Optimal and Adaptive Compensation of Voltage and Current Harmonics of Non- linear Loads under Non Stiff Voltage Conditions

Dr. S.M.R. Rafiei Faculty of Electrical Engineering,

Sahand University of Technology., Tabriz, Iran

Smr-Rafiei@yahoo.com

Abstrac-In this paper operation of OFC based active power filters under non stiff and distorted voltage conditions is investigated. Under these conditions there exist a feedback from the load currents to the load voltages hence this situation makes the OFC based system a closed loop adaptive system which may represent an unstable or poor transient response behavior. By identifying the source equivalent circuit an improved OFC algorithm OOFC) bas been proposed The propased control algorithm considers the load vollags variation due to the load current variation. Successful performance of the proposed algorithm has been investigated and proved hy several MATLAB based simulation studies. A new compensation strategy based on IOFC system which minimizes the load voltage harmonics has been proposed and investigated . Index Terms - Power Quality. Harmonics. Voltage Distortion, Power Factor, Optimal Compensation, Instability, Non-hear Optimization. IEEE-5 19.

1. INTRODUCTION

Widespread use of non-linear loads such as power electronic loads have increased voltage and cwrent harmonics in the utility power networks. Power quality improvement by active power filters [1,2] has been of interest for both industry and academic researchers in the passed three decades. Recently, an optimal and flexible control strategy (OFC) for shunt active filters and its simplified version using Neural networks (NNOFC) operating under distorted voltages conditions have been developed [3,4]. They are adaptive control systems which can compromise between the power factor, current harmonics and many of other power quality indexes . Some of the useful compensation strategies suggested are [3 ] : Maximizing the power factor subject to some constraint on the current harmonics (such as ieee-519 standards for harmonics[5]). Minimizing the current THD subject to a lower bound on the power factor. This paper is a further work on the application of OFC system by investigating the operation of the OFC based active filter in non stiff bus systems where the load current harmonics affect the load voltage harmonics. Figure 1 shows the block diagram of an under compensation load and

0-7803-7697-8/03/$17.00 0 2003 IEEE.

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Active Filter

Figure 1: Block diagram of an under compensation load

1 % h I

Figure 2: Block diagram of the system with the effect of the source equivalent impedance Z ( s ) .

Figure 2 shows the block diagram of the OFC system with considering the effect of the source equivalent impedance. As seen the source impedance Zi:s) makes the system a closed loop adaptive system which may represents unstable behavior. Due to effect of the source equivalent impedance viewed by the load, this subject can be considered as a very important problem in practice. Previous works in [3,4] are based on the constant load voltage harmonics assumption while in practice there exist considerable effect on the voltage harmonics due to the load current harmonics. Hence, once the optimization algorithm finished and filter bank parameters are determined [3] it changes the waveform of the load current and so it will cause some change in the load voltage harmonics. The new situation for the voltages makes the present filter banks sub optimal and so the optimization process should be run for the new voltage waveforms. This situation requires the optimization process to be run successively. Is the control algorithm convergent and provide stable current waveforms?. This is a very important question which will be addressed in this paper.

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Also, there are some interesting questions about the response of the OFC system under non stiff voltage conditions which will be considered. Suppose that the source voltage is harmonic free and the load voltage harmonics are due to the load current harmonics . What are the resulting voltage and current of the load where the maximum power factor compensation strategy is used? Also, a new important strategy will be presented which optimizes the load voltages harmonics . This strategy essentially considers the load voltages non stiff. Section 2 presents a short review of OFC while in Section 3 the transient behavior and stability analysis of the OFC system ares presented . Improved OFC system (IOFC) which guaranties the stability of the control system is proposed in section 4. The paper also proposes a compensation strategy for minimizing the load voltage harmonics by using parallel active filters in section 5. Section 6 concludes the paper.

2. REVIEW OF " O F C STRATEGY

A. OFC Structure The following show the OFC structure [3] :

i' = Y,'.e* (1)

e*(s) =G(s).e(s) (2) where I+I *. is a constant for balancing the active power of the compensated load and i' =[is., i'p , i', 1' is the desired load current vector in the a, $, and 0 coordinates [2] where the superscript "T" means the transpose of the vector.

