at SciVerse ScienceDirect

Energy 50 (2013) 232e244

Contents lists available

Energy

journal homepage: www.elsevier .com/locate/energy

An efficient scenario-based and fuzzy self-adaptive learning particleswarm optimization approach for dynamic economic emissiondispatch considering load and wind power uncertainties

Bahman Bahmani-Firouzi a, Ebrahim Farjah a,*, Rasoul Azizipanah-Abarghooee b

aDepartment of Power and Control Engineering, School of Electrical and Computer Engineering, Shiraz University, Shiraz, IranbDepartment of Electrical Engineering, Marvdasht Branch, Islamic Azad University, Marvdasht, Iran

a r t i c l e i n f o

Article history:Received 1 July 2012Received in revised form11 November 2012Accepted 12 November 2012Available online 5 January 2013

Keywords:Dynamic economic emission dispatchFuzzy adaptiveParticle swarm optimizationSelf-adaptive learning strategyStochastic optimizationWind power

* Corresponding author. Tel.: þ98 917 3116278; faxE-mail addresses: bahman_bah@yahoo.com (B

shirazu.ac.ir (E. Farjah), razizipanah@gmail.com (R. A

0360-5442/$ e see front matter � 2012 Elsevier Ltd.http://dx.doi.org/10.1016/j.energy.2012.11.017

a b s t r a c t

Renewable energy resources such as wind power plants are playing an ever-increasing role in powergeneration. This paper extends the dynamic economic emission dispatch problem by incorporating windpower plant. This problem is a multi-objective optimization approach in which total electrical powergeneration costs and combustion emissions are simultaneously minimized over a short-term time span.A stochastic approach based on scenarios is suggested to model the uncertainty associated with hourlyload and wind power forecasts. A roulette wheel technique on the basis of probability distributionfunctions of load and wind power is implemented to generate scenarios. As a result, the stochastic natureof the suggested problem is emancipated by decomposing it into a set of equivalent deterministicproblem. An improved multi-objective particle swarm optimization algorithm is applied to obtain thebest expected solutions for the proposed stochastic programming framework. To enhance the overallperformance and effectiveness of the particle swarm optimization, a fuzzy adaptive technique, q-searchand self-adaptive learning strategy for velocity updating are used to tune the inertia weight factor and toescape from local optima, respectively. The suggested algorithm goes through the search space in thepolar coordinates instead of the Cartesian one; whereby the feasible space is more compact. In order toevaluate the efficiency and feasibility of the suggested framework, it is applied to two test systems withsmall and large scale characteristics.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Nowadays, due to fast upgrading of renewable energy technol-ogies, large-scale wind power plant has reached significant levels ofpenetration in electrical power systems. In this regard, somecountries have installed significant levels of wind power forreducing carbon emissions and fuel cost for electricity generation[1]. However, the randomness nature of wind speed leads to theuncertainty of wind power output of wind power plants. So, thewind power output in each time interval for each wind power plantis stochastic. The aim of this study is to evaluate the DEED (dynamiceconomic emission dispatch problem) for power systems operatorswhich envisages with the high penetration of wind power plants.The requirements for DEED are at a higher level since comparing to

: þ98 711 2330766.. Bahmani-Firouzi), farjah@zizipanah-Abarghooee).

All rights reserved.

power output of conventional power generators, the wind powercharacteristics are extremely challengeable [2]. In this regard,a multi-layer feed-forward neural network and several approacheson the basis of neural network for different time horizon areproposed in Refs. [2,3], respectively. In addition, in order to studythe management of a variable speed wind turbine power produc-tion in a single bus under load disturbance, Ref. [4] has beenprovided. The DEED problem plays an essential task in powersystem operation [5e10]. Basu proposed a novel fuzzy interactivebased evolutionary algorithm and non-dominated sorting geneticalgorithm II for solving the DEED problem in Refs. [5,6], respec-tively. Liao integrated wind power plant in the DED (dynamiceconomic dispatch) problem in order to save energy and reducecarbon emission for electric power systems in Ref. [7]. Peng et al.modeled the wind speed of wind power plants by Weibull distri-bution functions and proposed a novel algorithm for solving wind-thermal DED problem [8]. Niknam et al. suggested a novel modifiedfirefly algorithm for solving the DEED problem in the presence ofcombined heat and power units in Ref. [9].

Nomenclature

Indicesf wind power plant index.i objective function index.iter iteration index.l load interval index.m particle index.s scenario index.t time interval index.u generating unit index.ug conventional generator index.uw wind power plant index.w wind turbine index.

Constantsaug, bug,cug, dug, eug cost coefficients of unit ug.Bu,u0 ,t loss coefficient relating the productions of units u and

u0 at time t (MW�1).B0,u,t loss coefficient associatedwith the production of unit u

at time t.B00,t loss coefficient parameter at time t (MW).c1, c2 cognitive and social parameters of PSO.RDug ramp-down rate of unit ug (MW/h).Fmin, Fmax estimated or real minimum/maximum fitness value,

respectively.Niter1 ; Niter

2 ;Niter3 ; Niter

4 number of particles which choose thevelocity updating method 1, 2, 3 and 4,respectively.

NF number of wind power plants.NG number of conventional units.NP number of particles.NS number of scenarios after scenario reduction.NT number of time intervals.NU number of generating units.NW number of wind turbines.PD,t,s load demand at time t in scenario s (MW).PforecastD;t forecast load demand at time t (MW).Pmaxug capacity of unit ug (MW).

Pminug minimum power output of unit ug (MW).

Pmaxuw;f capacity of unit uw in wind power plant f (MW).

Pforecastuw;f ;t forecast wind power for unit uw in wind power plant f

at time t (MW).ps normalized probability of scenario s.rand(.), r1, r2 random function generator in the range [0,1].sj,uw,f slope of segment j of unit uw in wind power plant f

(MWs/m).SRt,s SRR (spinning reserve requirement) at time t in

scenario s (MW).RUug ramp-up rate of unit ug (MW/h).

vci,uw,f cut-inwind speed of unit uw inwind power plant f (m/s).

vco,uw,f cut-out wind speed of unit uw in wind power plant f(m/s).

vj,uw,f breakpoint of segment j of unit uw inwind power plantf (m/s).

vr,uw,f rated wind speed of unit uw in wind power plant f (m/s).

wuw,f,t,s wind speed for unit uw in wind power plant f at time tin scenario s (m/s).

wforecastuw;f ;t forecast wind speed for unit uw in wind power plant f

at time t (m/s).WL

l;t;s binary parameter indicating whether the lth loadinterval is selected in scenario s (WL

l;t;s ¼ 1) or not(WL

l;t;s ¼ 0) at time t.WW

w;f ;t;s binary parameter indicating whether the wth windpower interval of wind power plant f is selected inscenario s (WW

w;f ;t;s ¼ 1) or not (WWw;f ;t;s ¼ 0) at time t.

al,t probability of the lth load interval at time t.aug, bug, gug, xug, lug emission coefficients of unit ug.bw,t probability of the wth wind power interval at time t.u inertia weight factor of PSO.DPD,t,s load demand forecast error at time t in scenario s

(MW).DPuw,f,t,s wind power forecast error for unit uw in wind power

plant f at time t in scenario s (MW).

Variablesf1(PG,s) total electrical energy costs in scenario s ($).f2(PG,s) total combustion emissions in scenario s (lb).bf i expected value of objective function i.fmini;s ; fmax

i;s minimum and maximum acceptable level ofobjective function i in scenario s, respectively.bfmin

i ; bf maxi expected maximum and minimum acceptable value

for objective function i.PG,s conventional production matrix for scenario s.PLoss,t,s total real power losses at time t in scenario s (MW).Pu,t,s generation output of unit u at time t in scenario s

(MW).Pug,t,s generation output of unit ug at time t in scenario s

(MW).Puw,f,t,s generation output of unit uw in wind power plant f at

time t in scenario s (MW).Worstiters worst vector among population in iteration iter and

scenario s.mfi;sðPG;sÞ membership function of objective function i in

scenario s.bmfi expected membership value for objective function i.

SetsUs non-inferior solutions for scenario s.

B. Bahmani-Firouzi et al. / Energy 50 (2013) 232e244 233

The DEED problem is an extension version of the complexmulti-objective EED optimization problem that takes into account of thecoupling effect of system at different time intervals like theramping limitation of the conventional generating units. Conse-quently, its computation procedure is more complicated than thatof the single-period EED (economic emission dispatch) problem,but its results are more in line with actual necessities. The DEEDdetermines the production levels of scheduled units so as to meetthe predicted load demand and SRRs (spinning reserve require-ments) over a time horizon satisfying various equality, inequality

and dynamic constraints. In the process of optimization, the totalelectrical energy costs and combustion emissions are both mini-mized [9]. It is notable that in the DEED problem, the value of windpower is the cost of fuel saved by the necessary support system. Inthe past few years, many researchers study the DED [11e26]comprising meta-heuristic techniques like SA (Simulated Anneal-ing) [12], APSO (Adaptive Particle Swarm Optimization) [13], AIS(Artificial Immune System) [14], TVAC-IPSO (Time-Varying Accel-eration Coefficients Improved PSO) [15], hybrid EP-SQP (Evolu-tionary Programming-Sequential Quadratic Programming) [16],

B. Bahmani-Firouzi et al. / Energy 50 (2013) 232e244234

MDE (Modified Differential Evolution) [17], HQPSO (HybridQuantum mechanics inspired PSO) [18], PSO-SQP (particle swarmoptimization-sequential quadratic programming) [19], DGPSO(Deterministically Guided PSO) [20], IPSO (Improved PSO) [21],AHDE (Adaptive Hybrid Differential Evolution) [22], CDE (ChaoticDE) [23], Improved Chaotic PSO [24], MHEP-SQP (Modified HybridEP-SQP) [25], ICA (Imperialist Competitive Algorithm) [26]; the EEDproblem [27,28]. In the recent years, the attention of theresearchers in the area of DEED is increased. A recent literaturesurvey on this topic is available in Ref. [29]. It should be noted thatthe DED and DEED problems are solved once the unit commitmentproblem has been solved, i.e., once the on/off status of each unit hasbeen determined.

