A Method for Assessing Customer Harmonic Emission Level Based on the Iterative Algorithm for Least Square Estimation *

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1 Engineering, 203, 5, Publihed Online September 203 ( A Method for Aeing Cutomer Harmonic Emiion Level Baed on the Iterative Algorithm for Leat Square Etimation * Runrong Fan, ianyuan an, Hui Chang, Xiaoning ong, Yunpeng Gao School of Electrical Engineering, Wuhan Univerity, Wuhan, China rrfran0426@whu.edu.cn, tty@whu.edu.cn Received June 203 ABSRAC With the power tem harmonic pollution problem becoming more and more eriou, how to ditinguih the harmonic reponibility accurately and olve the grid harmonic imply and effectively ha become the main development direction in harmonic control ubject. hi paper, baed on linear regreion analyi of baic equation and improvement equation, deduced the leat quare etimation (LSE) iterative algorithm and obtained the real-time etimate of regreion coefficient, and then calculated the level of the harmonic impedance and emiion etimate in real time. hi paper ued power tem imulation oftware Matlab/Simulink a analyi tool and analyzed the uer ide of the harmonic amplitude and phae fluctuation PCC (point of common coupling) at the harmonic emiion level, thu the reearch ha a certain theoretical ignificance. he development of thi algorithm combined with the intrument can be ued in practical engineering. Keyword: Harmonic Emiion Level; Harmonic Analyi; Leat Square Etimation; Iterative Algorithm. Introduction * Supported by the Fundamental Reearch Fund for the Central Univeritie (Grant No: ); Supported by NSFC (Grant No: ) In the modern power grid tem, the traditional power equipment ha gradually been replaced by mart device which were baed on power electronic and other nonlinear element. Meanwhile, nonlinear load of the uer ide were heavier and heavier. It made the harmonic pollution problem more eriou and harmful to the afe operation of the power tem. Beide, the uer faced a great power lo. With the public increaing emphai on harmonic problem, the uer reaonable aement of harmonic emiion level in public connection point became an important content of harmonic control []. he reearch of Emiion level etimation of harmonic ource wa focued on harmonic ource qualitative analyi and quantitative etimate at home and abroad: active power direction method [2] wa widely ued in engineering harmonic ource qualitative analyi method. However, thi method i affected deeply by the phae difference between the harmonic ource. And there i a big area of uncertainty, o it wa not uitable for complex power tem. Harmonic ource in quantitative analyi, alo known a harmonic ource emiion level aement, can be divided into intervention type and non-intervention type [3]. Intervention type need to inject into the tem diturbance artificially, which not only increaed the cot of harmonic analyi, alo limited the cope of it application. Non-intervention type wa different; it could ue harmonic ource fluctuation of itelf to etimate the harmonic impedance in tem without interfering with it normal operation. hi method wa imple to ue, ea to the development of equipment. It ha made a great difference in practice [4]. At preent, the main non-invaive method included fluctuation method [5,6], the linear regreion method [8,9] and the reference impedance method [7]. Among them, fluctuation method and linear regreion method were baed on the condition that tem harmonic impedance and equivalent tem invariant harmonic voltage ource and the load harmonic impedance and equivalent load harmonic current ource change greatly (or vice vera), it cannot be applied in a teady tate harmonic ource, and cannot applied in the condition while the tem and uer harmonic ource volatility at the ame [5-9]. he main diadvantage of the reference impedance method i that it need to get more accurate prior reference impedance [7]. hi paper propoed the iterative algorithm for leat quare etimation i baed on the linear regreion analyi of baic equation and the improvement equation. And Copyright 203 SciRe.

