POWER QUALITY ASSESSENT USING TWO STAGE NONLINEAR ESTIATION NUERICAL ALGORITH Vladiir Terzia ABB Gerany vadiir.terzia@de.abb.co Vladiir Stanoevic EPS Yugoslavia vla_sta@hotail.co artin axiini ABB Gerany artin.axiini@de.abb.co Hans-Jürgen Koglin Saarland University Gerany oglin@lev.ee.uni-sb.de Abstract This paper describes a new two stage Newton Type nuerical algorith for power quality assessent in electric power systes with sinusoidal, nonsinusoidal linear and non-sinusoidal nonlinear circuits. In order to estiate current and voltage spectra siultaneously with the fundaental frequency, in the first algorith stage, the Newton Type algorith is applied. In the second algorith stage power quality indices are calculated. Particularly the active, reactive, apparent and distortion powers are easured. As the ain advantage, the technique provides easureents not sensitive to frequency deviations. The algorith perforance is tested under laboratory conditions using voltage and current signals digitised during an AC-otor start. Keywords: power easureent, nonlinear estiation, power quality, laboratory and field testing INTRODUCTION The power quality (PQ assessent is raising a strong interest in the industry as deregulation sets tighter utility copetition for custoer []. Several instruents are developed in the past to easure soe properties of the power quality indicators. On the other hand, a nuber of papers are devoted to the definition of electrical power coponents [-]. Soe at the aret available devices are used for autoated PQ assessent [6]. In this paper a new two stage Newton Type nuerical algorith for PQ assessent is described. In the first stage of the algorith, in order to estiate current and voltage spectra, the signals are independent fro each other processed. By using suitable definitions, in the second algorith stage the unnown PQ indices, such as root ean square (RS values, the power factor (PF, the total haronic distortion factor (THD, distortion power etc. are calculated. In an electric power syste an increase or decrease in the power syste frequency occurs due to faults in the power transission syste. Large blocs of load are being connected or disconnected, or large sources of generation are going off-line. Frequency variations are uch ore liely to occur for loads that are supplied by a generator isolated fro the utility syste. Various nuerical algoriths for power easureents showed the sensitivity to frequency variations. A typical exaples are algoriths based on the Fast Fourier Technique (FFT, or algoriths based on the Least Error Squares Technique (LS and the assuption that the syste frequency is in advance nown and constant (0, or Hz. Nuerical algorith presented in this paper considers the syste frequency as an unnown signal-odel paraeter to be estiated. By this the proble of sensitivity to frequency variations is solved. With the introduction of the power frequency in the vector of the unnown odel paraeters, the signal-odel becoes nonlinear and the strategies of nonlinear estiation are used. For the purpose of the voltage and current spectra estiation, the Newton Type Algorith (NTA is developed and successfully ipleented in the PQ assessent. The features of the two stage nonlinear estiation algorith are presented by processing voltages and currents easured at an AC-otor during otor start. These conditions are selected in order to present the ain algorith properties. NTA ALGORITH Let us assue the following observation odel of the input voltage (or current signal, digitised at the easureent device location: v ( t = h(, t + ξ ( t x ( in which v(t is an instantaneous voltage at tie t, ξ ( t a zero ean rando noise, x a suitable paraeter vector h nonlinear function expressed as: and ( h( x, t = V0 + V sin( ω t + ϕ ( = For the generic odel (, a suitable vector of unnown paraeters is given by:
