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Simplified model for estimation of lightning induced transient transfer through distribution transformer
Electrical Power and Energy Systems 27 (2005) 241–253 www.elsevier.com/locate/ijepes
Simplified model for estimation of lightning induced transient transfer through distribution transformer M.J. Manyahia,b,*, R. Thottappillilb a Faculty of Electrical and Computer Systems Engineering, University of Dar es Salaam, P.O. Box 35131, Dar es Salaam, Tanzania Division for Electricity and Lightning Research, The Angstrom Laboratory, Uppsala University, Box 539, SE-751 21 Uppsala, Sweden
b
Accepted 25 February 2003
Abstract In this work a simplified procedure for the formulation of distribution transformer model for studying its response to lightning caused transients is presented. Simplification is achieved by the way with which the model formulation is realised. That is, by consolidating various steps for model formulation that is based on terminal measurements of driving point and transfer short circuit admittance parameters. Sequence of steps in the model formulation procedure begins with the determination of nodal admittance matrix of the transformer by network analyser measurements at the transformer terminals. Thereafter, the elements of nodal admittance matrix are simultaneously approximated in the form of rational functions consisting of real as well as complex conjugate poles and zeros, for realisation of admittance functions in the form of RLCG networks. Finally, the equivalent terminal model of the transformer is created as a p-network consisting of the above RLCG networks for each of its branches. The model can be used in electromagnetic transient or circuit simulation programs in either time or frequency domain for estimating the transfer of common mode transients, such as that caused by lightning, across distribution class transformer. The validity of the model is verified by comparing the model predictions with experimentally measured outputs for different types of common-mode surge waveform as inputs, including a chopped waveform that simulate the operation of surge arresters. Besides it has been verified that the directly measured admittance functions by the network analyser closely matches the derived admittance functions from the time domain impulse measurements up to 3 MHz, higher than achieved in previous models, which improves the resulting model capability of simulating fast transients. The model can be used in power quality studies, to estimate the transient voltages appearing at the low voltage customer installation due to the induced lightning surges on the high voltage side of the transformer. The procedure is general enough to be adapted for any two-port devices that behaves linearly in the frequency range of interest. q 2004 Elsevier Ltd. All rights reserved. Keywords: Distribution transformer model; Lightning induced transients; Admittance functions
1. Introduction Despite improvements in transient over-voltage protection over the years [1], lightning has continued to be a major cause of outage and equipment damage on low voltage distribution networks. Most modern electronics equipment/ systems found in consumer’s installation today are sensitive to the effects of transient over-voltages. Although, it is well known that there are various ways by which lightning as * Corresponding author. Address: Faculty of Electrical and Computer Systems Engineering, University of Dar es Salaam, P.O. Box 35131, Dar es Salaam, Tanzania. E-mail addresses: [email protected] (M.J. Manyahi), [email protected] (R. Thottappillil). 0142-0615/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2003.02.001
a source of transient can couple to electrical networks and induce transient over-voltages, the growing concern is how these voltage surges and transients are reaching consumer installations. Power distribution transformer is an important component of the distribution network by providing ultimate connection of low voltage (LV) customer installations to the medium voltage (MV) distribution network. Besides its role of providing isolation between the two circuits operating at different voltage levels, it also acts as a gateway for conveying the unwanted transient over-voltages in the MV network to the LV customers and vice versa. Consumer equipment that is connected to low voltage power network may have no direct exposure to lightning, however, lightning induced transients can be transferred to this
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equipment as a result of strikes on exposed lines on the primary side of the transformer. Even though transient protectors are incorporated on the primary side of the transformer, the transferred residual transients to secondary side may be a threat to connected equipment, a fact that prompts for a need of advanced understanding of transients propagation through distribution transformers. Even though efforts to understand the transfer of transients across the distribution transformer have been carried out for many years, it is widely accepted that, a comprehensive and generally acceptable transformer model for transient simulation does not exist. The reason behind this is that transformers come in different design and configurations, and with different parameters as well as different installation practices. There are two different approaches for modelling transformers. The first one, based on the detailed electromagnetic modelling of the windings and the core [2,3] are required for understanding the influence of transformer design parameters on