From the point of view of SO (System Operator), different sourcesof uncertainty such as the random nature of system demand andrenewable energy sources should be managed at the time of dis-patching. Consequently, researchers have begun to incorporateuncertainty modeling in the power system operation problem [30e35]. In Ref. [30], the uncertainty of the load demand and reservewas represented by fuzzy membership functions to solve the DEDproblem. A fuzzy optimizationmethodwas also presented inRef. [31]to model the hourly load demand, wind speed of wind power plants,solar radiation of photovoltaic generators and available water ofreservoirs randomness for DED. An incomplete gamma functionwasproposed in Ref. [32] to characterize the impact ofwind power plantson the output solution of the emission dispatch problem consideringthe cost of wind power plants and conventional power generators. Inrecent times, a probabilistic approach based on point estimatedmethod was implemented in Ref. [33] to model the wind speed andload demand uncertainties with the assist of Weibull and normaldistribution functions, respectively. In addition, a probabilisticapproach was proposed in Refs. [34,35] for the energy and operationmanagement of renewable micro-grids and distribution feederreconfiguration problem under uncertain environment, respectively.

Amajor feature of this paper is the deliberation of load andwindpower uncertainties in a multi-period and MOP (Multi-objectiveOptimization Problem) for the generation scheduling problem. Thereported approaches until now either (a) presented deterministicversion of the DEED problem [5,6,9] or (b) solved wind-constrainedDEED as a deterministic optimization problem with a singleobjective function comprising both generation and emission costs[7,8], or (c) proposed uncertainty modeling in a single-objectiveeconomic dispatch [36]. Though, no generation schedulingproblem incorporating uncertainty in a MOP is currently availablein the scientific literature. Another salient feature of this work isthat, unlike in Refs. [30,31], uncertainty is addressed through a SP(Stochastic Programming) framework [37] where load and windpower predicted errors are considered as random and uncontrol-lable variables and their probability distributions are representedby scenarios. MCS (Monte Carlo Sampling) and the RWM (Roulettewheel mechanism) are used for generation the desired number ofscenarios. Each scenario is characterized by an equivalent deter-ministic DEED problem.

The DEED model including wind energy has characteristics ofhigh dimensions, poly-constraints, stochastic variables, nonlinear,non-smooth, multi-modal, etc. It is difficult to solve the model byconventionalmathematical programmingmethod. This shortcomingmotivates the need for alternative methods such as those based onevolutionary optimization. To this end, the DEED problem is handledby an interactive fuzzy satisfying technique using a new FSALPSO(Fuzzy Self Adaptive Learning Particle Swarm Optimization) algo-rithm. It is worthwhile to note that the performance of the originalPSO highly relies on its parameters such as inertia weight factor,cognitive and social parameters. It suffers from the problem of beingtrapped in local optima. In this paper, a fuzzy adaptive control is used

for tuning the inertia weight factor based on Ref. [38] and cognitiveand social parameters are set to two. Meantime, in the proposedalgorithm the high dimensional search space is mapped from theCartesian coordinates to the polar one; thereby the smaller spaceshould be investigated which means the probability of finding theoptimal solution is increased. Also, this paper suggests a novel self-adaptive learning operator for velocity updating so that the perfor-mance of the algorithm would be improved effectively. Indeed, thisstrategy can improve the convergence property and thereforeenhance the quality of the solutions. The proposed approach issuitable to resolve the tradeoff between competing and conflictingobjective functions with different value orders.

Within the above framework, the main contributions of thisstudy are threefold:

1. The DEED problem is formulated in a scenario-based SPframework to consider the system uncertainties accurately.

2. An interactive fuzzy satisfying method based on the novelFSALPSO is proposed to solve the stochastic MOP.

3. A new self-adaptive learning strategy is presented for velocityupdating of PSO so that the performance of the original PSOalgorithm is enhanced efficiently.

4. Since the energy saving and emission reduction are the mainpurpose of this paper, some new economic and emission metricsare proposed for evaluating of wind power generations uncer-tainties in the SDEED (stochastic dynamic economic emissiondispatch) problem.

2. Uncertainty modeling

In the framework of SP, there are several types of techniques. Thestrategy which is used in this paper belongs to scenario generationand reduction approaches. A large number of scenarios are usuallygenerated in order to achieve an acceptable solution accuracy,which could increase the scale and the computation executionsignificantly. Then, a scenario reduction approach is implementedto reach the desired scenarios which are satisfied the stopping role.All the above procedures are explained in the following form.

2.1. Scenario generation

When dispatch decisions are made two sources of uncertaintyare present: load demands and wind power levels. The details ofhow to use the scenario generation considering the uncertainty ofboth factors can be found in Ref. [39]. Accordingly, load demand andwind power levels for each scenario can be expressed as follows:

Puw;f ;t;s ¼ Pforecastuw;f ;t þ DPuw;f ;t;s uw ¼ 1;.;NW;

f ¼ 1;.;NF; t ¼ 1;.;NT; s ¼ 1;.;NS ð1Þ

PD;t;s ¼ PforecastD;t þ DPD;t;s t ¼ 1;.;NT; s ¼ 1;.;NS (2)

Besides, each scenario has an occurrence probability whichassociate from all of the input random variables. The normalizedprobability of each generated scenario is obtained by using thefollowing equation:

ps ¼

YNTt¼1

0@X7l¼1

�WL

l;t;sal;t

�YNFf ¼1

X7w¼1

�WW

w;f ;t;sbw;t

�!1APNS

s¼1

YNTt¼1

0@X7l¼1

�WL

l;t;sal;t

�YNFf ¼1

X7w¼1

�WW

w;f ;t;sbw;t

�!1A (3)

B. Bahmani-Firouzi et al. / Energy 50 (2013) 232e244 235

2.2. Scenario reduction

For the sake of problem tractability, the size of the set of scenarios isconveniently reduced by scenario-reduction techniques [40]. In thisstudy to decrease the computation burden, a scenario reductionmechanism namely simultaneous backward reduction technique isapplied to reduce the number of scenarios, so that reasonably goodapproximation of the system uncertain behavior is preserved. To thisend, consider that there are NS different scenarios As(s¼ 1, ..., NS)withanoccurrenceprobabilityofps.Moreover,DTs;s0 is asCartesiandistancebetween scenario s and s0. The mechanism of this implementedscenario reduction can be summarized in the following form Ref. [40]:

Stage 1. An initial set of scenarios should be considered whichrepresented as S in this paper. Consider DS as the set of scenarioswhich must be deleted from the set of initial scenarios to reachactual realizations of the SP framework. Before the scenarioreduction process, the DS is null. Calculate the distances betweenall of the scenarios as follows DTs;s0 ¼ DTðAs;As0 Þ; s; s0 ¼ 1;.;NS

as DTs;s0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPd

i¼1ðasi � as0i Þ2q

. d is the dimension of each scenario

and asi is the ith member of scenario s.Stage 2. Determine other nearest scenario r to each scenario k

using DTk;r ¼ minDTk;s0 ; s0; k˛S and s0sk.Stage 3. Compute the PDk;r ¼ pk � DTk;r; k˛S for each pair of

scenario determined in the previous stage. Select d in whichPDd ¼ min PDk; k˛S.

Stage 4. After eliminating one scenario, add the probability of thedeleted scenario to the probability of the scenario which is closestto it as following form: S ¼ S � {d}, DS ¼ DS þ {d}, pr ¼ pr þ pd.

Stage5. Go to stage 2 and repeat stages 2e4 to eliminate scenariostill the desired number of final remaining scenarios is met.

2.3. Stopping rule

In this work, a criterion named CV (Coefficient of Variation) CVf isimplemented for choosing a number of scenarios to decide during thesimulation procedure, whether or not the selected realizations for theproposedproblemseemstobeaccurateenough. In thisregard,anumberof batches (a specific number of scenarios) are generated. As a result, ifthe results of the generated batch don’t satisfy the requirements of theSO, another batch should be generateduntil the value of CVf be less thana pre-specific tolerance. This procedure continues to reach the stoppingrule. This metric is calculated for each batch as follows [41]:

CVf ¼ sf

mfffiffiffiffiffiffiffiNS

p (4)

where, mf and sf are the mean and standard deviation values of theoutput random variable f, correspondingly.

3. Stochastic dynamic economic emission dispatch

This section presents the mathematical formulation of theSDEED (stochastic dynamic economic emission dispatch).

- Objective functions:

Minimization of the expected total electrical energy cost:

bf 1 ¼XNSs¼1

psf1�PG;s

�¼XNSs¼1

psXNTt¼1

XNGug¼1

�aug þ bugPug;t;s þ cugP2ug;t;s

þ���dugsin�eughPmin

ug � Pug;t;si����� ð5Þ

Minimization of the expected total combustion emissions

bf 2 ¼XNSs¼1

psf2�PG;s

�¼XNSs¼1

psXNTt¼1

XNGug¼1

�aug þ bugPug;t;s þ gugP

2ug;t;s

þ xugexp�lugPug;t;s

�� ð6Þ

It is noted that the fuel cost of each conventional unit (5) ischaracterized in the form of a quadratic function plus absolutevalue of sinusoidal term which refers to the valve point effects[39,42]. The emission of pollutant gases (6) is modeled in terms ofthe power output through a quadratic and an exponential termaccording to Ref. [39].

