Enhancement of electric arc furnace reactive power compensation using Grey–Markov prediction method

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Published in IET Generation, Transmission & DistributionReceived on 29th September 2013Revised on 4th January 2014Accepted on 2nd March 2014doi: 10.1049/iet-gtd.2013.0698

626The Institution of Engineering and Technology 2014

ISSN 1751-8687

Enhancement of electric arc furnace reactive powercompensation using Grey–Markov prediction methodHaidar Samet1, Aslan Mojallal1

1Department of Power and Control Engineering, School of Electrical and Computer Engineering,

Shiraz University, Shiraz, Iran

E-mail: [email protected]

Abstract: The time varying nature of electric arc furnace (EAF) gives rise to voltage fluctuations, which produces the effectknown as flicker. Employing reactive power compensation devices such as static VAr compensator (SVC) is one of the mainapproaches to mitigate this phenomenon. By utilising prediction methods to forecast EAFs reactive power consumption for ahalf-cycle ahead, performance of SVC can be enhanced substantially. This study proposes a rolling Grey model and a Grey–Markov method to predict the actual reactive power of Mobarakeh Steel Company, Isfahan/Iran. To investigate the efficiencyof the proposed methods the results are compared with the results of EAFs reactive power compensation when no predictionmethod is employed. Furthermore, autoregressive moving average (ARMA) models with updating coefficients, which arestudied in the literature are used to predict EAF reactive power. Various methods for updating ARMA coefficients includingnormalised least mean square, recursive least square method and an online genetic algorithm are used. By comparing theindices which are defined using the concept of flicker frequency and power spectral density, the superiority of Grey–Markovand rolling Grey model over the aforementioned prediction methods is investigated.

1 Introduction

Employing electric arc furnaces (EAFs) is a conventionalapproach in steel production industry since thesedevices have satisfactory performance. However, onecommon problem posed by employing these considerablepower consuming devices is flicker. The reactive powerconsumption of EAFs has a random and time varyingnature, which causes low-frequency (around 0.5–25 Hz)fluctuations in power grid voltage. These fluctuations involtage are called flicker. Owing to the importance ofpower grid power quality and notable utilisation of EAFs inpower networks, modelling of EAFs has been extensivelyinvestigated. For instance, in [1] adaptive neuro-fuzzynetworks are used in order to construct the model ofnon-linear v–i characteristic for EAFs whereas Chang et al.[2] employ a neural network based method for the samereason.Using SVC and other reactive power compensation devices

are considered as the best option for reducing the effects offlicker. However, the practical limitations such as thyristorignition delays can have a negative influence on the SVC’sperformance and reduce the ability of these devices tomitigate flicker [3]. Static compensators (STATCOMs) withfaster responses and an exceptional performance can beconsidered as a solution for these shortcomings of SVC;however, these devices are usually expensive. Therefore thenecessity of developing new methods to eliminate SVC’sdelay and enhance the performance of SVCs in flickerreduction applications becomes clear. These delays can be

minimised by predicting the reactive power of EAF for thenext half-cycle [3]. Reactive power consumption of EAFcan be considered as an auto regressive moving average(ARMA) process and can be predicted using ARMAmodels [4]. To this end, large number of samples in equalintervals should be collected from reactive power of EAF inorder to create time series. Afterwards, the past and presentvalues of reactive power can be employed to predict thefuture values. Statistical and artificial intelligenceapproaches are the main and the most conventional methodsfor time series prediction. Artificial neural networks basedprediction methods can be categorised as artificialintelligence based approaches. ARMA methods can beconsidered as an example for the statistical methods. Itshould be noted that the performance of these predictionmethods is dependent on the characteristics of the timeseries that are to be predicted and on the application ofprediction. For instance, in many applications, the statisticalmethods can be very inaccurate in comparison with artificialintelligence methods and also may become verycomplicated [5]. On the other hand, requiring a largenumber of training samples is the main shortcoming ofartificial intelligence methods.Since the first introduction of Grey system theory by Deng

in 1982, this method has been employed in numerousapplications fields of research including engineering,economics, agriculture and finance [6–10]. In powersystems studies, this method is used to predict the yearlypeak load power grids [11]. Various Grey system modelsare also used for the United States dollar to Euro parity

IET Gener. Transm. Distrib., 2014, Vol. 8, Iss. 9, pp. 1626–1636doi: 10.1049/iet-gtd.2013.0698

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prediction [12]. Furthermore, application of the Grey systemtheory in combination with linear regression methods andwavelet in order to mitigate the error of a dynamicallytuned gyroscope and using Grey system models inconjunction with fuzzification methods to predict the stockprices are examples of hybrid Grey methods [13, 14].Moreover, Grey–Markov and Grey system model alongwith singular spectrum analysis prediction method isemployed in [15] to predict energy consumption of India.Prediction of reactive power consumption of EAFs is also
investigated in the literature. In this regard, ARMA modelswhich employ normalised least mean square (NLMS),recursive least square (RLS) and genetic algorithm in orderto update model coefficients are used to forecast EAF’sreactive power absorption [16, 17].Additionally, in [18] an improved Grey system is used to

predict the flicker severity index level and it is shown thatGrey system can be more accurate than a neural networkbased method for this application. In [19], this methodis employed to forecast GDP, population and energyconsumption in Iran. Stock price prediction is anotherapplication of Grey–Markov method which is studied in [20].In the present study, superiority of Grey system model and