the transient response of the transformer. However, such detailed models are difficult to use if the primary aim is to predict the transfer of lightning transients across transformer windings in an existing distribution network. The second approach based on modelling the transformer using the voltage and current measurements at its terminals [4–6] is ideally suited for predicting the transients arriving at the consumer installation in the existing distribution network, even though it may not give as much insight into the influence of the design parameters on the transient response. Various transformer terminal models for high frequency or transient studies have been proposed in recent years [4–7], most of them for power transformers. Formulation of these models has been based on building up of equivalent linear network, from approximations of impedance or admittance function frequency responses. The principle differences between these models are on how the admittance functions are obtained and the specific methods used to approximate these frequency response functions to form the equivalent network of linear components. Different methods are used for obtaining the external terminal admittance functions, namely, sequence admittance measurement [5], impulse response measurement [4], and measurements at low frequencies with subsequent extrapolation to higher frequencies [6]. In spite of their differences in their formulation procedure, majority of these terminal models has been devoted for studying transients in power transformers, not in distribution transformers. Also, the models are bandwidth limited (upper frequency of several kilohertz to maximum 1 MHz), a limitation that makes them unsuitable for faithful simulation of the transient response to steep fronted or chopped pulses, which are crucial in studying response to lightning caused transients. On other hand, when distribution transformers have been covered, most of their models and transient response studies have been directed towards the main objective of protecting transformers from
damage [7–9]. Recently, studies are carried out on distribution transformer transient responses in view of their failures due to lightning induced transients in secondary (LV installations) [1,22]. Apparently for models derived by fitting admittance functions, accurate approximations (fitting) over a frequency range of interest and the choice of network representation of the model plays an important criterion for better numerical simulation and stability of the model. Fitting technique employed in some of these models [5] for approximation of rational functions used in realisation of the elements of model networks, involves many steps of mathematical optimisation performed by portioning the admittance function over various frequency ranges. Piecewise fitting have major influences in the portions being fitted, and is one of the reasons for reduction of the fitting accuracy and hence the accuracy of the resulting model. Nearby lightning strikes in transmission system may not be of concern to components operating at that level of voltage (200 kV and above), in contrast to switching transients (differential mode) generated in GIS (Gas insulated systems) in the same system. Meanwhile in distribution networks, both direct and indirect lightning strikes are of much concern to network components, and respective connected consumer’s installations. The problem of lightning caused transients becomes more pronounced in distribution network when overhead lines that are very long are used throughout the urban and rural areas or in both MV and LV network due to increased exposure to lightning strikes. In this work a simplified procedure for the formulation of distribution transformer model is presented. The paper is organised as follows. Section 2 describes the assumptions, along with its justification, used in the modelling. The model that has been implemented in this study is based on admittance network representation proposed in [5]. Section 3 describes the measurements performed on the transformer. Frequency domain measurements are undertaken with the use of Network Analyser method adopted by Caldecott et al. [10]. In their setup they made use of Network Analyser HP3577A along with a shunt resistor for measurement of current, which is essential for the computation of power transformer impedance. However, in this work the admittance functions are directly measured with a configuration consisting of Network Analyser HP4195A, Pearson probe 2877 for current measurement, and computer interlinked to Network Analyser through GPIB card for measured parameter storage. Acquired admittance frequency responses were thereafter, verified by comparing with corresponding responses that are derived from time domain (impulse response) measurements. Section 4 describes approximation (fitting) of the measured admittance function, and computation of equivalent RLCG network used in model formulation. Computed RLCG network responses are verified by comparing measured admittance function. An outline of computer program, which is used in this work for
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performing various steps of model formulation, is presented. Section 5 provides comparisons of model predictions with experimentally measured responses, for different stimulus surge applications. Chopped pulses and standard lightning impulse voltage have been used to verify the model response to transients that are encountered in practise. Section 6 summarises the work pointing out the benefits of the model and drawbacks on the use of the model.
