a stochastic programming to voltvartotal harmonic distortion control in distribution networks...

Electric Power Components and Systems, 43(7):733746, 2015Copyright C Taylor & Francis Group, LLCISSN: 1532-5008 print / 1532-5016 onlineDOI: 10.1080/15325008.2014.1002585

A Stochastic Programming to Volt/VAR/TotalHarmonic Distortion Control in DistributionNetworks Including Wind Turbines

Sajad Jashfar,1 Mohammad Mahdi Hosseini-Biyouki,2 and Saeid Esmaeili3,41Electrical Engineering Department, Graduate University of Advanced Technology, Kerman, Iran2Electrical Engineering Department, Amirkabir University of Technology, Tehran, Iran3Electrical Engineering Department, Shahid Bahonar University of Kerman, Kerman, Iran4Energy and Environmental Research Center, Shahid Bahonar University of Kerman, Kerman, Iran

CONTENTS

1. Introduction

2. Probabilistic Analysis of Load and WT Prediction

3. Harmonic Inclusion

4. Problem Formulation

5. Implementation of the Proposed HJP Method

6. Simulation Results

7. Results and Discussion

8. Conclusion

References

Keywords: volt/VAR control, distribution networks, genetic algorithm,harmonic distortions, renewable energy resources, stochastic programming,wind turbines

Received 20 October 2013; accepted 16 December 2014

Address correspondence to Mr. Sajad Jashfar, P.O. Box 76315-117,Electrical Engineering Department, Graduate University of AdvancedTechnology, Haft Baghe Alavi Highway, Mahan, Kerman, Iran 7631133131.E-mail: [email protected] versions of one or more of the figures in the article can be found onlineat www.tandfonline.com/uemp.

AbstractThis article presents a stochastic methodology forvolt/VAR/total harmonic distortion control to reduce power losseswhile satisfying the main recommended power quality standards andoptimizing dispatch schedules for the switchable shunt capacitor andon-load tap-changer in distribution networks. The main aim is tofind proper dispatch schedules for on-load tap-changer tap positions,substation capacitors, and along feeder capacitors. For this purpose,distribution network uncertainties, including load demand and windpower generation, are considered to provide a robust control scheme.A new scenario reduction method based on the highest potential clus-ter center is used to decrease the huge number of probable states. Anew scenario-based probabilistic time-interval division framework,over a 24-hr period on both load curve and wind power output, is in-troduced to reduce effects of forecast plan uncertainty and switchingoperations in the on-load tap-changer. A genetic algorithm solutionmethod is applied to find the best solution corresponding to variousscenarios. To improve search ability, a method guaranteeing the sup-pression of maximum allowable daily substation capacitors switchingand effectively correcting the convergence process is utilized. Theproposed stochastic approach is tested on an IEEE 123-bus distri-bution network containing a number of non-linear loads and windenergy generation systems.

1. INTRODUCTION

Daily off-line volt/VAR control of a large-scale power networkwill provide decision makers more possibilities to scheduleon-line volt/VAR control. Volt/VAR control at the distribu-tion system level have been widely employed to reduce energylosses and maintain voltage profiles within permissible lim-its [1]. High penetration of non-linear loads and renewableenergy sources (RESs) in distribution networks leads to morecomplexity of optimal operation scheduling of these networks.Propagation of harmonics through the system causes damageto devices and, consequently,more losses. Capacitorsmay have

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734 Electric Power Components and Systems, Vol. 43 (2015), No. 7

NOMENCLATURECtn = state of capacitor n at hour tCi,SC j = on/off states of secondary bus capacitorsCi,i+1 = capacitance of line segment between

buses i and i + 1,C (fh) = ratio of hth harmonic current to its funda-

mental currentdFC j = on time duration of feeder capacitorsdi,SC j = on/off states time duration for secondary

bus capacitorsD = number of optimization parametershmax = highest harmonic order of interesth0 = smallest harmonic order of interestI (h) = bus injection currents (by non-linear

loads) at hth harmonic orderIi (fh) = hth harmonic currents of the non-linear

lofad installed at bus iJ = junction of ith intervals end to start of (i

+ 1)th interval = index of time intervalLi,i+1 = reactance of line segment between buses i

and i + 1Losss = energy losses of compensated system for

sth scenarioLossagg = aggregated weighted energy losses of

compensated system for all scenariosLB = lower boundMKSC = maximum limit of substation capacitor

switchingMKC = maximum limit of capacitor switchingMKT = maximum allowable number of on-load

tap-changer daily switching operationsn = number of intervals for entire load periodNb = total number of busesNC = total number of capacitorsNL = total number of linesNFC = number of feeder capacitorsNSC = number of substation capacitorsNP = population sizeNL = number of loadsNWT = number of wind turbinesNS = number of reduced scenariosoi = ith offspringPlinear,i = active linear loads at bus iPt,sLoss = component of real power loss at hour t for

sth scenarioPt,sLoss,L (fh) = component of line real power loss in fre-

quency fh at hour t for sth scenario

Pt,sLoss, T (fh) = component of transformer real power lossin frequency fh at hour t for sth scenario