. Virtual voltage e' =[e',, e p, e 1' is generally a filtered version of the load voltage, e. Also, e'(s) and e(s) are the Laplace transforms of e' and e vectors respectively. Filter bank set G(s) can be designed based on any defined compensation strategy and by an optimization algorithm.

B. Coinpensation Strategv The following is one of the useful load compensation strategies that can be realized by OFC system. This strategy has already been investigated in [3,4]:

. .

Max II ((3 (3) Subject to: Constraints on the THD, or Harmonic Factors (HF) of the currents. The harmonic constraints can be chosen according to the IEEE standards for harmonics [5] and other power quality indexes such as THD of the compensated load currents can be considered as the cost function.

C. Control Alporithnr

for the a-axis load voltage e and e', we have Considering the first N terms of the Fourier series

where similar equations can be developed for $, and 0 axes. Scalars G a (i), G (i). and G ,, ( i ) are the gains of the

a. $, 0 coordinate filter hanks for the harmonic frequency. The phase response of the filters for all harmonic frequencies are set to zero. Neglecting the dc components, effective load voltage E, effective virtual voltage E., and& = (d.e'), can be represented by using the Fourier series coefficients of e and filter gains as shown in [3].

The desired current of the load i' = Yi.e' is calculated by solving the non-linear programming problem introduced by the compensation strategy for finding the optimal values of G = (i) ,cP(i) , G , ( i ) where the power factor 7) is described by the following and is a function of filter banks gains and load voltage harmonics. P,, is the active power of the load:

Also, after calculating the filter banks gains , is calculating by (7) to balance the active power:

(7)

Representation of the OFC system in a,b, c frames has been given in[4] and will be used in this paper.

3. STABILITY ANALYSIS

Figure 2 shows the effect of source impedance as a feedback element which provides a pass from the output(currents) to the input(vo1tages) and makes the control system closed loop. Analytical stability analysis of this adaptive closed loop system is very difficult. Several simulation studies show that the system is stable but the results can not be extended to other compensation strategies. Moreover it was sensed that in some cases the algorithm should run the optimization program for several times ( a i 12 times) successively which speed down the control algorithm response. As seen in Figure 1, the load voltages in a, b, and c phases can be represented as a function of the compensated load currents

i 'x(cjnw)= G, (,jnw).yie ,(, jnw) (x=a, b, c )as follows and w is the network angular frequency:

(8) e_ ( j n w ) e, ( jnw) = l + Z x ( j n w ) . G x ( j n w ) . ~ ;

Where W,' and G, ( j n w ) are the optimization program results and depend on the load voltage er(jnw) .

Transient Response ofthe OFC System: The following example shows the transient response of the OFC system operating under non stiff voltages. In the example, N is considered to be equal to 1 I .

1 04

Example: Consider a system with the source impedance Z(s) . lZQf)l (f is the continues frequency) has been shown in Figure 3 (equal for all phases) . As seen a resonance occurs near harmonics. Figure 4 shows the load phase "a" voltage and current before and after compensation with the resulting filter hank response after one step of operation of the control system . The compensation strategy is as shown in (3) and the harmonics constrains are based on IEEE-519 standards [3,4].The control system samples the load voltages and the optimization is done based on this voltage set. As seen, a new voltages waveforms are produced for the load and so the next optimization is done based on the new load voltages. After each optimization step the is calculated based on the sampled voltages to

balance the active power [3] . Applying the obtained Iy,'

and G, ( j n w ) will result in a new e,( jnw) as shown by

i iamnics order Figure 3: Frequency response of the source equivalent impedance Z(s).

C) Volts

e) Amps 0

r) Amps .,a

~ 4 0 I 0 3 2 U,> a m 03s .a- 0 3 7 0 , s

Figure 4: Results of OFC strategy after the first optimization process (phase a). a: Filter bank frequency response.,b Source voltage, c: Voltage of uncompensated load, d Voltage of

b) Volls m I

0.32 0 s 0.Y 0 , s 0.36 0.37 03a

Figure 5: Results of OFC strategy afler the 4' optimization step (phase a). a: Filter bank frequency response., b: Voltage ofcompensated load. E: Current of compensated load. Hoizonlal axes : HerWSeconds