Grey–Markov method over other methods investigated in theliterature is demonstrated. These methods are ARMA modelswhich use NLMS, RLS and genetic algorithm approaches toupdate ARMA coefficients. It is illustrated that advantages ofGrey system and Grey–Markov method made them excellentchoices for forecasting EAF’s reactive power consumption.To this end, actual reactive power data of Mobarakeh SteelCompany (MSC) is employed. Therefore in the secondsection overall definition and equations of Grey system andGrey–Markov are brought. Data obtained from MSC andEAF scheme are briefly discussed in Section 3. Indiceswhich will be used for comparison purposes are given inSection 4. Section 5 provides the definition and equationsof ARMA method using NLMS, RLS and the geneticalgorithms. Simulation results are presented in Section 6.

2 Grey system and Grey–Markov predictionmodels

The followings are the basic definitions and equations whichare employed in order to predict using Grey system theory andGrey–Markov method [21–23].

Fig. 1 Raw sequence of samples


2.1 Grey system theory

In order to have satisfactory results from Grey system andGrey–Markov methods, the randomness of input time seriesshould be eliminated. For this purpose, the time series willundergo a process called accumulated generating operation(AGO). The following example demonstrate the effects ofAGO on time series randomness reduction.Consider the following sequence of samples

X (0) = (2, 4, 7, 5, 9, 1)

According to Fig. 1, X(0) has huge amount of randomness.Now, Let’s define X(1) as the first-order AGO of t. It can bepresented as

X (1)(k) =∑ki=1

X (0)(i), k = 1, 2, . . . , n (1)

So X(1) would be

X (1) = (2, 6, 13, 18, 27, 28)

A comparison between Figs. 1 and 2 demonstrates thatAGO made a random sequence of data much smoother andaltered a random time series to a mono increasing sequence.According to Grey system theory GM(n, m) stands for a

Grey model which uses nth-order differential equation anduses m variables for prediction. However, simplicity, fastresponse and satisfactory accuracy of GM(1, 1) made thismodel the most conventional and appropriate model in mostof applications.It should be mentioned that, Grey system and subsequently

Grey–Markov method can merely be applied on time serieswith non-negative samples. In other words, after performingAGO on the initial sequence of data, the result of AGO(X(1)) must be a mono-increasing sequence. Taking thispoint into consideration, prediction using GM(1, 1) can bedone using the following equations.Assume the initial sequence of samples as

X (0) = X (0)(1), X (0)(2), . . . , X (0)(n)[ ]

(2)

1627& The Institution of Engineering and Technology 2014

Fig. 2 Sequence of data processed by AGO

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After employing AGO we have

X (1) = X (1)(1), X (1)(2), . . . , X (1)(n)[ ]

(3)

The mean sequence (Z(1)) can be presented as

Z(1)(k) = 0.5X (1)(k)+ 0.5X (1)(k − 1),

k = 2, 3, . . . , n(4)

The first order, one variable Grey model can be defined as

X (0)(k) + aZ(1)(k) = b (5)

Parameters a and b should be calculated. By employing theleast square method (5) can be solved as

ab

( )= BTB

( )−1BTY (6)

where B and Y are as follows

B =−Z(1)(2) 1−Z(1)(3) 1. . .

−Z(1)(n)

. . .

1

⎛⎜⎜⎜⎝

⎞⎟⎟⎟⎠ (7)

Y =X (0)(2)X (0)(3). . .

X (0)(n)

⎛⎜⎜⎜⎝

⎞⎟⎟⎟⎠ (8)

Using a and b the predicted value of X(1) can be calculatedusing (5)

X̂(1)(k + 1) = X (0)(1)− b

a

( )e−ak + b

a(9)

By applying the inverse of AGO, predicted values of X (0) canbe obtained as

X̂ (0)(k + 1) = X (0)(1)− b

a

( )e−ak(1− ea) (10)


The error of prediction using this method can be calculated by

Ei = x(0)(i)− x̂(0)(i) (11)

Conventionally, when the number of samples is too large,rolling Grey system will be employed which requires lesscomputation effort. In this method, the number of samplesused for prediction remains constant and when the latestsample enters, the last sample leaves the prediction window.This method is used in this study because of the largenumber of EAF’s reactive power samples. The number ofsamples employed in each window is 5, which results in anacceptable accuracy with a reasonable computation effort.