Pt,sLoss, Lines = total real power losses of lines at hour t forsth scenario

Pt,sLoss, Transformer = total real power losses of transformer athour t for sth scenario

Pt,sL = active load at tth hour for sth scenarioPNonlinear,i = active non-linear loads at bus iPt,sWT = active power output of wind turbine at

hour t for sth scenarioQlinear,i = reactive linear loads at bus iQNonlinear,i = reactive non-linear loads at bus iRi,i+1 = resistance of line segment between buses

i and i + 1t = index of time in a 24-hr periodts = start time of th time intervaltFC = start time of switching feeder capacitors

to on stateTAPt = on-load tap-changer tap position at hour tT HDV,max = maximum value of total harmonic distor-

tionT HDtVi = total harmonic distortion factor at bus i

and hour tUB = upper boundV (h) = bus voltage vector at hth harmonic orderVref = voltage reference valueV t,si (fh) = component of voltage in frequency fh for

bus i and hour t for sth scenarioV t,si = voltage deviation at bus i and hour t for

sth scenarioVmax = maximum allowable voltage deviation

valuey(h)capacitor,i = admittance of shunt capacitor banks in-

stalled at bus iY (h) = bus admittance matrix representing he

system at hth harmonic orderY (h)jj = Main-diagonal entries,Y (h)jk = off-diagonal entriesy(h)line,shunt = shunt parameter of linesy(h)line,longitudinal = longitudinal parameter of linesy(h)linear,i = admittance of linear load connected at bus

iXm = parent population = point of active power or reactive power at

th time interval = average of active power or reactive power

at th time interval

Jashfar et al.:AStochastic Programming toVolt/VAR/TotalHarmonicDistortionControl inDistributionNetworks IncludingWindTurbines 735

i = ith parentt,s = load demand or wind power generation

probability for tth hour for sth scenario

= start time vector = transformer voltage ratio = additional voltage ratio per tap

an important role in the propagation of harmonics in the net-works. The on/off capacitor switching does not introduce newharmonics into the network but may lead to amplify alreadypresent current and voltage harmonics due to possible reso-nance at one or more harmonic frequencies [24]. Harmonicsput power quality greatly at risk and lead to undesirable solu-tions at the operational level.

Significant improvement in the efficiency and quality ofpower system operation is achieved by coordinated operation[5]. Daily off-line volt/VAR control is a tool to coordinate thecentralized and local controllers in distribution managementnetworks [6, 7]. Additionally, volt/VAR control problems havebeen investigated in two different categories, including cer-tainty and uncertainty of input variables. In the first category,different volt/VAR control methods have been proposed to im-prove network conditions. The daily optimal volt/VAR controlintegrating distributed generators (DGs) under steady-state si-nusoidal operation condition was investigated in [8, 9]. Vi-awan and Karlsson [10] suggested a coordination strategy forvolt/VAR control in the presence of a DG and conventionalcontrol equipment, such as an on-load tap-changer (OLTC)and capacitors. A dynamic programming method under sinu-soidal operating system conditions across the real medium-voltage distribution system was presented in [11]. However,in large systems, the dynamic programming method is not ap-propriate due to the computational burden involved with it.A cost-based methodology for daily volt/VAR control with-out harmonic consideration in distribution systems, includingDGs, was presented in [1215]. Those authors used evolu-tionary methods, such as ant colony optimization (ACO) [12],honey bee mating optimization (HBMO) [13], particle swarmoptimization (PSO) [14], the gravitational search algorithm(GSA) [15], and the bacterial foraging algorithm (BFA) [15]to determine the solutions. The authors in [16] proposed a dis-patching schedule in a real distribution network regardless ofharmonics.Volt/VARcontrolwith harmonic considerationwasdiscussed in a few studies [1719]. In these works, the OLTCtap position planning and shunt capacitor on/off switchingstates have been done based on an optimal time-interval divi-sion for the forecasted daily load to decrease energy losses andimprove power quality; however, there are some distributionnetworks uncertainties, such as load demand (LD) and renew-

able energy electricity generation. In this regard, for the secondcategory, stochastic volt/VAR control with uncertain values forsome random variables regardless of harmonic considerationwere discussed in some works [2022]. A probabilistic analy-sis based on a 2m-point estimated method has been employedto solve the daily volt/VAR control problem in distributionsystems with uncertainty in LD and electrical power genera-tion [20, 21]. In [20], a scenario with an aggregation-basedsingle-objective solution considering environmental and eco-nomic aspects of volt/VAR control in the presence of RESswas presented. A fuzzy optimization approach to obtain theoptimal dispatching schedule under an uncertain environmentwas proposed in [22].

