Scheduling electric power production at a wind farm

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Innovative Applications of O.R.

Scheduling electric power production at a wind farm

Zijun Zhang a, Andrew Kusiak b, Zhe Song c,⇑a Department of Systems Engineering and Engineering Management, Room P6600, 6/F, Academic 1, City University of Hong Kong, Hong Kongb Department of Mechanical and Industrial Engineering, 3131 Seamans Center, The University of Iowa, Iowa City, IA 52242-1527, United Statesc School of Business, Nanjing University, 22 Hankou Road, Nanjing 210093, China

a r t i c l e i n f o

Article history:Received 18 May 2011Accepted 26 July 2012Available online 7 August 2012

Keywords:SchedulingEvolutionary computationsWind farmParticle swarm optimizationSmall world network

a b s t r a c t

We present a model for scheduling power generation at a wind farm, and introduce a particle swarmoptimization algorithm with a small world network structure to solve the model. The solution generatedby the algorithm defines the operational status of wind turbines for a scheduling horizon selected by adecision maker. Different operational scenarios are constructed based on time series data of electricityprice, grid demand, and wind speed. The computational results provide insights into management of awind farm.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

Environmental concerns and scarcity of fossil fuels are drivingthe interest in clean energy technologies. Technological maturity,safety, and cost competitiveness have made wind energy a widelyaccepted energy solution. Powered by the source of nature (thewind), wind turbines generate electricity, thus guaranteeing thereturn of investment. Due to variability of electricity price, grid de-mand, wind speed, as well as heterogeneity of wind turbines,determining a sequence of activating or deactivating turbines ina wind farm is a challenge.

A commercial wind farm usually involves a large number ofwind turbines (e.g., 100 or more) and each wind turbine performsuniquely even if all turbines are produced by the same manufac-turer. Various reasons could contribute to the heterogeneity (dif-ference in performance) of wind turbines. Factors, includinglocation of wind turbines, terrain, turbine–turbine interactions, dif-ferent component suppliers, and maintenance regimes, are con-tributing to this heterogeneity. Thus, for a wind farm operator, itis necessary to know which turbine could be turned on or off atappropriate times according to the efficiency of wind turbines.Power curves constructed from field data are good indicators of aturbine’s actual performance (Manwell et al., 2002) and thereforecould be incorporated into an optimization model to assist opera-tors in scheduling wind turbines.

This topic of scheduling wind turbines is new and it differs fromdispatching power generated by traditional power plants. One rea-son is that fuel (the wind) is free and therefore the desire to keepwind turbines running makes scheduling wind turbines seeminglyless important. Yet, as the number of wind farms is growing, oper-ating them at maximum capacity is not always possible and prof-itable, which was reflected in recent industrial practices (Rogerset al., 2010). Power curtailment, fluctuation of electricity price,and heterogeneity of wind turbines need to be considered whiledetermining the capacity at which a wind farm should be operated.Classical scheduling models usually have been studied in the con-text of manufacturing applications (Ouelhadj and Petrovic, 2009)and management problems (Kim et al., 2012; Cao et al., 2012;Yazid et al., 2011). This paper fills the research gap by bringingscheduling to the wind industry to reduce the cost of producingelectricity.

The published literature on wind farm scheduling has focusedon determining the power generation schedule of a wind farmintegrated with other power plants, such as coal-fired plants andhydroelectric plants, where the total grid demand usually is fixedor known before (Ren and Jiang, 2009; Siahkali and Vakilian,2009). In this paper, scheduling the power generation at a windfarm subject to minimizing the overall cost is studied. As a pio-neering study of wind-generated power scheduling, severalassumptions are considered to simplify the problem so that thefundamentals can be investigated. First, wind speed, electric powerdemand, and electricity price is assumed to be known. In addition,the characteristic function of wind turbine power generation ismodeled by a power curve and the maintenance schedule is notconsidered. A scheduling model is developed to determine the

0377-2217/$ - see front matter � 2012 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.ejor.2012.07.043

⇑ Corresponding author. Tel./fax: +86 25 83621041.E-mail addresses: [email protected], [email protected] (Z. Zhang),

[email protected] (A. Kusiak), [email protected] (Z. Song).

European Journal of Operational Research 224 (2013) 227–238

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activation/deactivation status of individual wind turbines to copewith the varying electricity prices, wind speed, and grid demand.In reality, the electricity price and wind speed is changing overtime. The grid demand of electric power produced by wind farmsis variable due to frequent power curtailments which are pre-determined by the grid operator according to the transmissionconstraints (Rogers et al., 2010). The model complexity warrantsthe application of a particle swarm optimization algorithm tosearch the optimal solution. The search algorithm is enhanced interms of exploration and exploitation through structural designaccording to small world theory (Watts and Strogatz, 1998).

Based on the historical time series of electricity prices, windspeed, and demand data, various scenarios are constructed to ana-lyze and evaluate the optimized schedules. Insights are obtainedinto managing heterogeneous wind turbines operating in a realisticenvironment. The proposed model is intended as a component of adecision-support system for wind farms.

2. Model for scheduling wind turbines

A model for wind turbine scheduling is developed. The modelminimizes the cost function defined in the next section subjectto constraints.

2.1. Objective function

Quantifying the total cost of running a wind farm is complexand the wind industry does not have a widely accepted standard.

In this study, the cost of running a wind farm is simply depictedas three major components: power shortage cost, wind turbineoperations and maintenance cost, and wind turbine start-up cost.

Power shortage occurs when the supply of power from a windfarm does not meet the grid demand as expressed as follows:

DPt ¼max 0;Dt �XI

i¼1

Pi;tsi;t

( )ð1Þ

In (1), DPt, is nonnegative because a power shortage only occurswhen the grid demand is greater than the energy generated by allturbines.

The power shortage cost can be categorized further into theopportunity cost of power shortage (Definition 2.1) and the com-pensation cost (Definition 2.2).

Definition 2.1. Once a power shortage occurs, the opportunitycost equals the benefits of selling the amount equivalent to thepower shortage, i.e., the product of the electricity price and thepower shortage at a given time window.

The total opportunity cost of a power shortage over thescheduling time horizon is expressed in (2).

Cps ¼XT

t¼1

BtDPt ð2Þ

Definition 2.2. The compensation cost is the penalty paid to com-pensate for the amount of power shortage.

Nomenclature

T scheduling horizont time unit (one hour in this study), t = 1, 2, 3, . . . , TI number of wind turbines installed at a wind farmi wind turbine indexDt grid demand at time tPi,t generated power of wind turbine i at time tPC rated power of wind turbinesfi(vt, hi) power curve (logistic function) of wind turbine ihi parameter vector of a logistic function, hi = (h1,i, h2,i, h3,i,

h4,i)vt average wind speed at time tvci cut-in wind speedvco cut-out wind speedI(�) indicator functionpt penalty for compensating the power shortage at time tsi,t operational status (on or off) of wind turbine i at time t,

denoted as 0 or 1.K energy needed to start up a wind turbineBt electricity spot price at time tDPt power shortage (the demand minus generated wind

power) at time tc estimated operations and maintenance cost of a wind

turbine to generate a unit of powerCps opportunity cost due to power shortageCpc compensation cost due to power shortageCom operations and maintenance cost of a wind farm (all tur-

bines in a wind farm)TC cost associated with a schedule: TCb is the cost of a base-

line schedule; TCc is the cost of a computed scheduleGain the gain of running an optimized schedule over the

baseline scheduleNe number of tribesNp number of particles in a tribem tribe index

h particle indexj iteration number in the particle swarm algorithmw dimensionality of the search spacexj

hm position of particle h in tribe m at iteration j

v jhm velocity of particle h in tribe m at iteration j

xjhm particle h is a local best in tribe m at iteration j

gjm the best of tribe m

f ðxjhmÞ fitness value of particle h in tribe m at iteration j

gjq the best of a random tribe q – m

f ðgjqÞ fitness value of a tribe’s best

gjb global best of all particles

rj1; rj

2 random vectors generated from uniform distribution inthe interval [0,1]

x inertia of the particle swarm algorithmd1, d2 constants used to update particles’ velocitya constant used in the communication functionq parameter controlling the structure of a small world

networkN (0,1) normal distribution with mean 0 and standard deviation

1Sm selected particles to communicate among tribes during

the search processk, k shape and scale parameters of Weibull distributionG1, G2 two groups of wind turbines, each including wind tur-

bines with identical power curvesXG1,t, XG2,t number of wind turbines turned on at time t in group