2.2 Grey–Markov prediction method

Grey–Markov prediction method is based on prediction usingGrey model and Markov chain. In this method, the initialprediction is performed using Grey system; furthermore,Markov chain is employed to predict the error of predictionintroduced by Grey system. By adding this predicted errorto the forecasted value of Grey system, overall error ofprediction reduces significantly and the accuracy ofprediction increases. Markov method forecasting process isas follows:

1. Determining state of error: In the first step, errors of Greysystem prediction method for a number of previouspredictions are considered (in this paper 40 samples ofprevious prediction errors are considered). Then thesesamples are divided into several states (here ten states areconsidered). The interval of these states are equal and arecalculated using the following equation

Ri = errmax − errmin

( )/NS (12)

where errmax and errmin are maximum and minimum values ofthe samples and NS is the number of states. Then starting fromminimum value, the whole interval is divided into ten stateswith equal lengths.2. Constructing probability transition matrix: Transition Tijis defined as a process by which the nth sample from state ichanges to n + 1th sample at state j. Higher orders oftransition T (m)

ij are defined when nth sample is in state ichanges to n +mth sample is in state j. Probability of thesetransitions from state i to state j can be calculated using the


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following equation
P(m)ij = M (m)

ij

Mi(13)

where i and j may vary from 1 to number of states.Mij(m) and Mi stand for number of transitions from state i tostate j and number of the total number of data in state i,respectively. The matrices constructed from various ordersof transition probability are regarded as transitionprobability matrices. Each array presents the probability ofm order transition from state i to state j

P(m) =

P(m)11 P(m)

12 . . . P(m)1n

P(m)21 P(m)

22 . . . P(m)2n

..

. ... ..

. ...

P(m)n1 P(m)

n2 . . . P(m)nn

⎡⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎦ (14)

3. Determining high probability transitions: Using several oflow-order probability transition matrices (here third-orderprobability matrix is the highest order considered) thehighest probability of transition from the last sample isobtained. This probability indicates that error of nextprediction using Grey model is most likely to be in whichstate.4. Forecasting prediction error: The state in which the nextsample is most likely to be in, is employed to forecast error.Using the following equation, which basically calculates theaverage of state determined in step 3, the amount ofpredicted sample, is obtained. Finally, by adding this valueto the predicted value from Grey model, the overallperformance of prediction method is enhanced

X̂ (0)GM(k + 1) = X̂ (0)

G (k + 1)+ (1/2)(A+ B) (15)

where X̂ (0)GM and X̂ (0)

G stand for the predicted value usingGrey–Markov method and using only Grey system model.Also, A and B are end points of the interval whichdescribes the future state of Grey model error and iscalculated in step 3.

Grey–Markov prediction method can be used to forecast asystem with severe randomly varying time series. ThereforeGrey–Markov method performs much better and accuratethan Grey system for time series with large amountof random fluctuations [23]. EAF’s reactive powerconsumption can be regarded as such time series. However,it should be noted that additional computation is needed inorder to predict the error, which may increase Grey–Markovcomputational effort. By increasing the number of statesand samples that are used for prediction, the accuracy ofGrey–Markov method can be improved. However, thisimprovement in accuracy is achieved at the cost ofincreasing in required computation. Therefore there shouldbe a trade-off between accuracy and complexity ofcalculations.Furthermore, owing to the fact that Grey–Markov method

is based on GM(1, 1), this method is precise for time serieswhich have small prediction errors for GM(1, 1). In GM(1,1) method after employing AGO on the raw data, afirst-order differential equation is used in order to predictthe future values. Therefore the predicted values will have


an exponential kind of growth. Hence, Grey system andsubsequently Grey–Markov methods will have the mostdesirable results for time series that after employing AGOhave characteristics close to exponential curve [23].

3 Data records

Since in this paper the main goal is the prediction of EAFreactive power, currents and voltages measured at the pointof common coupling in MSC are considered and reactivepower is calculated from their values. Fig. 3 presents asingle line diagram of the EAFs system used in thisindustrial site. Power system frequency is 50 Hz. Accordingto this figure, SVC is connected to 33 kV side of step-downtransformer whereas EAFs are connected to 63 kV output ofthis transformer trough EAF 63/0.718 kV transformers.Satisfactory power quality and harmonic reduction of eightEAFs are provided by two SVCs that each one is composedof a 97.2 MVAr capacitor bank and a 108 MVAr thyristorcontrolled reactor (TCR).Sampling frequency by which voltages and currents of

EAF’s at MSC are measured is 7812.5 Hz (sampling time is128 µs). Since, an accurate prediction can only be beneficialwhen the calculated reactive power is accurate, reactivepower is calculated at each half cycle with one cycleintegration period. This method of reactive powercalculation is proven to have desirable accuracy and speed[24]. Furthermore, the duration of each recorded voltagesand currents time series is 100 s. Considering thatintegration interval is half cycle (0.01 s) and each data setcovers 100 s, the number of samples in each sequence ofreactive power is 10 000 samples.