This article extends the precise mutual impact of powerquality constraints and a stochastic methodology for volt/VARplanning that has not been considered in previous research.Consideration of uncertainty in the power output of a windturbine (WT) as well as power demand of load leads toprocurement of probabilistic optimal time-interval division.Also, the control possibility should be performed in switchablecapacitor banks and the OLTC. The active power outputs areoften specified by characteristics of energy resources [23].To perform precise calculations, hybrid joint programming(HJP) to volt/VAR/total harmonic distortion (THD) control isdeveloped and implemented utilizing the integration of MAT-LAB (The MathWorks, Natick, Massachusetts, USA) andDIgSILENT (DIgSILENT GmbH, Gomaringen, Germany)[19]. In this regard, a new trend in programming procedureto alleviate the probabilistic computational burden of theproblem is adopted.

This article is organized as follows. The probabilistic anal-ysis of load and WT prediction to model the occurrence of aspecific power is described in Section 2. A harmonic load flowcalculation (HLFC) procedure and formulation is explainedin Section 3. Section 4 presents the problem formulation. Im-plementation of an HJP method to determine the optimal dis-patch schedules for all capacitors and OLTC tap positions isproposed in Section 5. Simulation results of applying the sug-gested control scheme to three test cases is demonstrated inSection 6, while a detailed discussion of the obtained resultsare presented in Section 7. Finally, the major contributions andconclusions are summarized in Section 8.


2. PROBABILISTIC ANALYSIS OF LOAD ANDWTPREDICTION

2.1. Distribution

One of the major problems in real-world electrical power sys-tems is the uncertainty of LD forecast and RES power produc-tion variability. In this article, both LD and wind power (WP)have been considered as random variables with a probabilitydistribution function (PDF). To simplify the problem, it is as-sumed that there is no correlation between LD and WP [24].To model the occurrence of a specific power value, the betafunction can be much more appropriate than other PDFs [25]:

fPPred (x) = x1. (1 x)1 .Nf . (1)For each hour, the historical data are used to apply proba-

bilistic model for LD and WP. Hence, to predict LD and WPproduction at each hour, different shape parameters must beconsidered. The beta function shape parameters and canbe derived from the mean and variance of historical data, asexpressed by Eqs. (2) and (3); here, PPred is in per unit formand its range is from 0 to 1:

PPred = + , (2)

2 = ( )( + )2 ( + +1) . (3)

The relation between variance and predicted power is ob-tained as [20, 25]:

= 0.215 PPred + 0.0285. (4)

2.2. Scenario Generation

Considering continuous probabilistic distribution parametersgenerally leads to no specific way of solving the mathematicalproblems. The general method is discrete approximation of thecontinuous PDF. In other words, there is a finite scenario set foreach of the random variables [26]. Each scenario includes a 24-hr period of LD and WP production. Monte Carlo simulationis used as an effective approach to generate a discrete set ofhourly LD and WP:

Scenarios :{t,sL ,

t,sWT

}.

2.3. Scenario Reduction

Although large number of scenarios increases modeling ac-curacy, it is time consuming and has a high computationalburden in real-world power systems [27]. To overcome thisdifficulty, the number of scenarios should be reduced to an ac-ceptable amount. If the initial scenario involves N-scenario set{S1, . . . , SN } in them-dimensional space (xi=[xi,1, . . . , xi,m]),there are potential cluster centers according to the distance of

each scenario with respect to other scenarios. The potentialvalue for each data point (Si) is determined as [28]

(Si) =N

j =1j = i

e(SiSj2

(ri/2)2

), i =1, ...,N, (5)

where ri=mk=1 (x2i,k ). The scenarios outside of the nearestneighbor radius have little impact on the potential value.Hence, the initial scenario sets are decreased using the sce-nario reduction technique, which is described in the followingsteps.

Step 1: Find the most probable scenario Si with the greatestpotential:

max=max { (Si)} . (6)Step 2: Find the finite scenario sets to the nearest neighbor

Si that is located in the radius (ri/2). After scenario reduction,the occurrence probability of each scenario is calculated viathe following equation [20]:

S =24t=1

(NLL=1

Lt,s NWTWT=1

WTt,s

)/ NRs=1

24t=1(

NLL=1

Lt,s NWTWT=1

WTt,s

). (7)

In this article, the considered mean values of the hourly LDand WT production for the statistical analysis have been bor-rowed from [18, 21], respectively. To consider the uncertaintyin forecast plan, five desired scenarios are achieved for a 24-hrtime period by alternating the network conditions, which aregiven in Table 1.

3. HARMONIC INCLUSION

At higher frequencies, the system is modeled by a harmonicfrequency admittance matrix and harmonic current sources.The admittance matrix Y (h)is computed by modifying the con-ventional fundamental admittance according to the harmonicorder. The distribution lines are considered as lumped param-eter elements connected in a -model. The shunt and lon-gitudinal parameters of the lines are included in the diagonaland off-diagonal entries of the harmonic frequency admittancematrix Y (h), respectively:

y(h)line,longitudinal = y(h)i,i+1 =1

Ri,i+1 + j 2 fh Li,i+1 , (8)

y(h)line,shunt = y(h)i,i = y(h)i+1,i+1 = j 2 fh Ci,i+1. (9)