G1 or group G2lw1, lw2 average wind speeds of data sets 1 and 2, respectivelyle1, le2 average electricity prices of data sets 1 and 2ld1, ld2 average grid demands of data sets 1 and 2

228 Z. Zhang et al. / European Journal of Operational Research 224 (2013) 227–238


The wind farm can compensate for the power shortage byactivating an alternative power generation resource or purchasingpower from other utilities. The total compensation is expressed asfollows:

Cpc ¼XT

t¼1

ptDPt ð3Þ

where pt is the penalty cost per power unit.The operations and maintenance (O&M) cost of all wind

turbines in a farm is formulated as follows:

Com ¼XT

t¼1

XI

i¼1

cPi;tsi;t ð4Þ

where c is the cost in generating a unit of power.The O& M cost is then determined as a function of the generated

power and on/off decision variable.The last major component is the start-up cost of wind turbines.

The start-up cost refers to the energy consumed by wind turbinesduring the start-up process. Activating a wind turbine requires acertain amount of electricity from the grid as stated as follows:

Cstart ¼XT

t¼1

XI

i¼1

BtKi;tsi;tð1� si;t�1Þ ð5Þ

By considering (2)–(5), the total operating cost of a wind farm isexpressed as follows:

TC ¼XT

t¼1

BtDPt þXT

t¼1

ptDPt þXT

t¼1

XI

i¼1

cPi;tsi;t

þXT

t¼1

XI

i¼1

BtKi;tsi;tð1� si;t�1Þ ð6Þ

It is obvious that the first two components of the total cost arenot considered when the grid demand is met (i.e., DPt = 0). How-ever, satisfying the grid demand increases both the O& M andthe start-up costs. To minimize TC in (6), it is necessary to turnoff some wind turbines once the grid demand is met.

2.2. Constraints

The scheduling model built in this paper calls for constraints. Awind turbine operates if, and only if, the wind speed is between thespecified cut-in and cut-out wind speed. Therefore, the wind speedcondition in (7) is needed.

vci < v t < vco ð7Þ

Another constraint is related to the performance of wind tur-bines expressed by power curves. A power curve describes a map-ping between wind speeds and wind energy produced by a windturbine. Various functions have been investigated in the literatureto model power curves. Boukhezzar and Siguerdidjane (2009)developed a power-curve function to design a non-linear controllerto optimize the power of the DFIG (Doubly Fed Induction Genera-tor). The power curve was simply expressed as a product of a con-stant, a power coefficient function, and wind speed cube. Ustüntasand Sahin (2008) investigated a new approach, cluster center fuzzylogic, to model the power curve of a wind turbine. Kusiak et al.(2009a,b) utilized the logistic function to model the power curveand developed a wind turbine monitoring method.

In this study, a power curve modeled with the logistic functionpresented in Kusiak et al. (2009b) is employed (see the followingequation).

Pi;t ¼ fiðv t; hiÞ ¼ h1;i1þ h2;ie�vt=h4;i

1þ h3;ie�vt=h4;i; hi ¼ ðh1;i; h2;i; h3;i; h4;iÞ ð8Þ

The expression (8) is transformed into (9) by considering thecut-in and cut-out wind speeds.

Pi;t ¼ fiðv t; hiÞ ¼ h1;i1þ h2;ie�vt=h4;i

1þ h3;ie�vt=h4;iIðvci ;vcoÞðv tÞ;

hi ¼ ðh1;i; h2;i; h3;i; h4;iÞ ð9Þ

As the generated power cannot exceed the maximum capacityof a wind turbine, the power generation constraint is expressed as:

Pi;t 6 PC ð10Þ

2.3. Scheduling model formulation

The objective function and constraints discussed in Sections 2.1and 2.2 lead to the wind farm scheduling model as:

Min TC

s:t:si;t 2 f0;1g

ð11Þ

The solution of the model (11) is a schedule represented by var-iable si,t.

The model expressed in (11) is generic. It can be extended byincorporating different objectives and additional constraints, suchas curtailment of wind power and maintenance activities.

3. Particle swarm small world optimization algorithm

Solving model (11) is challenging as the number of decisionvariables si,t increases with the number of wind turbines and thelength of scheduling horizon. For example, assume the schedulinghorizon is 24 hours; the number of binary decision variables in-creases from 240 to 480 when the number of wind turbines in-creases from 10 to 20. Similarly, assume the number of windturbines is 10; the number of binary decision variables changesfrom 240 to 480 if the scheduling horizon is extended from24 hours to 48 hours. A Particle Swarm Small World Optimization(PSSWO) is applied to solve model (11) due to its complexity.

The PSSWO algorithm is inspired by two streams of research:Particle Swarm Optimization (PSO) discussed in Kennedy and Eber-hart (1995) and Small World Theory reported in Milgram (1967).

The development of PSO was simulated by the social behaviorof bird flocks and fish schools. The survey paper by AlRashidi andEl-Hawary (2009) addressed eight advantages of PSO that are sum-marized as follows:

(1) Ease of implementation: Less tuned parameters, simple logicoperations in search and derivative free property.

(2) Ability of escaping local optima: The stochastic nature of thesearch function, low sensitivity to the form of an objectivefunction, and slight dependence on population initialization.

(3) Compatibility: Ease of integration with other algorithms.

The PSO algorithm has been widely applied. Park et al. (2005)applied PSO to solve an economic dispatch problem with a non-smooth cost function. Park et al. (2010) improved the PSO algo-rithm to handle more complex economic dispatch problems withnon-convex cost functions. Kusiak and Zhang (2011) developedan adaptive PSO by integrating fuzzy logic to optimize perfor-mance of wind turbine systems. Kusiak and Li (2010) applied amodified PSO algorithm to optimize a heating, ventilating, andair conditioning system. Seo et al. (2006) studied PSO in electro-magnetic applications.

Although the original PSO algorithm has performed well inmany applications, a drawback is that its search trajectory is

Z. Zhang et al. / European Journal of Operational Research 224 (2013) 227–238 229


uncontrollable (de Castro, 2002; Lee and Park, 2006; Saxena andVora, 2008). This drawback can be handled by balancing the explo-ration and exploitation in the PSO algorithm. Numerous techniqueshave been developed to improve performance of PSO. Clerc andKennedy (2002) studied convergence and the search trajectoriesof PSO. The ability of PSO to find an optimal solution was improvedby controlling particle’s velocities. Liu et al. (2007) combined PSOwith the memetic algorithm to enhance its performance. Ratnawe-era et al. (2004) presented a study of improving PSO by controllinglocal search and convergence to global optima. To control the localsearch and convergence, a mutation operator and re-initializationof particles’ velocities were added to PSO.