4 Performance evaluation indices

In this paper, the prediction methods are employed to reducethe flicker caused by EAFs by improving the performance ofSVC. Three indices are introduced to evaluate the predictionmethods [4]. The defined indices are based on power spectraldensity (PSD) [25] of the prediction error (e) which is thedifference between the actual and predicted value ofreactive power signal

PSD( f ) = 1

nfs

∑nt=1

e(t)e−i2pft

∣∣∣∣∣∣∣∣∣∣2

(16)

In the above equation PSD ( f ), fs and N represent the value ofPSD at frequency f, the sampling frequency and the datarecord length, respectively.Flicker mitigation factor (FMF), which uses weighted

prediction error for each data record j and is defined asfollowing

FMFj =∑25

f=1 c( f )PSDqSj ( f )∑25

f=1 c( f )PSDqj ( f )

(17)

where PSDqj ( f ) stands for the PSD for the jth source reactive

power sample in the absence of SVC and PSDqsj ( f ) presents

the power density spectral for the jth source reactive powersample by using SVC. c( f )s are the weighting flickerfactors which are defined by IEC [26].Since prediction can be considered as a high-pass filter,

high-frequency mitigation factor (HMF) is defined toconsider and expand the influence of high frequency


Fig. 3 Single line diagram of EAFs used in MSC

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components [24]. This index concerns about how good aprediction method performs in high frequencies around 16–25 Hz. HMF can be obtained as

HMFj =∑25

f=16 PSDqSj (f )∑25

f=16 PSDqj (f )

(18)

The third index which is used to assess the performance ofprediction methods is standard deviation (STD) of error.

5 Updating ARMA coefficients with adaptivemethods

If ARMA models are being used for prediction, theircoefficients should be updated [16]. In this study, threedifferent approaches are employed to update these coefficients.

5.1 NLMS

One of the approaches for updating the ARMA models isusing NLMS. Prediction using NLMS can be done usingthe following equations [27, 28]

q̂(t) = WT(t)x(t) (19)

e(t)− q(t)− q̂(t) (20)

W (t + 1) = W (t)+ m

xT(t)x(t)+ ce(t)x(t) (21)

In (19), q̂ denotes the value of forecasted sample, q is theactual value of future sample, x stands for vector ofprevious samples, e(t) shows the error signal and w(t)represents the coefficients vector. In (21), μ is an importantparameter in the performance of algorithm since smallvalues of μ can slow algorithm convergence speed, whereas


excessively large values of this parameter can make thealgorithm unstable [27].Here, ARMA(2,1) is considered for predicting the EAF

reactive power consumption. Prediction using this modelcan be done using the following equations

q̂(t + 1) = k1q(t)+ k2q(t − 1)+ k3e(t) (22)

e(t) = q(t)− q̂(t) (23)

5.2 RLS

RLS is the second algorithm which is employed here toupdate ARMA models coefficients. It should be noted thatRLS is generally faster than NLMS, however, equationsdescribing RLS are more complex and this method requiresmore computational effort than NLMS

k(t) = l−1p(t − 1)x(t)

1+ l−1xH(t)p(t − 1)x(t)(24)

q̂(t) = WH(t − 1)x(t) (25)

e(t) = q(t)− q̂(t) (26)

W (t) = W (t − 1)+ k(t)e(t) (27)

P(t) = l−1P(t − 1)− l−1k(t)xH(t)P(t − 1) (28)

where P stands for the inverse of signal correlation matrix,λ is a constant close to unity and k represents the gainvector [16].

5.3 Online genetic algorithm

In online genetic algorithm, L latest samples are employed toobtain the error function. Errors function (also called fitting


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function) then will be used for prediction purposes [17].Fitting function can be defined as
ft(i) =∑t

m=t−L

e2i (m) (29)

In (29), error function for the chromosome i at time t isrepresented by ft(i) whereas ei(m) is the prediction error andL denotes the number of samples which are used to acquirethe fitting function. The prediction using combination ofARMA models and online genetic algorithm can be defined

q̂(t + 1) = k1q(t)+ k2q(t − 1)+ k3e(t)+ k4 (30)

e(t) = q(t)− q̂(t) (31)

Fig. 4 Reactive power absorption of one EAF

Fig. 5 Compensation error of SVC, as a result of prediction by

a Grey (GM (1, 1))b Grey–Markov method andc Without any prediction methods


The number of chromosomes used in this study is 30 and thenumber of samples used to calculate the fitting function is 100samples.

6 Simulation results

In this study, a rolling grey model with sampling windowequal to 5 and a Grey–Markov method with ten states and40 samples for constructing probability transition matrix areemployed to predict the reactive power consumption ofEAFs. The accuracy of these prediction methods and theability of the proposed methods for mitigation of flicker arecompared with the conventional methods which use ARMAmodels for prediction. FMF, HMF and STD error are usedto evaluate the ability of proposed methods. Additionally,the beneficial influence of reactive power prediction on


Table 1 STD of source reactive power using different prediction methods

STD1 STD2

No pred. Grey Grey-Markov RLS NLMS Gen alg. No pred. Grey Grey-Markov RLS NLMS Gen alg.

