Partial Linear Regression Method in the Application of the Accurate Calculation of PQDIF Harmonic Responsibility

Transcription

1 International Conference on Modelling, Simulation and Applied Mathematics (MSAM 05) Partial Linear Regression Method in the Application of the Accurate Calculation of PQDIF Harmonic Responsiility Aiqiang Pan and Xingang Yang Electric Power Research Institute of SG Shanghai Electric Power Company Hongkou District, Shanghai 00437,China Yijun Liu School of Electrical Engineering & Information Sichuan niversity Chengdu 60065, Sichuan Province, China Astract Quantitatively evaluating the harmonic responsiility of harmonic source at the point of common coupling (PCC) is an important task of power quality management. Considering the change of ackground harmonic and the fact that the current monitoring system can only provide the voltage and current statistical values, a method called partial linear regression method for accurately calculating harmonic responsiility under PQDIF is proposed. By means of measuring the information of PCC, the proposed method uilds a more accurate partial linear regression model of calculation of the equivalent harmonic impedance, and then design two-stage method to calculate the uniased estimation of the equivalent harmonic impedance and the ackground harmonic voltage, finally implement the accurate calculation of harmonic responsiility under PQDIF. Simulation results of IEEE 4-us system show the effectiveness and accuracy of the proposed method. Keywords-power quality; harmonic responsiility; power quality data interchange format; partial linear regression method I. INTRODCTION With the development of the industry, all kinds of power electronic devices and nonlinear loads are widely used in power system. The prolem of harmonic pollution in distriution network is increasingly seriously. Therefore, the reasonale assessment of harmonic voltage responsiility of power system and user at the common coupling point, is the precondition of power quality management and controlling[]. Nowadays,methods for determination of harmonic responsiility at home and aroad is mainly aout the harmonic impedance estimation of system and user, mainly divided into "intervention" [-4] method and "nonintrusive" method[5-7]. "Intervention" method mainly through the artificial injection of into the system to measure the harmonic impedance of system, this method may change the operation of the power system, and has large estimation error." The intervention method" mainly use the measured value of the harmonic voltage and current at PCC to determine harmonic responsiility quantitatively, such as the method of linear regression, including roust regression method [5], the inary regression method [6], partial least squares[7], etc. The essence of these methods is ased on the Thevenin equivalent circuit or Norton equivalent circuit to constructs the corresponding regression equation, and divides the harmonic voltage and current phasor at the PCC into real part and imaginary part to otain the harmonic impedance. However, due to the ackground harmonic voltage is a variale in the regression equation, and the relationship etween independent variale and dependent variale is not strict linear, so the linear regression will cause greater error. Besides, most of the aove methods is ased on the voltage and current instantaneous value, and it s not conform to the current situation that the daily power quality monitoring system mainly provide statistics such as voltage, current RMS value. Considering the actual situation that the power grid power quality monitoring data is often expressed as power quality data interchange format (PQDIF),so it is necessary to study the division of harmonic responsiility for PQDIF data [8]. Literature [8] just put forward a rough estimate of harmonic responsiility under PQDIF, the algorithm is simple and easy to implement, ut the method cannot meet the accuracy requirement in the case of precise harmonic responsiility. On account