In this research, PSO is integrated with the small world theoryto balance the exploration and exploitation. Although the originalsmall world experiment conducted by Milgram (1967) aimed atexamining the mean length of a path between two unknown peo-ple in a social network, the research was then extended. March(1991) applied it to organizational learning and discussed the myo-pia of learning. Watts and Strogatz (1998) studied the collectivedynamics of small world networks based on various structures ofnetworks. Fang et al. (2010) argued that semi-isolated subgroupsmay help the balance of exploration and exploitation, and the sim-ulation results supported the argument. This semi-isolated net-work structure then was utilized in the proposed PSSWO toassist the balance of exploration and exploitation in search.

In the PSSWO algorithm discussed in this paper, particles are or-ganized into semi-isolated, equal-size groups. Each group in theswarm presents a tribe. In each tribe, the best position of particlesis treated as the tribe’s best.

Definition 3.1. Assume xjhm presents a particle in a tribe and gj

m

presents the tribe’s best at iteration j, the gjm equals to xj

hm, iff ðxj

hmÞ 6 f ðgj�1m Þ and f ðxj

hmÞ attain minimum in the tribe.The network structure can vary with the level of connections

between groups. Tribes without connections to others are isolatedsubgroups. The semi-isolated structure is a network structure inwhich individuals in the network are highly clustered while thereare some connections between the groups. The network becomes arandom network, if individuals in the network are randomlyconnected to others.

Fig. 1 illustrates the semi-isolated structure of PSSWO algo-rithm. The network structure is established through the commu-nication between tribes. After each flight, a random size of particlesin a tribe is selected to communicate with one of the other tribe’sbests. Definition 3.2 is concerned with the communication process.

Definition 3.2. Assume Sm presents the set of randomly selectedparticles in a tribe and gq presents another tribe’s best. Iff ðgqÞ 6 f ðxhmÞ and xhm 2 Sm, the position of the local best of xhm willbe updated as follows:

xhm ¼ axhm þ ð1� aÞgq ð12Þ

where a � U[0,1] and the size of Sm is determined by the product ofa random number q � U[0,1] and the size of tribe m. Both a and qimpact the semi-isolated network structure.

Assume the number of tribes in a swarm is Ne, the number ofparticles in each tribe is Np,j represents the index of iterations forimplementing the PSSWO algorithm, and w is the dimension ofsearch space; then the PSSWO algorithm is expressed next (seealso Fig. 1).

Step 1. Initialize Ne tribes of particles, Np particles for each tribe,the position of each particle, xj

hm 2 Rw, and the associatedvelocity, v j

hm 2 Rw, where h = 1, 2, . . . , Np, m = 1, 2, . . . , Ne

and j = 0.Step 2. Initialize the local best xj

hm for each particle by xjhm xj

hm ineach tribe, and estimate the initial tribe’s best gj

m bygj

m arg minðf ðxjhmÞÞ, where h = 1, 2, . . . , Np and j = 0.

Step 3. Repeat until the stopping criterion is satisfied.For each tribe 1 6m 6 Ne and for each particle 1 6 h 6 Np

Step 3.1. Create random vectors rj1 and rj

2 2 Rw whererj

1; rj2 � U½0; 1�.

Step 3.2. Update the velocities of particles by v jhm xv j

hmþc1rj

1ðxjhm � xj

hmÞ þ c2rj2ðg

jm � xj

hmÞ and update the particlepositions by xj

hm xjhm þ v j

hm.Step 3.3. Update the local best by xj

hm xjhm if f ðxj

hmÞ 6 f ðxjhmÞ.

Step 3.4. Update the tribe’s best by gjm xj

hm if f ðxjhmÞ 6 f ðgj

mÞ.Step 3.5. Determine the value of q by q � U [0,1]Step 3.6. Communication: Assume Sm presents a set of selected

particles from each tribe, then xjhm axj

hmþð1� aÞgj

q; xjhm 2 Sm if f ðgj

qÞ 6 f ðxjhmÞ; q – m.

Step 3.7. Update the global best by gjb gj

m if f ðgjmÞ has the low-

est valueStep 4. Terminate the algorithm when the stopping criterion

(number of iterations in this paper) is satisfied.

To implement the PSSWO, some parameters need to be deter-mined. Based on Shi and Eberhart (1998), the inertia weight x isset to 0.5 because their experiments have shown that the inertiaweight less than 0.8 improved convergence. The c1 and c2 are bothset to 2 as suggested by Shi and Eberhart (1998).

4. Description of simulation experiments

In this section, two simulation experiments, Experiments 1 and2, are conducted to investigate the wind farm scheduling model indifferent operating scenarios.

4.1. Scheduling scenarios

In this paper, the wind turbine schedules are determined usingeight scenarios constructed based on three parameters: windspeed, electricity price, and grid demand (see Table 1). These threeparameters are categorized into two levels, high and low.

Global best Tribe’s best Local best Particle’s position

Region contains global optima

Search direction of tribes

Tribe 1

Tribe 2

Tribe 3

Communication

Fig. 1. Structure of the PSSWO algorithm.



In this research, hourly average wind speed data of a wind farmis used. Weibull distribution is employed to simulate the 24 hourswind speed for scheduling. Table 2 presents the five combinationsof scale and shape parameters of the Weibull distribution used todescribe wind speeds in five different areas in Taiwan (Yeh andWang, 2008).

In this study, the scale and shape parameters of Weibull distri-bution in combination 4 is selected to generate wind speed data.Two sets of wind speed data then are generated based on Weibulldistribution. Then, before categorization, two sets of generatedwind speed data need to be examined to make sure they are signif-icantly different. Categorizing two data sets that were almost iden-tical would be meaningless. A t-test is utilized to determinewhether two data sets are significantly different. Table 3 illustratesthe mean and standard deviation of two data sets. It is obvious that

the mean of data set 1 and mean of data set 2 are not equal (t-testalso confirms this difference). This indicates that the two sets ofdata are different. Fig. 2 shows the run-chart of data sets 1 and 2.Next, based on Table 3, we can conclude that data set 1 of windspeed data presents a high wind speed condition and data set 2 de-scribes a low wind speed condition.

The web site of the European Energy Exchange (EEX) offers real-time European Electricity Index (ELIX) and describes the hourlyelectricity price. Two sets of ELIX data are collected to representdifferent conditions of electricity price for wind farm scheduling.Data set 1 contains hourly ELIX data on October 29, 2010, and dataset 2 includes the hourly ELIX data on October 31, 2010. Table 4and Fig. 3 summarize these two data sets. t-test confirms thedifference in the mean prices.

Table 1Scheduling scenarios.

Scenario no. Description

Scenario 1 High electricity price, high grid demand, high wind speedScenario 2 High electricity price, high grid demand, low wind speedScenario 3 High electricity price, low grid demand, high wind speedScenario 4 High electricity price, low grid demand, low wind speedScenario 5 Low electricity price, high grid demand, high wind speedScenario 6 Low electricity price, high grid demand, low wind speedScenario 7 Low electricity price, low grid demand, high wind speedScenario 8 Low electricity price, low grid demand, low wind speed

Table 2Scale and shape parameters of the Weibull distribution.

Combination Scale (k) Shape (k)

Combination 1 11.01 1.96Combination 2 10.66 1.92Combination 3 11.91 1.77Combination 4 11.09 1.61Combination 5 10.42 2.05

Table 3Summary of two sets of wind speed data.

