reduction of thd in grid using recursive newton method · microgrid and non -linear load is consi...

Reduction of THD in Grid using Recursive Newton Method

1Srihari Mandava, 2Ramesh Varadarajan

1,2SELECT, VIT University, Vellore - 14

Abstract

With the advance in technology, the usage of non-linear loads and the role of electronic devices are increasing in the power system network. These devices injects higher amount of harmonics into the grid and reduces the quality of power. It has become a challenge to a power system engineer to identify the harmonics more accurately, fastly and precisely to compensate them in the network for increasing the power quality. Many methods exist in the literature to compensate the harmonics by shunt and series method. But, the identification of harmonics is not exact and hence the power quality is still a problem in the network. In this work, the recursive newton method is used to identify the power system harmonics in less time and compensate them in the network so that the total harmonic distortion (THD) in the grid is less than 1%. This compensation method can be applied to grid in grid connected as well as in islanded mode of operation. Key Words—Grid, Microgrid, total harmonic distortion (THD), Harmonics, Newton, Recursive

1.INTRODUCTION The demand for electrical power and the use of non-linear electrical loads is growing at higher rate with the changes in digital technology. The use of non-linear loads is injecting current harmonics and reduces the quality of power in the network. With this the quality of power has become a great challenge [1]. Similarly, the use of renewable energy sources for green energy and to meet the power demand are increasing in the network connected to it using power electronic converters is another source of harmonics into the network to reduce the power quality. Various techniques are proposed so far for improving the quality of power. The accurate measurement of harmonics in the power signal is the basic step to increase the power quality of it. The measured harmonics are injected in to the network by 180o in series or in shunt mode. The first and frequently used method is the Fourier Transformion (FFT)[2]. But, the change in fundamental value of frequency is not shown properly by this. The drawback of FFT is reduced by using discrete wavelength transform (DWT) [3] in which the input signal is classified into different frequency bands uniformly present in the signal. In this, the accuracy of measurement depends on the use of mother wavelet which is a great challenge. The other methods like kalman filtering [4], least square methods[5], recursive mmethods [6] are also present for measuring the parameters of harmonics signals. Kalman filter method[4] shows better performance of estimation in normal as well as under abnormal conditions. Least square method[5] also reduces the difference between the measured and observed values effectively. These methods are better than FFT but are not good in performance when the signal parameters changes suddenly. There are some hybrid algorithms to estimate the harmonics of power system. Genetic algorithm (GA)[7], bacterial foraging(BF)[8], particle swam optimization(PSO)[9], artificial bee colony (ABC)[10] and fire fly algorithm[11] are some of them. These methods are based on evolutionary optimization technique to estimate the phase and least square method to estimate the amplitude. Fire fly algorithm convergence is better [11] than other methods as it looks like very natural random process and use real random numbers. Singular Value Decomposition (SVD) [12] is used on Linear Time Varying (LTV) system having

International Journal of Pure and Applied MathematicsVolume 118 No. 18 2018, 4095-4106ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.euSpecial Issue ijpam.eu

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dynamics in phase and in amplitude in order to extract the harmonic components. In this, subspace technique is used and the varying transition matrix is calculated for every two successive Hankel matrices. Vibration response measurement [13] is also used to measure the parameters of harmonics in time varying system. A vector of unknown parameters are estimated by using Newton type algorithm (NTA) [14] which takes more time due to computational issues on jaccobian matrix. This problem is solved in Recursive type newton algorithm proposed in [15] for estimation of frequency and amplitude. The computational time in real time for identifying the unknown parameters is reduced by making the NTA recursive in nature. But, IRNTA [15] , the stopping condition is not mentioned and hence the loop should carry for the iterations mentioned during the initial conditions. But, there are many chances for the unknown parameters to converge at a very lesser number of iterations than the maximum iterations considered. In this paper, the IRNTA [15] is considered as base and it is modified by proposing the stopping criterion by finding the eigen values of Hessain matrix, and updating the parameters until the difference of eigen values falls below a threshold value. This helps in identifying the unknown parameters in lesser number of iterations which reduces the computational time much better than IRNTA. Also, the same method is expanded for identifying the harmonics up to 31st order of a microgrid signal. The proposed method measures the harmonics of grid in both grid connected mode and stand-alone mode. The harmonics are then compensated using the harmonics reduction unit connected in series to the grid. In the remaining part of paper, section B describes the proposed algorithm and equations involved, section C discusses on the implementation of the proposed algorithm for identifying the harmonics upto 31st order from a microgrid (with solar and wind energy sources) signal using MATLAB/Simulink, section D shows the validation of the output comparing with the original signal from microgrid and section E gives the conclusion on the total work.