Eq. (8) and so this new voltage set requires a new Iy,' for

power balancing , The calculating program (Eq. (7) ) is

run successively to reach a stable condition for Iy,' . In this example both the source voltage and load current are rich in harmonics. Figure 4 shows that the resulting uncompensated load voltage is very distorted due to both the voltage source harmonics and the voltage drop of the load harmonics through the Z ( s ) . Because of the tight constraints on the load current, used in the optimization program the load current harmonics have been reduced and so the load voltage waveform has been approached to the source voltage. Figure 5 shows the compensated load voltage and current with the filter bank response after 4 steps of running the optimization program. Figure 6 shows the resulting power factor after each step. Results obtained show that the system is stable but considering tens sub steps for each

Power

optimization step .+I

Figure 6: optimizulion . Final power factor IS 0.9585

The resulting power factor afler each step of

compensated load. e: Current of uncompensated load f

Honzontal axes : HerWSecoods Current of compensated load. step of the iterative optimization program shows the low

speed behavior of the system . In the next section, an

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improved control algorithm which provides the final results in one step is proposed.

4. IMPROVED CONTROL ALGORITHM

The control algorithm has been modified to avoid the instability. By knowing the source equivalent circuit which can be identified easily , the effect of the source impedance has been considered within the optimization program. Due to such modification, a new unknown parameter appears in the optimization process which is solved by a separate iterative algorithm successfully.

A) Source Equivalent Circuit Identification: The new control algorithm which will be presented in the next section needs source equivalent circuit (i.e voltage and impedance) viewed from the load point. Though the equivalent circuit might he obtained from the nominal data of the power network but the exact data can be obtained experimentally using the OFC based active filter system. In the OFC based system the compensated load current is equal to i 'x(jnw)= G, (jnw).y,'e r(jnw) and sothe load is linear (hut not necessary resistive) . Also, the compensated load voltage and current can he measured easily. Using Eq. (8) and measuring the compensated load voltage ex(jnw) for two different and suitable values of

G, (jnw).y, we have two linear equations with two

unknown values e,(jnw) and Z(jnw) . So the unknown parameters can be calculated for each n=1,2, ... N. As presented in the previous section, normal operation of OFC system in non stiff voltage condition provides a set of

successive values for G, ( j n w ) y ; that can be used for identification.

B) Improved OFC system (IOFC) In the OFC system, GI ( j n w ) s are calculated using a non linear optimization program and based on the load voltage harmonics e1(jnw) (n=1,2, ..., N) which are independent

constants and then the is calculated using Eq. (7). In non stiff voltage conditions , at each iteration of the optimization process, a set of G, ( jnw)s is given to calculate the power factor and constraints. Also e,(jnw) s

are functions of G, ( jnw) and yi . Hence yi can not be calculated separately. Moreover it can not he considered as an optimization variable since if we add it to the set of optimization variables {GI (jnw) }, then we should add an equality constraint to guaranty the active power balancing condition . As experienced by the author, this constraint will speed down the optimization program dramatically.

In the proposed algorithm, the optimization program starts with an initial set of G, ( j n ~ ) s which are the filter banks gains in a,b,and ,c frames.. After each iteration the cost function and constraints are calculated and a new set {

G, ( j n w ) ) is produced for the next iteration. In each

iteration, for a given set { G, ( jnw)) and considering an

initial value for y i , and knowing the values of the {

e,(jnw) } and { Z ( j n w ) } (n=1,2, ..., N) we can calculate the load voltage harmonics set { e,(.jnw) }. This values can

be used in Eq. (7) for finding a new value for yi. Sub procedure ends if difference between two successive values of is less than a given small number y . This sub procedure has been tested for many different values of initial yi and some different optimization conditions and has represented a good convergent behavior. It often provides the result after 5 or 6 iterations.

C) Simulation Results The results obtained by simulation studies of the improved OFC (IOFC) using MATLAB programs[6] showed the successful operation of the control algorithm. Several cases with different compensation strategies were tested. The final results that is the compensated current waveforms are the same as those obtained from the classic OFC system via several optimization steps . Figure 7 shows the results obtained from IOFC system. y has been chosen to be .0001 and as seen the results obtained are very close to the previously obtained results from OFC after 12 steps. The resulting power factor is same as previous and is equal to 0.9878.