Data set No. ofdatapoints

Mean(m/h)

Std. dev.

Data set 1 24 10.85 6.13Data set 2 24 7.25 5.22

Table 4Summary of two sets of electricity price data.

Data set No. ofdatapoints

Mean(Eur/MW hour)

St. dev.

Data set 1 24 52.8 10.1Data set 2 24 43.46 6.94

Table 5Summary of two sets of grid demand data.

Data Set No. ofdatapoints

Mean(MW)

St. dev.

Data set 1 24 11.28 2.13Data set 2 24 9.67 1.41

0

5

10

15

20

25

30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Win

d sp

eed

(m/s

)

Scheduling horizon (h)

Data set 1 Data set 2

Fig. 2. Two sets of wind speed data.

0

10

20

30

40

50

60

70

80

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Ele

ctric

ity p

rice

(Eur

o/M

WH

)



Fig. 3. Two sets of electricity price data.

0

2

4

6

8

10

12

14

16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Grid

dem

and

(MW

)



Fig. 4. Two sets of grid demand data.



The hourly grid demand data are derived based on national griddemand data from the web site of the National Grid UK. The website offers data of electricity demand in the United Kingdom (UK)at 30 minutes intervals. Then, 30-minutes demand data are trans-formed into hourly demand. In this paper, the wind farm is as-sumed to satisfy 1/3000 of national grid demand. Therefore,value of Dt is set to 1/3000 of the electricity demand data collectedfrom National Grid UK. Data set 1 is the demand data on July 1,2010, and data set 2 is the demand data on July 3, 2010. Table 5shows the summary of the two sets. The t-test is applied to deter-mine if the two demand data sets are different.

The means of two samples are significantly different. Based onthe summary in Table 5, data average demand for data set 1 ishigher than that of data set 2. Fig. 4 depicts the two sets.

4.2. Experiment 1

In Experiment 1, the schedules are developed for a wind farmbased on the eight scenarios. The wind turbines are categorizedinto two groups with all turbines identical within each group. Inthis experiment, wind turbines in Group 1 are assumed to be lessefficient than wind turbines in Group 2. Definition 4.1 expressesthe condition that wind turbines in one group have higher effi-ciency than the turbines in another group.

Definition 4.1. If a wind turbine of Group 2 generates more powerthan a wind turbine of Group 1 under any wind speed conditions,the turbine in Group 2 is assumed to have a better performingpower curve. This definition is formularized as follows:

h1;G21þ h2;G2e�v t=h4;G2

1þ h3;G2e�v t=h4;G2Iðvci ;vcoÞðv tÞ

> h1;G11þ h2;G1e�vt=h4;G1

1þ h3;G1e�vt=h4;G1Iðvci ;vcoÞðv tÞ ð13Þ

To ensure that a wind turbine of Group 2 has a more efficientpower curve than a wind turbine of Group 1, a lower value of h

4,G2 than h4,G1. Fig. 5 introduces one example of power curves ofwind turbines in Group 1 and Group 2. To construct a power curve,the values of vector h need to be provided. Kusiak et al. (2009b)applied logistic function to model power curves and suggested

h = {103.33,20.53,1190.73,1.14}. The logistic function needs to bescaled to reflect the power curve of a 1.5 Mega Watt wind turbine.For the power curve in Fig. 5, h4,G1 follows the value of 1.14suggested in Kusiak et al. (2009b). The value of h4,G2 is arbitrarilyset to 1.05.

Another interesting question to investigate is the ratio of windturbines between Groups 1 and 2. In this experiment, five ratios(i.e., the number of turbines in Group 1 divided by the number ofturbines in Group 2), Ratio 1 = (100%/0), Ratio 2 = (70%/30%), Ratio3 = (50%/50%), Ratio 4 = (30%/70%), Ratio 5 = (0/100%) are used toconduct sensitivity analysis of the schedule for this wind farm. Forsimplicity, the total number of wind turbines in the wind farm isassumed to be 10. As the wind turbines in each group are assumedto be identical, the scheduling model in (11) is expressed in (14),where XG1,t and XG2,t denote the number of wind turbines in eachgroup at time t and I is the total number of wind turbines.

The binary programming problem described in (11) is trans-formed into an integer programming model (14) which can besolved by the proposed PSSWO. Note that the number of decisionvariables is significantly reduced, from 240 binary variables to 48.In (14), the decision variables at each time window are the numberof operated wind turbines in Groups 1 and 2. The position of eachparticle in PSSWO is a 2 � 24 array which includes all decisionvariables over the scheduling horizon.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 5 10 15 20 25

Gen

erat

ed p

ower

(MW

)

Wind speed (m/s)

Group 1 Group 2

Fig. 5. Power curve models in Groups 1 and 2.

Table 6High wind speed data and the generated power over the scheduling horizon.

Scheduling horizon 1 2 3 4 5 6 7 8 9 10 11 12

High vt 5.24 13.55 11.22 7.92 1.31 7.08 6.12 3.04 4.86 8.53 9.88 7.49PG1,t 0.14 1.49 1.42 0.72 0 0.46 0.25 0 0.11 0.91 1.25 0.58PG2,t 0.20 1.50 1.47 0.94 0 0.65 0.37 0 0.15 1.13 1.38 0.79


High vt 14.35 23.22 14.68 16.42 9.28 25.10 2.80 10.08 13.44 11.14 19.61 14.13PG1,t 1.50 1.50 1.50 1.50 1.12 0 0 1.29 1.49 1.41 1.50 1.50PG2,t 1.50 1.50 1.50 1.50 1.29 0 0 1.40 1.50 1.46 1.50 1.50

Table 7Low wind speed data and the generated power over the scheduling horizon.


Low vt 8.12 8.20 12.29 12.25 2.60 14.27 9.28 23.79 3.03 7.77 6.21 6.89PG1,t 0.78 0.81 1.47 1.47 0 1.50 1.12 1.50 0 0.67 0.27 0.41PG2,t 1.01 1.03 1.49 1.49 0 1.50 1.29 1.50 0 0.89 0.39 0.59


Low vt 2.58 2.98 4.34 12.73 5.65 1.12 2.33 10.99 6.00 3.38 5.22 1.91PG1,t 0 0 0.08 1.48 0.18 0 0 1.40 0.23 0 0.14 0PG2,t 0 0 0.11 1.50 0.27 0 0 1.46 0.34 0 0.20 0



Min TC

s:t:

TC ¼XT

t¼1

BtDPt þXT

t¼1

ptDPt þXT

i¼1

ctPtðXG1;t þ XG2;tÞ

þXT

i¼1

BtKtjðXG1;t þ XG2;tÞ � ðXG1;t�1 þ XG2;t�1ÞjIð0;þ1Þ

ððXG1;t þ XG2;tÞ � ðXG1;t�1 þ XG2;t�1ÞÞ ð14ÞDPt ¼maxf0;Dt � ðPG1;tXG1;t þ PG2;tXG2;tÞg

Pb;t ¼ fbðtt ; hbÞ ¼ h1;b1þ h2;be�vt=h4;b

1þ h3;be�vt=h4;bIðvci ;vcoÞðv tÞ;

hb ¼ ðh1;b; h2;b; h3;b; h4;bÞ and b ¼ G1 or G2PIt ;t 6 PC ; XG1;t þ XG2;t 6 I; XG1;0 ¼ XG2;0 ¼ 0

Table 6 presents the high wind speed data and the correspond-ing power generated by a single wind turbine in Group 1 or Group2. Table 7 addresses similar information as Table 6 except that thelevel of wind speed in Table 7 is low.