2 R. R. FAN E AL. 7 we ue thi method to etimate harmonic emiion level and obtain the real-time etimate of regreion coefficient, and then calculate the level of the harmonic impedance and emiion etimate in real time. 2. he Principle and Baic Equation of Linear Regreion Analyi Method he etimated value of the harmonic impedance and the cutomer emiion level can be obtained by tatitical analyi with linear regreion analyi method, which can be divided into bilinear regreion and 2 element linear regreion. 2.. Bilinear Regreion Analyi Method he premie of bilinear regreion method i that auming the tem operation i table, harmonic impedance i purely inductive impedance; the reitance component i ignored, which mean I, X remain the ame and R i 0. he principle diagram of the harmonic emiion level at uer ide i hown in Figure. he harmonic voltage equation at point PPC i: = pcc + Ipcc () Divide the vector into the real and imaginary part, which are: = pccx + Ipccx Ipccy (2) = pccy + Ipccy + Ipccx (3) he variation of harmonic current at point PCC lead to the change of harmonic voltage at uer ide, and the value of pccx, pccy, I pccx, I pccy are obtained by meaurement and analyi. Ue linear regreion analyi with thee value to obtain the etimated value of,,, and then equivalent harmonic voltage ource and the harmonic impedance can be got, o harmonic emiion level of uer ide i: c c _ pcc = pcc c + = pcc (4) + / c pcc PCC I pcc pcc c I c Figure. he principle diagram of the harmonic emiion level at uer ide wo Element Linear Regreion Analyi Method Bilinear regreion ignore reitance component of harmonic impedance of tem ide, which may lead to the etimated reult different from the actual harmonic impedance largely. In order to reduce the deviation of etimation value and the actual value, ue the analyi method with the leat quare method of type (2) and (3) to obtain the regreion coefficient. he 2 element linear regreion model i a follow: y= b0+ bx + bx + ε 2 ε ~ N ( 0, σ ) 3. he Improvement of Linear Regreion Analyi A howed in type (2) and (3), when the change of load harmonic ource i only amplitude fluctuation, the real and imaginary part of harmonic current at point PCC will nchronize increae or decreae. In order to reduce the correlation and improve the reliability of etimated value, the type (2) and (3) hould tranform, a follow: pccx + Ipccx = (6) I Put the type (6) into (3): pccy I + I pccx pccx pccy pccy ( Ipccx Ipccy ) Ipccx Ipccy = (5) (7) he type (3) i a follow: Ipccx pccy = (8) I Put the type (8) into (2): pccy I I pccy pccx pccx pccy ( Ipccx Ipccy ) Ipccy ( Ipccx ) = he matrix form of type (7) i: ( Ipccx + Ipccy ) Ipccx I pccy ( Ipccx2 + Ipccy2) Ipccx2 Ipccy2 X = ( Ipccxn + Ipccyn) Ipccxn Ipccyn he matrix form of type (3) i: ( Ipccx + Ipccy ) Ipccy I pccx ( Ipccx2 + Ipccy2) Ipccy2 Ipccx2 X = ( Ipccxn + Ipccyn) Ipccyn Ipccxn (9) (0) () Copyright 203 SciRe.

3 8 R. R. FAN E AL. Ue the tranformed Equation (6) and (8) to olve the regreion coefficient, in the invere proce; the correlation i reduced, o the reliability of reult i improved. he above two equation are obtained after a erie of data, through the matrix tranformation and it invere, and a complete algorithm for leat quare. Completing algorithm in practical ue need large amount of memory and cannot be ued in on-line identification. 4. he Leat Square Etimation (LSE) Recurive Algorithm he baic idea of LSE recurive algorithm i that the new etimator i the um of the lat etimation and correction, the computation and torage of each tep i mall, offline or on-line identification can be allowed, and ha the track time-varying parameter ability. When the n meaurement date (ample) obtained, ue the leat quare complete algorithm to obtain the type B= ( X X) X Y, where: X n β = (2) β n y Yn = (3) y n Make P ( n = Xn Xn), next moment, after a et of new data β n +, y n + i obtained: n = ( ) = ( n, β ) ( n n βn β ) = + + β P X X X X X X (4) Y X Y ( X, β ) X Y y n = n = n n + β y (5) Suppoe A i n-dimenional array, B and C are n dimenional vector. A, A + BC and Im + C A B are full rank matrixe, and then: ( A+ BC ) = A A B( Im + C A B) C A (6) When P n + meet the above condition, the type can be got: Pβ = (7) + n n Pn n nβ + P hu, new etimate can be got: B X X X Y P Pβ X Y y n n = ( ) = n n n + + n nβ + ( β ) = P X Y Pβ X Y Pβ + P y Pβ Pnβ y + β n n n n n n n n n n β + n nβ n n nβ n = Bn β B n + + n nβ + n (8) Above i the derivation proce of recurive method. hi paper adopt the following recurive formula: Bn ( + ) = Bn ( ) + γ ( ) PnX ( ) ( ) Yn ( + ) X ( ) Bn ( ) Pn ( + ) = Pn ( ) γ ( ) PnX ( ) ( ) X ( ) Pn ( ) γ ( ) = / + X ( ) PnX ( ) ( ) (9) 5. Simulation Analyi hi article etablihed imulation model to meet the precondition of algorithm, in order to verify the linear regreion analyi of three kind of algorithm in the MALAB oftware (baic equation, modified equation, the iteration algorithm). It alo etimated the harmonic impedance and harmonic emiion level of the uer. o facilitate the analyi, thi article elect 5th harmonic a Copyright 203 SciRe.