1 0.1999 0.0691 0.0590 0.1993 0.1758 0.1559 0.2066 0.0729 0.0616 0.2050 0.1771 0.16302 0.2073 0.0765 0.0633 0.2159 0.1922 0.1625 0.2245 0.0847 0.0710 0.2328 0.2047 0.17553 0.1648 0.0522 0.0451 0.1568 0.1481 0.1262 0.1722 0.0547 0.0473 0.1608 0.1433 0.13104 0.2012 0.0654 0.0569 0.1672 0.1548 0.1394 0.2098 0.0631 0.0521 0.1496 0.1391 0.13175 0.1885 0.0630 0.0552 0.1625 0.1490 0.1337 0.1914 0.0592 0.0506 0.1419 0.1330 0.12396 0.1572 0.0533 0.0423 0.1313 0.1224 0.1091 0.1687 0.0489 0.0419 0.1220 0.1140 0.10797 0.2190 0.0753 0.0559 0.1293 0.1373 0.1164 0.2059 0.0743 0.0528 0.1292 0.1231 0.11448 0.2242 0.0834 0.0641 0.1598 0.1537 0.1375 0.2003 0.0763 0.0587 0.1489 0.1364 0.12699 0.2026 0.0745 0.0564 0.1252 0.1334 0.1120 0.1801 0.0690 0.0513 0.1089 0.1049 0.098310 0.1523 0.0529 0.0394 0.1000 0.1100 0.0895 0.1685 0.0544 0.0426 0.1139 0.1121 0.102411 0.1468 0.0521 0.0412 0.1031 0.1074 0.0909 0.1722 0.0591 0.0477 0.1194 0.1128 0.105512 0.1399 0.0512 0.0394 0.0876 0.1142 0.0792 0.1793 0.0557 0.0438 0.1048 0.1030 0.095913 0.1530 0.0464 0.0363 0.0885 0.0972 0.0798 0.1593 0.0498 0.0388 0.0910 0.0898 0.082414 0.1068 0.0353 0.0280 0.0722 0.0768 0.0638 0.1094 0.0358 0.0284 0.0719 0.0676 0.063815 0.1407 0.0417 0.0333 0.0820 0.0922 0.0741 0.1651 0.0494 0.0393 0.0941 0.0951 0.085416 0.1355 0.0500 0.0342 0.0726 0.0922 0.0672 0.1275 0.0484 0.0334 0.0710 0.0706 0.065317 0.1955 0.0891 0.0571 0.0995 0.1142 0.0937 0.1974 0.0950 0.0597 0.0989 0.0980 0.093118 0.2188 0.1004 0.0659 0.1178 0.1273 0.1099 0.2190 0.0995 0.0679 0.1207 0.1156 0.111919 0.2225 0.0763 0.0605 0.1660 0.1610 0.1414 0.2323 0.0716 0.0569 0.1558 0.1508 0.141420 0.2770 0.1589 0.1126 0.2040 0.1866 0.1740 0.2911 0.1582 0.1074 0.1804 0.1653 0.162921 0.2021 0.0635 0.0496 0.1183 0.1220 0.1076 0.1883 0.0571 0.0440 0.1013 0.1018 0.094322 0.1800 0.0498 0.0393 0.1012 0.1035 0.0926 0.1681 0.0447 0.0349 0.0856 0.0866 0.080423 0.2027 0.0605 0.0463 0.1093 0.1093 0.1004 0.1983 0.0572 0.0442 0.0980 0.0946 0.091224 0.2677 0.0877 0.0683 0.1420 0.1503 0.1316 0.2670 0.0873 0.0687 0.1409 0.1355 0.130625 0.1678 0.0504 0.0389 0.0843 0.1048 0.0793 0.1724 0.0521 0.0401 0.0844 0.0849 0.080026 0.2007 0.0578 0.0455 0.0996 0.1243 0.0940 0.2080 0.0592 0.0456 0.1020 0.1032 0.096327 0.2737 0.0899 0.0678 0.1441 0.1550 0.1346 0.2711 0.0883 0.0661 0.1406 0.1351 0.132328 0.1682 0.0499 0.0384 0.0839 0.1094 0.0780 0.1683 0.0493 0.0380 0.0819 0.0835 0.076329 0.1986 0.0623 0.0465 0.0993 0.1189 0.0933 0.1965 0.0614 0.0462 0.0992 0.0963 0.0937avg. 0.1904 0.0703 0.0536 0.1266 0.1306 0.1103 0.1946 0.0683 0.0523 0.1249 0.1183 0.1105