2.FORMULATION OF THE PROPOSED ALGORITHM

Let the signal (voltage or current) at a given power system node is

(1) )t( +)t),t(x(h=)t(y

where )t( is a random noise of zero mean, )t(x is a time-varying parameter vector and ) (h is given

in (2).

(2) )+t).t(ksin()t(A=)t),t(x(h k∑h

1=kk

The unknown parameters from (2) are [ ] Tkkk , ,A for each harmonic signal where kA is the

magnitude, k is the angular frequency and k is the phase of each harmonic order k. As the

parameters are dynamic in nature the frequency is calculated using (3).

(3) ))t((dt

d+))t((

dt

d*t+)t(=))t(+t*)t((

dt

d

Let the frequency and phase angle do not have any change in a data window, then

(4) )t(=))t(+t*)t((dt

d

Neglecting temporarily, (1) can be rewritten as

(5) 0=)x(F=y-)x(h

Where )x(F is an mx1 linear vector of a suitable non expressions and 0 is the mx1 zero vector.

Vector x is then obtained through iterations. Given initial guess ix , let 0≠)x(F . Then there

exists a correction factor such that

(6) 0=)x+x(F ii

Applying taylors series on (6),

(7) x*)x(J+)x(F=)x+x(F iiiii

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Where J is a Jacobin matrix of order mx3 given in (8)

(8)

v∂

F ∂

∂

F ∂

v∂

F ∂

∂

F ∂

∂

F ∂

v∂

F ∂

=)x(J

mmm

111

i

Where )+tsin(= v∂

F ∂p

k

)+tcos(At= ∂

F ∂ pp

k

)+tcos(A= ∂

F ∂ p

k

Considering the possible error after neglecting the higher order terms in the Taylor series expansion,

(9) v+)xh(-A=xJ iii

Where v is an error vector because of neglecting higher order terms in Taylor's series. To minimize

the sum of square errors in (9), one obtains the expression for the unknown correction vector ix as

(10) )]xh(-A[J=)]xh(-A[J)JJ(=x i#ii

Ti

-1i

Tii

Where H is a pseudo inverse of iJ . By adding the correction vector to the original guess ix , we

obtain the following relationship

(11) )]xh(-A[J +x=x+x=x i#iiii1+i

From (11), the inverse of )JJ( iTi called hessian matrix H is calculated to find 1+ix in each iteration.

The inverse of H takes more time for calculation and hence eigen values of H are calculated and compared with a threshold value. The same step is used a stopping criteria for the updating of unknown parameter values. The flow chart of the proposed algorithm is presented in Fig.2.

3.THD REDUCTION UNIT As per the standards of IEEE 519, the individual harmonic component should be less than 3% and the total harmonic distortion (THD) of the signal should be less than 5% for a good power quality. Fig.1 shows the block diagram of harmonics reduction unit[16]. The THD of the grid current is compared with the desired amount of THD as per IEE standards 519. If the difference is not zero, then the harmonics measured from the signal are given as input to the PI controller else a zero input is provided to it. During the absence of harmonics of any order the reduction unit will not inject any other harmonics in to the system, The PI controller make sure that the THD value of the power signal is less than the specified value and once the THD value is at acceptable value, the PI controller gets stabilizes and continuous to inject the same harmonics at which it is stabilized.

Fig. 1. Block diagram of THD reduction unit

Harmonics measurement

unit Switch

THD

Comparison Desired THD

Harmonics

measured from grid

Zero

PI

Controller

PWM

generator


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Fig.1. Harmonics estimation using Recursive Newton Method

4.APPLICATION OF PROPOSED ALGORITHM A microgrid having a solar and wind energy source connected to non-linear load is designed in Matlab/Simulink as shown if Fig.3. The voltage (or current) signal from the bus connecting the microgrid and non-linear load is considered. Let the signal be (12) which has k number of harmonics.

(12) )+tsin(A=)t(y kkkk

Using (8), find the Jacobean matrix (J), for each harmonic order k .Then, the eigen values of matrix H are calculated for each iteration i. If the eigen values do not meet the threshold criteria, update H using the damping factor λ and the correct vector is calculate using (11). This is then added to the

If eigen values of (H(i) - H(i-1)) <

threshold

Update the forgetting factor λ

Maximum

iterations reached End

start

Input f(x) and initial guess values

If error <

threshold

Evaluate jacobian matrix J (xi) using (8) to

obtain F(xi + Δ xi) parameters

Update Δ xi in very iteration using (10)

xi + 1 = xi + Δ xi Compute Eigen values of H=

If eigen values of

(H(i) - H(i-1)) <

threshold

Update H using damping

factor λ

No

Yes No

Yes

Yes

No


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unknown parameters ),,A( kkk and calculate the signal )t('ky using (13) with the new values and

compare it with )t(yk .