- Constraints:

XNGPug;t;s þ

XNF XNWPuw;f ;t;s ¼ PD;t;s þ PLoss;t;s

ug¼1 f ¼1 uw¼1

t ¼ 1;.;NT; s ¼ 1;.;NS ð7Þ

PLoss;t;s ¼XNUu¼1

XNUu0 ¼1

Pu;t;sBu;u0;tPu0;t;s þXNUu¼1

B0;u;tPu;t;s þ B00;t

t ¼ 1;.;NT; s ¼ 1;.;NS ð8Þ

Pug;t;s � Pug;t�1;s � RUug ug ¼ 1;.;NG; t ¼ 1;.;NT;

s ¼ 1;.;NS ð9Þ

Pug;t�1;s � Pug;t;s � RDug ug ¼ 1;.;NG; t ¼ 1;.;NT;

s ¼ 1;.;NS ð10Þ

Pminug � Pug;t;s � Pmax

ug ug ¼ 1;.;NG; t ¼ 1;.;NT;

s ¼ 1;.;NS ð11Þ

0@Dt;s ¼XNGug¼1

min�Pmaxug � Pug;t;s;URug=6

�� SRt;s

1A � 0

t ¼ 1;.;NT; s ¼ 1;.;NS ð12Þ

where PG, s ¼ [P1, s P2, s . PNT, s] and Pt, s ¼ [P1, t, s P2, t, s . PNG, t, s]T.Constraints (7) represent the power balance equation in

each period. Using the approximation based on the B-coeffi-cients [10], transmission network losses are expressed in (8).Constraints (9) and (10) model up and down ramp rate limits,respectively. Generation limits are set in (11). Similar to[9,30], 10 min spinning reserve requirements are consideredin (12).

The generated power of each wind turbine is directly per-tained to the wind speed level at the height of unit hub.The produced power of each wind turbine uw in wind powerplant f for scenario s at time t can be calculated asPuw;f ;t;s ¼ 1=2rArcpw3

uw;f ;t;s where ris the air density [Kg/m3]; Ar

is the area covered by the rotor [m2]; and cpis the performancecoefficient or power coefficient [8]. Since the non-linear coef-ficient has little effect and can be ignored, a simplified piece-wise linear approximation is adopted from Ref. [43] to

B. Bahmani-Firouzi et al. / Energy 50 (2013) 232e244236

represent the relationship between wind power generation andwind speed as depicted in the following equation:

Puw;f ;t;s ¼

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

Pmaxuw;f

�s1;uw;f wuw;f ;t;s� vci;uw;f

�; vci;uw;f �wuw;f ;t;s � v1;uw;f

Pmaxuw;f

�s1;uw;f

�v1;uw;f � vci;uw;f

�þ s2;uw;f

�wuw;f ;t;s� v1;uw;f

��; v1;uw;f �wuw;f ;t;s � v2;uw;f

Pmaxuw;f

�s1;uw;f

�v1;uw;f � vci;uw;f

�þ s2;uw;f

�v2;uw;f � v1;uw;f

�þ s3;uw;f

�wuw;f ;t;s� v2;uw;f

��; v2;uw;f �wuw;f ;t;s � vr;uw;f

Pmaxuw;f ; vr;uw;f �wuw;f ;t;s � vco;uw;f

0; otherwise

uw ¼ 1;.;NW; f ¼ 1;.;NF; t ¼ 1;.;NT; s ¼ 1;.;NS

(13)

- DEED Procedure under load and wind power uncertainties

If the under study power system is not elastic enough toincorporate wind power plants, this will have to be irremediablycurtailed [44]. The occurrence of wind spillage events (suddendrops in wind power or sudden increases [[45], Fig. 1]) should bestudied and considered as a comprehensible signal that activeactions according to the abovementioned features need to be usedto suitably accommodate larger number of wind power producers.Some of these actions are demand-side management, integratingnewcontrollable loads in the system, developing electricity storage,wind technology upgrading, optimal use and expansion of thetransmission networks, wind speed prediction upgrading, flexibleconventional generation and considering the spinning reservecapacity in the system. As you seen, three last techniques or actionshave been implemented in this paper. It should be noted that the SPproblems are characterized by the existence of some decisionvariables that have to be taken before some kind of uncertaintydisclose. These variables, called first-stage or here-and-now vari-ables, need to be the same for all scenarios. These variables are theon and off statues of generating units which are specified in the unitcommitment procedure and the DEED problem should be run withthese available units. So, according to the two-stage SP [37], thesame here-and-now variables are considered for all scenarios.Then, the wait and see variables are determined for each scenario.These variables are the power output of generating units for eachscenario. According to these second-stage variables the objectivefunctions are calculated for each scenario and the expected objec-tive functions are assigned to each particle in the FSALPSO proce-dures. Next, we describe the proposed approach for SDEED.

4. Solution methodology

4.1. Interactive fuzzy satisfying method

The details of how to apply the interactive fuzzy satisfyingmethod to simultaneously optimize both objective function ofSDEED can be found in Ref. [39]. By the way, an expected linear

(a) (b) (c)

Fig. 1. Membership functions: (a) NFks , (b) uk, (c) Duk.

membership function is defined for each expected objective func-tion as follows [39,41]:

bmfi ¼

8>>>>>>>>>>><>>>>>>>>>>>:

1; bf i � bfmini

bfmaxi � bf ibfmax

i � bfmini

; bfmini � bf i � bfmax

i

0; bf i � bfmaxi

(14)

The best solution, i.e., the closest to the requirements of the DM(decision maker), is the optimal solution to the following min-maxproblem [46]:

min�maxi¼1;2

mreffi

� bmfi

�(15)

\It is necessary to note that there are several methods which caninstitute the trade-off between all objective functions and extract thedesired solution between a set of non-inferior points. One of thesemethods extract only one solution; i.e. weighted coefficient method;another extract a set of point as the solutionof theproblem; i.e. Paretooptimal solution. The interactive fuzzy satisfyingmethod determinesseveral solutions between a set of points based on the preferences ofDM or SO. The SO can determine the condition of the system opera-tion by changing the RMV (reference membership values). RMVdefines the importance of each objective function. To compute theinteractive fuzzy satisfying method, a real value between 0 and 1 isassigned to each objective functionwhich represents the importanceandpreferenceofeachobjective functionforDMorSO. Indeed,he/sheruns the problem by a set of his/her desired RMV, if this situationcannot satisfy his/her preference, the problem is run by a new RMV.Furthermore, different situations for power system operation as wellas the conflicting and similarity of the objective functionmanner canbe investigated clearly during the interactive process.

4.2. Overview of PSO and qePSO

The PSO algorithm is one of the modern evolutionary compu-tation techniques for optimization problems which the details dataabout it can be established in Refs. [39,47]. Using a notationconsistent with that of Section 3, PSO is characterized by Piter

G;m;s,Viterm;s , and Pbestiterm;s which respectively represent the position, the

velocity, and the best position matrices of each particle m at iter-ation iter in scenario s, and can be defined as Ref. [47]:

PiterG;m;s ¼ ½Piter

m;1;s Piterm;2;s . Piter

m;NT;s � (16)

B. Bahmani-Firouzi et al. / Energy 50 (2013) 232e244 237

Viterm;s ¼ ½Viter

m;1;s Viterm;2;s . Viter

m;NT;s � (17)

Pbestiterm;s ¼ ½Pbestiterm;1;s Pbestiterm;2;s . Pbestiterm;NT;s � (18)

In addition, each particle m at iteration iter in scenario s isassociated with a fitness value:

FðPiterG;m;sÞ ¼ max

i¼1;2

hmreffi;s

� mfi;sðPiterG;m;sÞ

i(19)

The position with the best fitness value among all Pbestkm;s isdenoted by Gbestks . According to Ref. [39], the velocity and theposition of each particle are updated according to the followingequations:

Viterþ1m;s ¼uViter

m;s þ c1r1ðPbestiterm;s � PiterG;m;sÞ þ c2r2

� ðGbestiters � PiterG;m;sÞ m ¼ 1;.;NP; s ¼ 1;.;NS

ð20Þ

Piterþ1G;m;s ¼ Piter

G;m;s þ Viterþ1m;s m ¼ 1;.;NP; s ¼ 1;.;NS (21)

Despite many acceptable features of PSO [39], it may experienceinappropriate convergence, which motivates the three mainmodifications characterizing our proposed FSALPSO as describednext.