2. Transformers in power distribution network 2.1. Dominance of indirect strikes In most cases the three-phase MV overhead line conductors are either spaced in horizontal or triangular construction, with or without a grounded shielding wire placed at the pole-top. The clearance distances of line conductors to ground and phase conductor spacing, besides other influencing factors, may depend mainly on the level of operating voltage. Due to their low height and presence of other tall objects within their proximity, distribution lines do not attract many direct lightning strikes. On the other hand, indirect strikes are considered to be more frequent and are thought to be capable of inducing transient over-voltages of up to 300 kV [11]. 2.2. Common-mode versus differential mode Generally, the lateral distance (for strikes in proximity of line) is large compared to the distance of separation between the phase conductors. Therefore, the phase conductors can be assumed to be at equidistant from the point of strike, without any appreciable error. This assumption is realistic for overhead distribution lines, which are of concern/interest to this work, because the ratio of phase conductor separation to the height above ground level is smaller compared to that of HV transmission lines. In a study by Ericksson et al. [11] on 9.9-km experimental MV overhead line, it was found that waveshapes of induced surges are identical on all the three phases. In their recent work, Rachidi et al. [12] have confirmed that the induced voltages are practically proportional to the conductors height above ground level. In particular reference to horizontally spaced conductors, they have shown that almost same voltage is induced on each of the three-phase conductors, and the effect of overhead grounded wire is to reduce the induced voltages on phase conductors. With above review, it is reasonable to assume that the induced surges in primary distribution lines have the same magnitude and waveshape for all the three-phase conductors (Common Mode, CM). This generalized assumption holds, because even if the conductors are not arranged in horizontal configuration, i.e. in equilateral triangle or vertical configuration, their vertical separation is very small compared to the pole height. In spite of the generalized assumption of common mode induced voltages, it is possible also that during unbalanced operation of
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protective devices (arresters), a transient voltage between phase conductors may be produced (Differential Mode, DM). Therefore, the lightning caused transients are predominantly CM even though small DM transients may appear at the transformer terminals. 2.3. Frequency content of lightning induced transient Lightning caused transient voltages are rarely having standard waveshape [13], they can have very fast components depending on the ground conductivity, connected loads, and reflections from terminations, etc. [14,15], and hence the need for consideration of higher frequencies in modelling network components. This has been confirmed by measurements at distribution arresters performed by Barker et al. [16] and Fernandez et al. [17]. They recorded lightning caused surges that were quite steep, with rise times of first stroke caused surges in the order of few microseconds, while, for subsequent strokes the surges had rise times that are less than 0.5 ms. From the rise times of lightning caused transient overvoltages [14–17], it can be safely assumed that the upper frequency content of these transients are several megahertz. In transfer of transients across transformers, high frequency range is very important because they are less affected by the nominal turns ratio of the transformer. The transformer model developed in this paper is tested for its accuracy up to 3 MHz and hence well suited to evaluate conducted transients arriving at consumer installations in a distribution network. 2.4. Linearity of transformer at high frequencies Distribution transformers are characterised by high turns ratio between primary and secondary winding, and have high ratio between the basic insulation level and the rated voltage of the low voltage winding [18]. The impedance of high voltage winding is higher than that of corresponding low voltage winding. It has been shown experimentally [4] that transformers behave linearly above 1 kHz and iron core does not play any significant role in the voltage and current transfer at high frequencies. Woivre et al. [7] found that transformer windings behave as linear system (without saturation effects) for lightning caused over-voltages. Exploiting the linearity property of transformers at high frequencies, we can write general two-port nodal admittance equations that relates the voltages and currents at the accessible transformer terminals as " # " #" # I1 V1 Y11 Y12 Z (1) I2 Y21 Y22 V2 2
Iia
3
6 7 ½Ii Z 4 Iib 5 Iic
(2.1)
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2
Via
3
6 7 ½Vi Z 4 Vib 5
(2.2)
Vic 2
3
Yij;aa
Yij;ab
Yij;ac
6 ½Yij Z 4 Yij;ba
Yij;bb
Yij;bc 7 5
Yij;ca
Yij;cb
Yij;cc
(2.3)
where [Ii], [Vi] and [Yij] are sub-matrix 3!1 and 3!3, respectively, relating to three-phase transformer terminals.
