

+10038161003816100381610038161003816119890119894sin (119891

119894(119875

TU119894min minus 119875

TU119894119905))10038161003816100381610038161003816

(1)

where 119886119894 119887119894 and 119888

119894are positive constants 119890

119894and 119891

119894are the

coefficients of conventional thermal unit 119894 reflecting valve-point effects 119875TU

119894119905is the power generation of conventional

thermal unit 119894 during the 119905th time interval [119905 minus 1 119905) 119875TU119894min

is the minimum capacity of conventional thermal unit 119894 and119862TU119894(119875

TU119894119905) is the fuel cost of conventional thermal unit 119894 to

produce 119875TU119894119905

CHP Units A CHP unit has a convex cost function in bothpower and heat The form of the fuel cost function of CHPunits can be given by [5 19]

119862CHP119895

(119875CHP119895119905

119867CHP119895119905)

= 119886119895+ 119887119895119875CHP119895119905

+ 119888119895(119875

CHP119895119905

)2

+ 119889119895119867

CHP119895119905

+ 119890119895(119867

CHP119895119905)2

+ 119891119895119875CHP119895119905

119867CHP119895119905

(2)

where 119862CHP119895(119875

CHP119895119905

119867CHP119895119905) is the generation fuel cost of CHP

unit 119894 to produce power 119875CHP119895119905

and heat 119867CHP119895119905

Constants119886119895 119887119895 119888119895 119889119895 119890119895 and 119891

119895are the fuel cost coefficients of CHP

unit 119895

Heat-Only Units Cost The cost function of heat-only unitscan take the following form [5 19]

119862119867

119896(119867119867

119896119905) = 119886119896+ 119896119867119867

119896119905+ 119888119896(119867119867

119896119905)2

(3)

Mathematical Problems in Engineering 3

where 119886119896 119896 and 119888

119896 are the fuel cost coefficients of heat-only

unit 119896 and they are constantsLet 119873 be the number of dispatch intervals and let 119873

119901+

119873119888+ 119873ℎbe the number of committed units where119873

119901is the

number of conventional thermal units 119873119888is the number of

the CHP units and119873ℎis the number of the heat-only units

Then the total fuel cost over the dispatch period [0119873] is givenby

119862 (PH)

=

119873

sum

119905=1

(

119873119901

sum

119894=1

119862TU119894(119875

TU119894119905) +

119873119888

sum

119895=1

119862CHP119895

(119875CHP119895119905

119867CHP119895119905)

+

119873ℎ

sum

119896=1

119862119867

119896(119867119867

119896t))

(4)

where PH = (PH1PH2 PH119905 PH119873)

1015840 PH119905= (PTU119905

PCHP119905HCHP119905H119867119905)1015840 PTU119905

= (119875TU1119905 119875

TU2119905 119875

TU119873119901119905)1015840 PCHP119905

=

(119875CHP1119905

119875CHP2119905

119875CHP119873119888 119905

)1015840 HCHP119905

= (119867CHP1119905 119867

CHP2119905 119867

CHP119873119888119905)1015840

andH119867119905= (119867119867

1119905 119867119867

2119905 119867

119867

119873ℎ119905)1015840

The CHPDED problem can be mathematically formu-lated as a nonlinear constrained optimization problem as

minPH119862 (PH) (5)

subject to the constraintsPower production and demand balance

119873119901

sum

119894=1

119875TU119894119905+

119873119888

sum

119895=1

119875CHP119895119905

= 119875119863119905+ Loss

119905 119905 = 1 119873 (6)

where 119875119863119905

and Loss119905are the system power demand and

transmission line losses at time 119905 (ie the 119905th time interval)respectively The B-coefficient method is one of the mostcommonly used methods by power utility industry to calcu-late the network losses In this method the network losses areexpressed as a quadratic function of the unitrsquos power outputsthat can be approximated in the following

Loss119905=

119873119901+119873119888

sum

119894=1

119873119901+119873119888

sum

119895=1

PL119894119905119861119894119895PL119895119905

119905 = 1 119873 (7)

where

PL119894119905=

119875TU119894119905 119894 = 1 119873

119901

119875CHP119894minus119873119901 119905

119894 = 119873119901+ 1 119873

119901+ 119873119888

(8)

and 119861119894119895is the 119894119895th element of the loss coefficient squarematrix

of size119873119901+ 119873119888

Heat production and demand balance

119873119888

sum

119895=1

119867CHP119895119905

+

119873ℎ

sum

119896=1

119867119867

119896119905= 119867119863119905 119905 = 1 119873 (9)

where119867119863119905

is the system heat demand at time 119905Capacity limits of conventional thermal units

119875TU119894min le 119875

TU119894119905le 119875

TU119894max 119894 = 1 119873

119901 119905 = 1 119873 (10)

where 119875TU119894min and 119875TU

119894max are the minimum and maximumpower capacities of conventional thermal unit 119894 respectively

Capacity limits of CHP units

119875CHP119895min (119867

CHP119895119905) le 119875

CHP119895119905

le 119875CHP119895max (119867

CHP119895119905)

119895 = 1 119873119888 119905 = 1 119873

119867CHP119895min (119875

CHP119895119905

) le 119867CHP119895119905

le 119867CHP119895max (119875

CHP119895119905

)