4.3. Experiment 2

The wind farm in Experiment 2 is realistic as the total numberof wind turbines is 10 and the power curve of each wind turbine is

unique. To generate different power curves, a normal distributionwith the mean 1.14 and the standard deviation 0.114 is utilizedto produce 10 random values of h4,i. Fig. 6 illustrates the powercurves of 10 wind turbines, with T1–T10 representing the windturbine No. 1–10.

Solving (11) directly is computationally expensive since thereare 240 binary decision variables. In addition, solving (11) (a binaryprogramming formulation) with the proposed PSSWO is not

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 5 10 15 20 25

Gen

erat

ed p

ower

(MW

)

Wind speed (m/s)

T1 T2 T3 T4 T5 T6 T7 T8 T9 T10

Fig. 6. Power curve models of wind turbines.

1 11100011 0001

3 -34-3

Fig. 7. Transformation of a binary string into duty-cycle based variable.

Table 8High wind speed and the generated power over the first half of the scheduling horizon.


High vt 5.24 13.55 11.22 7.92 1.31 7.08 6.12 3.04 4.86 8.53 9.88 7.49P1,t 0.12 1.48 1.37 0.59 0 0.37 0.21 0 0.09 0.78 1.15 0.47P2,t 0.52 1.50 1.50 1.40 0 1.25 0.91 0 0.38 1.45 1.49 1.34P3,t 0.13 1.49 1.39 0.65 0 0.42 0.23 0 0.10 0.84 1.20 0.52P4,t 0.31 1.50 1.49 1.22 0 0.96 0.58 0 0.23 1.34 1.46 1.10P5,t 0.18 1.50 1.46 0.88 0 0.60 0.33 0 0.13 1.07 1.35 0.74P6,t 0.22 1.50 1.48 1.03 0 0.74 0.42 0 0.16 1.20 1.41 0.89P7,t 0.26 1.50 1.49 1.13 0 0.85 0.49 0 0.19 1.28 1.44 0.99P8,t 0.09 1.45 1.25 0.43 0 0.27 0.15 0 0.08 0.58 0.95 0.34P9,t 0.73 1.50 1.50 1.46 0 1.38 1.13 0 0.55 1.48 1.50 1.43P10,t 0.15 1.50 1.44 0.79 0 0.52 0.28 0 0.12 0.98 1.30 0.64

Table 9High wind speed and the generated power over the second half of the scheduling horizon.


High vt 14.35 23.22 14.68 16.42 9.28 25.10 2.80 10.08 13.44 11.14 19.61 14.13P1,t 1.49 1.50 1.50 1.50 1.00 0 0 1.19 1.48 1.36 1.50 1.49P2,t 1.50 1.50 1.50 1.50 1.48 0 0 1.50 1.50 1.50 1.50 1.50P3,t 1.50 1.50 1.50 1.50 1.06 0 0 1.24 1.49 1.39 1.50 1.49P4,t 1.50 1.50 1.50 1.50 1.43 0 0 1.47 1.50 1.49 1.50 1.50P5,t 1.50 1.50 1.50 1.50 1.25 0 0 1.37 1.50 1.45 1.50 1.50P6,t 1.50 1.50 1.50 1.50 1.34 0 0 1.43 1.50 1.48 1.50 1.50P7,t 1.50 1.50 1.50 1.50 1.39 0 0 1.45 1.50 1.49 1.50 1.50P8,t 1.48 1.50 1.48 1.50 0.79 0 0 1.01 1.45 1.23 1.50 1.47P9,t 1.50 1.50 1.50 1.50 1.50 0 0 1.50 1.50 1.50 1.50 1.50P10,t 1.50 1.50 1.50 1.50 1.18 0 0 1.33 1.50 1.43 1.50 1.50



feasible. In this study, binary string variables are transformed intoduty-cycle based variables (Pappala et al., 2009; Pappala andErlich, 2008) to reduce the model complexity as illustrated inFig. 7. The position of each particle is a vector including all duty-cycle based variables in PSSWO which is similar to the algorithmpresented in Pappala et al. (2009) and Pappala and Erlich (2008).

The electricity price and grid demand data used in Experiment 2are the same as in Figs. 3 and 4. Tables 8 and 9 present the highwind speed data and the corresponding generated power of eachwind turbine of a wind farm. Tables 10 and 11 present data similarto those of Tables 8 and 9 for low wind speed scenario.

5. Simulation results

To solve the wind farm scheduling model, the values of param-eters, pt, ct, and K, need to be determined. Parameter pt is arbitrarilyset to 0.8. ct is conservatively set to 50 Euro/Mega Watt houraccording to Wind Energy-The Facts and AWEA. The parameterK is set to 0.06 Mega Watt based on the advice of wind farmmanagers. To make the wind farm scheduling problem tractableby the PSSWO algorithm, a round-off approach for variables(particle’s positions) is utilized. A more accurate model ofestimating these three parameters needs to be investigated infuture research. The Visual Basic language is used to code thePSSWO algorithm and a desktop with Intel i7-2600 processor,8 Giga Byte memory and 64-bit windows operating system isutilized to compute the simulations in this research.

5.1. Simulation results of Experiment 1

To reduce computational cost, a stopping criterion, the numberof iterations, needs to be determined. The convergence of thePSSWO algorithm is examined by solving (14) for Scenario 1. Thenumber of tribes and particles are set to 5 and 30 separately forimplementing PSSWO algorithm. Fig. 8 shows the convergence of

the PSSWO algorithm attaining minimum at the 50th iteration.However, this number of iterations (stopping criterion) may nothold for all scheduling scenarios. Therefore, a more conservativestopping criterion, 100 iterations, is adopted.

Sample simulation results of Experiment 1 are illustrated inTable 12, where the first column lists the scheduling scenarios.The index of scheduling time period is addressed in the first andtenth rows. The entries in Table 12 are the computed schedules,the number of operated wind turbines in Groups 1 and 2 (sepa-rated by a dot), at each time period.

The PSSWO does not guarantee global optimality. To evaluatethe quality of the solutions, a comparative analysis is performedbased on a baseline schedule. In the baseline schedule, all windturbines are activated once the measured wind speed is betweenthe cut-in and the cut-out values. Table 13 includes the costs ofoperating a wind farm with the baseline schedule and the com-puted schedules. It is obvious that the computed schedules outper-form the baseline schedule as indicated by the increasing trend of

Table 10Low wind speed and the generated power over the first half of the scheduling horizon.


Low vt 8.12 8.20 12.29 12.25 2.60 14.27 9.28 23.79 3.03 7.77 6.21 6.89P1,t 0.65 0.67 1.44 1.44 0 1.49 1.00 1.50 0 0.55 0.22 0.33P2,t 1.42 1.43 1.50 1.50 0 1.50 1.48 1.50 0 1.38 0.94 1.19P3,t 0.71 0.74 1.46 1.46 0 1.50 1.06 1.50 0 0.60 0.24 0.37P4,t 1.27 1.28 1.50 1.50 0 1.50 1.43 1.50 0 1.18 0.62 0.89P5,t 0.95 0.97 1.49 1.49 0 1.50 1.25 1.50 0 0.83 0.35 0.54P6,t 1.10 1.12 1.50 1.50 0 1.50 1.34 1.50 0 0.98 0.44 0.67P7,t 1.19 1.21 1.50 1.50 0 1.50 1.39 1.50 0 1.09 0.52 0.77P8,t 0.47 0.49 1.38 1.38 0 1.48 0.79 1.50 0 0.39 0.16 0.24P9,t 1.47 1.47 1.50 1.50 0 1.50 1.50 1.50 0 1.45 1.16 1.34P10,t 0.85 0.88 1.48 1.48 0 1.50 1.18 1.50 0 0.73 0.30 0.46

Table 11Low wind speed and the generated power over the second half of the scheduling horizon.