Scenario 1 Scenario 2 Scenario 3 Scenario 4 Scenario 5

Hour LD WP LD WP LD WP LD WP LD WP

1 31.95 92.58 32.39 94.46 31.63 89.56 32.68 93.54 33.65 91.232 30.59 97.25 29.79 96.58 29.1 94.81 29.5 99.36 28.85 94.983 32.22 89.24 32.48 88.38 33.47 89.79 32.74 90.16 32.1 90.454 27.95 78.66 29.21 80.43 27.94 79.89 29.54 80.43 29.75 76.535 29.02 74.71 29.87 72.17 28.22 74.69 28.25 71.16 28.36 74.356 42.81 73.04 40.85 71.73 40.95 73.53 40.06 73.83 40.94 71.897 49.2 74.04 47.36 74.37 49.67 75.63 48.98 70.87 47.71 71.368 69.96 75.41 72.65 74.48 71.03 78.13 70.99 73.91 71.98 78.129 88.22 78.21 89.53 78.42 83.82 80.32 86.31 79.87 87.34 80.7610 91.55 79.54 90.41 79.12 90.42 78.17 94.43 78.86 93.76 79.9311 96.09 83.04 93.54 81.51 94.35 85.27 96.34 82.93 93.6 80.3712 94.62 78.24 93.55 77.53 90.18 80.39 94.44 79.32 89.62 80.7213 94.14 77.97 91.14 78.83 90.39 82.62 93.59 81.63 94.49 78.1514 91.6 70.14 89.49 72.61 91.89 69.74 92.53 71.68 86.91 69.8615 81.34 53.52 79.75 54.69 81.63 51.37 81.28 51.84 81.51 51.4616 61.14 43.52 63.21 42.88 63.73 42.47 63.61 42.28 63.09 43.1117 51.86 41.35 49.76 41.71 52.34 41.58 49.31 39.87 52.25 40.3918 45.96 35.58 46.65 38.02 47.21 36.67 46.45 35.93 45.18 37.8719 40.37 40.08 40.39 41.34 40.35 40.98 41.23 41.33 42.21 42.5220 41.92 47.69 40.34 47.56 41.15 49.14 41.88 48.66 42.27 49.2721 39.56 54.49 39.45 55.53 38.21 54.97 39.69 53.63 40.66 53.4422 37.38 71.35 37.96 69.14 37.26 69.96 37.83 67.09 38.4 68.5323 33.83 84.22 34.86 80.49 32.57 85.29 33.96 81.36 32.01 82.4124 33.32 88.24 33.82 87.64 32.96 89.34 34.66 84.69 32.29 86.76

TABLE 1. Achieved scenarios for 24-hr time period LD and WP generation

Shunt capacitor banks are represented as shunt-connectedelements:

y(h)capacitor,i = h y(1)capacitor,i. (10)

The admittance of the linear load connected at bus i isdefined as [17, 18]

y(h)linear,i =Plinear,i

|Vi (f1)|2 j Qlinear,i

h |Vi (f1)|2. (11)

Non-linear loads are modeled as harmonic current sourcesthat inject harmonic currents into the system. The fundamentaland the hth harmonic currents of the non-linear load installedat bus i with active power and reactive power are expressed by[17, 18]

Ii (f1) =(PNonlinear,i + j QNonlinear,i

V i (f1)

), (12)

Ii (fh) = C (fh) Ii (f1) . (13)

A conventional NewtonRaphson algorithm is used to solvethe power flow at fundamental frequency, and harmonic volt-ages are calculated by solving the following load flowequation,

which is derived from the node equations, as follows:

Y (h) V (h) = I (h). (14)

4. PROBLEM FORMULATION

The control variables include tap positions of the OLTC aswell as substation capacitor (SC) and feeder capacitor (FC)on/off switching states. The aim is to find the minimum valueof objective function while satisfying the operational and prac-tical constraints. To overcome the stochastic nature of system,the normalized scenario probability is used in the computationof objective function. Considering all reduced scenarios, themathematical formulation of the optimization problem givennext.

4.1. Objective Function

4.1.1. Energy Losses over a 24-hr Period

Total real power losses at all frequency components over a24-hr period for all possible scenarios has to be minimized.The active power losses at hour t can be defined as the sum of


losses in each line and transformer, as follows:

PsLoss, Transformer = PsLoss,T (f1)+hmaxh=h0

PsLoss,T (fh), (15)

PsLoss, Lines =NlL=1

PsLoss,L (f1)+NlL=1

hmaxh=h0

PsLoss,L (fh), (16)

PsLoss = PsLoss, Transformer+PsLoss, Lines. (17)The objective function of the proposed stochastic-

based volt/VAR/THD control problem considers aggregatedweighted energy losses of the system for possible scenarios asfollows:

Objective: min

Lossagg=

NSi=1

SNS

i=1i

24t=1

Pt,sLoss

,

(18)

4.2. Operational and Practical Constraints

4.2.1. Voltage Deviation at Each Bus

Corresponding to scenario s, the RMS value of bus i voltageat hour t is defined by

Vt,si,rms=Vt,si (f1)2 + hmax

h=h0

Vt,si (fh)2. (19)To minimize the difference between bus voltages from the

actual operating voltage (|Vti|=1p.u.)and enhance voltage se-curity, voltage deviation can be calculated as