119895 = 1 119873119888 119905 = 1 119873

(11)

where 119875CHP119895min(119867

CHP119895119905) and 119875CHP

119895max(119867CHP119895119905) are the minimum and

maximum power limits of CHP unit 119895 respectively and theyare functions of generated heat (119867CHP

119895119905) 119867CHP119895min(119875

CHP119895119905

) and119867

CHP119895max(119875

CHP119895119905

) are the heat generation limits of CHP unit 119895which are functions of generated power (119875CHP

119895119905)

Capacity limits of heat-only units

119867119867

119896min le 119867119867

119896119905le 119867119867

119896max 119896 = 1 119873ℎ 119905 = 1 119873 (12)

where 119867119867119896min and 119867119867

119896max are the minimum and maximumheat capacities of heat-only unit 119896 respectively

Upperdown ramp rate limits of conventional thermalunits

minus119863119877TU119894le 119875

TU119894119905+1

minus 119875TU119894119905le 119880119877

TU119894

119894 = 1 119873119901 119905 = 1 119873 minus 1

(13)

where 119880119877TU119894

and 119863119877TU119894

are the maximum ramp updownrates for conventional thermal unit 119894 [16]

Upperdown ramp rate limits of CHP units

minus119863119877CHP119895

le 119875CHP119895119905+1

minus 119875CHP119895119905

le 119880119877CHP119895

119895 = 1 119873119888 119905 = 1 119873 minus 1

(14)

where 119880119877CHP119895

and 119863119877CHP119895

are the maximum ramp updownrates for CHP unit 119895 [19]

3 Differential Evolution Method

DE is a simple yet powerful heuristicmethod for solving non-linear nonconvex and nonsmooth optimization problemsDE algorithm is a population-based algorithm using threeoperators mutation crossover and selection to evolve fromrandomly generated initial population to final individualsolution [20] In the initialization a population of119873119875 targetvectors (parents) 119883

119894= 1199091119894 1199092119894 119909

119863119894 119894 = 1 2 119873119875

is randomly generated within user-defined bounds where119863 is the dimension of the optimization problem Let 119883119866

119894=

119909119866

1119894 119909119866

2119894 x119866

119863119894 be the individual 119894 at the current generation


Table 1 Data of the CHP units and heat-only unit of the eleven-unit system

CHP units 119886119895

119887119895

119888119895

119889119895

119890119895

119891119895

DRCHP119895

= URCHP119895

119895 = 1 2650 145 00345 42 0030 0031 70

119895 = 2 1250 36 00435 06 0027 0011 50

Heat-only units 119867119867

119896max 119867119867

119896min 119886119896

119896

119888119896

119896 = 1 26952 0 950 20109 0038

Table 2 Heat load demand of the eleven-unit system

Time (h) Demand (MWth)1 3902 4003 4104 4205 4406 4507 4508 4559 46010 46011 47012 48013 47014 46015 45016 45017 42018 43519 44520 45021 44522 43523 40024 400

119866 Amutant vector119881119866+1119894

= (V119866+11119894 V119866+12119894 V119866+1

119863119894) is generated

according to

119881119866+1

119894= 119883119866

1199031+F times (119883

119866

1199032minus 119883119866

1199033)

1199031= 1199032= 1199033= 119894 119894 = 1 2 119873119875

(15)

with randomly chosen integer indexes 1199031 1199032 1199033isin 1 2

119873119875 HereF is the mutation factorAccording to the target vector 119883119866

119894and the mutant

vector 119881119866+1119894

a new trial vector (offspring) 119880119866+1119894

= 119906119866+1

1119894

119906119866+1

2119894 119906

119866+1

119863119894 is created with

119906119866+1

119895119894=

V119866+1119895119894

if (rand (119895) le 119862119877) or 119895 = 119903119899119887 (119894)

119909119866

119895119894otherwise

(16)

where 119895 = 1 2 119863 119894 = 1 2 119873119875 and rand(119895) is the119895th evaluation of a uniform random number between [0 1]

119862119877 isin [0 1] is the crossover constant which has to bedetermined by the user 119903119899119887(119894) is a randomly chosen indexfrom 1 2 119863 which ensures that 119880119866+1

119894gets at least one

parameter from 119881119866+1119894

[20]The selection process determines which of the vectors

will be chosen for the next generation by implementingone-to-one competition between the offsprings and theircorresponding parents If 119891 denotes the function to beminimized then

119883119866+1

119894= 119880119866+1

119894if 119891 (119880119866+1

119894) le 119891 (119883

119866

119894)

119883119866

119894otherwise

(17)

where 119894 = 1 2 119873119875 The value of 119891 of each trial vector119880119866+1

119894is compared with that of its parent target vector 119883119866

119894

The above iteration process of reproduction and selectionwillcontinue until a user-specified stopping criteria is met

In this paper we define the evaluation function forevaluating the fitness of each individual in the population inDE algorithm as follows

119891 = 119862 + 1205821

119873

sum

119905=1

(

119873119901

sum

119894=1

119875TU119894119905+

119873119888

sum

119895=1

119875CHP119895119905

minus (119875119863119905+ Loss

119905))

2

+ 1205822

119873

sum

119905=1

(

119873119888

sum

119895=1

119867CHP119895119905

+

119873ℎ

sum

119896=1

119867119867

119896119905minus 119867119863119905)