Low vt 2.58 2.98 4.34 12.73 5.65 1.12 2.33 10.99 6.00 3.38 5.22 1.91P1,t 0 0 0.07 1.46 0.15 0 0 1.34 0.19 0 0.12 0.00P2,t 0 0 0.24 1.50 0.70 0 0 1.50 0.85 0 0.51 0.00P3,t 0 0 0.07 1.47 0.17 0 0 1.37 0.21 0 0.13 0.00P4,t 0 0 0.15 1.50 0.42 0 0 1.49 0.54 0 0.30 0.00P5,t 0 0 0.09 1.49 0.24 0 0 1.45 0.30 0 0.17 0.00P6,t 0 0 0.11 1.50 0.30 0 0 1.47 0.38 0 0.22 0.00P7,t 0 0 0.13 1.50 0.35 0 0 1.48 0.45 0 0.25 0.00P8,t 0 0 0.06 1.41 0.12 0 0 1.21 0.14 0 0.09 0.00P9,t 0 0 0.34 1.50 0.93 0 0 1.50 1.08 0 0.72 0.00P10,t 0 0 0.09 1.49 0.20 0 0 1.42 0.26 0 0.15 0.00

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

1 5 10 20 30 40 50 60 70 80 90 100 2000

Stan

dard

ized

fitn

ess v

alue

No. of iterations

Fig. 8. Convergence of PSSWO algorithm in Experiment 1.



the total cost gain from ratio 1 to ratio 5, as shown in Table 13,which indicates that the wind farm performs better.

The data in Table 14 illustrate the total cost gain for the com-puted schedules for five ratios. A metric, Gain, is utilized to evalu-ate the gain of total cost reduction.

Gain ¼ TCb � TCc

TCb� 100% ð15Þ

As shown in Table 14, on average, total cost can be reduced 10%through the computed schedules. The potential benefits of sched-uling have been validated. The schedules in Table 12 reveal that

the efficient wind turbines do not need always run. The reason isthat wind turbines balance the power shortage cost, operationsand maintenance cost, and start-up cost.

5.2. Simulation results of Experiment 2

The stopping criterion of PSSWO algorithm in Experiment 2 isdetermined in a similar way. The settings of tribe number and

Table 13Comparison between baseline and computed schedules.

Scenario Ratio 1 Ratio 2 Ratio 3 Ratio 4 Ratio 5

TCb TCc TCb TCc TCb TCc TCb TCc TCb TCc

Scenario 19941.0 18415.3 19800.1 18301.0 19725.1 18149.2 19683.4 18067.2 19628.1 17906.31 2 1 8 1 2 7 5 7 7Scenario 23323.1 21957.9 23222.7 21849.9 23178.9 21760.8 23165.3 21690.5 23144.9 21529.92 5 3 8 5 2 8 2 2 1 2Scenario 17981.2 15445.1 17942.0 15365.7 17933.6 15372.6 17925.1 15290.2 17918.4 15095.53 9 8 6 9 8 4 7 9 7Scenario 20407.7 18515.1 20333.4 18528.5 20306.9 18370.6 20293.3 18345.2 20272.9 18049.34 4 4 7 4 7 8 7 2 7 7Scenario 17793.4 16272.3 17738.8 16378.5 17719.7 16200.8 17726.2 16071.0 17740.8 16002.85 8 9 1 7 9 9 3 8 9 1Scenario 20602.8 19193.5 20559.5 19114.3 20553.1 19122.1 20575.7 19096.6 20609.6 18939.76 2 4 4 5 6 7 5 9 1Scenario 16330.9 13763.9 16356.3 13806.1 16389.6 13780.4 16422.9 13750.6 16477.1 13597.27 4 1 7 1 9 2 5 3 1Scenario 18205.7 16293.7 18188.0 16260.3 18198.3 16156.2 18220.9 16119.1 18254.9 16018.68 5 7 9 4 6 8 7 6

Table 14The cost gain for Experiment 1.

Gain1 (%) Gain2 (%) Gain3 (%) Gain4 (%) Gain5 (%)

Scenario 1 8 8 8 8 9Scenario 2 6 6 6 6 7Scenario 3 14 14 14 15 16Scenario 4 9 9 10 10 11Scenario 5 9 8 9 9 10Scenario 6 7 7 7 7 8Scenario 7 16 16 16 16 17Scenario 8 11 11 11 12 12Average 10 10 10 10 11

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

1 5 10 20 30 40 50 60 70 80 90

100

110

120

130

140

150

2000

Stan

dard

ized

fitn

ess v

alue

No. of iterations

Fig. 9. Convergence of the PSSWO algorithm in Experiment 2.

Table 12Scheduling results for ratio 2.


Scenario 1 7.3 4.2 6.0 7.3 0.0 7.3 7.2 0.0 7.3 7.3 7.3 7.3Scenario 2 7.3 5.3 4.1 4.1 0.0 2.3 5.3 7.1 0.0 7.3 7.3 7.3Scenario 3 7.2 3.2 4.1 7.2 0.0 5.3 7.3 0.0 7.3 7.3 7.2 7.3Scenario 4 7.1 6.3 4.1 4.1 0.0 2.3 5.2 5.1 0.0 7.3 7.3 7.1Scenario 5 7.0 4.1 5.0 6.0 0.0 7.1 7.1 0.0 7.3 7.1 7.3 7.2Scenario 6 7.3 6.3 2.3 2.3 0.0 2.3 6.2 6.1 0.0 7.3 7.3 7.3Scenario 7 7.3 3.2 4.1 6.1 0.0 5.2 7.3 0.0 7.3 7.2 6.3 7.2Scenario 8 7.2 6.3 4.1 3.2 0.0 2.2 4.2 5.1 0.0 7.3 7.3 7.2


Scenario 1 7.2 7.2 7.2 7.1 7.3 0.0 0.0 7.3 7.1 7.1 7.0 4.2Scenario 2 0.0 0.0 7.1 7.2 7.3 0.0 0.0 6.3 7.3 0.0 7.3 0.0Scenario 3 5.2 6.1 4.3 5.2 6.3 0.0 0.0 6.2 6.1 5.2 5.1 5.1Scenario 4 0.0 0.0 6.1 6.1 7.2 0.0 0.0 5.2 7.3 0.0 7.2 0.0Scenario 5 7.2 7.1 7.1 7.1 7.3 0.0 0.0 6.3 7.1 6.2 5.2 5.1Scenario 6 0.0 0.0 7.0 7.2 7.3 0.0 0.0 7.2 7.3 0.0 7.3 0.0Scenario 7 4.3 5.2 6.1 5.2 6.3 0.0 0.0 5.3 7.0 6.1 6.1 5.1Scenario 8 0.0 0.0 7.0 6.1 7.3 0.0 0.0 4.3 7.3 0.0 7.1 0.0



particle number of PSSWO algorithm are the same as settings inExperiment 1. The number of duty-cycle variables and their valuesare randomly initialized in the initialization of PSSWO algorithm.After the convergence of PSSWO algorithm, the optimal numberof duty-cycle variables and their values can be obtained. Fig. 9shows that the PSSWO definitely converges within 150 iterations.The stopping criterion of the PSSWO algorithm in Experiment 2is conservatively set to 300.