[ V ω, V, K,, ϕ,, ϕ ] T x = (3 0, V K where V 0 is the agnitude of the decaying DC cop o- nent at t = 0, is the highest order of the haronics presented in the signal, ω is the fundaental angular velocity, equal to πf, f being frequency, V is the agnitude of the -th haronic and ϕ is the phase angle of the -th haronics. The equivalent signal odel can be assued for the current signal The adopted signal odel is a highly nonlinear function of the unnown frequency, so the application of nonlinear estiation is required. In coparison to the linear estiation, this is uch ore serious proble. The expected benefit of introducing power syste frequency (e.g. the frequency of the signal processed into the vector of unnown signal paraeters is the algorith insensitivity to frequency changes, both sall and large. Large interconnected electric power systes are norally operated in a such a state that its frequency changes are in the very sall range (+/- 0.0 Hz. At the sae tie the rate of these changes are next to negligible. Contrary, during and after faults in the syste, i.e. during the large power ibalances in the syste, the changes of frequency and its rate of change are bigger and this influences the accuracy of the existing algoriths for power easureent and easureent generally. Frequency changes in the sall power systes not connected to the large interconnections, in the so called sall islands, are ore often and uch bigger. If the input signal is uniforly sapled with the sapling frequency f s and the sapling period T s = /f s, then the value of t at a discrete tie index is given by t = T s and the following discrete representation of the signal odel can be used: v ( t + ξ = h x, ; =,,3, K (4 = 0 = (, t V + V sin ( ω t + ϕ h x ( where all unnown paraeters fro (3 now have a subscript. The nuber of unnowns, i.e. the odel order, is n = +. The odel order can be reduced by using soe siplified odels, i.e. by decreasing the value of. In the siplest case, the odel containing only the fundaental haronic has the order n = 3 and x = [ ω, ϕ ] T V,. Such a odel can be used for the processing of a pure sinusoidal input signals and under nonsinusoidal conditions the strong input signal prefiltering should be required. The odel selection depends on the application, i.e. on the features of the input signal processed. The input signals are sapled during a finite period of tie, called data window. By providing N saples on the data window, a set of N ( N n = + nonlinear equations, given by ( and (, in n unnowns are obtained. The proble is to solve the overdeterined syste of nonlinear equations, i.e. to deterine the unnown odel paraeters. In [7] the Newton Type algorith for the siultaneously estiation of voltage phasors and power syste frequency is presented. It is derived under the assuption that the input voltage is a pure sine wave. In this paper, in which a proble of power easureent is considered, the distortion of the input signals is taen into account, as it was done during the signal odel developent. NTA algorith belongs to non-recursive nonlinear estiators. The ey relation of the NTA algorith is given by [7]: T T ( J J J v h( ˆ t ( xˆ ˆ i + = xi + i i i xi, (6 where i is an iteration index, xˆ is estiated vec- T T tor, ( J # i Ji J i = J is referred to as a left pseudoinverse of Jacobian J i, v is an ( tor, h ( ˆx i, t is an ( i N easureent vec- N vector of nonlinear functions to the data window. Jacobian atrix J is an ( deterined by the assued atheatical odel of the input signal and N is the nuber of saples belonging N n atrix, having as its eleents the partial derivatives of the signal odel (. Let us denote with an arbitrary row of the Jacobian, where [,, 3,, + ] h( x = L (7 = = V 0 ( x h = = V t ω cos h ( x = + = = sin V h ( x ( ω t + ϕ + + = = V cos ϕ ( ω t + ϕ ( ωt + ϕ (8 (9 (0 ( where =, K,. The eleents of Jacobian are calculated in ters of the estiates estiated in the step before, i.e. by processing the data belonging to the preceding data window. The new algorith design procedure requires the appropriate choice of sapling frequency, the length of data window T dw and the initial guess for the vector of unnown paraeters x 0. For exaple, the initial vector x 0 can be calculated by using FFT. In [7] it is shown that the nuber of iterations provided at one data window, i, can be reduced to a single iteration, i.e. to set i =. By this the estiate fro the preceding iteration is used as an input to the next iteration. This siplification significantly reduces CPU requireents, and at the sae tie does not influence the algorith features. It can be