A problem with the numerous wait and see variables, likeSDEED, has a big search space. Thus, finding an optimum solutionfor the problem needs to cover a big space during the optimizationprocess which means a large number of initial-population anditerative search process are needed. Here, thewait and see variablescan be transformed to the polar coordinationwhere they defined bya magnitude and an angle. Since each wait and see variable has onedimension, defining the below equation remove the dependency ofpolar coordination from the variable’s magnitude. The followingprocess can be done for converting the search space from PSO to a qsearch space i.e. (qePSO):

Pm;t;ug;s ¼ Pol�Pm;t;ug;s;polar

�¼0@Pm;t;ug;s � Pm;t;ug;s

2� sin

�Pm;t;ug;s;polar

�þ Pm;t;ug;s þ Pm;t;ug;s

2

1Am ¼ 1;.;NP; t ¼ 1;.;NT; ug ¼ 1;.;NG; s ¼ 1;.;NS

(22)

where, Pm;t;ug;s;polar is the angle of the ugth unit at hour t for themthparticle in q-space. Above equation define a bijective mappingwhich means it is both onto and one-to-one. For getting to theaccelerate qePSO, the following velocity and the position updatingare applied to each particle as follows:

Viterþ1m;s;polar ¼ uViter

m;s;polar þ c1r1ðPbestiterm;s;polar � PiterG;m;s;polarÞ þ c2r2ðGbe

m ¼ 1;.;NP; s ¼ 1;.;NS

Piterþ1G;m;s;polar ¼ Piter

G;m;s;polar þViterþ1m;s;polar m ¼ 1;.;NP; s ¼ 1;.;NS

(24)

The q-space is mapped to the real space and vice versa. Applingthis technique give the following benefits:

a) Despite the real space, the upper and lower limits of the waitand see variables are the same; i.e. [�p/2,p/2], which simplifythe calculations.

b) As mentioned before, the q-space is compared to the real one.Thus the probability of finding an optimal solution near theglobal one is increased in this space. On the other word, for thealgorithm with the constant iteration and initial-populationa small search space can be investigated more watchfulcompared to the big one; so, searching in q-space preparea more suitable optimization condition.

c) When a big space is mapped on a smaller one, themovement insmall space has more acceleration. Indeed, a small movementin a real space corresponds to a big change in q-space. Here, thesmall change in q-space may not model in the real space andthe small and big tumble have better meaning in the q-space.

4.3. Fuzzy adapting of the inertia weight factor u

In the past years, many researchers have been tended towardsadapting suitable parameters for PSO. In some early works, theyanalyzed the impact of the inertia weight factor in PSO. Theyexposed that different combinations of u may result in differentperformance on the optimization functions. According to theimplemented empirical analysis, it was concluded that a time-varying u can result to a better performance for PSO algorithm. Inthis study, the u is iteratively adapted as follows:

uiterþ1 ¼ uiter þ Duiter (25)

Duiter is adequately determined using a fuzzy adapting systemon the basis of the following normalized fitness value [38]:

NFiters ¼ FðGbestiters Þ � FminFmax � Fmin

(26)

It is necessary to note that NFiters lies in the range [0, 1]. In theproposed fuzzy adapting system, NFiters and uiter comprise the inputvariables and the change in u i.e. Duiter is the output. A range of

stiters;polar � PiterG;m;s;polarÞ (23)

B. Bahmani-Firouzi et al. / Energy 50 (2013) 232e244238

[�0.1, 0.1] is chosen for Duiter. The fuzzy rules and membershipfunctions of this fuzzy system are depicted in Table 1 and Fig. 1,respectively. As it can be seen from Ref. [38], three fuzzy sets,indicated as large (L), medium (M) and small (S) are designed foreach input variable. In addition, three fuzzy sets like the positive(P), negative (N) and zero (Z) are set for the output variable.According to above discussion, there are nine “IF-THEN” rules. Forinstance, the first rule can be extracted as follows:

IF (NFiters is S) and (uiter is M) THEN Duiter is N.

4.4. Pseudo-code of the proposed FSALPSO algorithm

Input all required date.

1. Set the initial reference membership valuesmreffi ;s

¼ 1; i ¼ 1;2 and mreffi¼ 1

Begin

2. Scenario generation: generate NS scenarios.2.1. Initialization:

For m ¼ 1 to NPFor s ¼ 1 to NSFor t ¼ 1 to NTGenerate P0

m;t;s;polar randomly. Then, calculate P0m;t;s while

satisfying eqs. (7)e(13). To this end, the constraints handlingscheme which is described in Ref. [48] has been implemented.

End For (it refers to index t)Calculate f1ðP0

G;m;sÞ; f2ðP0G;m;sÞ from eqs. (5) and (6).

Calculate the membership values of both objectives and thenobtain max

i¼1;2½mreffi;s

� mfi;sðP0G;m;sÞ� according to Ref. [39].

End For (it refers to index s)Calculate the expected membership values of both objectives

using eq. (14) and then obtain maxi¼1;2

½mreffi� bmfi �:

End For (it refers to index m)Initialize:

Pbest0m;s ¼ P0G;m;s and Pbest0m;s;polar ¼ P0

G;m;s;polar:

Gbest0s and Gbest0s;polar: the position with the best fitness valueamong all Pbest0m;s and Pbest0m;s;polar, respectively.

V0m;s ¼ 0 and V0

m;s;polar ¼ 0.iter ¼ 1.

2.2. While iter � itermax

For m ¼ 1 to NPFor s ¼ 1 to NS.Modification on PSO (1): Update the inertia weight factor using

fuzzy rules on the basis of Subsection 4.3.Modification on PSO (2): SALPSO (Self-Adaptive Learning PSO).Update the velocity through the following self-adaptive

learning procedure:

Table 1Fuzzy rules for the inertia weight factor tuning.

uk

S M L

NFks S Z N NM P Z NL P Z N

Select three solutions q1 s q2 s q3 s m from the existingpopulation, randomly and choose one of the following velocityupdating strategy based on the Roulette wheel mechanismaccording to Ref. [49] to generate the new solution Pk

G;m;s;new asfollows:

Velocity updating strategy 1:

Viterm;s;polar;1 ¼randð:Þ1ðGbestiters;polar�Piter

G;m;s;polarÞþrandð:Þ2ðGbestiters;polar�Worstiters;polarÞ m¼ 1;.;Niter

1

ð27ÞVelocity updating strategy 2:

Viterm;s;polar;2 ¼ �0:3randð:Þ1 þ 0:2

�ðPiterG;q1;s;polar � Piter

G;q2;s;polarÞþ �0:3randð:Þ2 þ 0:2

�ðPbestiterm;s;polar � PiterG;m;s;polarÞ

m ¼ 1;.;Niter2 ð28Þ

Velocity updating strategy 3:

Viterm;s;polar;3 ¼uiterViter�1

m;s;polar;a þ�0:3randð:Þ1 þ 0:2

�randð:Þ2

� ðPbestiterq1;s;polar � PiterG;m;s;polarÞ;

m ¼ 1;.;Niter3 ; a ¼ 1;.;4 ð29Þ

Velocity updating strategy 4:

Viterm;s;polar;4 ¼uiterViter�1

m;s;polar;a þ 0:5�0:3randð:Þ1 þ 0:2

�randð:Þ2

� ðPbestiterq1;s;polar � PiterG;m;s;polar þ Pbestiterm;s;polar

� PiterG;m;s;polarÞ m ¼ 1;.;Niter

4 ; a ¼ 1;.;4 ð30ÞMove each particle according to eq. (24) to generate

PiterG;m;s;polar;new and Piter

G;m;s;new. To satisfy the constraints, theconstraints handling scheme which is described in Ref. [48] hasbeen implemented.

Calculate maxi¼1;2

½mreffi;s� mfi;sðPiter

G;m;s;new�:End For (it refers to index s)Calculate max

i¼1;2½mreffi

� bmfi �:End For (it refers to index m)Update the population using Piter

G;m;s:

Determine Pbestiterm;s ;Gbestiters :

Update Pbestiterm;s ;Gbestiters :

iter ¼ iter þ 1End While (it refers to index iter)

2.3. Return the best solution found PG;s; f1ðPG;sÞ; f2ðPG;sÞ; bf 1; bf 2:3. If the DM is satisfied with the current bf i, then stop.

Otherwise ask the DM to update mreffi ;s; i ¼ 1;2 and mreffi

and goto Begin.

5. Simulation and numerical results

The suggested SDEED scheme is illustrated next using two testsystems. The first test system includes 10 conventional units andlosses are considered. Loss coefficients are assumed to stayunchanged over the time intervals. The dispatch span is selected asone day with 24 dispatch intervals of each 1 h. The data for this testsystem is adapted from Refs. [5,6,12]. The second system is a large-scale and consists of 100 conventional units and time span is 24 hwhich obtained by making tenfold the first test system. In order tomeasure better, the performance of the suggested approach, thescalability study is conducted by using the second test system. The

Fig. 2. Forecasted load demand for 10-unit test system.

Table 2Results obtained by different method for DED1of Case I.

Solution technique Total electricalenergy costs ($)

Scaled CPUtime

Bestvalue

Meanvalue

Worst value (Min)

10-unit (without loss)SQP [16] 1,051,163 NA NA 0.421EP [16] 1,048,638 NA NA 15.049EP-SQP [16] 1,031,746 1,035,748 NA 7.264MDE [17] 1,031,612 1,033,630 NA 4.417HQPSO [18] 1.031,559 1,033,837 1,036,681 0.773PSO-SQP [19] 1,030,773 1,031,371 1,053,983 6.364MHEP-SQP [25] 1,028,924 1,031,179 NA 21.23DGPSO [20] 1,028,835 1,030,183 NA 4.809PSO-SQP (C) [19] 1,027,334 1,028,546 1,033,983 7.219SALPSO [21] 1,023,807 1,026,863 NA 0.050AIS [14] 1,021,980 1,023,156 1,024,973 25.346AHDE [22] 1,020,082 1,022,476 NA 1.10CDE method3 [23] 1,019,123 1,020,870 1,023,115 0.32ICPSO [24] 1,019,072 1,020,027 NA 0.350

B. Bahmani-Firouzi et al. / Energy 50 (2013) 232e244 239

forecast hourly load demand for test system 1 and 2 are shown inFigs. 2 and 3, respectively. The SRR is considered as a 5% of the loaddemand ineachhour for both test cases.Asa result, these test systemsare changed to incorporate wind power plants to the power system.For evaluation purposes, the single objective prescription of theSDEED is solved, namely the DED1 (dynamic economic dispatch) andthe DED2 (dynamic emission dispatch). Similarly, the deterministicversion of SDEED, referred to as DDEED (deterministic dynamiceconomicemissiondispatch), has alsobeen solved. The settings of theFSALPSO technique are as follows: The size of thepopulations is equalto50and150 for test system1and2, respectively.Maximumiterationnumber (itermax) is 200 and 500 for test system 1 and 2, corre-spondingly. To obtain the statistical results from the FSALPSO algo-rithmand compared itwith other techniques, 30 runs are carried out.The simulation procedures are carried out on a Pentium P4, Core 2Duo 2.4 GHz personal computer with 1 GB of RAM. The software isdeveloped via MATLAB 7.8.