3. Experiment for parameter measurement The elements of the nodal admittance matrix given in Section 2.4 may be obtained from short-circuit tests at the terminals of distribution transformer. All terminals except one that is connected to test signal source are grounded. Currents are measured in all terminals along with the stimulus voltage in turns or all measurements are done at once depending on channels available for oscilloscopes or data acquisition system. These terminal measurements of admittance parameter may be performed in either frequency or time domain. In the time domain method, impulse voltage and current waveforms at the transformer terminals are measured, and the corresponding frequency domain admittance function is computed using Fourier transform. The frequency domain measurements of admittance function is realised by utilising a tracking generator, which sweeps the defined frequency range, and at sampled points within the range, complex admittance is computed. Since, all the admittance function measurements are carried out at reduced voltage levels (nondestructive voltage levels); the corresponding measured currents are comparatively low. It is important, therefore, that true currents are measured rather than the true voltages, by inserting current probe on the load side and voltage probe on the source side. This setup is of special importance in eliminating recording equipment loading effects, especially when performing the measurements of driving point impedance. Accurate and easy determination of admittance functions, whether it is obtained from time domain derivation or direct frequency domain measurements, are important factors for accurate and simple model formulation. The distribution transformer used in this work is a three-phase, 11/0.4 kV, 50 kVA, with Dzn10 winding connections.
condition for impulse response (time domain) measurement is that the impulse source should produce impulses that are steep enough to contain sufficient high frequencies that will excite the transformer resonances adequately in relevant frequency range. It is essential to have a well-designed impulse source that is capable of providing same test impulse voltages and waveshape within small tolerances for various transformer winding connection and different admittance parameter measurements. Moreover, due to wide difference in primary and secondary circuit impedances, it is sometimes necessary to design impulse source circuits for tests carried on each transformer circuit (i.e. primary and secondary). However, if the measurements have to be carried on a number of distribution transformers of different sizes and rating, it will be difficult to modify impulse source circuit for each measuring case by changing component values. Impulse response measurements have been preferred from the early days [19] because it has been easier to construct impulse sources for variety of voltage levels, and more importantly the generated impulses replicates the transient nature of surges encountered in practice. Furthermore, an impulse under consideration contains all frequency spectra, with the highest frequency content determined by its rise time or steepness. The accuracy of impulse waveform recordings will depend on the settings and bandwidth of the equipment used, i.e. storage oscilloscopes, voltage and current probes. Fig. 1 shows the setup for impulse response measurement of distribution transformer voltage and currents used for deriving admittance functions. The impulse generator used in this setup was designed inhouse and is capable of providing a non-standard open circuit impulse of 0.4/20 ms with highest value of peak voltage up to 2.5 kV. Lecroy oscilloscope LC540, with corresponding high voltage probe and Pearson current probes 2877, were the equipment used for the setup. MATLAB routines were compiled for the purpose of performing FFT function of acquired waveforms and hence compute the respective admittance function. The need for impulses that have higher rise time (for high frequency responses), which have substantial long tail (decay to zero), would require higher sampling rate on the recording oscilloscope, with a requirement of long
3.1. Time domain parameter measurement As mentioned earlier, time domain measurements has been performed in this work for the purpose of obtaining parameters for computation of admittance functions, as a comparison reference to corresponding frequency domain measured functions, to be described later. The necessary
Fig. 1. Experimental setup for measuring impulse response voltage and current values, for computation of driving point and mutual admittance functions.
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recording length for a given sweep, in order for the waveforms to be correctly acquired. The requisites for higher sampling rate with long records have a drawback of forfeiting the waveform smoothness due to oversampling noise on the impulse tail. Opting for short record length would result to uncompleted recording of voltage and current waveforms, i.e. they will not have sufficiently decayed to zero, which will consequently cause false interpretation on performing FFT. Accordingly, windowing function need to be applied to last part of the tail or acquired waveforms to ascertain smooth transition of the waveform tail to zero at record end. Therefore, the requirements of broad bandwidth impulses, accurate parameter recordings, efficient waveform processing and computation of admittance functions are the essential features in time domain measurements. Care should be taken to avoid noise contamination to the measuring signal caused by the use of unshielded cables, which will inevitably lead to errors in magnitude and phase of the resulting admittance function. This, however, will depend very much on the level or magnitude of test voltage and the impedance of the system. 3.2. Frequency domain parameter measurement Frequency domain parameter measurement is achieved basically by using a variable frequency signal source (which replaces an impulse source in time domain), accompanied with the use of voltage and current probes. The frequency source generates signals at various frequencies within the bandwidth of interest, and at each frequency, admittance value is computed from measured respective values of voltage and current. In this work, Network analyser has been used in transmission parameter measurement mode. The parameter measured is a complex transfer coefficient T, which is defined as the ratio of the transmitted voltage to the incident voltage expressed as: TZ
Vtransmitted Z t:f Vincident
(3)
The signal source terminal S is connected directly with the reference terminal R (incident terminal) and thereafter connected to ungrounded transformer terminal. Transmission terminal T on the Analyser is connected to current probe, which has an output ratio of 1.0 V/A, which is a mirror conversion of current to transmitted voltage. Fig. 2 illustrates the use of Network Analyser in direct measurement of distribution transformer admittance functions. With its internal operating software the network analyser computes the complex transfer coefficient, which is equivalent to admittance function derived from measured transmitted voltage that is proportional to current, and the incident voltage, which is equal to the source voltage. The analyser has a signal source (tracking generator), that sweeps a broadband of frequencies, and at sampled frequencies measurement is performed with subsequent
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Fig. 2. Experimental setup for measuring admittance frequency responses in transmission parameter mode of the network analyser.