2

(18)

where 1205821and 120582

2are penalty values Then the objective is to

find119891min theminimumevaluation value of all the individualsin all iterations The penalty term reflects the violation of theequality constraints Once the minimum of 119891 is reached theequality constraints are satisfied

4 Sequential Quadratic Programming Method

SQP method can be considered as one of the best nonlinearprogramming method for constrained optimization prob-lems [28] It outperforms every other nonlinear program-mingmethod in terms of efficiency accuracy and percentageof successful solutions over a large number of test prob-lems The method closely resembles Newtonrsquos method forconstrained optimization just as is done for unconstrainedoptimization At each iteration an approximation is madeof the Hessian of the Lagrangian function using Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton updatingmethod The result of the approximation is then used togenerate a Quadratic Programming (QP) subproblem whose


Table 3 Hourly heat and power schedule obtained from CHPDED using DE-SQP for eleven-unit system

H 119875TU1

119875TU2

119875TU3

119875TU4

119875TU5

119875TU6

119875TU7

119875TU8

119875CHP1

119875CHP2

Loss 119867CHP1

119867CHP2

119867119867

1

1 1500000 1350000 745372 720784 1245129 1244302 200000 100000 2368041 1101974 215630 573450 1355994 19705562 1500000 1350000 981135 1220784 1222113 1016179 482025 100000 2368011 1101974 242248 573614 1355994 20703923 1500000 1350000 1781135 1720784 1207640 987468 782025 100000 2353275 1101974 304319 656496 1355994 20875094 1500000 1350000 1880106 2185077 1600000 1263142 800000 400000 2352182 1101974 372496 662643 1355994 21813635 1500000 1350000 2680106 2447145 1280292 1299179 800000 422707 2332313 1101974 413736 774390 1355994 22696166 1500000 1350000 3344706 2947145 1600000 1300000 800000 480931 2356609 1101974 501383 637746 1355994 25062607 1500000 1991593 3400000 3000000 1600000 1300000 800000 497990 2380991 1101974 552549 500614 1355994 26433928 1897336 2295497 3400000 3000000 1600000 1300000 800000 550000 2422569 1101974 607377 266766 1355994 29272409 2653596 3095497 3400000 3000000 1600000 1300000 800000 550000 2470000 1101974 731068 00 1355994 324400610 3036024 3785162 3400000 3000000 1600000 1300000 800000 550000 2469410 1101974 822580 03317 1355994 324068911 3688317 4056648 3400000 3000000 1600000 1300000 800000 550000 2470000 1101974 906945 00 1355994 334400612 3677179 4554472 3400000 3000000 1600000 1300000 800000 550000 2470000 1101974 953624 00 1355994 344400613 3520071 3850034 3400000 3000000 1600000 1300000 800000 550000 2470000 1101974 872079 00 1355994 334400614 2720071 3050034 3400000 3000000 1600000 1300000 800000 550000 2449090 1101974 731169 117604 1355994 312640215 1936233 2250034 3400000 3000000 1600000 1300000 800000 550000 2429121 1101974 607362 229917 1355994 291408916 1500000 1450034 2968330 2508703 1600000 1299573 800000 434626 2332660 1101974 455900 772439 1355994 237156717 1500000 1350000 2600109 2500000 1600000 1000000 800000 409143 2353888 1101974 415121 653046 1355994 219095918 1500000 1510646 3194485 3000000 1600000 1300000 800000 400577 2374722 1101974 502419 535869 1355994 245813719 2294141 2310646 3133779 3000000 1600000 1300000 800000 460360 2370065 1101974 610988 562062 1355994 253194320 3094141 3110646 3400000 3000000 1600000 1300000 800000 550000 2470000 1169757 774552 00 907694 359230621 2724577 3008037 3400000 3000000 1600000 1300000 800000 550000 2470000 1118344 730959 00 1247723 320227722 1924577 2208037 2606669 2500000 1600000 1241397 800000 459763 2346724 1101974 509154 693338 1355994 230066823 1500000 1408037 1806669 2000000 1276584 1300000 500000 400000 2364213 1101974 337482 594980 1355994 204902624 1500000 1350000 1006669 1770362 1232649 1286636 423316 100000 2346572 1095624 271834 694196 1350513 1955291Cost ($) = 25257 times 106 total loss (MW) = 13443 times 103

solution is used to form a search direction for a line searchprocedure Since the objective function of the CHPDEDproblem is nonconvex and nonsmooth SQP ensures a localminimum for an initial solution In this paper DE is used as aglobal search and finally the best solution obtained from DEis given as initial condition for SQP method as a local searchto fine-tune the solution SQP simulations can be computedby the fmincon code of the MATLAB Optimization Toolbox

5 Simulation Results

In this section we present an eleven-unit test system Thehybrid DE-SQPmethod is applied to the CHPDED problemwhere three types of generating units conventional thermalunits CHP units and heat-only units are considered In DE-SQP method the control parameters are chosen as 119873119875 =

80 F = 0423 and 119862119877 = 0885 The maximum numberof iterations is selected as 20 000 The results represent theaverage of 30 runs of the proposedmethod All computationsare carried out by MATLAB program

Eleven-Unit System This system consists of eight conven-tional thermal units two CHP units and one heat-only unitThe CHPDED problem is solved by hybrid DE-SQPmethodThe technical data of conventional thermal units the matrix119861 and the power demand are taken from the ten-unit systempresented in [27] The 5th and 8th conventional units in [27]were replaced by twoCHPunitsThe technical data of the two