For a better illustration of the performance of the suggestedmethod in the stochastic optimization, six batches of scenarios areconsidered. They consist of 6 � 30, 5 � 30, 4 � 30, 3 � 30, 2 � 30,and 30 samples of scenarios, correspondingly. As previouslymentioned, a scenario is constructed by samples of the load

Fig. 3. Forecasted load demand 100-unit test system.

demand and wind power for all wind power plants for all hours ineach time period. It is clear that the computational burden of thescenario-based technique is straightforwardly proportional to thenumber of realizations in each batch. Hence, the simultaneousbackward scenario reduction method would reduce the totalnumber of realizations considering a trade-off between solutionprecision and CPU time.

A. Case I: solving deterministic conventional DED1 and DED2 forevaluating the proposed FSALPSO algorithm

To validate the use and efficiency of the proposed optimizationmethod, the deterministic DED1 and DED2 with only using theconventional generating units are also solved and the results arecompared with that of the results reported in the literature byavailablemethods [5,12e26].Moreover, three additionalmethods i.e.PSO, FAPSO (Fuzzy Adaptive Particle Swarm Optimization) andSALPSO (Self-Adaptive Learning Particle Swarm Optimization) havealso been applied to investigate the computational gain provided bythree new features of FSALPSO as described in Section 4. Theproblem is solved for both conditions (considering and neglectinglosses). From Tables 2 and 3, it can be seen that, the FSALPSOmethod

ICA [26] 1,018,467 1,019,291 1,021,796 NATVAC-IPSO [15] 1,018,217 1,018,965 1,020,418 2.718PSO 1,020,607 1,021,781 1,023,210 0.232FAPSO 1,019,851 1,020,522 1,021,463 0.256PSO-method1 1,017,892 1,018,406 1,019,008 0.184PSO-method4 1.017,748 1,018,341 1,018,947 0.173PSO-method3 1,017,411 1,018,036 1,018,500 0.170PSO-method2 1,017,285 1,017,541 1,017,826 0.165SALPSO 1,016,971 1,017,157 1,017,248 0.163FSALPSO 1,016,818 1,016,952 1,017,003 0.15410-unit (with loss)EP [25] 1,054,685 1,057,323 NA 47.23EP-SQP [25] 1,052,668 1,053,771 NA 27.53IPSO [21] 1,046,275 1,048,154 NA 0.150AIS [14] 1,045,715 1,047,050 1,048,431 30.973TVAC-IPSO [15] 1,041,066 1,042,118 1,043,625 3.155ICA [26] 1,040,758 1,041,665 1,043,174 NAPSO 1,048,903 1,049,780 1,050,996 0.498FAPSO 1,046,870 1,047,964 1,049,109 0.549SALPSO 1,038,104 1,038,610 1,039,158 0.304FSALPSO 1,037,698 1,037,814 1,038,049 0.300100-unitPSO 10,209,005 10,214,497 10,219,855 4.280FAPSO 10,200,860 10,207,245 10,211,461 4.441SALPSO 10,171,951 10,174,762 10,176,223 2.451FSALPSO 10,171,034 10,172,356 10,174,012 2.412

NA: not available in the literature.

Table 3Results obtained by different method for DED2 of Case I.

Solution technique Total combustion emissions (lb) Scaled CPUtime (min)

Best value Mean value Worst value

10-unit (without loss)PSO 270,003 270,044 270,071 0.071FAPSO 269,981 269,998 270,027 0.083SALPSO 269,936 269,945 269,956 0.058FSALPSO 269,930 269,941 269,948 0.05410-unit (with loss)PSO 289,309 289,367 289,413 0.162FAPSO 289,287 289,341 289,389 0.178SALPSO 289,223 289,234 289,240 0.101FSALPSO 289,208 289,219 289,228 0.099100-unitPSO 2,702,349 2,703,466 2,704,981 0.945FAPSO 2,701,133 2,701,781 2,702,068 0.978SALPSO 2,699,846 2,700,140 2,700,477 0.502FSALPSO 2,699,754 2,700,031 2,700,228 0.481

NA: not available in the literature.

B. Bahmani-Firouzi et al. / Energy 50 (2013) 232e244240

obtains lower best, mean and worst generation cost through 30calculation trails than the PSO, FAPSO, SALPSO, EP (evolutionaryprogramming) [5], AIS [14], TVAC-IPSO [15], EP [16], SQP [16], EP-SQP[16], MDE [17], HQPSO [18], PSO-SQP [19], PSO-SQP (C) [19], DGPSO[20], IPSO [21], AHDE [22], CDEmethod3 [23], ICPSO [24], MHEP-SQP[25] and ICA [26], thus, resulting in the higher quality solution thanother methods in this area. In this study, the Scaled CPU time isimplemented instead of average CPU time according to Ref. [17] inorder to normalize the CPU time of other methods in the imple-mented computer for simulation process.

In order to check whether the constraints of the problem aresatisfied or not, the obtained dispatch results of the FSALPSOsolution for 10-unit test system (with and without loss) are given inTables 4 and 5, respectively. The total loss for the 10-unit testsystem is 808.5MW. In Fig. 4, the best convergence performance forPSO, PSO with velocity updating method1, PSO with velocityupdating method2, PSO with velocity updating method3, PSO withvelocity updating method4 and FSALPSO for 10-unit test systemneglecting losses is depicted. This figure indicates that the FSALPSOconsistently converges faster than other methods.

Table 4Best dispatch found by FSALPSO for DED1 (10-unit without loss) of Case I.

Hour Load P1 (MW) P2 (MW) P3 (MW) P4 (MW) P5 (

1 1036 150.0000 309.5329 73.0000 60.0000 732 1110 150.0000 309.5329 143.4268 60.0000 733 1258 226.6242 309.5329 214.8026 60.0000 734 1406 303.2484 309.5329 286.1784 60.0000 735 1480 303.2484 314.3697 305.4750 60.0000 1226 1628 379.8726 394.3697 296.8508 60.0000 1227 1702 379.8726 396.7994 318.5546 60.0000 1728 1776 379.8726 396.7994 297.2511 105.4369 2229 1924 456.4968 396.7994 297.3995 146.6643 22210 2072 456.4968 396.7994 325.0782 190.8316 22211 2146 456.4968 396.7994 317.0211 240.8316 22212 2220 456.4968 460.0000 323.1204 241.1356 22213 2072 456.4968 396.7994 318.3210 191.1356 22214 1924 456.4968 396.7994 289.9282 141.1356 22215 1776 379.8726 396.7994 297.4553 116.7872 17216 1554 303.2484 396.7994 281.9461 66.7872 12217 1480 226.6242 396.7994 299.6696 60.0000 12218 1628 303.2484 396.7994 321.1788 60.0000 17219 1776 379.8726 396.7994 297.3876 105.3005 22220 2072 456.4968 460.0000 340.0000 121.3131 22221 1924 456.4968 396.7994 316.1140 120.3997 22222 1628 379.8726 392.2880 238.6663 70.3997 17223 1332 303.2484 312.2880 159.5567 60.0000 12224 1184 226.6242 309.5329 90.9360 60.0000 122

B. Case II: DED1 and DED2 with wind power plants: stochastic vs.deterministic

In this case, the 10-unit system is modified by replacingconventional unit 9 by 40 � 2-MW wind turbines, and bysubstituting conventional unit 10 by 55 � 1-MW wind turbines.Wind speed forecasts are depicted in Fig. 5 where w1 and w2 areused for respective first and second wind power plants. The large-scale 100-unit test system is implemented by making tenfold themodified 10-unit test system. This test system is consisted of 80conventional units and 20 wind power plants. Based on Ref. [33],the parameters for wind turbines are: vci ¼ 4 m/s, vr ¼ 14 m/s,vco ¼ 25 m/s, v1 ¼ 7 m/s, v2 ¼ 12 m/s, s1 ¼ 0.2/(v1 � vci),s2 ¼ (0.96 � 0.2)/(v2 � v1), s3 ¼ (1 � 0.96)/(vr � v2). For the sake ofsimplicity, the same parameters are considered for all windturbines. In addition, it should be pointed out that the configurationof the wind power plants is important and affect the wind speedforecast. In the proposedmodel, we disregard this issue for the sakeof simplicity. However, note that incorporating it in the proposedmodel is straightforward.