of the shortcomings of the aove methods, this paper proposes the partial linear regression method to estimate the harmonic responsiility precisely under PQDIF data. Firstly, use the PQDIF data to construct the partial linear regression model. And then, use the Kernel Smoother method to get the system harmonic impedance and the iased estimation of ackground harmonic voltage. Finally, the ordinary least squares solution is used to get the real-time accurate harmonic responsiility. Experimental results prove the validity and the accuracy of this method. II. DEFINITION AND CALCLATION FORMLA OF HARMONIC RESPONSIBILITY As shown in figure, usar X is the point of common coupling (PCC).The side connected to the grid system is known as the system side; and the other side connected to several loads is known as the user side. Suppose D as a major source of nonlinear harmonic load, now we calculate the harmonic responsiility of load D at the usar X. Set I hd is the h time that load D inject into X, and the h time harmonic voltage measured at X is hx. So the following relation can e estalished: 05. The authors - Pulished y Atlantis Press 36

2 = Z I + hx hx,d hd () power Q hx, h time harmonic voltage RMS hx and current RMS I hx. There are relationships etween aove four: Bus X Load A P = I θ (4) cos hx hx hx h System side Load B Load C Load D Q = I sinθ (5) hx hx hx h θ h is the power factor angle that hx advances I hx, and it can e ackstepped y measurement data: FIGRE I. THE PCC POINT WITH SEVERAL LOADS MODEL Where Z hx, D is the equivalent harmonic impedance of the system and the rest of the loads except for load D. hx, 0 is the harmonic voltage contriution at us X of the system and the rest of the loads esides load D, that is the ackground harmonic voltage. The right side of equation () is the harmonic voltage produced y load D at X, and we recorded it as hx, D. There is: = Z I hx,d hx,d hd The vector diagram of hx, D, hx, 0 and hx as shown in figure. () θ = (6) h tan ( QhX / Ph X) B. Partial Linear Regression Model of System Harmonic Impedance Calculation In equation () there is: So () can e written as: = + j Zh X,D = ZR + jzx Ih D = IDR + jidi = + j hx XR XI 0R 0X (7) + j = Z I Z I + XR XI R DR X DI 0R + j( Z I + Z I + ) R DI X DR 0I (8) FIGRE II. THE VECTOR DIAGRAM OF hx, D, hx, 0 AND hx That is: Define μ h is the harmonic voltage contriution of load D at X, and μ h can e written as the equation expressed y Z hx, D and hx, 0 according to [9]: = Z I Z I + XI = ZR IDI + ZXIDR + 0I XR R DR X DI 0R (9) h hx + hx,d hd hx Z I μ = (3) Therefore, the key to compute harmonic voltage responsiility under PQDIF is to use PQDIF harmonic measurement data to calculate h time system equivalent harmonic impedance Z hx and ackground harmonic voltage hx, 0 accurately. Rewrite (9) into matrix form: R X ( XR XI ) ( DR DI ) Z Z = I I + ( 0R 0I ) ZX ZR (0) The system harmonic impedance can e seen as a constant approximately in a certain period of time, ut the ackground harmonic voltage changes over time, so equation (0) is a partial linear regression model. Get m times sampling values in a certain period of time, there are: III. PARTIAL LINEAR REGRESSION MODEL APPLIED TO PQDIF A. Calculation of Phase Angle Difference In figure, we can use PQDIF harmonic measurement data to get the h time harmonic active power P hx and reactive Where: y = Eω + f () 363

3 y XR () XI () IDR () IDI () () () IDR () IDI () E = M M M M XR XI IDR IDI 0R () 0I () 0R () 0I () ω ZR Z f = X M M = ZX ZR 0R 0I XR XI = In the existing power quality monitoring system, the phase angles of h harmonic voltage in matrix y and the phase angles of h in matrix E is unknown, so the harmonic measurement data under PQDIF cannot e represented as ().However, θ h can e otained y Ⅲ.A. So in this paper we use phase angle difference θ h to get the transformation of () applied to PQDIF. Suppose at a measurement time t, the actual phase angle of I hd is α t. The phase angle difference etween harmonic voltage and current at us X is θ h. Devide the oth