180

247

Heat (MWth)

Pow

er (M

W)

C

B

A

1048

D

215

81

988

Figure 1 Heat-power feasible operating region for CHP unit 1 of theeleven-unit system

CHP units and the heat-only unit are taken from [19] and aregiven in Table 1 The heat demand for 24 hours is given inTable 2 The feasible operating regions of the two CHP unitsare taken from [3] and are given in Figures 1 and 2

The best solution of the CHPDED problem obtained byDE-SQP algorithm is given in Table 3 The best cost andtransmission line losses are also given in Table 3


1356

1102

Heat (MWth)

Pow

er (M

W)

A

75324159

40

C

E44 F

D

B1258

Figure 2 Heat-power feasible operating region for CHP unit 2 ofthe eleven-unit system

6 Conclusion

This paper presents a hybrid method combining DE and SQPfor solving the CHPDED problem with valve-point effectsIn this paper DE is first applied to find the best solutionThis best solution is given to SQP as an initial condition tofine-tune the optimal solution at the final The feasibility andefficiency of the DE-SQP were illustrated by conducting casestudy with eleven-unit test system

Acknowledgments

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah Saudi ArabiaThe authors therefore acknowledge the DSR technical andfinancial support The authors are grateful to the anonymousreviewers for constructive suggestions and valuable com-ments which improved the quality of the paper

References

[1] A VasebiM Fesanghary and SM T Bathaee ldquoCombined heatand power economic dispatch by harmony search algorithmrdquoInternational Journal of Electrical Power andEnergy Systems vol29 no 10 pp 713ndash719 2007

[2] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof combined heat and power economic dispatch using selfadaptive real-coded genetic algorithmrdquo Applied Energy vol 86no 6 pp 915ndash921 2009

[3] T Guo M I Henwood and M van Ooijen ldquoAn algorithm forcombined heat and power economic dispatchrdquo IEEE Transac-tions on Power Systems vol 11 no 4 pp 1778ndash1784 1996

[4] A Sashirekha J Pasupuleti N H Moin and C S Tan ldquoCom-bined heat and power (CHP) economic dispatch solved usingLagrangian relaxation with surrogate subgradient multiplierupdatesrdquo International Journal of Electrical Power amp EnergySystems vol 44 pp 421ndash430 2013

[5] A M Jubril A O Adediji and O A Olaniyan ldquoSolving thecombined heat and power dispatchproblem a semi-definite

programming approachrdquoPowerComponent Systems vol 40 pp1362ndash1376 2012

[6] V N Dieu and W Ongsakul ldquoAugmented lagrangehopfieldnetwork for economic load dispatch with combined heat andpowerrdquo Electric Power Components and Systems vol 37 no 12pp 1289ndash1304 2009

[7] E Khorram and M Jaberipour ldquoHarmony search algorithmfor solving combined heat and power economic dispatchproblemsrdquo Energy Conversion and Management vol 52 no 2pp 1550ndash1554 2011

[8] C-T Su and C-L Chiang ldquoAn incorporated algorithm forcombined heat and power economic dispatchrdquo Electric PowerSystems Research vol 69 no 2-3 pp 187ndash195 2004

[9] Y H Song C S Chou and T J Stonham ldquoCombined heatand power economic dispatch by improved ant colony searchalgorithmrdquo Electric Power Systems Research vol 52 no 2 pp115ndash121 1999

[10] S S Sadat Hosseini A Jafarnejad A H Behrooz and A HGandomi ldquoCombined heat and power economic dispatch bymesh adaptive direct search algorithmrdquo Expert Systems withApplications vol 38 no 6 pp 6556ndash6564 2011

[11] M Behnam M Mohammad and R Abbas ldquoCombined heatand power economic dispatch problem solution using particleswarm optimization with time varying acceleration coeffi-cientsrdquo Electric Power Systems Research vol 95 pp 9ndash18 2013

[12] V Ramesh T JayabarathiN Shrivastava andA Baska ldquoAnovelselective particle swarm optimization approach for combinedheat and power economic dispatchrdquo Electric Power Componentsand Systems vol 37 no 11 pp 1231ndash1240 2009

[13] M Basu ldquoArtificial immune system for combined heat andpower economic dispatchrdquo International Journal of ElectricalPower amp Energy Systems vol 43 pp 1ndash5 2012

[14] K PWong and C Algie ldquoEvolutionary programming approachfor combined heat and power dispatchrdquo Electric Power SystemsResearch vol 61 no 3 pp 227ndash232 2002

[15] B Bahmani-Firouzi E Farjah and A Seifi ldquoA new algorithmfor combined heat and power dynamic economic dispatchconsidering valve-point effectsrdquo Energy vol 52 pp 320ndash3322013

[16] X Xia and A M Elaiw ldquoOptimal dynamic economic dispatchof generation a reviewrdquo Electric Power Systems Research vol 80no 8 pp 975ndash986 2010

[17] P Attaviriyanupap H Kita E Tanaka and J Hasegawa ldquoAhybrid EP and SQP for dynamic economic dispatch withnonsmooth fuel cost functionrdquo IEEE Transactions on PowerSystems vol 17 no 2 pp 411ndash416 2002