Table 2 FMF index of source reactive power using different prediction methods

FMF1 FMF2


1 0.0846 0.0079 0.0049 0.0133 0.0213 0.0267 0.0832 0.0083 0.0050 0.0142 0.0165 0.02752 0.0814 0.0084 0.0055 0.0159 0.0223 0.0274 0.0941 0.0102 0.0068 0.0191 0.0199 0.03203 0.0524 0.0043 0.0029 0.0084 0.0143 0.0158 0.0565 0.0047 0.0030 0.0090 0.0099 0.01604 0.0919 0.0081 0.0047 0.0139 0.0195 0.0207 0.0844 0.0066 0.0033 0.0106 0.0123 0.01455 0.0879 0.0080 0.0051 0.0137 0.0178 0.0211 0.0778 0.0064 0.0036 0.0106 0.0106 0.01426 0.0765 0.0079 0.0037 0.0112 0.0177 0.0171 0.0673 0.0051 0.0027 0.0085 0.0095 0.01187 0.0858 0.0101 0.0038 0.0091 0.0164 0.0104 0.0834 0.0110 0.0038 0.0097 0.0105 0.01138 0.1209 0.0149 0.0065 0.0173 0.0226 0.0206 0.1232 0.0162 0.0071 0.0190 0.0174 0.02269 0.0900 0.0117 0.0047 0.0105 0.0187 0.0125 0.0804 0.0118 0.0047 0.0093 0.0101 0.011210 0.0773 0.0092 0.0035 0.0096 0.0162 0.0121 0.0688 0.0069 0.0030 0.0091 0.0101 0.011911 0.0914 0.0106 0.0047 0.0130 0.0182 0.0164 0.0870 0.0095 0.0044 0.0121 0.0121 0.015612 0.0762 0.0102 0.0041 0.0091 0.0205 0.0109 0.0722 0.0069 0.0030 0.0077 0.0089 0.009513 0.1055 0.0088 0.0037 0.0087 0.0194 0.0115 0.1027 0.0092 0.0038 0.0084 0.0105 0.011014 0.1225 0.0116 0.0051 0.0151 0.0349 0.0183 0.1208 0.0116 0.0050 0.0146 0.0192 0.017615 0.0990 0.0080 0.0035 0.0083 0.0208 0.0112 0.0984 0.0082 0.0036 0.0081 0.0126 0.010716 0.0570 0.0085 0.0028 0.0053 0.0117 0.0062 0.0635 0.0094 0.0032 0.0064 0.0065 0.007417 0.0827 0.0182 0.0051 0.0072 0.0131 0.0081 0.0829 0.0204 0.0056 0.0071 0.0079 0.008018 0.1013 0.0215 0.0063 0.0091 0.0148 0.0105 0.1113 0.0226 0.0072 0.0103 0.0104 0.011919 0.1287 0.0123 0.0055 0.0151 0.0280 0.0232 0.1256 0.0101 0.0042 0.0122 0.0144 0.016920 0.1615 0.0526 0.0169 0.0180 0.0268 0.0271 0.1538 0.0437 0.0130 0.0143 0.0158 0.018821 0.0909 0.0081 0.0035 0.0096 0.0143 0.0112 0.0831 0.0072 0.0029 0.0080 0.0085 0.009322 0.0658 0.0047 0.0021 0.0065 0.0097 0.0079 0.0584 0.0041 0.0018 0.0054 0.0064 0.006023 0.0926 0.0076 0.0030 0.0084 0.0115 0.0095 0.0879 0.0069 0.0027 0.0076 0.0072 0.008024 0.1265 0.0122 0.0051 0.0119 0.0188 0.0131 0.1283 0.0123 0.0053 0.0119 0.0133 0.013025 0.0612 0.0056 0.0024 0.0058 0.0113 0.0063 0.0617 0.0057 0.0024 0.0057 0.0061 0.006126 0.0875 0.0070 0.0030 0.0081 0.0154 0.0085 0.0863 0.0068 0.0028 0.0078 0.0086 0.008227 0.1271 0.0123 0.0047 0.0120 0.0197 0.0133 0.1231 0.0118 0.0046 0.0112 0.0120 0.012628 0.0638 0.0059 0.0023 0.0062 0.0142 0.0066 0.0627 0.0057 0.0022 0.0060 0.0063 0.006229 0.0843 0.0083 0.0030 0.0080 0.0149 0.0083 0.0857 0.0082 0.0031 0.0082 0.0084 0.0084avg. 0.0946 0.0152 0.0058 0.0112 0.0192 0.0151 0.0913 0.0113 0.0045 0.0104 0.0114 0.0136

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Fig. 6 STD of source reactive power