sides of equation () y α t at the same time we can get: Where: 0R () 0I () 0R () 0I () ω ZR Z f = M M X = ZX Z R 0R 0I real( hx ( θh + αc)()) imag( hx ( θh + αc)()) real( hx ( θh + αc)()) imag( hx ( θh + αc)()) y = M M real( hx ( θh + αc)) imag( hx ( θh + αc)) real( IhD αc()) imag( IhD αc()) real( IhD αc()) imag( IhD αc()) E = M M real( Ih D αc) imag( Ih D αc) IV. THE SOLTION OF THE LINEAR REGRESSION MODEL AND THE ACCRATE CALCLATION OF HARMONIC RESPONSIBILITY I Z hx hd = hx,d + αt αt αt () A. The Principle of the Partial Linear Regression Model The expression of the common linear regression model is: That is: Z I 0 hx θh = hx,d hd + (3) αt The phase angle of I hd in (3) is zero, so the second column of matrix E is 0.Then using equation (3) to calculate the parameters may produce singularity. To avoid this situation, multiply equation (3) y α c at the same time. α c is the reference phase angle of I hd and the value of α c is without any influence on the calculation of system harmonic impedance Z hx, D. For ease of calculation in this paper we set α c = 0, there are: ( θ + α ) = Z I α + α hx h c hx,d hd c c α t Set the last item of the right side of (4) is transformation of () expressed y θ h is: hx h c hx,d hd c (4), so the ( θ + α ) = Z I α + (5) Get m times sampling values in a certain period of time, and the partial linear regression model applied to PQDIF fomat data is: y = Eω + f (6) y = ax + + ε ( i m) (7) i i i In (7),a, is a constant, so it s easy to use least-square method to solve the value of a,. However, in the equation (6), the system harmonic impedance ω is a constant, ut the ackground harmonic voltage is a variale, so (6) is a partial linear regression model and cannot e solved y the common linear regression model. On account of this model, this paper puts forward twostage method to solve it. In the first stage we introduce kernel smoothing estimation method to calculate the iased estimation of ω. Because we hope the result is uniased, so the second stage is to use iased value f to construct common plural regression model, then use the plural least-square method to otain the accurate uniased value ω u.finally we can get the uniased value f u y ω u and equation(6). B. se Kernel Smoothing Method to Get Biased Value Split matrice E into two matrices as follows: E = g+ η (8) where E=( e,,e p ),g=( g,,g p ),η=(η,, η p ).And there is E i =( e i,,e mi ),g i =( g i,,g mi ),η i =(η i,, η mi ). For the partial linear regression model as (6), hypothesis is given: ()m - η η V, V is a positive definite matrix; 364

4 m m () tr( K K) = Kst = O ( ) ; s= t= (3) ηi ( ) K = O = K η ( i p) ; * / v (4) n η f = O( n ) ; i 0.75( t ) t wt () = 0 t > (3) () Secondly, calculate the matrix E * and y * which are the residual error matrix of E and y : (5) For any continuous function z (t), there is a proaility density function p (t) on [0, ] that when m there is: m ( i ) () () (9) 0 i= m z t z t p t dt * E = E KE * y = y Ky (3) Finally get the iased estimate value of ω and f: (4) Where K is a Kernel smoothing matrix. And there is theorem aout using the kernel smoothing method to solve the partial linear regression model. Theorem : If the partial linear regression model meets the assumption,then when the m, 0 there is: * v ( v) ω = 0 (0) v / E( ω ) V gt () f th() tptdt () + Om ( ) By the theorem, the kernel smoothing estimation method is a iased estimate, the error of the estimate result and the real value is a constant. So the process of using the kernel smoothing method to calculate ω and f which are the iased estimate of ω and f is: () Firstly, construct a smooth nuclear matrix K m m, in which the element K (s, t) is as follows : Kst (, ) = w(( s t)/ h)/ w(( s t)/ h) m t = () Here h is the window width, and the cross verification method is availale for the calculation of window width, here we set h=m -/5. In () w is the kernel function and kernel function is a limited support function, and we require kernel function has the following properties: ( l = ) 0 l wttdt () = 0 ( l v ) c ( l = v) () In which v is the smooth order of f. Many functions can e selected as the kernel function, and choosing a different kernel function makes negligile influence on the result. We choose kernel function here is: ω = ( E E ) E y * * * * f = K( y Eω ) (5) C. se the Plural Least Squares Method for niased Estimate According to the theorem, the error of the iased estimate f and the real value f u is a constant, and we set the error is β 0, so there is : That is: y = Eω + f = Eω+ f + β0 (6) y f = Eω + β (7) In (7), matrix y-f and E are known matrix, and ω,β 0 is an unknown constant. Therefore equation (7) is a common linear regression model. Now we rewrite (7) into the form of the plural matrix. r 0 = Qξ (8) In which r=(r,r,,r m ),ξ=[z hx,d c],and c is the plural corresponding to the mean value of the constant β 0.And there is: Q IhD αc() Ih D αc()... IhD αc =... i hx h c 0R( ) hx θh αc 0I( ) (9) r = ( real( ( θ + α )( i)) ( i)), i=,,,m (30) + j( imag( ( + )( i)) ( i)) and 0R( ) 0I( ) is the iased value of 0R and 0I respectively. Equation (8) is the plural linear regression model, and the plural least squares solution to (8) is: 365

5 ξ = ( QQ ) Qr (3) The first element of ξ vector is the accurate calculation result of the h time system harmonic impedance Z hx,d. D. Accurate Calculation of Harmonic Responsiility The injection of load D I hd and the ackground harmonic voltage is variale as time changes. So every time the corresponding harmonic voltage responsiility value μ h is also a variale. Therefore when Z hx,d is known, at the moment i there is: hx + hx,d hd hx ( i) () i Z I () i () i μ () i = (3) h V. THE SIMLATION VERIFICATION In this paper, we use the IEEE 4 us system as figure 3 as an example with Digsilent software simulation. In the case of 5th harmonic,we select us 5 to us X,and HL is the harmonic source load of concern at us X. The ackground harmonic voltage at X is produced y the harmonic source load HS at us 3 and the generators in the system. Now we calculate Z 5X,D which is the rest of the system harmonic impedance except for HL and the ackground harmonic voltage 5X,0. (a) the real part of 5 th () the imaginary part of 5 th FIGRE V. 5 th HARMONIC CRRENT OF HARMONIC SORCE HS We use the classic curve to simulate the injection harmonic current of load HL and HS as shown in figure 4 and 5 respectively. Besides,let the measured interval e min and measured duration e 3s,and take 3s average value for each measurement result. According to the aove algorithm process to measure instantaneous harmonic voltage and harmonic source ranch current at us 5.Then we can get a set of 5 th harmonic voltage vector at us 5 and 5th harmonic source current vector of harmonic source through FFT transformation as figure 6 and 7 respectively. And there are 380 measurement points. (a) The amplitude of () The angle of harmonic voltage harmonic voltage FIGRE VI. 5 th HARMONIC VOLTAGE MEASREMENT VALE AT BS 5 FIGRE III. IEEE-4 BS SYSTEM (a) the real part of 5 th () the imaginary part of 5 th FIGRE IV. 5 th HARMONIC CRRENT OF HARMONIC SORCE HL (a) The amplitude of () The angle of FIGRE VII. 5 th HARMONIC CRRENT MEASREMENT VALE OF HL According to the steps in Ⅲ.B to construct the partial linear regression model, and use the method in Ⅳ.B to get the iased estimation of the ackground harmonic voltage as figure 8. In figure 8, the solid lines and the dashed lines show the real value and the estimate value of the ackground harmonic voltage respectively. It can e found that the difference of the iased estimate and the actual value is a constant as theorem. 366