[18] A M Elaiw X Xia and A M Shehata ldquoApplication of modelpredictive control to optimal dynamic dispatch of generationwith emission limitationsrdquo Electric Power Systems Research vol84 no 1 pp 31ndash44 2012

[19] T Niknam R Azizipanah-Abarghooee A Roosta and BAmiri ldquoA new multi-objective reserve constrained combinedheat and power dynamic economic emission dispatchrdquo Energyvol 42 no 1 pp 530ndash545 2012

[20] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[21] M Basu ldquoCombined heat and power economic dispatch byusing differential evolutionrdquo Electric Power Components andSystems vol 38 no 8 pp 996ndash1004 2010


[22] A M Elaiw X Xia and A M Shehata ldquoDynamic economicdispatch using hybrid DE-SQP for generating units with valve-point effectsrdquoMathematical Problems in Engineering vol 2012Article ID 184986 10 pages 2012

[23] A M Elaiw X Xia and A M Shehata ldquoHybrid DE-SQPand hybrid PSO-SQP methods for solvin dynamic economicemission dispatch problem with valve-point effectsrdquo ElectricPower Systems Research vol 84 pp 192ndash200 2013

[24] X Xia J Zhang and A Elaiw ldquoAn application of modelpredictive control to the dynamic economic dispatch of powergenerationrdquo Control Engineering Practice vol 19 no 6 pp 638ndash648 2011

[25] A M Elaiw X Xia and A M Shehata ldquoMinimization of fuelcosts and gaseous emissions of electric power generation bymodel predictive controlrdquo Mathematical Problems in Engineer-ing vol 2013 Article ID 906958 15 pages 2013

[26] C K Panigrahi P K Chattopadhyay R N Chakrabarti andMBasu ldquoSimulated annealing technique for dynamic economicdispatchrdquo Electric Power Components and Systems vol 34 no5 pp 577ndash586 2006

[27] M Basu ldquoDynamic economic emission dispatch using non-dominted sorting genetic algorthim-IIrdquo International Journal ofElectrical Power amp Energy Systems vol 30 no 2 pp 140ndash1492008

[28] P T Boggs and JW Tolle ldquoSequential quadratic programmingrdquoActa Numerica vol 3 no 4 pp 1ndash52 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


the committed units so as to meet the predicted heat andpower load demands over a time horizon at minimumoperating cost under ramp rate constraints and other con-straints [15] CHPDED has a look-ahead capability whichis necessary to schedule the load beforehand so that thesystem can anticipate sudden changes in power and heatdemands in the near future The ramp rate constraint isa dynamic constraint which is important to maintain thelife of the generators [16] Since the ramp rate constraintcouples the time intervals the CHPDEDproblem is a difficultoptimization problem If the ramp rate constraint is notincluded in the optimization problem theCHPDEDproblemis reduced to a set of uncoupled CHPED problems that caneasily be solved The traditional dynamic economic dispatch(DED) problem which considers only conventional thermalunits that provide only electric power has been studied byseveral authors (see eg [17 18] and the review paper [16])However the CHPDED problem has only been consideredin [15 19]

Differential Evolution (DE) algorithm which was pro-posed by Storn andPrice [20] is a population-based stochasticparallel search technique DE uses a rather greedy and lessstochastic approach to problem solving compared to otherevolutionary algorithms DE has the ability to handle opti-mization problems with nonsmoothnonconvex objectivefunctions [20] Moreover it has a simple structure and a goodconvergence property and it requires a few robust controlparameters [20] DE has been applied to the CHPEDproblemwith nonsmooth and nonconvex cost functions in [21]

The DE shares many similarities with evolutionary com-putation techniques such as Genetic Algorithms (GA) tech-niquesThe system is initialized with a population of randomsolutions and searches for optima by updating generationsDE has evolution operators such as crossover and mutationAlthough DE seems to be a good method to solve theCHPDED problem with nonsmooth and nonconvex costfunctions solutions obtained are just near global optimumwith long computation timeTherefore hybrid methods suchas DE-SQP can be effective in solving the CHPDED problemwith valve-point effects Hybrid DE-SQP method has beenused for solving the DED problem in [22 23]

The aim of this paper is to propose a hybrid DE-SQPmethod for solving the CHPDED problem with nonsmoothand nonconvex objective function DE is used as a base levelsearch for global exploration and SQP is used as a local searchto fine-tune the solution obtained fromDEThe effectivenessof the proposed method is shown for test system

2 Problem Formulation

In this section we formulate the CHPDED problem Thesystem under consideration has three types of generatingunits conventional thermal units (TU) CHP units andheat-only units (119867) The power is generated by conventionalthermal units and CHP units while the heat is generatedby CHP units and heat-only units The objective of theCHPDED problem is to minimize the systemrsquos productioncost so as tomeet the predicted heat and power load demands

over a time horizon under ramp rate and other constraintsThe following objectives and constraints are taken intoaccount in the formulation of the CHPDED problem

21 Objective Functions In this section we introduce thecost function of three types of generating units conventionalthermal units CHP units and heat-only units

ConventionalThermalUnitsThe cost function curve of a con-ventional thermal unit can be approximated by a quadraticfunction [24 25] Power plants commonly have multiplevalves which are used to control the power output of theunit When steam admission valves in conventional thermalunits are first open a sudden increase in losses is registeredwhich results in ripples in the cost function [16 26] Thisphenomenon is called as valve-point effects The generatorwith valve-point effects has very different input-output curvecompared with smooth cost function Taking the valve-pointeffects into consideration the fuel cost is expressed as the sumof a quadratic and sinusoidal functions [17 19 27]Thereforethe fuel cost function of the conventional thermal units isgiven by