Table 15 illustrates one sample of the scheduling strategiesinterpreted by duty-cycle based variables. In Table 15, the numberof duty-cycle variables is 74 while the number of binary variablesis 240. The number of variables in the computation is significantlyreduced by using duty-cycle variables. Table 16 presents the base-line schedule of wind turbines under high and low wind speed con-ditions. In Tables 15 and 16, the positive number describes theoperational hours of a wind turbine and the negative number rep-resents the idle hours of a wind turbine. In this experiment, thecomparative analysis of total costs between the baseline scheduleand computed schedules is performed. Table 17 discusses the gainin terms of cost reduction. As presented in Table 17, the averagegain is 10%. These results reveal that even though the power curvesof wind turbines are different, the model can produce appropriateschedules to minimize the total costs of running the wind farm. Atthe same, the produced schedules outperform the baselineschedule.

5.3. PSSWO versus PSO

To assess performance of the PSSWO algorithm, a canonical PSOis implemented with 2000 iterations to solve the models in

Scenario 1 for the two experiments. Performance of the two algo-rithms is compared in Table 18.

As shown in Table 18, the PSO converged faster than the PSSWOalgorithm. In Experiment 1, PSO and PSSWO both converged with-in 2 minutes. In Experiment 2, PSO took 15 minutes to convergewhile PSSWO use 25 minutes. However, the PSSWO produced abetter quality solution while PSO was trapped in local optima. In-deed, the semi-isolated network structure improved the perfor-mance of the canonical PSO.

Besides comparing with the classic PSO algorithm, the PSSWOalgorithm was applied to compute the 24 hours operational sche-dule of 100 wind turbines based on Scenario 1 in Experiment 2. An-other 90 power curves were produced by generating 90 randomvalues of h4,i. The total cost of a baseline schedule is 253494.81while the total cost of a computed schedule by PSSWO is186429.80. The total runtime of computing the schedule is about2 hours, which is not increasing as much as the problem size dueto the usage of duty-cycle variables. To apply the proposed model,the 2 hours computational time can be compensated by a longerforecasting of wind speed, electricity price and electric power de-mand. In the future research, improving the computational timeof solving large size problems needs to be investigated. It can beachieved by either using more efficient programming languagesand techniques, such as C++ with parallel computations, or usingbetter computers with more computational power.

On the other hand, as stated in Rogers et al. (2010), the responsetime to meet the power curtailment request of a wind farm in realpractice is 5–15 minutes. If the electric power demand is predictedor pre-determined by considering factors including planned powercurtailment practices, the wind farm can be operated by followingthe computed schedule. If an emergent curtailment event occursand plant operator needs an extra schedule quickly, our modelcan be used to do one step computation to generate power produc-

Table 15Wind turbine schedule in Scenario 1 of Experiment 2.

Turbine index 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Turbine 1 4 �1 2 �1 9 �7 0 0 0 0 0 0 0 0 0Turbine 2 4 �1 2 �1 4 �4 1 �2 3 �2 0 0 0 0 0Turbine 3 4 �1 2 �1 9 �2 5 0 0 0 0 0 0 0 0Turbine 4 1 �2 1 �1 2 �1 9 �2 5 0 0 0 0 0 0Turbine 5 �5 2 �1 9 �2 1 �4 0 0 0 0 0 0 0 0Turbine 6 �5 2 �1 9 �2 5 0 0 0 0 0 0 0 0 0Turbine 7 4 �1 2 �1 9 �2 5 0 0 0 0 0 0 0 0Turbine 8 4 �1 2 �1 9 �2 5 0 0 0 0 0 0 0 0Turbine 9 4 �1 2 �1 9 �2 5 0 0 0 0 0 0 0 0Turbine 10 4 �1 2 �1 9 �2 4 �1 0 0 0 0 0 0 0

Table 16Baseline schedule.

Wind speed 1 2 3 4 5 6 7 8 9 10 11 12High wind speed 4 �1 2 �1 9 �2 5 0 0 0 0 0Low wind speed 4 �1 3 �1 3 �2 3 �2 2 �1 1 �1

Table 17Cost reduction for Experiment 2.

TCb TCc Gain (%)

Scenario 1 19399.10 17756.54 8Scenario 2 22775.85 21121.55 7Scenario 3 17657.43 15042.15 15Scenario 4 19903.90 17716.81 11Scenario 5 17594.80 16045.10 9Scenario 6 20352.17 18660.32 8Scenario 7 16304.70 13582.13 17Scenario 8 17997.38 15754.51 12Average 11

Table 18Comparison of PSO and PSSWO algorithms.

Algorithm No. ofiterations ofconvergence inExperiment 1

Best fitnessinExperiment1

No. ofiterations ofconvergence inExperiment 2

Best fitnessinExperiment2

PSO 37 18513.36 50 17802.54PSSWO 50 18415.31 150 17756.54



tion schedule for next hour. Computing one step schedule of awind farm with 100 wind turbines can be completed within 5 min-utes (5-minutes per step = 120-minutes/24-steps). The responsetime of modern wind turbines is very prompt, usually no morethan 10 seconds.

5.4. Evaluation of computational results

In this section, a lower bound of minimizing model (11) isoffered to validate the quality of solutions computed by PSSWO.To obtain the lower bound, model (11) is firstly relaxed to an inte-ger programming problem by discarding the quadratic term in theobjective function (the start-up cost). Since start-up cost is qua-dratic and nonnegative, the new objective function becomes linearand the minimal value of the new objective function cannot belarger than that of the original objective function. A linear pro-gramming (LP) relaxation (Danzig et al., 1954) problem of therelaxed model (11) is than obtained by relaxing the variables fromintegral to fractional. Solution of the LP relaxation problem ofthe relaxed model (11) is the lowest bound of minimizing model(11). To solve the LP relaxation problem of the transformed model(11), AMPL with CPLEX solver is utilized. Table 19 compares thecost of relaxed and the original problem in Experiments 1 and 2based on Scenario 1. In Table 19, the costs of relaxed and originalsolutions as well as their differences in Experiments 1 and 2 areprovided. The difference is calculated according to the followingequation:

Difference¼ðCost of Original Solution�Cost of Relaxed SolutionÞCost of Relaxed Solution

ð16Þ

A small difference indicates a high quality of the PSSWO solu-tion. As shown in Table 19, it is clear that the PSSWO solution isapproaching the extreme lower bound of minimizing model (11).

6. Conclusion

A model for scheduling wind turbines was presented. The totalcost of operating a wind farm was minimized. The optimized costincluded the power shortage cost, operational and maintenancecost, and start-up cost. Each schedule represents the operationalstatus of wind turbines over a scheduling horizon. To solve themodel formulated in this paper, a PSSWO algorithm was developedby integrating the traditional PSO with the small world networkstructure. A swarm of particles was divided into tribes, with eachtribe having its own search trajectory. A communication mecha-nism was introduced to exchange information among the semi-iso-lated tribes. The goal of the communication was to balance theexploration and exploitation of the search process by introducingdiversity.

Two experiments were designed to analyze the optimizedschedules produced by the models. Experiment 1 represented asimple wind farm structure with wind turbines having two typesof power curves, efficient and less efficient. In Experiment 2, a windfarm contained 10 wind turbines, each with a different power

curve. To evaluate the quality of the computed schedules, abaseline schedule was used. The comparative analysis indicatedthat the cost of operating a wind farm was reduced by the com-puted schedules over the baseline schedule. Moreover, the com-puted schedules indicated that the wind turbines with moreefficient power curves were not always run due to a tradeoffamong the power shortage cost, operations and maintenance cost,and the start-up cost. As the gains in cost reduction were estimatedbased on the actual data, the proposed model has a potential to beused in practice.