(13) )+tsin(A=)t(y 'k

'k

'k

'k

If )t(y-)t(y k'k is not meeting the threshold criteria, update the damping factor λ and go for the next

iteration. The input signal taken from the microgrid is shown in Fig.4 and the harmonics up to 31st

order measured from the input signal using the proposed method are shown in the Fig.5 and Fig.6 respectively. The x-axis in the Fig.5 and Fig.6 is the time in sec and y-axis is the magnitude in per unit. The changes in magnitude of each harmonic signal in Fig.4 and and Fig.5 indicate the impact of each harmonics for every cycle.

Fig.3. Structure of grid network considered.

Fig. 4. Input signal from microgrid to measure harmonics

5.RESULT

The harmonics measured shown in Fig.4 and Fig.5 are all summed up and compared with the input signal as shown in Fig.7. The two signals are approximately equal in magnitude and there exists some difference between the two. The difference is because of the presence of error in the estimation of phase angle using the newton method. In [ 15], it is said that the guess for initial values is done easily for amplitudes as 1/3, 1/5, 1/7 etc.… respectively and frequencies as 50, 150, 250, 350, etc... for each odd harmonic signal but the estimation of angles of each harmonic order is not so easy which depends on the non-linearity of the load. So if the initial guess of angles is not approximate, it would result in wrong angles. In this work, the accuracy considered in estimating the unknown parameters for each harmonic signal is 1e-5. The proposed method has taken 4 cycles of input for estimating the harmonics with more accuracy which is observed from the Fig.7. The

Solar

Arrays

Wind

Turbines

Conv

erters

Power

System

Networ

k

Point

of common

coupling

Non- linear

load


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maximum number of iterations considered is 100 and the proposed method has taken 3 to 15 iterations for estimating each harmonic signal. This shows the impact of adding the stopping criteria in the IRNTA method [15] using the eigen values of hessian matrix (H).

Fig. 5. Odd Harmonics of input signal upto 15

th order

Fig. 6. Odd Harmonics of input signal from 17

th order to 31

st order

Fig7. Comparison of input signal with the sum of all harmonics and fundamental measured using proposed method.


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The harmonics obtained are then given as input to the THD reduction block shown in Fig.1. Before injecting the harmonics in to network, the THD in the grid connected mode is 12%. The harmonics are added to network at exactly T=0.2 s. The grid current and the voltage drop across the coupling impedance is shown in Fig.8 and Fig.9. From Fig.8 , the grid current is distorted more and then the distortion comes down after injecting the harmonics. The amplitude of the current also changes in Fig.8 from T=0.2s from where the harmonics are injected into grid. The THD of the grid current in grid connected mode is shown in Fig.9. The THD of the current changes from 5% to <1.5% after the harmonics reduction unit is switched on. The voltage drop across the coupling impedance also changes once the harmonics is injected as shown in Fig.10.The voltage at point of common coupling (PCC) is shown in Fig.11. Its changes are clearly seen in magnitude and also in distortions after harmonics are added into network. The THD of the voltage at PCC is shown in Fig.12 in which the THD was 12% without harmonics compensation and once the harmonics are injected , the PI controller adjusts and gets stable which is seen as reduction of the THD starts dropping down from T=0.2 sec and finally it stabilizing to 1% at T=0.5 S. The harmonics that are getting injected into the grid using the harmonics reduction unit is shown in Fig. 13.

Fig. 8. Grid current before and after harmonics injection

Fig. 9.THD of voltage at PCC


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Fig. 10. Voltage drop over coupling impedance before and after harmonics injection

Fig.11. Voltage at PCC before and after harmonics injection


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Fig. 12.THD of voltage at PCC

Fig.13. Harmonics injected by THD reduction unit

6. CONCLUSION A microgrid having wind and solar sources are designed and connected to the power grid. The network is connected to a non-linear load. A new method is proposed for measuring the harmonics of microgrid. Odd harmonics up to order 31 are measured and then the same harmonics are added with fundamental and compared with the input signal. The difference between the two is very less which gives the accuracy of harmonics measurement. The harmonics measured using the proposed methods are then added in series to the network using the PI controller. The PI controller aids in helping the reduction of THD as per the difference in THD value of actual grid current and desired value. The value of desired THD considered is 2% in this work.


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[15] Pravat Kumar Ray, Pratap Sekhar Puhan, Gayadhar Panda, ― Improved Recursive Newton Type Algorithm based Power Systemm Frequency Estimation‖,International Journal of Electrical Power & Energy Systems, 2015, Vol. 65, pp: 231-237.


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reduction of thd in grid using recursive newton method · microgrid and non -linear load is consi...

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