119862TU119894(119875

TU119894119905)

= 119886119894+ 119887119894119875TU119894119905+ 119888119894(119875

TU119894119905)2

+10038161003816100381610038161003816119890119894sin (119891

119894(119875

TU119894min minus 119875

TU119894119905))10038161003816100381610038161003816

(1)

where 119886119894 119887119894 and 119888

119894are positive constants 119890

119894and 119891

119894are the

coefficients of conventional thermal unit 119894 reflecting valve-point effects 119875TU

119894119905is the power generation of conventional

thermal unit 119894 during the 119905th time interval [119905 minus 1 119905) 119875TU119894min

is the minimum capacity of conventional thermal unit 119894 and119862TU119894(119875

TU119894119905) is the fuel cost of conventional thermal unit 119894 to

produce 119875TU119894119905

CHP Units A CHP unit has a convex cost function in bothpower and heat The form of the fuel cost function of CHPunits can be given by [5 19]

119862CHP119895

(119875CHP119895119905

119867CHP119895119905)

= 119886119895+ 119887119895119875CHP119895119905

+ 119888119895(119875

CHP119895119905

)2

+ 119889119895119867

CHP119895119905

+ 119890119895(119867

CHP119895119905)2

+ 119891119895119875CHP119895119905

119867CHP119895119905

(2)

where 119862CHP119895(119875

CHP119895119905

119867CHP119895119905) is the generation fuel cost of CHP

unit 119894 to produce power 119875CHP119895119905

and heat 119867CHP119895119905

Constants119886119895 119887119895 119888119895 119889119895 119890119895 and 119891

119895are the fuel cost coefficients of CHP

unit 119895

Heat-Only Units Cost The cost function of heat-only unitscan take the following form [5 19]

119862119867

119896(119867119867

119896119905) = 119886119896+ 119896119867119867

119896119905+ 119888119896(119867119867

119896119905)2

(3)


where 119886119896 119896 and 119888



119901+


119901is the




119862 (PH)

=

119873

sum

119905=1

(

119873119901

sum

119894=1

119862TU119894(119875

TU119894119905) +

119873119888

sum

119895=1

119862CHP119895

(119875CHP119895119905

119867CHP119895119905)

+

119873ℎ

sum

119896=1

119862119867

119896(119867119867

119896t))

(4)


1015840 PH119905= (PTU119905

PCHP119905HCHP119905H119867119905)1015840 PTU119905

= (119875TU1119905 119875

TU2119905 119875

TU119873119901119905)1015840 PCHP119905

=

(119875CHP1119905

119875CHP2119905

119875CHP119873119888 119905

)1015840 HCHP119905

= (119867CHP1119905 119867

CHP2119905 119867

CHP119873119888119905)1015840

andH119867119905= (119867119867

1119905 119867119867

2119905 119867

119867

119873ℎ119905)1015840


minPH119862 (PH) (5)


119873119901

sum

119894=1

119875TU119894119905+

119873119888

sum

119895=1

119875CHP119895119905

= 119875119863119905+ Loss

119905 119905 = 1 119873 (6)

where 119875119863119905



Loss119905=

119873119901+119873119888

sum

119894=1

119873119901+119873119888

sum

119895=1

PL119894119905119861119894119895PL119895119905

119905 = 1 119873 (7)

where

PL119894119905=

119875TU119894119905 119894 = 1 119873

119901

119875CHP119894minus119873119901 119905

119894 = 119873119901+ 1 119873

119901+ 119873119888

(8)


of size119873119901+ 119873119888


119873119888

sum

119895=1

119867CHP119895119905

+

119873ℎ

sum

119896=1

119867119867

119896119905= 119867119863119905 119905 = 1 119873 (9)

where119867119863119905


119875TU119894min le 119875

TU119894119905le 119875

TU119894max 119894 = 1 119873

119901 119905 = 1 119873 (10)




119875CHP119895min (119867

CHP119895119905) le 119875

CHP119895119905

le 119875CHP119895max (119867

CHP119895119905)

119895 = 1 119873119888 119905 = 1 119873

119867CHP119895min (119875

CHP119895119905

) le 119867CHP119895119905

le 119867CHP119895max (119875

CHP119895119905

)

119895 = 1 119873119888 119905 = 1 119873

(11)

where 119875CHP119895min(119867

CHP119895119905) and 119875CHP



119895119905) 119867CHP119895min(119875

CHP119895119905

) and119867

CHP119895max(119875

CHP119895119905


119895119905)