V t,si =1Vt,si,rms , i =1, ...,Nb, t =1, ..., 24. (20)

A well-operated distribution system must keep the voltagesat all nodes within the allowed limits:

V t,si (%) Vmax. (21)

4.2.2. THD at Each Bus

Corresponding to scenario s, the voltage THD at bus i at hourt is expressed by

THDt,sV i =(

Vt,si,rms)2 Vt,si (f1)2Vref

i = 1, ...,Nb;t = 1, ..., 24. (22)

The amount of RMS voltage improvement not only relieson fundamental voltage but also on harmonic components,which play an important role in the improvement [19]:

Vt,si,rms=Vt,si (f1)

1 + (THDt,sV i)2. (23)

THDV,max for all scenarios should be within acceptable op-erating limits through the optimization process. Therefore,THDt,sVi is limited to a maximum value as follows:

THDt,sV i (%) THDV ,max. (24)

The steady-state voltage deviation in Eq. (20) and voltageTHD in Eq. (22) are restricted according to Standard IEEE-519[29]. Frequent switching operations may reduce switchablecapacitor banks and OLTC lifetime. It is necessary to considerlife expectancies of them. Practical constraints, such as themaximum allowable number of switching operations in a dayfor OLTCs [30], SCs, and FCs (over all scenarios) are definedin what follows.

4.2.3. Maximum Switching Operations of OLTC

24t=1

TAPt TAPt1 MKT . (25)The maximum permissible operating times of an OLTC in

a day is considered to be 30 [18, 22].

4.2.4. Maximum Switching Operations of Capacitors

24t=1

CtnCt1n MKC, n =1, 2, ...,Nc. (26)The maximum permissible switching operating times for

the capacitors installed at the secondary bus in a day is consid-ered to be 6, and those installed through the feeder is assumedto 2 [19, 22].

5. IMPLEMENTATION OF THE PROPOSED HJPMETHOD

The proposed scheme comprises two outstanding features.First, the average curve for available scenarios is divided intoseveral levels. Second, the combination of optimal dispatchschedule of all control devices, such as OLTCs, SCs, and FCs,simultaneously besides considering stochastic framework isperformed using a genetic algorithm (GA). A feasible solutionfor volt/VAR/THD control includes OLTC and capacitor 24-hrsettings at each hour where the operational and practical con-straints are within the acceptable limits. The control variablesfor each interval include 17 OLTC tap position states ([8, . . .,1, 0, 1, . . ., 8]) and 2 capacitor states (0 = off; 1 = on) foreach capacitor at each hour [19].


5.1. Probabilistic-based Time-interval Method

The optimal probabilistic time intervals can be determined tosatisfy the maximum switching operations of the OLTC andreduce the effect of uncertainty and slight variations in the fore-cast plan. The OLTC tap position during all hours at each inter-val remains constant [17, 18]. The presented method has beenpromoted for both electrical LD and alternative energy gener-ation, simultaneously. To perform the proposed probabilistic-based time-interval method while considering uncertainties inLD and RES forecasting, an average curve for both generationand consumption for available scenariosmust be procured. Thecurve is obtained using the reduced scenarios from Table 1 andthe calculation of Ptnorm, for each of 24 hr using Eq. (27):

Ptnorm, =1

NS

NSs=1

(Pt,s t,s

). (27)

In the next step, because of the interaction between gen-eration and consumption curves in the time-interval method,the obtained average curves are simultaneously divided intointervals. Regarding to the considered number of intervals, ev-ery chromosome contains the start time of each time interval.The solution structure of the time-interval division problem,which represents the start time of each interval, is formed as = [ts1 , ts2 ,..., tsn](1n). A flowchart diagram of the proposedtime-interval method is shown in Figure 1. The value of anygenerations suggests the start time of each time interval (tsl ).The number of time intervals is chosen 4, and a GA is em-ployed to determine the start time of each time interval [19].The start time of each interval can be obtained from minimiz-ing the fitness function F(). This formula consists of twosub-equations, as follows:

l {1, 2, ..., n} = [ts1 ts2 ...tsn](1n): 1 tsl 24 & tsl1 < tsl ,

l {Ptnorm,L, Ptnorm,WT} = [12...n](124) :

1= [ (ts1) (ts1+1) , , (ts21) ](1(ts2ts1 ))2= [ (ts2) (ts2+1) , , (ts31) ](1(ts3ts2 ))...

n=[(tsn), , (24)

(1) , , (ts11)](1(tsnts(n1)

))

,

(28)

F () = minn

l=1

l l2. (29)

FIGURE 1. Flowchart diagram of the proposed time-intervalmethod.

5.2. HJP Applied to Volt/VAR/THD Control

The optimization problem is solved using two separatemodules. The flowchart in Figure 2 demonstrates the calcu-lation process of the optimization problem. The GA is imple-mented in MATLAB, while DIgSILENT programming lan-guage (DPL) is utilized to perform the objective functionscalculations. The proposed procedure for daily volt/VAR/THDcontrol is iterated utilizing the combinatorial method until con-vergence is achieved [19]. The modules are described next.