4 R. R. FAN E AL. 9 analyi object. Harmonic impedance of tem ide i uually maller than the uer ide, if impedance at the uer ide i unable to get accurate numerical, harmonic voltage formula emiion level of uer ide can be obtained by the above equation. Ue the imulation of the modulu value level evaluation index for launch, namely c _ pcc pcc. In formula (9), Bn ( ) tand for the etimate value of lat moment, B ( ) tand for new etimate of the current moment, the initial value of B and P elect the firt few data point to olve or et to B (0) = 0 and P(0) = ρi effectively, ρ take a lot of poitive calar 5 8 ( 0-0 ), I tand for the Unit matrix, the influence of the initial value drop with the recurive number increaing, ρ i et 0 in the 6 imulation. 5.. Baic Equation and the Modified Equation Algorithm Simulation he baic equation and the modified equation algorithm imulation model i hown in Figure 2, the tem ide harmonic ource i et a contant 5th harmonic ource, the uer ide et fluctuated 5th harmonic ource, reitance, inductance are et a: R = 5, L = 0.002H, Rc = 5, Lc = 0.02H. Set 5th harmonic amplitude of tem ide i 5A; the 5th harmonic amplitude of the uer ide (coefficient) varie with time a hown in Figure 3. he real and imaginary part of the 5th harmonic current and the real and imaginary part of the 5th harmonic voltage are hown in Figure 4. Ue the imout module to tore real and the imaginary part of 5th harmonic wave to work pace, and then according to the baic Equation (2) and (3) and modified Equation (7) and (9) algorithm to write the procedure of the regreion coefficient, the data hown in able and 2. a) When the uer ide only harmonic amplitude fluctuation, reult were obtained a follow: Figure 2. Linear regreion analyi of complete imulation graph algorithm. Copyright 203 SciRe.

5 0 R. R. FAN E AL. Figure 3. 5th harmonic amplitude variation curve of uer ide (coefficient). Figure 4. Real part and the imaginary part of 5th harmonic current and the real and imaginary part of 5th harmonic voltage. able. he uer ide only harmonic amplitude fluctuation data. Baic equation Set value Etimated value Deviation (%) Modified equation Set value Etimated value Deviation (%) () ( Ω ) ( Ω ) () ( Ω ) () () ( Ω ) ( Ω ) () ( Ω ) () able 2. Data when the uer ide of harmonic amplitude and phae fluctuation. Baic equation Set value Etimated value Deviation (%) Modified equation Set value Etimated value Deviation (%) () ( Ω ) ( Ω ) () ( Ω ) () () ( Ω ) ( Ω ) () ( Ω ) () Copyright 203 SciRe.

6 R. R. FAN E AL. Improved regreion coefficient equation algorithm to obtain etimate of the value of deviation i maller, and put the average value (6) to emiion level of the 5th harmonic wave the harmonic voltage at PCC (amplitude) i: c _ pcc pcc ( ) = ( j6.2 + j6.02) / 2 = b) Harmonic amplitude and phae fluctuation in the uer ide Etimation of the baic equation algorithm average calculation 5th harmonic wave, harmonic voltage in the PCC emiion level (amplitude) i: ( ) c _ pcc pcc = ( j6.04 = ; Improved etimation equation algorithm average calculation 5th harmonic voltage in the PCC emiion level (amplitude) i: c _ pcc pcc ( j6.04 j6.05 ) / ( ) = = he data how, when in uer ide harmonic amplitude fluctuate only and not phae fluctuation, regreion coefficient obtained from the baic equation algorithm etimate a large deviation, the deviation i maller by the etimation improved equation algorithm, namely improved equation algorithm can be ued; when the amplitude and phae of the harmonic ource uer ide fluctuate at the ame time from the baic equation, and improved equation algorithm etimate the deviation i relatively mall, the baic equation and the improved algorithm can be applied to equation Leat Square Etimation Iterative Algorithm Simulation When LSE Iterative algorithm i applied (Set iteration coefficient matrix according to the improvement equation mentioned above), made uer ide harmonic ource have only amplitude fluctuation, and the etting condition of harmonic on both wa the ame to the algorithm that linear regreion analyi complete dipoable. Simulation Model i hown in Figure 5. Figure 5. LSE iterative algorithm imulation model. Copyright 203 SciRe.