a By discarding the first 500 samplesb By discarding the first 5000 samples

Table 3 HMF index of source reactive power using different prediction methods

HMF1 HMF2


1 1.3357 0.3172 0.1750 0.8821 0.5242 0.5851 1.2989 0.3105 0.1617 0.8534 0.5275 0.56432 1.3881 0.3281 0.1447 0.9211 0.5743 0.5943 1.3814 0.3272 0.1415 0.8710 0.5665 0.56983 1.3611 0.3092 0.1602 0.9077 0.5619 0.5842 1.2453 0.2815 0.1392 0.7419 0.4856 0.47554 1.3479 0.3093 0.1495 0.9019 0.5197 0.5900 1.3215 0.2958 0.1297 0.8216 0.5135 0.54485 1.3269 0.2994 0.1520 0.8366 0.5366 0.5804 1.3259 0.2960 0.1389 0.8556 0.5581 0.54866 1.3396 0.3266 0.1558 0.9053 0.5362 0.6147 1.3036 0.3033 0.1594 0.9034 0.5787 0.63427 1.3135 0.3379 0.1584 0.7912 0.4472 0.5370 1.2868 0.3306 0.1442 0.8014 0.4939 0.53098 1.3105 0.3194 0.1290 0.7558 0.4541 0.4642 1.3002 0.3116 0.1342 0.7663 0.4984 0.46269 1.3003 0.3391 0.1603 0.7896 0.4464 0.5125 1.2462 0.3300 0.1765 0.7695 0.4686 0.516910 1.2769 0.3127 0.1488 0.7840 0.4594 0.5499 1.2728 0.3050 0.1378 0.7872 0.4898 0.574211 1.2833 0.3037 0.1478 0.8011 0.4788 0.5270 1.2903 0.2989 0.1518 0.8090 0.4888 0.546612 1.2910 0.3316 0.1878 0.8071 0.5692 0.5662 1.2839 0.3026 0.1586 0.8014 0.5056 0.609113 1.2299 0.3018 0.1688 0.7218 0.4887 0.6017 1.2175 0.3005 0.1783 0.7065 0.4760 0.590314 1.2957 0.2976 0.1288 0.7559 0.5533 0.4839 1.2770 0.2908 0.1347 0.7259 0.4477 0.477515 1.2921 0.3068 0.1713 0.7495 0.4761 0.6216 1.2739 0.3011 0.1759 0.6938 0.4062 0.591516 1.2937 0.3455 0.2088 0.8035 0.4813 0.6070 1.1851 0.3119 0.1850 0.8079 0.5091 0.578017 1.2965 0.4537 0.2721 0.7684 0.4507 0.5489 1.2089 0.4831 0.2900 0.7642 0.4854 0.544218 1.2794 0.4808 0.2674 0.7167 0.4420 0.5052 1.2952 0.4976 0.2880 0.7464 0.4954 0.520319 1.3099 0.3044 0.1404 0.7699 0.5066 0.5250 1.2942 0.2932 0.1559 0.7668 0.4781 0.573520 1.3162 0.4914 0.2853 0.7351 0.4478 0.4854 1.2995 0.4736 0.2879 0.7179 0.4405 0.473721 1.3204 0.3176 0.1472 0.7754 0.4451 0.5048 1.3034 0.3153 0.1544 0.7566 0.4702 0.520622 1.3109 0.2993 0.1304 0.7718 0.4570 0.5137 1.3100 0.3000 0.1404 0.7657 0.4855 0.524123 1.2945 0.3051 0.1376 0.7273 0.4375 0.4932 1.2944 0.3078 0.1460 0.7090 0.4709 0.503724 1.2576 0.3141 0.1421 0.6566 0.3718 0.4540 1.2608 0.3115 0.1494 0.6698 0.4063 0.464325 1.2349 0.2728 0.1513 0.7495 0.4290 0.5387 1.1903 0.2612 0.1418 0.7171 0.4473 0.525726 1.2626 0.2932 0.1391 0.6957 0.4036 0.5006 1.2605 0.2918 0.1344 0.7138 0.4311 0.511627 1.2689 0.3196 0.1437 0.6516 0.3714 0.4497 1.2669 0.3237 0.1487 0.6711 0.4102 0.465228 1.2042 0.2873 0.1499 0.6839 0.4122 0.5015 1.2914 0.3134 0.1632 0.7547 0.4550 0.559829 1.2696 0.3211 0.1536 0.6979 0.4143 0.5034 1.2372 0.3122 0.1410 0.7141 0.4424 0.5184avg. 1.2976 0.3510 0.1788 0.7772 0.4750 0.5346 1.2792 0.3276 0.1674 0.7681 0.4824 0.5356

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flicker reduction is investigated by comparing the flickerindices when using prediction methods with the results ofcompensation that does not use any prediction method.The positive effect of prediction methods on reactive power
compensation can be investigated by comparing thecompensation error of SVC for two scenarios. In the firstscenario, the reference signal of SVC is provided by theprediction made by Grey system model. In the secondscenario, this reference signal is the present value ofreactive power. Although giving the present value ofreactive power as a reference signal to SVC assures thetracking of reactive power, this method neglects thyristorignition delays and causes large variations at source reactivepower. For instance, Fig. 4 shows reactive powerconsumption of one of the eight EAFs. Adequatecompensation of this reactive power can reduce flickersignificantly. Therefore the aforementioned scenarios aredeveloped. Fig. 5 shows the compensation error of SVC.According to this figure, using GM(1,1) as a referencesignal for SVC can greatly reduce the compensation error,which validates the assumption that prediction methods canbe beneficial in reactive power compensation applications.Tables 1–3 present the same results.Now the results of proposed methods are compared with

those obtained by ARMA models. As illustrated earlierARMA models use NLMS, RLS and online geneticalgorithm to update their coefficients. Rolling Grey modeluses five samples to predict the value of future sample andGrey–Markov method uses ten states and 40 previous

Fig. 7 FMF of source reactive power



samples. STD of compensation error for each of thesemethods is given in Table 1. In this study, 29 sequences ofEAF reactive power with duration of 100 s are considered.Tables 2 and 3 present the FMF and HMF of theseapproaches, respectively. In these tables, indices are shownin two forms. In STD1, FMF1 and HMF1 the first 500samples (first 5 s) are neglected whereas in STD2, FMF2and HMF2 the first 5000 samples (first 50 s) are neglected.The first form assesses the transient performance ofprediction methods whereas the second form shows theability of predicted methods in flicker reduction in steadystate.According to these tables, Grey–Markov method shows an

outstanding performance and reduces the flicker more thanARMA models and Grey model. However, the Grey systemmethod reduces flicker indices significantly and incomparison with ARMA models and compensation withoutprediction shows better performance.Figs. 6–8 show the performance of prediction methods.