5.3. MATLAB Module

The output of the MATLAB module is utilized as the ini-tial values for the DIgSILENT module. Figure 3 illustratesthe solution structure for the volt/VAR/THD problem for theexisting control variables. As expressed by Eq. (30), eachchromosome of population(Np)generated in a random man-ner by MATLAB is composed of two parts. The first partis related to SC on/off switching modes with dimension(NSC (2 MKSC 1))as well as the FC on/off switchingmodes with dimension (2NFC), while the second part is re-lated to OLTC tap position modes with dimension (n);

X{NPD} = [SC{NP(NSC(2MKSC1))} FC{NP(2NFC)}OLTC{NP(n)}]. (30)


FIGURE 2. Flowchart of the proposed algorithm for optimal scheduling.

FIGURE 3. Solution structure for volt/VAR/THD problem for the existing control variables.


Therefore, if the constraint of maximum OLTC switchingoperations is satisfied, MATLAB writes on the chromosomeavailable in a text file [19].

5.4. DIgSILENT Module

The outputs of this module are used in a cyclic procedureas the initial values for the previously described MATLABmodule. DIgSILENT reads the chromosome data as inputand applies them to perform hourly on/off capacitor switch-ing and OLTC tap position. The dispatch of shunt capaci-tors to perform hourly on/off capacitor switching is presentedin next subsection. Afterwards, HLFC is run based on as-signed hourly optimal scheduling, and the objective functionvalues are calculated. Again, DIgSILENT exports the objec-tive function values through a text file into MATLAB as inputdata [19].

5.5. Dispatch of Shunt Capacitors

Considering themaximumallowable number of capacitor dailyswitchings, these capacitors should be programmed in a waythat the constraints in switching capacitors become implicit.Such a programming procedure would lead to appropriate con-vergence despite the complexity and computational burden. Inthis article, a method that guarantees the suppression of max-imum allowable daily FCs and SCs switching and effectivelycorrects the convergence process is utilized. Figure 4 illus-trates the programming of capacitors installed at a substation.The red line shows the change in the state of switching. Theunchanged switching blue line is followed by a reduction of theswitching states. At each interval, the values 0 or 1 representthe on/off state of the capacitor. The maximum time intervalis achieved by dividing 24 hr to MKSC, and its minimum is0. Therefore, it is obvious that if each di is 0 or a value intwo consecutive intervals, the number of capacitors switchingwill decrease. This idea would satisfy maximum allowable ca-pacitors switching as well as on/off periods of capacitors (seeFigure 4(a)). For example, Figure 4(b) presents the sample dataof a chromosome representing the scheduling of a substationscapacitor that is fed into the DIgSILENTmodule. Consideringthese data for Figure 4(a) implies that the capacitor would stayon from hour 1 for three hours. Since d2 is assigned by 0, theon/off state is not determined in this interval, and there wouldonly be one switching reduction. In hour 4, the capacitor isswitched off for two hours. In hour 6, on state is scheduledfor two consecutive periods of three and four hours. This statealso represents a switching reduction. In the remaining hours,the off state is scheduled for the capacitor (see Figure 4(c))[19].

FIGURE 4. Programming of capacitors installed in the sub-station [24].

5.6. Optimization Process of the GA

5.6.1. Initial Population

Considering the bounds on the decision variables, a new ran-dom chromosome (power system variables) consists of an op-erating point of the capacitor on/off switching modes and theOLTC tap position modes scheduling (see Figure 3). The re-lated part of the capacitor on/off switching modes in eachgenerated solution is spontaneously restricted by operationalconstraints. In this article, the population size (NP) is 15 timesthe number of optimization parameters (D).

5.6.2. Evaluation of Population

With the initial random values of control variables, energylosses at each hour, voltage deviation, and voltage THD at eachbus is calculated by HLFC. Total energy losses are calculatedby combining the energy losses of all 24 hr. Maximum voltagedeviation andmaximumvoltage THDof the entire 24-hr powersystem operation are also calculated.


5.6.3. Crossover and Mutation

New solutions (power system variables) of size NPare gener-ated by crossover and mutation applied to X0. The mathematicdescription of crossover is

o1 = 1 + rand (2 1) ,o2 = 2 + rand (2 1) ,

The mathematic description of mutation is o = + rand (UB LB), where rand is a random number in therange of 0 to 1, while and are scalar parameters.

5.6.4. Save Best Solution

The best solution is retained that complies with the lowest totalenergy losses during a 24-hr period.

5.6.5. Stopping Criteria

Stopping criteria is decided based on experience, and in thisvolt/VAR/THD problem, the value of iteration number is cho-sen as 300.

6. SIMULATION RESULTS

The proposed stochasticmethodology for daily volt/VAR/THDcontrol is applied to a 4.16-kV, IEEE 123-bus distribution testsystem [31]. The IEEE 123-bus test system considered in thisarticle contains non-linear loads besides a WT (see Figure 5).