7 2 R. R. FAN E AL. Figure 6. 5th harmonic emiion level of uer ide at point of common coupling. Figure 7. 5th harmonic regreion coefficient etimate of recurive algorithm. he 5th harmonic emiion level (voltage amplitude) of uer ide at point of common coupling i hown in Figure 6. he imulation reult of LES recurive algorithm hown in Figure 7. Follow from top to bottom they are,,,,,. We can ee from the reult of Iterative algorithm, the etimate of the third tationary of iterative algorithm became relatively table, and it deviation i mall compared with etting. Compared the two value (the etimate of the third tationary and the etting), we can get the Deviation range: : 0.00% %; : 0.002% %; : 2.9% %; : 2.94% %; : 0.006% %; : 2.93% %. According to the data and figure above, we can conclude that, when the uer ide of the harmonic ource fluctuation i mall, iterative algorithm can get a mall deviationit etimate and real-time, table harmonic emiion level. What more, with a mall amount of data tored, running fat, online or offline running to get real-time harmonic etimate, iterative algorithm i ideal algorithm in the condition. 6. Concluion hi paper derived iterative algorithm baed on the baic equation and improved equation. hi algorithm revied recognition reult contantly on the advantage of new data; it got real-time harmonic impedance and harmonic emiion level etimate. It calculated peed with mall amount of data torage by thi way. In addition, the iterative algorithm wa without inverion proce, o were the uer ide harmonic amplitude fluctuation and not only the phae fluctuation. here wa no correlation between cauing erroneou reult and obtaining invere problem. hi algorithm will be improved and combined with the intrument development, it can be ued in practical engineering, real-time monitoring of the PCC harmonic level. REFERENCES [] International Electric technical Commiion (IEC), Sub- Copyright 203 SciRe.

8 R. R. FAN E AL. 3 Committee 77A, Electric magnetic Compatibility (EMC) Part 3-6: Limit Aement of Emiion Limit for the Connection of Ditorting Intallation to M, H and EH Power Sytem, Britih Standard Intitution, United Kingdom, [2] H. Yang, P. Porotte and A. Robert, Aeing the Harmonic Emiion Level from One Particular Cutomer, Proceeding of the 3rd International Conference on Power Quality, ol. 2, No. 8, 994, pp [3] W. A. Omran, H. S. K. EI-Goharey, M. Kazerani and M. M. A. Salama, Identification and Meaurement of Harmonic Pollution for Radial and Nonradial Sytem, IEEE ranaction on Power Delivery, ol. 24, No. 3, 2009, pp [4] W. Xu, Power Direction Method Cannot Be Ued for Harmonic Source Detection, IEEE Power Engineering Society Summer Meeting, 6-20 July 2000, ol. 2. [5] W. Xu and Y. L. Lin, A Method for Determining Cu- tomer and Utility Harmonic Contribution at the Point of Common Coupling, IEEE ranaction on Power Delivery, ol. 5, No. 2, [6] C. Li, W. Xu and. ayjaanant, A Critical Impedance Baed Method for Identifying Harmonic Source, IEEE ranaction on Power Delivery, ol. 9, No. 2, 2004, pp [7] Y. Xiao, J.-C. Maun, H. B. Mahmoud,. Detroz and D. Stephane, Harmonic Impedance Meaurement Uing oltage and Current Increment from Diturbing Load, 9th International Conference on Harmonic and Quality of Power, ol., Orlando, -4 Oct. 2000, pp [8] W. hang and H.-G. Yang, A Method for Aeing Harmonic Emiion Level Baed on Binary Linear Regreion, Proceeding of the CESS, ol. 24, No. 6, 2004, pp [9] Q. Che, H.-G, Yang, Aeing the Harmonic Emiion Level Baed on Robut Regreion Method, Proceeding of the CSEE, ol. 24, No. 4, 2004, pp Copyright 203 SciRe.