119867119867

119896min le 119867119867

119896119905le 119867119867

119896max 119896 = 1 119873ℎ 119905 = 1 119873 (12)

where 119867119867119896min and 119867119867



minus119863119877TU119894le 119875

TU119894119905+1

minus 119875TU119894119905le 119880119877

TU119894

119894 = 1 119873119901 119905 = 1 119873 minus 1

(13)

where 119880119877TU119894

and 119863119877TU119894



minus119863119877CHP119895

le 119875CHP119895119905+1

minus 119875CHP119895119905

le 119880119877CHP119895

119895 = 1 119873119888 119905 = 1 119873 minus 1

(14)

where 119880119877CHP119895

and 119863119877CHP119895




119894= 1199091119894 1199092119894 119909

119863119894 119894 = 1 2 119873119875


119894=

119909119866

1119894 119909119866

2119894 x119866




CHP units 119886119895

119887119895

119888119895

119889119895

119890119895

119891119895

DRCHP119895

= URCHP119895

119895 = 1 2650 145 00345 42 0030 0031 70

119895 = 2 1250 36 00435 06 0027 0011 50


119896max 119867119867

119896min 119886119896

119896

119888119896

119896 = 1 26952 0 950 20109 0038



119866 Amutant vector119881119866+1119894

= (V119866+11119894 V119866+12119894 V119866+1


according to

119881119866+1

119894= 119883119866

1199031+F times (119883

119866

1199032minus 119883119866

1199033)

1199031= 1199032= 1199033= 119894 119894 = 1 2 119873119875

(15)




vector 119881119866+1119894


= 119906119866+1

1119894

119906119866+1

2119894 119906

119866+1


119906119866+1

119895119894=

V119866+1119895119894

if (rand (119895) le 119862119877) or 119895 = 119903119899119887 (119894)

119909119866


(16)




parameter from 119881119866+1119894



119883119866+1

119894= 119880119866+1

119894if 119891 (119880119866+1

119894) le 119891 (119883

119866

119894)

119883119866

119894otherwise

(17)



119894



119891 = 119862 + 1205821

119873

sum

119905=1

(

119873119901

sum

119894=1

119875TU119894119905+

119873119888

sum

119895=1

119875CHP119895119905

minus (119875119863119905+ Loss

119905))

2

+ 1205822

119873

sum

119905=1

(

119873119888

sum

119895=1

119867CHP119895119905

+

119873ℎ

sum

119896=1

119867119867

119896119905minus 119867119863119905)

2

(18)

where 1205821and 120582







H 119875TU1

119875TU2

119875TU3

119875TU4

119875TU5

119875TU6

119875TU7

119875TU8

119875CHP1

119875CHP2

Loss 119867CHP1

119867CHP2

119867119867

1







180

247

Heat (MWth)

Pow

er (M

W)

C

B

A

1048

D

215

81

988





1356

1102

Heat (MWth)

Pow

er (M

W)

A

75324159

40

C

E44 F

D

B1258


6 Conclusion


Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where 119886119896 119896 and 119888



119901+


119901is the




119862 (PH)

=

119873

sum

119905=1

(

119873119901

sum

119894=1

119862TU119894(119875

TU119894119905) +

119873119888

sum

119895=1

119862CHP119895

(119875CHP119895119905

119867CHP119895119905)

+

119873ℎ

sum

119896=1

119862119867

119896(119867119867

119896t))

(4)


1015840 PH119905= (PTU119905

PCHP119905HCHP119905H119867119905)1015840 PTU119905

= (119875TU1119905 119875

TU2119905 119875

TU119873119901119905)1015840 PCHP119905

=

(119875CHP1119905

119875CHP2119905

119875CHP119873119888 119905

)1015840 HCHP119905

= (119867CHP1119905 119867

CHP2119905 119867

CHP119873119888119905)1015840

andH119867119905= (119867119867

1119905 119867119867

2119905 119867

119867

119873ℎ119905)1015840


minPH119862 (PH) (5)


119873119901

sum

119894=1

119875TU119894119905+

119873119888

sum

119895=1

119875CHP119895119905

= 119875119863119905+ Loss

119905 119905 = 1 119873 (6)

where 119875119863119905



Loss119905=

119873119901+119873119888

sum

119894=1

119873119901+119873119888

sum

119895=1

PL119894119905119861119894119895PL119895119905

119905 = 1 119873 (7)

where

PL119894119905=

119875TU119894119905 119894 = 1 119873

119901

119875CHP119894minus119873119901 119905

119894 = 119873119901+ 1 119873

119901+ 119873119888

(8)


of size119873119901+ 119873119888


119873119888

sum

119895=1

119867CHP119895119905

+

119873ℎ

sum

119896=1

119867119867

119896119905= 119867119863119905 119905 = 1 119873 (9)

where119867119863119905


119875TU119894min le 119875

TU119894119905le 119875

TU119894max 119894 = 1 119873

119901 119905 = 1 119873 (10)




119875CHP119895min (119867

CHP119895119905) le 119875

CHP119895119905

le 119875CHP119895max (119867

CHP119895119905)

119895 = 1 119873119888 119905 = 1 119873

119867CHP119895min (119875

CHP119895119905

) le 119867CHP119895119905

le 119867CHP119895max (119875

CHP119895119905

)

119895 = 1 119873119888 119905 = 1 119873

(11)

where 119875CHP119895min(119867

CHP119895119905) and 119875CHP



119895119905) 119867CHP119895min(119875

CHP119895119905

) and119867

CHP119895max(119875

CHP119895119905


119895119905)