FIGURE 6. (a) Transformer with tap setting ratio ; trans-former presented by -model with: (b) principal tap ratio and(c) a new voltage ratio.

FIGURE 5. IEEE 123-bus distribution network.


Figure 6(a) demonstrates a transformer connected betweenbuses 119 and 120 with admittance (y) and principal tap ratio( ). The tap-changer is installed at the high-voltage or low-current side of the transformer winding. The OLTC is used tomaintain the secondary bus voltage profile close to the nominalvalue under all load conditions. Impedance of the transformeris 0.01 + j0.08 p.u. Base values of transformer are consid-ered to be 5 MVA and 4.16/115 kV [31]. This transformeris described by -model with indirect representation of thetransformer tap ratio in series and parallel admittances, as de-picted in Figure 6(b). Figure 6(c) demonstrates the transition oftransformer tap from to + . Transformer voltage ratiodepends on the tap position and additional voltage ratio per tap( = 1 + TAP p.u.). It must be taken into considerationthat the transformer tap position should be an integer. In thisarticle, the OLTC can regulate voltage deviation (TAP )from 5% to +5%, considering = 0.625 %. The data ofshunt capacitors installed in the distribution system are givenin [17]. Two 200-kW WTs are installed at buses 89 and 108.The system includes five types of non-linear loads with the

harmonic spectrum given in [17]. The harmonic spectrum forWTs is taken from [32]. In the presence of harmonics, threedifferent cases are considered to investigate the effectivenessof the proposed method.

Case 1: System initial condition with no control scheme.Case 2: Control scheme without harmonic consideration.Case 3: Control scheme considering harmonic.

The computation is carried out on an Intel core i73.40-GHz CPU 8-GB-RAM PC (Intel, Santa Clara, Cali-fornia, USA). Optimal dispatch scheduling results of theIEEE 123-bus system under non-sinusoidal operating con-dition by the proposed control method are demonstrated inTable 2. The dissimilar schedule of shunt capacitors and OLTCtap positions generated by the proposed method are also pre-sented in Table 2 for different cases. Also, the results of theprobabilistic-based time-interval approach can be seen at theOLTC tap position dispatch schedules in Table 2. Correspond-ing to the five achieved scenarios, THD reduction and voltage

Optimal dispatch schedule of OLTC Optimal dispatch schedule of OLTCand shunt capacitors for case 2 and shunt capacitors for case 3

C C C C C C C C C C C C C C C C C C C C C C C C C C C CHour OLTC 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Hour OLTC 1 2 3 4 5 6 7 8 9 10 11 12 13 14

1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 0 1 0 0 0 1 0 0 0 0 0 0 0 02 3 0 0 0 0 0 0 1 0 0 0 0 0 0 0 2 2 0 1 0 0 0 1 0 0 1 0 0 0 0 03 3 0 1 0 0 0 0 1 0 0 0 0 0 0 0 3 2 0 0 0 0 0 1 0 0 1 0 0 0 0 14 3 1 1 0 0 0 1 1 0 0 0 0 1 0 0 4 2 1 0 0 0 0 1 0 0 1 0 0 0 0 15 3 1 1 0 1 0 1 1 0 0 0 0 1 0 0 5 2 1 0 0 0 0 1 0 0 1 0 0 0 0 16 3 1 0 0 1 1 1 1 1 0 1 0 1 0 0 6 2 1 1 0 0 0 1 0 0 1 0 0 0 0 17 3 0 0 1 1 1 1 1 1 0 1 0 1 1 0 7 2 0 0 0 0 0 1 0 1 1 0 0 0 1 18 1 0 0 1 1 1 1 1 1 0 1 0 1 1 0 8 3 0 0 0 1 1 1 1 1 1 0 0 0 1 19 1 1 0 1 1 1 1 1 1 0 1 0 1 1 1 9 3 0 0 0 1 1 1 1 1 1 0 0 1 1 110 2 1 0 1 1 1 1 1 1 1 1 1 1 1 1 10 4 0 0 1 1 1 1 1 1 1 0 0 1 1 111 2 1 0 1 1 1 1 1 1 1 1 1 1 1 1 11 4 0 1 1 1 1 1 1 1 1 1 0 1 1 112 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 12 4 0 1 1 1 1 1 1 1 1 1 0 1 1 113 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 13 4 1 1 1 1 1 1 1 1 1 1 0 1 0 114 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 14 4 1 1 1 1 1 1 1 1 1 1 0 1 0 115 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 15 3 1 0 1 1 1 1 1 1 1 1 0 1 0 016 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 16 3 1 0 1 1 1 1 1 1 1 1 0 1 0 017 3 1 1 1 1 1 1 1 1 1 0 1 1 1 1 17 3 1 0 1 0 1 1 1 1 1 1 0 1 0 018 3 1 1 1 1 1 1 1 1 1 0 1 1 1 1 18 3 1 0 1 0 1 1 0 1 1 1 0 1 0 019 3 1 1 1 0 1 1 1 1 1 0 1 1 1 1 19 3 1 0 1 0 1 1 0 1 1 0 0 1 0 020 3 1 1 1 0 1 1 1 1 1 0 0 1 1 1 20 3 1 0 1 0 0 1 0 1 1 0 0 1 0 021 3 1 1 1 0 1 1 1 1 1 0 0 1 0 1 21 3 0 0 1 0 0 1 0 0 1 0 0 1 0 022 3 1 0 0 0 0 1 1 0 1 0 0 1 0 0 22 2 0 0 0 0 0 1 0 0 1 0 0 1 0 023 3 1 0 0 0 0 1 1 0 1 0 0 1 0 0 23 2 0 0 0 0 0 1 0 0 0 0 0 0 0 024 3 1 0 0 0 0 0 0 0 1 0 0 0 0 0 24 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0