proved that NTA algorith has the second order convergence. This feature allows us to set i =. Fro the frequency estiation point of view, one of the good features of NTA algorith is that it estiates frequency directly, not fro the already estiated paraeters, e.g. as the first derivative of phase angle. By this the algorith sensitivity to rando noise is not so critical, as it is nown that the nuerical differentiation is very sensitive to rando noise and requires additional postfiltering. The presented NTA algorith is adaptive in nature. That allows us to provide high easureent accuracy over a wide range of frequency changes. With the initial guess x 0 correct deterined, the true estiates can be obtained in the frequency range of / to + /. Given a step change of one (or all odel paraeter(s, after a short convergence period t cnv, the true estiates are obtained. Convergence period is approxiately equal to the length of data window, i.e. t T [7]. Since the approach is based on the suitable linearization and on the ordinary Least Error Squares Estiation, it does not require a priori nowledge of the noise statistics. This is iportant, because it is soeties difficult to get a reliable inforation of the noise statistics of the signal processed. cnv 3 TWO STAGE NUERICAL ALGORITH The second stage algorith calculates PQ indicators fro the current/voltage spectra estiated. The basic forulas used by this are given in the next text. Voltage and current root ean square (RS values: f s dw f s Active and reactive powers of the fundaental haronic are expressed as: P = VI cos( ϕ ψ ( Q = V Isin( ϕ ψ (6 Several definitions of power coponents in linear sinusoidal and non-sinusoidal systes [-] are proposed. The average power is expressed as: P av = V I + V I cos ( ϕ ψ 0 0 (7 = where ϕ and ψ are voltage and current -th haronics phase angles. There is not an universal definition for the reactive power. The following Budeanu s reactive power definition [] will be used in this paper: Q Bud = = V I sin ( ϕ ψ Apparent haronic power: (8 S = V RS I RS (9 Budeanu s distortion power: D = S P av Q Bud (0 V RS = V = Total power factor (PF: PF = = V = I I cos I RS = I ( ϕ ψ = V = ( (3 4 DATA ACQUISITION DURING OTOR START Under laboratory conditions the otor voltages and currents are recorded during the otor start test. An AC-otor, coupled to a loaded DC-achine was switched to the power syste at t s =.6 s. In Fig. the scheatic structure of the test-achine unit is given. Total haronic distortion (THD, e.g. for voltage: power syste induction achine coupling d.c. achine load V = THD v = 00 % (4 V = Figure : Scheatic structure of the test-achine unit. The AC-otor, used in the test has a double squirrel cage rotor and is air-cooled. The ain rated data of the otor are given in the Table.
U n I n P n f el 380/0 V. A W 0 Hz cos ϕ n 0.86 n n 470 rp 7 N T n Table : Rated data of the AC-otor under test. The DC-achine fro Figure is separately excited, with P n = W, and variable speed in the range of 0-00 rp. Signals are digitised before and after the beginning of the otor start (t s =.6 s. The total observation tie was t o =. s. Data are sapled with the sapled frequency f Hz. Data window size used in NTA s = algorith was s. In Figures and 3 the recorded current and voltage of one phase are presented. The starting current is obvious. The otor start lasted approxiately t = 0. 3 s. In Figure 3 a slight voltage drop occurred during the otor start. Under these severe circustances PQ indicators are calculated. Fro the aplitude estiation point of view, the fast aplitude changes ae the estiation coplex. Such a fast change is possible to be traced only with the estiators possessing the excellent convergence properties. ESTIATION RESULTS In this Section of the paper the ain results of the application of the two-stage nonlinear estiation nuerical algorith will be given. Firstly, the results obtained directly by the application of NTA algorith are given: current RS values for all three phases (Figure 4, voltage RS values for all three phases (Figure, and voltage frequency (Figure 6. Before the otor start, the input signals are practically zero, presenting a easuring noise (see zero RS estiated values of currents and voltages. Even under these conditions, frequency is estiated, but these values do not have any practical eaning. During the otor start both the current and voltage RS values in all three phases are correctly estiated. During the testing soe independent easuring device for the chec of the results presented was not available. i (A 300 0 00 0 00 0 0-0 -00-0 -00..6.7.8.9...3.4. Figure : otor current before, during and after the start. I RS (A 0 00 80 0..6.7.8.9...3.4. Figure 4: RS values of otor currents. 300 00 0 00 80 u (V 00 0-00 -00 U RS (V 0 00 80 0-300..6.7.8.9...3.4. Figure 3: otor voltage before, during and after the start...6.7.8.9...3.4. Figure : RS values of otor voltages.