Due to the randomicity and uncontrollability of load demandsand wind power outputs of all wind power plants, the proposed SPtechnique can obtain a more exact response for providing theacceptable and optimum total electrical energy costs andcombustion emissions for the SO. The stochastic DED1&2, namedSDED1&2 are firstly solved by usage of the scenario generationscheme described in Section 2.1. 1000 scenarios consisting of theload demands and wind powers of all wind turbines for a time spanof 24 h are generated and accordingly decreased to six batches ofscenarios. According the stopping rule discussion, the number ofscenarios with the less value of CV is usually selected [41]. Table 6shows the CV for the aforementioned batches for test system 1.According to this table, the total electrical energy costs are reducedwith the increasing of the total number of the realizations forselected batches. The statistical metrics like the expected, SD(Standard Deviation), mean, relative error [40], 95% confidenceinterval [40], CV [41], and the VSS (Value of the Stochastic Solution)[33] are also depicted in this Table. The values of all metrics exceptSD are decreased slightly when the number of realizations isincreased. The mean value of the set of the scenario’s output

MW) P6 (MW) P7 (MW) P8 (MW) P9 (MW) P10 (MW)

.0000 118.8766 129.5904 47.0000 20.0000 55

.0000 122.4498 129.5904 47.0000 20.0000 55

.0000 122.4498 129.5904 47.0000 20.0000 55

.0000 122.4498 129.5904 47.0000 20.0000 55

.8666 122.4498 129.5904 47.0000 20.0000 55

.8666 122.4498 129.5904 47.0000 20.0000 55

.7331 122.4498 129.5904 47.0000 20.0000 55

.5997 122.4498 129.5904 47.0000 20.0000 55

.5997 122.4498 129.5904 77.0000 20.0000 55

.5997 160.0000 129.5904 85.6039 50.0000 55

.5997 160.0000 129.5904 115.6039 52.0571 55

.5997 160.0000 129.5904 120.0000 52.0571 55

.5997 160.0000 129.5904 120.0000 22.0571 55

.5997 122.4498 129.5904 90.0000 20.0000 55

.7331 122.4498 129.5904 85.3121 20.0000 55

.8666 122.4498 129.5904 55.3121 20.0000 55

.8666 122.4498 129.5904 47.0000 20.0000 55

.7331 122.4498 129.5904 47.0000 20.0000 55

.5997 122.4498 129.5904 47.0000 20.0000 55

.5997 160.0000 129.5904 77.0000 50.0000 55

.5997 160.0000 129.5904 47.0000 20.0000 55

.7331 122.4498 129.5904 47.0000 20.0000 55

.8666 122.4498 129.5904 47.0000 20.0000 55

.8666 122.4498 129.5904 47.0000 20.0000 55

Table 5Best dispatch found by FSALPSO for DED1 (10-unit with loss) of Case I.

Hour Load (MW) P1 (MW) P2 (MW) P3 (MW) P4 (MW) P5 (MW) P6 (MW) P7 (MW) P8 (MW) P9 (MW) P10 (MW) Loss (MW)

1 1036 226.6277 135.0000 179.0508 60.0000 73.0000 122.4505 129.5904 47.0000 20.0000 55 11.71942 1110 226.6269 135.0000 204.7742 60.0015 122.8663 122.4532 129.5905 47.0000 20.0000 55 13.31263 1258 226.6464 215.0000 278.7182 60.0000 122.8093 122.4467 129.5904 47.0000 20.0000 55 19.21094 1406 303.2483 222.2638 297.3397 60.0000 172.7291 122.4498 129.5904 47.0000 20.0000 55 23.62115 1480 379.7613 222.2667 297.3928 60.0000 172.7335 122.4517 129.5905 47.0000 20.0000 55 26.19646 1628 379.7794 302.2667 292.4475 60.0000 222.5989 122.4301 129.5910 77.0000 20.0000 55 33.11367 1702 379.8906 309.5421 301.5586 109.9999 222.6051 122.6619 129.5901 85.3165 20.0000 55 34.16488 1776 456.5095 312.2950 298.5570 159.9946 172.7674 122.4543 129.5959 85.3138 20.0000 55 36.48759 1924 456.4982 392.2946 297.5968 188.5292 222.6006 122.5548 129.5918 85.50469 20.0000 55 46.170610 2072 456.5001 396.8003 326.4138 238.5292 222.6135 160.0000 129.5936 115.5047 20.0000 55 48.955211 2146 456.5002 396.8007 339.9988 288.5292 230.5890 160.0000 129.5913 120.0000 20.0104 55 51.019612 2220 456.4977 460.0000 325.0709 299.9990 222.5997 160.0000 129.5904 120.0000 50.0104 55 58.768113 2072 456.4971 396.7993 306.4969 291.5919 222.6035 122.4758 129.5903 120.0000 20.0104 55 49.065314 1924 456.4848 316.7998 310.5458 241.5919 222.5975 122.4539 129.5904 90.0000 20.0000 55 41.064115 1776 379.8705 309.534 295.4981 191.5919 222.6526 122.4515 129.5905 85.3122 20.0000 55 35.501316 1554 303.2253 229.534 296.9255 163.6765 172.7177 122.4476 129.5905 85.3120 20.0000 55 24.429117 1480 226.6271 222.2791 299.5011 120.5255 222.6786 122.5036 129.5913 85.3127 20.0000 55 24.019118 1628 303.2294 302.2791 293.1260 125.4348 222.5467 122.4499 129.5891 85.3125 20.0000 55 30.967519 1776 379.8732 312.6511 308.6025 175.4334 222.6057 122.4944 129.5911 85.3127 20.0003 55 35.564520 2072 456.7663 392.6511 339.9996 225.4334 225.7206 160.0000 129.5932 115.3127 20.0000 55 48.476921 1924 456.5514 390.8432 306.4618 180.8225 222.6052 122.7249 129.5901 85.3127 20.0024 55 45.914222 1628 379.8752 310.8432 283.6122 130.8225 172.8075 122.6767 129.5932 55.3128 20.0000 55 32.543223 1332 303.2107 230.8432 203.6122 118.7253 122.8630 122.4500 129.5911 47.0000 20.0000 55 21.295424 1184 226.6257 222.2673 184.5950 120.4182 73.0000 122.4633 129.5905 47.0000 20.0000 55 16.9600

B. Bahmani-Firouzi et al. / Energy 50 (2013) 232e244 241

objective function is their sum divided by the number of scenarios.The SD is the positive square root of the mean of the squares of thedeviations of the given scenario’s output objective function valuesfrom their mean. The 95% confidence interval is a confidenceinterval surrounding the mean value of a set of observations. Theselection of a confidence level for an interval determines theprobability that the confidence interval produced will contain thetrue parameter value. A common choice for the confidence level is0.95. This level corresponds to percentage of the area of the normaldensity curve. A 95% confidence interval covers 95% of the normalcurve and the probability of observing a value outside of this area isless than 0.05. According to Refs. [40,41], a 95% confidence intervalfor the given mean value is Mean� 1:96SD=

ffiffiffiffiffiffiNS

p. The smaller the

value of 1:96SD=ffiffiffiffiffiffiNS

p, the more accurate output objective functions

of the scenarios is Ref. [41]. The difference value between theoptimal output solutions of the single objectives DDEED and SDEEDis represented as VSS on the basis of Ref. [33]. In addition, therelative error is computed as ((95% confidence interval)/expected

Fig. 4. Convergence graphs of FSALPSO, PSO-method1, PSO-method2, PSO-method3,PSO-method4 and PSO for DED1 (10-unit without loss) for Case I.

value) � 100% [41]. The smaller amount for this metric shows theefficiency of the implemented SP framework. It is worthwhile tonote that the main drawback of the huge scenarios in each batch isthe enormous CPU execution time requirement. It is clear fromTable 6, the tradeoff between the total electrical energy costs andthe CPU execution time depicts the superiority of the proposedbackward scenario reduction technique. As a result, the 30scenarios are considered for the SP problems.

The corresponding deterministic problem has been solved byconsidering the most probable scenario, i.e., that corresponding tothe forecast values shown in Figs. 2, 3 and 5 for wind speed and loaddemand levels. The results for both test systems are listed in Table 7.It is worth noting that modeling load andwind power uncertaintiesincrease the cost and emission values as well as CPU time withrespect to the deterministic scheme.

With the considering of the batch with 30 scenarios, the firstand second test system with respective (8 conventional units andtwo wind power plants) and (80 conventional units and 20 wind

Fig. 5. Forecasted wind speed profiles.

Table 6Comparison of the results obtained by different scenarios for SDED1 (10-unit without loss) of Case II.

No. of scenarios Total operation costs ($) Relative error (%) Coefficient of variation (%) CPU time (min)

Mean SD 0.95% Confidence intervals Expected VSS

1 � 30 897,875 27,437 9818 900,253 26,580 1.0905 0.5579 4.6902 � 30 897,338 28,691 7260 900,144 26,471 0.8037 0.4128 8.2583 � 30 898,724 29,028 5997 900,126 26,453 0.6662 0.3405 12.5194 � 30 896,887 30,132 5391 900,071 26,398 0.5989 0.3067 15.6515 � 30 897,200 30,275 4845 900,063 26,390 0.5383 0.2755 19.1566 � 30 897,295 31,378 4584 900,049 26,376 0.5093 0.2606 22.066

Table 7Results for the stochastic problems of Case II.