Acknowledgements

The authors thank the reviewers for providing careful reviewsand valuable comments. This research has been supported by theNatural Science Foundation of China, Grant No. 71001050, andthe Iowa Energy Center, Grant No. 07-01.

References

AlRashidi, M.R., El-Hawary, M.E., 2009. A survey of particle swarm optimizationapplications in electric power systems. IEEE Transactions on EvolutionaryComputation 13 (4), 913–918.

AWEA. <http://www.awea.org/> (accessed 2011).Boukhezzar, B., Siguerdidjane, H., 2009. Nonlinear control with wind estimation of a

DFIG variable speed wind turbine for power capture optimization. EnergyConversion and Management 50 (4), 885–892.

Cao, X., Jouglet, A., Nace, D., 2012. A multi-period renewable equipment problem.European Journal of Operational Research 218 (3), 838–846.

Clerc, M., Kennedy, J., 2002. The particle swarm – explosion, stability, andconvergence in a multidimensional complex space. IEEE Transactions onEvolutionary Computation 6 (1), 58–73.

Danzig, G., Fulkerson, R., Johnson, S., 1954. Solution of a large-scale traveling-salesman problem. Operations Research 2 (4), 393–410.

de Castro, L.N., 2002. Immune, swarm, and evolutionary algorithms part II:philosophical comparisons. In: Proceedings of the ICONIP Conference(International Conference on Neural Information Processing), Workshop onArtificial Immune Systems, pp. 1469–1473.

Fang, C., Lee, J., Schilling, M.A., 2010. Balancing exploration and exploitation throughstructural design: the isolation of subgroups and organizational learning.Organization Science 21 (3), 625–642.

Wind Energy-The Facts. <http://www.wind-energy-the-facts.org/en/part-3-eco-nomics-of-wind-power/> (accessed 2011).

Kennedy, J., Eberhart, R.C., 1995. Particle swarm optimization. IEEE InternationalConference on Neural Networks, 1942–1948.

Kim, B., Kim, S., Park, J., 2012. A school bus scheduling problem. European Journal ofOperational Research 218 (2), 577–585.

Kusiak, A., Li, M.Y., 2010. Reheat optimization of the variable-air-volume box.Energy 35 (5), 1997–2005.

Kusiak, A., Zhang, Z., 2011. Adaptive control of a wind turbine with data mining andswarm intelligence. IEEE Transactions on Sustainable Energy 2 (1), 28–36.

Kusiak, A., Zheng, H.Y., Song, Z., 2009a. Models for monitoring wind farm power.Renewable Energy 34 (3), 583–590.

Kusiak, A., Zheng, H.Y., Song, Z., 2009b. On-line monitoring of power curves.Renewable Energy 34 (6), 1487–1493.

Lee, K.Y., Park, J.B., 2006. Application of particle swarm optimization to economicdispatch problem: advantages and disadvantages. Power Systems Conferenceand Exposition, 188–192.

Liu, D., Tan, K.C., Goh, C.K., Ho, W.K., 2007. A multiobjective memetic algorithmbased on particle swarm optimization. IEEE Transactions on Systems, Man, andCybernetics: Part B 37 (1), 42–50.

Manwell, J.F., McGowan, J.G., Rogers, A.L., 2002. Wind Energy Explained: TheoryDesign and Application, first ed. John Wiley and Sons, London.

March, J.G., 1991. Exploration and exploitation in organizational learning.Organization Science 2 (1), 71–87.

Milgram, S., 1967. The small world problem. Psychology Today 1 (1), 60–67.

Table 19Validation of the PSSWO schedules.

Problems Experiment 1 Experiment 2

Ratio 1 Ratio 2 Ratio 3 Ratio 4 Ratio 5

Cost of relaxed solution 18080.53 17893.90 17774.84 17673.79 17524.10 17401.17Cost of original solution 18415.31 18301.01 18149.27 18067.27 17906.37 17756.54Difference 0.02 0.02 0.02 0.02 0.02 0.02



Ouelhadj, D., Petrovic, S., 2009. A survey of dynamic scheduling in manufacturingsystems. Journal of Scheduling 12 (4), 417–431.

Pappala, V.S., Erlich, I., 2008. A new approach for solving the unit commitmentproblem by adaptive particle swarm optimization. In: Proceedings of IEEEPower and Energy Society General Meeting—Conversion and Delivery ofElectrical Energy in the 21st Century, pp. 1–6.

Pappala, V.S., Erlich, I., Rohrig, K., Dobschinski, J., 2009. A stochastic model for theoptimal operation of a wind-thermal power system. IEEE Transactions on PowerSystems 24 (2), 940–950.

Park, J.B., Lee, K.S., Shin, J.R., Lee, K.Y., 2005. A particle swarm optimization foreconomic dispatch with nonsmooth cost functions. IEEE Transactions on PowerSystems 20 (1), 34–42.

Park, J.B., Jeong, Y.W., Shin, J.R., Lee, K.Y., 2010. An improved particle swarmoptimization for nonconvex economic dispatch problems. IEEE Transactions onPower Systems 25 (1), 156–166.

Ratnaweera, A., Halgamuge, S.K., Watson, H.C., 2004. Self-organizing hierarchicalparticle swarm optimizer with time-varying acceleration coefficients. IEEETransactions on Evolutionary Computation 8 (3), 240–254.

Ren, B., Jiang, C., 2009. A review on the economic dispatch and risk managementconsidering wind power in the power market. Renewable and SustainableEnergy Reviews 13 (8), 2169–2174.

Rogers, J., Fink, S., Porter, K., 2010. Examples of Wind Energy Curtailment Practices(SR-550-48737). National Renewable Energy Laboratory, Denver.

Saxena, A.K., Vora, M., 2008. Novel approach for the use of small world theory inparticle swarm optimization. In: 16th International Conference on AdvancedComputing and Communications, pp. 363–366.

Seo, J.H., Im, C.H., Heo, C.G., Kim, J.K., Jung, H.K., Lee, C.G., 2006. Multimodal functionoptimization based on particle swarm optimization. IEEE Transactions onMagnetics 42 (4), 1095–1098.

Shi, Y., Eberhart, R.C., 1998. A modified particle swarm optimizer. In: Proceedings ofIEEE International Conference on Evolutionary Computation, pp. 69–73.

Siahkali, H., Vakilian, M., 2009. Electricity generation scheduling with large-scalewind farms using particle warm optimization. Electric Power Systems Research79 (5), 826–836.

Ustüntas, T., Sahin, A.D., 2008. Wind turbine power curve estimation based oncluster center fuzzy logic modeling. Journal of Wind Engineering and IndustrialAerodynamics 96 (5), 611–620.

Watts, D.J., Strogatz, S.H., 1998. Collective dynamics of ‘small-world’ networks.Nature 393 (6684), 440–442.

Yazid, M., Stphane, D., Chams, L., 2011. A general approach for optimizing regularcriteria in the job-shop scheduling problem. European Journal of OperationalResearch 212 (1), 33–42.

Yeh, T., Wang, L., 2008. A study on generator capacity for wind turbines undervarious tower heights and rated wind speed using weibull distribution. IEEETransactions on Energy Conversion 23 (2), 592–602.


Scheduling electric power production at a wind farm

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