According to these figures and tables, rolling Grey modelperformance is very effective and average value of 29 seriesis reduced in comparison with other approaches, however,in some cases ARMA models yield more satisfactory resultsin term of defined indices. For instance, according toFigs. 6b, in the 17th set of second form of data (STD2),ARMA model using genetic algorithm performs better interm of STD. In the case of FMF1, FMF2, HMF1 andHMF2 also in 12, 11, 3 and 2 series of data, ARMAmodels predict more accurately than GM(1,1) and hence,


Fig. 8 High-frequency mitigation factor of source reactive power


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reduce the aforementioned indices more markedly. However,all of the results report the superiority of Grey–Markovprediction method for EAF reactive power compensationapplication. According to Figs. 6–8, Grey–Markov methodis distinctly superior to other methods in all of data sets.Average of STD without using prediction 4 and by usingARMA methods are more than two times larger thanaverage of STD when employing Grey–Markov predictionapproach. This proportion becomes even smaller whentaking HMF and FMF into consideration.Furthermore, a comparison between Grey–Markov method

and Grey system prediction results indicates that Grey–Markov is much more accurate and effective for reactivepower prediction application. However, one should notethat Grey–Markov method is based on Grey systemprediction method and therefore the accuracy of Grey

Table 4 Grey–Markov performance for 8, 10 and 12 states and for 30

Number of states and samples STD1 STD2

8 states, 30 samples 0.0639 0.066310 states, 30 samples 0.0609 0.063612 states, 30 samples 0.0575 0.05988 states, 40 samples 0.0627 0.065510 states, 40 samples 0.0590 0.061612 states, 40 samples 0.0555 0.05888 states, 50 samples 0.0601 0.063310 states, 50 samples 0.0566 0.059012 states, 50 samples 0.0537 0.0568


system approach directly affects the Grey–Markovperformance. Also, since Grey–Markov method usesadditional calculation in order to predict the amount of errorintroduced by Grey system, it would require more time andmore computational effort. However, Grey–Markov can stillbe considered as a fast prediction approach and can beregarded as the most effective method of prediction as longas flicker reduction is concerned.Finally, influence of the number of states and the number of

samples for constructing probability transition matrix onprediction accuracy is demonstrated in Table 4. In thisTable, Grey–Markov method is employed on the first set ofEAF’s reactive power data. In addition to using ten states,eight and 12 states are also considered. Furthermore, 30 and50 samples are considered for building probability transitionmatrix.

, 40 and 50 samples window

FMF1 FMF2 HMF1 HMF2

0.0061 0.0059 0.1980 0.18680.0054 0.0054 0.1826 0.16930.0047 0.0047 0.1577 0.15120.0057 0.0057 0.1940 0.18210.0049 0.0050 0.1750 0.16170.0043 0.0044 0.1453 0.13720.0052 0.0052 0.1723 0.16320.0044 0.0044 0.1556 0.14520.0039 0.0040 0.1376 0.1317


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According to Table 4, as the number of states increases the
accuracy of prediction method and the ability to reduce flickerincreases too. The number of samples used for Markov hasthe same behaviour and as the number of samples increasesthe prediction becomes more precise. However, thisimprovement in accuracy will increase the computationalburden.

7 Conclusions

SVC is regarded as one of the best solutions for EAF reactivepower compensation. However, because of several practicalshortcomings of SVC, the performance of these devices inflicker mitigation may be unsatisfactory. To address thisproblem, the prediction methods can be employed toenhance the performance of the SVC in flicker reduction. Inthis paper, a rolling Grey system and a Grey–Markovmethod is employed to predict the reactive power of EAFs.To this end, the actual data of MSC is used. Furthermore,to assess the results of the proposed methods, thesemethods are compared with the results obtained fromARMA models which have been investigated in theliterature. Also, the three indices that evaluate theperformance of prediction methods for flicker mitigationapplications are introduced and employed. The resultsstrongly suggest the superiority of Grey–Markov methodover other prediction methods. Furthermore, Grey systemshows a considerable potential for EAF reactive powerprediction and in the majority of cases performs better andmore accurate than ARMA models. Finally, a comparisonbetween Grey system and Grey–Markov is drawn andadvantages and disadvantages of these methods over eachother are demonstrated.

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Enhancement of electric arc furnace reactive power compensation using Grey–Markov prediction method