5. MINIMIZING THE LOAD VOLTAGE HARMONICS.

Load voltage harmonics might be due to source voltage harmonics, load current harmonics or both. Voltage harmonics minimization can not clearly performed by using a strategy which minimizes power factor or current harmonics . Based on the Improved OFC system a new compensation strategy which minimizes the voltage THD subject to some constraints on the level of the current harmonics and power factor has been proposed and investigated. The strategy can consider the voltage harmonics situation beside the current harmonics, power factor and other desired parameters. This is very important since in some cases a major part of the voltage harmonics are due to the current harmonics of the uncompensated load .In another cases, increase in a load current harmonic might reduce the relevant load voltage harmonic due to filtering effect of the source impedance. Figure 8 shows the results obtained from the proposed compensation strategy for minimizing the voltage harmonics while a lower band of

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a) Gam

h) Volts

c ) VOl lS

d) Volts

Figure 7: Results of IOFC strategy after the oplimication process (phase a). a: Filter banlt frequency responss., b: Source voltage. E: Voltage of uncompensated load, d Voltage of compensated load. Cumnl of uncompensated load, and current of the uucompensated load are as shown if Fig. llonzontal axes : HertziSeconds/ Harmonic order

a) Gain

b) Volts

E ) VOllS

d j Volts

e j Alnps

D AmPS

Figure 8: Results of IOFC strategy &er first oplimlzation process (phase a). a: Filter bank frequency response., b Source voltage, c: Voltage of uncompensated load, d: Voltage of compensatrd loa& e: Current of uncompensated load, f. Cum01 of the uncompensated load, Ho"mnta1 axes : HelWSeconds

0.95 has been considered for the power factor and the load current THD constraints have been chosen to be 10%. Also the initial value of y,' is set to be equal to 0.1. The resulting power factor is equal to 0.9987. The voltage THD is equal to 11.24 % while the source voltage THD is 14.18%.

6. CONCLUSION

In this paper transient response of the OFC based active filters operating under non stiff and distorted voltage condition was investigated , It was shown that under this practical condition the feedback from the compensated load currents to the load voltages makes the OFC system a closed loop control system . Simulation results showed the poor transient response of the system. A new control algorithm called IOFC was proposed which provides the optimal response in one step of running the optimization program. One application of the IOFC was presented by introducing a new compensation strategy which minimizes the load voltage harmonics subject to some constraints on the load current harmonics and power factor. Control system presented in this paper opens the way for putting the OFC based active power filters in practical use for achieving the desired power quality performance.

7. ACKNOWLEDGMENT

The author gratefully acknowledge the financial support of the Sahand University of Technology.

REFERENCES

1. F. Peng, H. Aka&?, and .A. Nabae, "A Study of Active Power Filters Using Quad-Source P W M Convaten for Harmonic Compensation," IEEE, Trans. on PowerEloctronics Vol. 5, No. I , January 1990.

2. H. Akagi, "New Trends in Active Filters for Line Condmming," IEEE, Trans. on Industry App., Vol. 32, No. 6, Nov.iDec. 1996.

3. S.M.R. Rafiei, H.A. Toliyat, R. Ghazi, and T. Gopalarathmm, "An Optimal and Flexible Control Slralegy For Adive Filtering and Power Fador Correction Under NonSinusoiQl Line Voltages," : IEEE Transadionson Powerlkliuely. Vol. 16, No. 2, A p d 2001, pp 207-305. S.M.R. Rafiei, R. Ghzi , and H.A. Toliyat, "IEEE-519 Bawd Real Tim and Optimal Control of Active Filters Under Non-Sinusoidal Line Voltages Using Neural Networks, " IEEE Transactions on Power Delivery, Vol. 17, No. 3. July 2002, pp 815-822.

5. IEEE Recommended Practices and Requirements for Harmonic Control inElectncPowerSystems, IEEEstd 519-1992, NewYork, 1993.

6. MATLAB Optimization Tool BoxUser's Guide, Mathworks. 1998.

Mobammad-Rem &Sei (M03) received BSc. degree with honor from the Sistan and Baluchistan University, Z a h A o . Iran, in 1991, M.Sc., and Ph.D degrees from the Ferdowa Univenily of Mashhad. Mashhad, Iran. in 1995 and 2000 respectively all in electrical engineering. Since January 2001 he has been an assistant Prof$ssor at the Facully of Electrical En@neenng of Sahand Univenity of Technology, Tabnz, Iran . His research interests are Control Systems, Power Electronics., Power @ality and computer Scienw.

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