6 TABLE I. THE COMPARISON OF SYSTEM HARMONIC IMPEDANCE ESTIMATION RESLT BY THE FOR METHODS Method Amplitude Error (%) Angle ( ) Error (%) Real value (a) The real part of the harmonic voltage () The imaginary part of the harmonic voltage /V FIGRE VIII. THE BIASED ESTIMATION OF THE BACKGROND HARMONIC VOLTAGE The plural regression model as (7) can e constructed y the iased value which is otained in the first stage, so we can use the plural least-square method to otain β 0 in (7) which is the difference of the iased estimation and the actual value of the ackground harmonic voltage. Here, we can get the uniased value of the ackground harmonic voltage y adding the deviation valueβ 0 to the iased value of the ackground harmonic voltage f as figure 9.It can e seen that the ackground harmonic voltage estimation curve and the actual curve is essentially the same. The inary linear regression Dominating fluctuating quantity The maximum likelihood estimation Proposed method At the same time we can calculate the estimation value of the system harmonic impedance is As shown in tale, y comparing the estimate result of the method in this paper with the traditional algorithm we can found that the result of the proposed method is more accurate. Besides, the simulation results show that when the ackground harmonic voltage fluctuates at 8%, the result still can meet the precision requirement. As shown in figure 0, we can use equation (3) to calculate the load harmonic responsiility at every measuring point on the asis of the accurate calculation of the ackground harmonic voltage and the system harmonic impedance. (a) The real part of the harmonic voltage FIGRE X. THE RESLT OF THE LOAD HARMONIC VOLTAGE RESPONSIBILITY In figure 0, the averaging value for each measurement point result is the load side harmonic voltage responsiility: 380 μ = ( μi ) / % = 5.3% (33) i= () The imaginary part of the harmonic voltage FIGRE IX. THE COMPARISON OF NBIASED ESTIMATION AND THE REAL VALE OF THE BACKGROND HARMONIC VOLTAGE VI. CONCLSION Considering the change of the ackground harmonic, this paper proposed a method called partial linear regression model for accurately calculating harmonic responsiility under PQDIF. The main conclusions are as follows: 367

7 () On account of the PQDIF data format harmonic information, this paper proposed a method of computing the phase angle difference asolute value etween harmonic voltage and at PCC; () sing the harmonic voltage, current and the phase angle difference asolute data to construct the partial linear regression model for accurately calculating harmonic responsiility under PQDIF; (3)On account of the partial linear regression model, this paper designed the two-stage method to calculate the uniased estimation of the equivalent harmonic impedance and the ackground harmonic voltage, finally achieved the accurate calculation of harmonic responsiility under PQDIF format. REFERENCES [] Jia Xiufang, Hua Huichun, Cao Dongsheng, et al. Determining harmonic contriutions ased on complex least squares method[j]. Proceedings of the CSEE, 03, 33(4): 49-55(in Chinese). [] Ahmed E E,Xu W. Assessment of potential harmonic prolems for systems with distriuted or random harmonic sources[j]. IEE Proceeding on Generation,Transmission & Distriution,007,(3): [3] Hou Lili. The orientation of harmonic sources indistriution network[d]. Jinan: Shandong niversity,009(in Chinese). [4] Huang Shun. Method research of the orientation and detection of harmonic sources in distriution network[d]. Beijing: North China Electric Power niversity,006(in Chinese). [5] Che Quan, Yang Honggeng. Assessing harmonic emission level ased on roust regression method[j]. Proceedings of the CSEE,004,4(4): 39-4(in Chinese). [6] Zhang Wei,Yang Honggeng. A method for assessing harmonic emission level ased on inary linear regression[j]. Proceedings of the CSEE,004,4(6): 50-53(in Chinese). [7] Huang Shun, Xu Yonghai. Assessing harmonic impedance and the harmonic emission level ased on partial least-squares regression method[j]. Proceedings of the CSEE, 007, 7(): 93-97(in Chinese). [8] Hua Huichun, Jia Xiufang, Cao Dongsheng, et al. Harmonic contriution estimation under power quality data interchange format[j]. Power System Technology, 03, 37(): 30-37(in Chinese). [9] HA Huichun, JIA Xiufang, ZHANG Shaoguang. Neighorhood Multi- Point Measurement Method for Harmonic Contriution Determination[J]. Power System Technology, 04, 38(): (in Chinese). 368