Problem Method Best solution Worst solution Average solution CPU time (min)

10-units (without loss) DED1 ($) FSALPSO 900,253 901,235 900,897 4.659SALPSO 900,499 901,615 901,103 4.728FAPSO 912,339 916,117 914,233 8.166PSO 913,734 918,465 915,899 7.739

DED2 (lb) FSALPSO 241,083 241,097 241,088 1.511SALPSO 241,088 241,115 241,104 1.694FAPSO 243,477 245,841 244,759 2.380PSO 248,412 250,852 249,536 2.013

10-units (with loss) DED1 ($) FSALPSO 921,070 923,616 922,381 8.213SALPSO 921,605 923,804 922,793 8.448FAPSO 928,277 931,812 930,423 14.760PSO 929,153 932,539 930,991 14.128

DED2 (lb) FSALPSO 262,458 262,528 262,493 2.377SALPSO 262,770 262,980 262,857 2.469FAPSO 264,491 264,982 264,704 3.702PSO 266,198 267,449 266,732 3.317

100-units DED1 ($) FSALPSO 9,071,297 9,072,891 9,072,026 22.334SALPSO 9,072,225 9,074,710 9,073,451 22.456FAPSO 9,138,803 9,166,824 9,149,722 43.037PSO 9,142,637 9,169,557 9,155,108 42.722

DED2 (lb) FSALPSO 2,417,113 2,418,870 2,418,001 13.189SALPSO 2,418,145 2,420,072 2,419,341 13.375FAPSO 2,445,759 2,450,106 2,447,981 25.557PSO 2,489,920 2,498,005 2,493,146 25.081

B. Bahmani-Firouzi et al. / Energy 50 (2013) 232e244242

power plants), the SDED1&2 are solved. The results obtained byFSALPSO have been assessed with those obtained by SALPSO,FAPSO, and the classical PSO. Expected values of costs and emis-sions over the 30 scenarios are calculated in each run. Table 8 liststhe best, worst, and average expected values of cost and emissionsover the 30 runs which each run consists of SP based on a numberof 30 scenarios. Furthermore, the average computing time is re-ported. As can be seen, the suggested FSALPSO provides lower ex-pected electrical energy costs and lower expected combustionemissions than the other methods. Therefore, FSALPSO is effectivein providing better solutions and shows amore robust performancein the stochastic optimization problems.

Rising concern about the environmental and economical issues,two contradictory properties of integrating wind power plants tothe electric power networks should be considered and evaluateswhich include as follows: (i) Substituting the wind power plantswith conventional power generators in the aforementioned test

Table 8Results for the deterministic problems of Case II.

No. of units DED1 DED2

Cost ($) CPU time (min) Emission (lb) CPU time (min)

10 (without loss) 873,673 0.298 232,355 0.04910 (with loss) 895,480 0.573 253,892 0.101100 8,973,518 3.811 2,336,902 0.502

systems leads to a reduction of the total electrical energy costs andtotal combustion emissions due to the zero-cost and zero-emissionof wind power producers [44]. (ii) Since the produced wind powerof wind power plants is unmanageable and hard to forecastprecisely, it contributes to characterize the uncertainty of windpower variables and requires the SO to consider SRR or take furtherpower from the conventional power generators in order to keep thesecurity of the power network. This raise in the power output of theconventional units means that the total electrical energy costs andcombustion emissions will increase. Since the energy saving andemission reduction are the main purpose of this paper, severaleconomic and emissionmetrics are came into view for evaluating ofwind power generation in the following form.

1. Average cost and emission saving: These metrics in $/MWh andlb/MWh are the measures of the reducing in expected totalelectrical energy costs and emissions according to injecting anextra MWh of wind power into the power systems test cases.They can be defined as follows:

without wind b

ACS

�Average Cost Saving

� ¼

f1 � f 1PNTt¼1

PNFf ¼1

PNWuw¼1 P

forecastuw;f ;t

NT(31)

Table 9Results of the interactive fuzzy satisfying procedure for SDEED.

Interaction mreff1mreff2

10-Units (without loss) 10-Units (with loss) 100-Unitsbf 1 ($) bf 2 (lb) CPU time (min) bf 1 ($) bf 2 (lb) CPU time (min) bf 1 ($) bf 2 (lb) CPU time (min)

0 1.00 1.00 914,579 264,289 2.9595 935,596 284,568 4.4315 9,167,239 2,703,154 25.63351 1.00 0.95 912,655 273,470 2.9635 934,178 290,031 4.4805 9,149,713 2,828,691 26.39052 1.00 0.90 911,209 283,711 2.9775 933,042 298,120 4.4875 9,129,825 2,912,333 26.56503 1.00 0.85 909,363 292,879 2.9835 931,836 306,405 4.4945 9,110,496 2,996,912 26.72704 0.95 1.00 916,623 255,426 2.9505 938,210 276,137 4.4225 9,187,622 2,631,476 25.49105 0.90 1.00 920,073 248,748 2.9445 941,629 269,762 4.4140 9,219,519 2,549,817 24.95606 0.85 1.00 925,311 245,208 2.9395 945,103 267,329 4.4050 9,276,431 2,500,438 24.6185

B. Bahmani-Firouzi et al. / Energy 50 (2013) 232e244 243

fwithout wind2 � bf 2

AES�Average Emission Saving

� ¼ PNTt¼1

PNFf ¼1

PNWuw¼1 P

forecastuw;f ;t

NT(32)

where, fwithout wind1 and fwithout wind

2 are the deterministic totalelectrical energy costs and combustion emissions of the systemwithout wind power generation, respectively. The ACS (AverageCost Saving) and AES (Average Emission Saving) are (1.566 $/MWhand 0.387 lb/MWh), (1.565 $/MWh and 0.359 lb/MWh), (1.477$/MWh and 0.379 lb/MWh) for 10-unit without loss, 10-unit withloss and 100-unit test systems, respectively. It can be seen thatusing wind power generations can greatly save the total costs andreduce the total emissions of power system test cases.

2. Average uncertainty costs and emissions: These metrics in$/MWh and lb/MWh can be defined as follows:

bf 1 � fwith wind1P P P

AUC�Average Uncertainty Cost

� ¼ NTt¼1

NFf ¼1

NWuw¼1 P

forecastuw;f ;t

NT(33)

AUE�AverageUncertaintyEmission

�¼bf 2�fwithwind

2PNTt¼1PNF

f¼1PNW

uw¼1Pforecastuw;f ;t

NT(34)

where, fwith wind1 and fwith wind

2 are the deterministic total electricalenergy costs and combustion emissions of the system with windpower generation, respectively. The AUC (Average UncertaintyCost) and AUE (Average Uncertainty Emission) are (0.357 $/MWhand 0.117 lb/MWh), (0.344 $/MWh and 0.115 lb/MWh), (0.131$/MWh and 0.108 lb/MWh) for 10-unit without loss, 10-unit withloss and 100-unit test systems, respectively.

It should be noted that the four performance metrics (ACS, AES,AUC, and AUE) have positive amounts and take on nearly the samevaluesdespite thenatureof test cases.However, underconsidering lossin the first test system, the ACS and AES metrics decreases since thesystem cannot fully save electrical energy cost and combustion emis-sion from the wind potential due to the transmission loss restrictionsimposed by the power network. The AUC and AUE also reduce due to

Table 10Results for interaction 0 of DDEED.

No. of units f1 ($) f2 (lb) CPU time (min)

10 (without loss) 913,626 259,752 0.11610 (with loss) 934,877 279,334 0.223100 9,160,027 2,669,315 1.080

the power network constraint and accordingly reducing, the uncertainvariability of wind power generation of wind power plants.

C. Case III: DDEED and SDEED with uncertainty on wind power andload demand

In this case, wind power production data are equal to thoseimplemented in Case III. Analogously, the number of 24-h scenariosafter scenario reduction and the number of runs are both equal to30. The results of all interactions analyzed are shown in Table 9. Itcan be seen that the DM can consider its past experience to changethe reference membership values to reach the desired solutions.

Table 10 provides the results of the suggested interactive fuzzysatisfying procedure for DDEEDwith mreffi;s

¼ 1 and mreffi¼ 1 where

s is equal to the index of the most probable scenario.It can be concluded from the analysis of the simulation results

that characterizing of the uncertainties increases the total electricalenergy costs and combustion emissions amounts since differentprobable scenarios are considered in the SP instead of one scenariowhich is considered for the deterministic scheme. As a result, thedeterministic scenario doesn’t give an acceptable solution by itself.On the other hand, using the obtained scenarios from scenarioreduction procedure, all 30 scenarios contribute into determiningthe SDEED results according to their probability values. The 30representative scenarios totally capture more of the uncertaintyspectrumof thewindpowerofwindpower plants and loaddemand,which is close to three times more than that of the deterministicscenario. So, the multi-objective SDEED results of the stochasticanalysis are more realistic than the deterministic analysis.

6. Conclusions

This paper provides a novel FSALPSO algorithm to cope witha stochastic multi-objective DEED problem in which the valve-point effects, ramping limits, spinning reserve requirements, andwind power constraints are modeled. In this regard, a self-adaptivelearning framework is used to probabilistically guide four effectivevelocity updating strategies with different features from thecandidate strategy pool in parallel to optimize the suggested non-linear non-smooth and non-convex problem. The use ofa scenario-based approach to characterize the stochastic nature ofthe load and wind power forecast errors renders appropriateresults for the DEED problem. The stochastic approach allows thedispatcher to account for the likelihood of uncertainties therebyleading to amore efficient use of energy resources. Since the energysaving and emission reduction are the main purpose of this paper,several economic and emission metrics like ACS and AES areappeared for evaluating of wind power generation. It can be seenthat by injecting an extra MWh of wind power into the powersystems test cases, the respective total costs and emissions ofpower system test cases are reduced 1.5 $/MWh and 0.38 lb/MWh,approximately.

B. Bahmani-Firouzi et al. / Energy 50 (2013) 232e244244

Moreover, the proposed multi-objective method can resolve thetradeoff between the conflicting objectives of the SDEED problemthrough an interactive procedure by which the DM can update thesolutions based on its preferences through the optimizationprocess. Numerical experiments back the effective performance ofthe suggested approach.