TABLE 2. Non-sinusoidal IEEE 123-bus radial network operating condition


FIGURE 7. Voltage THD of all buses under non-sinusoidaloperating condition.

improvements of themost distorted buses are plotted in Figures7(a), 7(b), 7(c), 8(a), and 8(b), respectively.

7. RESULTS AND DISCUSSION

The proposed stochastic methodology is applied to the IEEE123-bus distribution test-system. The impact of energy savingfor the compensated network is summarized in column 8 ofTable 3. As can be observed, the energy saving without har-monic consideration (Case 2) is less than with consideration

FIGURE 8. Voltage improvement of bus 66.

of this restriction (Case 3). Also, hourly voltage improvementsindicate that average voltage deviations decrease from 9.756%to about less than 0.921% for different cases. Considering in-evitable propagation of harmonics in distribution networks,the harmonic distortion levels should be kept within the per-mitted limits. Non-linear load level increase results in THDvalue increment in the network. However, after compensa-tion, the compensating capacitors play more important roles.Corresponding to different scenarios, outputs of HLFC beforeoptimization show the maximum voltage THD for this systemas given in Figure 7 for Case 1. Results show that a maximum

Computation Minimum system Maximum system Average system Energy losses Energy savingCase time (sec) Scenarios voltage (p.u.) voltage (p.u.) voltage (p.u.) (MWh) (%)

1 1 0.8759 1 0.8997 4.5607 2 0.8701 1 0.9089 4.5574 3 0.8739 1 0.9024 4.5621 4 0.8698 1 0.8943 4.5576 5 0.8793 1 0.9015 4.5591

2 451.24 1 0.9708 1.0329 0.9878 4.1793 8.36282 0.9624 1.0349 0.9893 4.1475 8.99423 0.9620 1.0428 0.9971 4.1538 8.93984 0.9607 1.0293 0.9859 4.1520 8.89945 0.9666 1.0371 0.9980 4.1787 8.4338

3 3082.42 1 0.9534 1.0338 0.9901 4.1968 7.97902 0.9556 1.0299 0.9926 4.2061 7.70833 0.9548 1.0413 0.9869 4.2012 7.91084 0.9716 1.0319 0.9891 4.1975 7.90115 0.9542 1.0376 0.9912 4.2013 7.8480

TABLE 3. Summary results of proposed control method in IEEE 123-bus test system


voltage THD of 14.6397% is limited to 3.4843% for Case 3.After applying the proposed control scheme, the distortion lev-els are effectively suppressed below the permitted level of 5%.The reduction in the maximum THD level in Case 3 in com-parison with Cases 1 and 2 justifies the inclusion of harmonicsin the optimal planning. Scheduling without taking harmon-ics into account causes a severe harmonic distortion problem,i.e., 17.2731% in Case 2, which is higher than standard limit.These facts are also demonstrated in Figure 7(a) to 7(c). Thevoltage profile of bus 66, which is the bus with lowest volt-age in the network, is shown through 24 hr in Figure 8. Theinclusion of harmonics in the proposed dispatch algorithmresults in different schedules for the OLTC and switch ca-pacitors under non-sinusoidal conditions (see Table 2). TheOLTC tap positions and capacitor on/off switching status varyat substations and along feeders through 24 hr. Total OLTCand capacitor switching operation numbers per day satisfy theconstraints. In Case 2, network losses are reduced in compar-ison with Case 3. This is a sacrifice for harmonic reduction inCase 3.

8. CONCLUSION

This article proposes a new trend of programming to mitigatethe computation burden of stochastic-based volt/VAR/THDcontrol action. The proposed scheme comprises two outstand-ing features. First, the average of different LD and WT poweroutput day-ahead generation scenarios, which are divided intoseveral levels, is obtained. Second, the combination of opti-mal dispatch schedule of all control devices, such as OLTCs,SCs, and FCs, considering harmonics for all correspondingscenarios is performed. This control scheme leads to energyloss reductions and voltage profile improvements. The controlsystem applies regulation of its action while considering con-straints related to maximum voltage and THD violation. Theapplication of the conventional optimal dispatch schedulingfor non-sinusoidal operating conditions is not acceptable, asit leads to high THD voltage distortions. A proper coordina-tion between OLTCs, SCs, and FCs has also been treated. It isconcluded from the study results that the proposed stochasticprogramming method is very efficient and robust in obtainingthe solution of the volt/VAR/THD control problem.

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