f (Hz 6 0 4 In Figure 9 the total power factors per phase are presented. As expected, during the otor start it is essentially larger value than in the steady state. In Figures 0-3 active, reactive, apparent and distortion powers are presented. All powers present the total otor powers (su of the single powers per phase. As expected, coparing to the steady state, during the otor start, the greater values are obtained. The steady state started approxiately at. s. 3 30 0.8..6.7.8.9...3.4. Figure 6: Frequency of the supplying voltage. Based on the estiates obtained in the first algorith stage, other for the PQ assessent iportant indicators are derived. In Figure 7 and 8, THD factors for currents and voltages are presented. As entioned, before the otor start the input signals were practically a noise, so THD factors before the otor start are relatively large values. During the steady state the current THD factor is greater than the voltage THD factor. cos(phi 0.7 0.6 0. 0.4 0.3 0. 0...6.7.8.9...3.4. Figure 9: Total power factors per phase. x 0 4 90. 80 4. 70 4 I THD (% 0 30 0 P tot (W 3. 3.. 0 0. 0..6.7.8.9...3.4. Figure 7: THD values of otor currents...6.7.8.9...3.4. Figure 0: 3-phase average active power. x 0 4 6 U THD (% 0 Q Bud (VAr 4 3 0..6.7.8.9...3.4. Figure 8: THD values of otor voltages...6.7.8.9...3.4. Figure : Boudeanu s reactive power.
S (VA 0 x 04 9 8 7 6 4 3..6.7.8.9...3.4. Figure : 3-phase apparent power. Algorith for the siultaneously frequency and spectru estiation of distorted input signals. The algorith is not sensitive to the changes of the frequency of the input signal. These are typical for power systes during faults and abnoral operations. The algorith is successfully applied in the processing of currents and voltages recorded during start of an AC W otor. During the otor start both the currents and voltages were distorted. Particularly the aplitude of otor current was changing during the transient. The results obtained offered a hope that the algorith presented could be efficiently applied in the praxis. The technique is not liited only to the easureent applications in power syste. It ay be used in designing the algoriths for the applications in other fields of engineering. D (VA 9000 8000 7000 00 000 00 3000 000 000..6.7.8.9...3.4. Figure 3: 3-phase Budeanu s distortion power. 6 CONCLUSION In this paper a two stage nuerical algorith for the estiation of power quality indicators is presented and successfully applied in the test exaple of otor starting. It is based on the application of the Newton Type REFERENCES [] R.C.Dugan, Electrical Power Systes Quality, New Yor, cgraw-hill, 996. [] C.I.Budeanu, Puissances Reactives at Fictives, Institut Roain de l Energie, Publ., Bucharest, Roania, 97. [3] S.Fryze, Active, Reactive and Apparent Power in Non-sinusoidal Systes, Przegled Eletrote. No 7, pp. 93-03, 93. [4] H.L.Kusters, W.J..oore, On the Definition of Reactive Power Under Non-Sinusoidal Conditions, IEEE Trans. on Power Apparatus and Systes, Vol. PAS-99, pp. 84-84, Oct. 980. [] W.Shepherd, P.Zaihani, Suggested Definition of Reactive Power for Non-Sinusoidal Systes, IEE Proceedings, Vol. 9, pp. 36-36, Sept. 97. [6] DFR AssistantT Product Description, Test Laboratories Int. 996. [7] V.Terzia et all, Voltage Phasor and Local Syste Frequency Estiation Using Newton Type Algorith, IEEE Trans. on Power Delivery, Vol. 9, No. 3, pp. 368-374, July 994.