Future work will include an estimate of the number andfrequency of “time out” hours for wind power. In addition, in thefuture study an estimate of the return on investment in awind farmfor savings of $1.50/MWh might be included.

References

[1] The European Wind Energy Association website. Available from: www.ewea.org; July 2011.

[2] Hong Y-Y, Chang H-L, Chiu C-S. Hour-ahead wind power and speed fore-casting using simultaneous perturbation stochastic approximation (SPSA)algorithm and neural network with fuzzy inputs. Energy 2010;35:3870e6.

[3] Giorgi MGD, Ficarella A, Tarantino M. Assessment of the benefits of numericalweather predictions in wind power forecasting based on statistical methods.Energy 2011;36:3968e78.

[4] Masmoudi A, Abdelkafi A, Krichen L. Electric power generation based on vari-able speed wind turbine under load disturbance. Energy 2011;36:5016e26.

[5] Basu M. Dynamic economic emission dispatch using evolutionary program-ming and fuzzy satisfyingmethod. Int J Emerg Electric Power Syst 2007;8. art 1.

[6] Basu M. Dynamic economic emission dispatch using nondominated sortinggenetic algorithm-II. Int J Electr Power Energ Syst 2008;30:140e9.

[7] Liao G-C. A novel evolutionary algorithm for dynamic economic dispatch withenergy saving and emission reduction in power system integrated windpower. Energy 2011;36:1018e29.

[8] Peng C, Sun H, Guo J, Liu G. Dynamic economic dispatch for wind-thermalpower system using a novel bi-population chaotic differential evolutionalgorithm. Int J Electric Power Energ Syst 2012;42:119e26.

[9] Niknam T, Azizipanah-Abarghooee R, Roosta A, Amiri B. A newmulti-objectivereserve constrained combined heat and power dynamic economic emissiondispatch. Energy 2012;42:530e45.

[10] Wood AJ, Wollenberg BF. Power generation, operation, and control. 2nd ed.USA, New York: Wiley; 1996.

[11] Ross DW, Kim S. Dynamic economic dispatch of generation. IEEE Trans PowerAppar Syst 1980;PAS-99:2060-8.

[12] Panigrahi CK, Chattopadhyay PK, Chakrabarti RN, Basu M. Simulated anneal-ing technique for dynamic economic dispatch. Electr Power Compon Syst2006;34:577e86.

[13] Panigrahi BK, Pandi VR, Das S. Adaptive particle swarm optimization approachfor static and dynamic economic load dispatch. Energ Convers Manag 2008;49:1407e15.

[14] Hemamalini S, Simon SP. Dynamic economic dispatch using artificial immunesystem for units with valve-point effect. Electr Power Energ Syst 2011;33:868e74.

[15] Mohammadi-ivatloo B, Rabiee A, Ehsan M. Time-varying acceleration coeffi-cients IPSO for solving dynamic economic dispatch with non-smooth costfunction. Energ Convers Manag 2012;56:175e83.

[16] Attaviriyanupap P, Kita H, Tanaka E, Hasegawa J. A hybrid EP and SQP fordynamic economic dispatch with nonsmooth fuel cost function. IEEE TransPower Syst 2002;17:411e6.

[17] Yuan X, Wang L, Yuan Y, Zhang Y, Cao B, Yang B. A modified differentialevolution approach for dynamic economic dispatch with valve point effects.Energ Convers Manag 2008;49:3447e53.

[18] Chakraborty S, Senjyu T, Yona A, Saber AY, Funabashi T. Solving economic loaddispatch problem with valve-point effects using a hybrid quantum mechanicsinspiredparticle swarmoptimization. IETGener TransmDistrib 2011;5:1042e52.

[19] Victoire TAA, Jeyakumar AE. Reserve constrained dynamic dispatch of unitswith valve-point effects. IEEE Trans Power Syst 2005;20:1273e82.

[20] Victoire TAA, Jeyakumar AE. Deterministically guided PSO for dynamic dispatchconsidering valve-point effects. Electr Power Syst Res 2005;73:313e22.

[21] Yuan X, Su A, Yuan Y, Nie H, Wang L. An improved PSO for dynamic loaddispatch of generators with valve-point effects. Energy 2009;34:67e74.

[22] Lu Y, Zhou J, Qin H, Li Y, Zhang Y. An adaptive hybrid differential evolutionalgorithm for dynamic economic dispatch with valve-point effects. ExpertSyst Appl 2010;37:4842e9.

[23] Lu Y, Zhou J, Qin H, Wang Y, Zhang Y. Chaotic differential evolution methodsfor dynamic economic dispatch with valve-point effects. Eng Appl Artif Intel2011;24:378e87.

[24] Wang Y, Zhou J, Qin H, Lu Y. Improved chaotic particle swarm optimizationalgorithm for dynamic economic dispatch problem with valve-point effects.Energ Convers Manag 2010;51:2893e900.

[25] Victoire TAA, Jeyakumar AE. A modified hybrid EP-SQP approach for dynamicdispatch with valve-point effect. Electr Power Energ Syst 2005;27:594e601.

[26] Mohammadi-ivatloo B, Rabiee A, Soroudi A, Ehsan M. Imperialist competitivealgorithm for solving non-convex dynamic economic power dispatch. Energy2012;44:228e40.

[27] Vahidinasab V, Jadid S. Joint economic and emission dispatch in energymarkets: a multiobjective mathematical programming approach. Energy2010;35:1497e504.

[28] Panigrahi BK, Ravikumar Pandi V, Das S, Das S. Multiobjective fuzzy domi-nance based bacterial foraging algorithm to solve economic emission dispatchproblem. Energy 2010;35:4761e70.

[29] Xia X, Elaiw AM. Optimal dynamic economic dispatch of generation: a review.Electr Power Syst Res 2010;80:975e86.

[30] Attaviriyanupap P, Kita H, Tanaka E, Hasegawa J. A fuzzy-optimizationapproach to dynamic economic dispatch considering uncertainties. IEEE TransPower Syst 2004;19:1299e307.

[31] Liang R-H, Liao J-H. A fuzzy-optimization approach for generation schedulingwith wind and solar energy systems. IEEE Trans Power Syst 2007;22:1665e74.

[32] Liu X, Xu W. Minimum emission dispatch constrained by stochastic windpower availability and cost. IEEE Trans Power Syst 2010;25:1705e13.

[33] Azizipanah-Abarghooee R, Niknam T, Roosta A, Malekpour AR, Zare M.Probabilistic multiobjective wind-thermal economic emission dispatch basedon point estimated method. Energy 2012;37:322e35.

[34] Niknam T, Golestaneh F, Malekpour A. Probabilistic energy and operationmanagement of a microgrid containing wind/photovoltaic/fuel cell generationand energy storage devices based on point estimate method and self-adaptivegravitational search algorithm. Energy 2012;43:427e37.

[35] Niknam T, Kavousi Fard A, Baziar A. Multi-objective stochastic distributionfeeder reconfiguration problem considering hydrogen and thermal energyproduction by fuel cell power plants. Energy 2012;42:563e73.

[36] Bunn DW, Paschentis SN. Development of a stochastic model for the economicdispatch of electric power. Eur J Oper Res 1986;27:179e91.

[37] Birge JR, Louveaux F. Introduction to stochastic programming. USA, New York:Springer-Verlag; 1997.

[38] Niknam T, Mojarrad HD, Nayeripour M. A new fuzzy adaptive particleswarm optimization for non-smooth economic dispatch. Energy 2010;35:1764e78.

[39] Azizipanah-Abarghooee R, Aghaei J. Stochastic dynamic economic emissiondispatch considering wind power. IEEE Power Eng Auto Conf (PEAM) 2011;1:158e61.

[40] Wu L, Shahidehpour M, Li T. Stochastic security-constrained unit commit-ment. IEEE Trans Power Syst 2007;22:800e11.

[41] Niknam T, Azizipanah-Abarghooee R, Narimani MR. An efficient scenario-based stochastic programming framework for multi-objective optimalmicro-grid operation. Appl Energ 2012;99:455e70.

[42] Niknam T, Mojarrad HD, Meymand HZ, Firouzi BB. A new honey bee matingoptimization algorithm for non-smooth economic dispatch. Energy 2011;36:896e908.

[43] Slootweg JG, de Haan SWH, Polinder H, Kling WL. General model for repre-senting variable speedwind turbines in power system dynamics simulations.IEEE Trans Power Syst 2003;18:144e51.

[44] Morales JM. Impact on system economics and security of a high penetration ofwind power. Ph.D. dissertation, Dept Electr Eng, Univ Castilla-La Mancha;Ciudad Real, Spain: 2010.

[45] Belanger C, Gagnon L. Adding wind energy to hydropower. Energ Policy 2002;30:1279e84.

[46] Niknam T, Narimani MR, Jabbari M, Malekpour AR. A modified shuffle frogleaping algorithm for multi-objective optimal power flow. Energy 2011;36:6420e32.

[47] Nafar M, Gharehpetian GB, Niknam T. Improvement of estimation of surgearrester parameters by using modified particle swarm optimization. Energy2011;36:4848e54.

[48] Niknam T, Aizipanah-Abarghooee R, Roosta A. Reserve constrained dynamiceconomic dispatch: a new fast self-adaptive modified firefly algorithm. IEEESyst J 2012. http://dx.doi.org/10.1109/JSYST.2012.2189976.

[49] Niknam T, Azizipanah-Abarghooee R, Narimani MR. Reserve constraineddynamic optimal power flow subject to valve-point effects, prohibited zonesand multi-fuel constraints. Energy 2012;47:451e64.