119867119867

119896min le 119867119867

119896119905le 119867119867

119896max 119896 = 1 119873ℎ 119905 = 1 119873 (12)

where 119867119867119896min and 119867119867



minus119863119877TU119894le 119875

TU119894119905+1

minus 119875TU119894119905le 119880119877

TU119894

119894 = 1 119873119901 119905 = 1 119873 minus 1

(13)

where 119880119877TU119894

and 119863119877TU119894



minus119863119877CHP119895

le 119875CHP119895119905+1

minus 119875CHP119895119905

le 119880119877CHP119895

119895 = 1 119873119888 119905 = 1 119873 minus 1

(14)

where 119880119877CHP119895

and 119863119877CHP119895




119894= 1199091119894 1199092119894 119909

119863119894 119894 = 1 2 119873119875


119894=

119909119866

1119894 119909119866

2119894 x119866




CHP units 119886119895

119887119895

119888119895

119889119895

119890119895

119891119895

DRCHP119895

= URCHP119895

119895 = 1 2650 145 00345 42 0030 0031 70

119895 = 2 1250 36 00435 06 0027 0011 50


119896max 119867119867

119896min 119886119896

119896

119888119896

119896 = 1 26952 0 950 20109 0038



119866 Amutant vector119881119866+1119894

= (V119866+11119894 V119866+12119894 V119866+1


according to

119881119866+1

119894= 119883119866

1199031+F times (119883

119866

1199032minus 119883119866

1199033)

1199031= 1199032= 1199033= 119894 119894 = 1 2 119873119875

(15)




vector 119881119866+1119894


= 119906119866+1

1119894

119906119866+1

2119894 119906

119866+1


119906119866+1

119895119894=

V119866+1119895119894

if (rand (119895) le 119862119877) or 119895 = 119903119899119887 (119894)

119909119866


(16)




parameter from 119881119866+1119894



119883119866+1

119894= 119880119866+1

119894if 119891 (119880119866+1

119894) le 119891 (119883

119866

119894)

119883119866

119894otherwise

(17)



119894



119891 = 119862 + 1205821

119873

sum

119905=1

(

119873119901

sum

119894=1

119875TU119894119905+

119873119888

sum

119895=1

119875CHP119895119905

minus (119875119863119905+ Loss

119905))

2

+ 1205822

119873

sum

119905=1

(

119873119888

sum

119895=1

119867CHP119895119905

+

119873ℎ

sum

119896=1

119867119867

119896119905minus 119867119863119905)

2

(18)

where 1205821and 120582







H 119875TU1

119875TU2

119875TU3

119875TU4

119875TU5

119875TU6

119875TU7

119875TU8

119875CHP1

119875CHP2

Loss 119867CHP1

119867CHP2

119867119867

1







180

247

Heat (MWth)

Pow

er (M

W)

C

B

A

1048

D

215

81

988





1356

1102

Heat (MWth)

Pow

er (M

W)

A

75324159

40

C

E44 F

D

B1258


6 Conclusion


Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











CHP units 119886119895

119887119895

119888119895

119889119895

119890119895

119891119895

DRCHP119895

= URCHP119895

119895 = 1 2650 145 00345 42 0030 0031 70

119895 = 2 1250 36 00435 06 0027 0011 50


119896max 119867119867

119896min 119886119896

119896

119888119896

119896 = 1 26952 0 950 20109 0038



119866 Amutant vector119881119866+1119894

= (V119866+11119894 V119866+12119894 V119866+1


according to

119881119866+1

119894= 119883119866

1199031+F times (119883

119866

1199032minus 119883119866

1199033)

1199031= 1199032= 1199033= 119894 119894 = 1 2 119873119875

(15)




vector 119881119866+1119894


= 119906119866+1

1119894

119906119866+1

2119894 119906

119866+1


119906119866+1

119895119894=

V119866+1119895119894

if (rand (119895) le 119862119877) or 119895 = 119903119899119887 (119894)

119909119866


(16)




parameter from 119881119866+1119894



119883119866+1

119894= 119880119866+1

119894if 119891 (119880119866+1

119894) le 119891 (119883

119866

119894)

119883119866

119894otherwise

(17)



119894



119891 = 119862 + 1205821

119873

sum

119905=1

(

119873119901

sum

119894=1

119875TU119894119905+

119873119888

sum

119895=1

119875CHP119895119905

minus (119875119863119905+ Loss

119905))

2

+ 1205822

119873

sum

119905=1

(

119873119888

sum

119895=1

119867CHP119895119905

+

119873ℎ

sum

119896=1

119867119867

119896119905minus 119867119863119905)

2

(18)

where 1205821and 120582







H 119875TU1

119875TU2

119875TU3

119875TU4

119875TU5

119875TU6

119875TU7

119875TU8

119875CHP1

119875CHP2

Loss 119867CHP1

119867CHP2

119867119867

1







180

247

Heat (MWth)

Pow

er (M

W)

C

B

A

1048

D

215

81

988





1356

1102

Heat (MWth)

Pow

er (M

W)

A

75324159

40

C

E44 F

D

B1258


6 Conclusion


Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











H 119875TU1

119875TU2

119875TU3

119875TU4

119875TU5

119875TU6

119875TU7

119875TU8

119875CHP1

119875CHP2

Loss 119867CHP1

119867CHP2

119867119867

1







180

247

Heat (MWth)

Pow

er (M

W)

C

B

A

1048

D

215

81

988





1356

1102

Heat (MWth)

Pow

er (M

W)

A

75324159

40

C

E44 F

D

B1258


6 Conclusion


Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1356

1102

Heat (MWth)

Pow

er (M

W)

A

75324159

40

C

E44 F

D

B1258


6 Conclusion


Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article hybrid de-sqp method for solving combined...

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