display of complete amplitude and phase. Coaxial cables are essential for connecting the device under test to Network analysers, which is a 50 U terminated equipment at all its terminals. However, due to the nature of connection to transformer terminals, with a need of placing in a measuring current probe, short length of unshielded connecting leads cannot be avoided. These leads may have a significant influence at high frequencies. In order to reduce the cable influences on all measured parameters, same set of cables of less than 1 m in length were calibrated with the instrument before commencing measurements, and used in various connection setup of equipment under test. Major drawback with the use of Network Analyser (in particular one used in this work) is in short record length of measured parameter. The maximum number of samples over frequency bandwidth of interest is 401 samples, with an option of spreading the sampled points linearly or logarithmically. Therefore, broad bandwidth measurement of parameter will result to poor sampling and hence inaccurate measurement of admittance function frequency response. Additional drawback on measuring admittance function is that its source gives an output test signal with a maximum value of 15 dbmV at 50 U, which limits some measurement of high impedances, although, this problem has not been experienced in measurements of this work. 3.3. Comparison of time and frequency domain parameter measurement In both measuring schemes, Pearson current probes 2877 have been used. For the purpose of substantiating that the current probe has the same influence (identical attenuation) over all frequencies within the bandwidth of interest, probe frequency response was tested to verify their constant amplitude and linear phase behaviour over the bandwidth. Fig. 3 shows the measured current probe amplitude and phase frequency response over a frequency range of 200 Hz– 10 MHz. The current probe frequency responses in Fig. 3 are significantly flat over the measuring bandwidth of interest in this work, i.e. 1 kHz–3 MHz. Conversion factor of the probe is unity over the bandwidth and there is no phase angle shift,
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Fig. 3. Magnitude and phase frequency response of Pearson current probe 2877.
i.e. phase angle is zero over the range. Even at 200 Hz the magnitude response drops by 6% only. Since both admittance-measuring schemes have some advantage and disadvantages, their comparison on suitability of use depends on practical aspect of using the method on field, namely, accuracy and easiness with which the scheme can be applied for complete measurement of parameters. Figs. 4 and 5 show the comparison of mutual and driving point (self) admittance functions obtained from frequency domain measurements and that computed from time domain acquisitions. There is a good correlation in the admittance functions obtained from two measuring schemes, in respect of magnitude and phase angle, with only small discrepancies
Fig. 4. Comparison of frequency domain and time domain derived frequency response of mutual admittance function Y21 between high and low voltage winding.
Fig. 5. Comparison of frequency domain and time domain derived frequency response of low voltage winding self-admittance function Y22.
in phase angles at higher frequencies. However, due to limitation of our signal source for time domain measurements we could not realise impulses that have broad frequency bandwidth considered in this work, i.e. 1 kHz– 3 MHz, for parameter measurement on low voltage winding. That can be noted from Fig. 5, where the bandwidth of computed admittance functions is 40 kHz–4.5 MHz. As mentioned earlier, the signal source for time domain measurements need to be of very good design, so that it is capable of providing same test impulse voltages and waveshape for various transformer winding (high and low voltage) terminal measurements.
4. Model setup Each admittance function is measured or determined independently of other functions in the case of network analyser method because only one channel can be used at one time for that purpose. However, with time domain derived admittance functions a number of these can be obtained in single test depending on number of available channels on the recording devices, e.g. oscilloscopes, and also on the number of admittance function that need to be determined by particular terminal source injection. Regardless of whether these admittance functions are measured/obtained separately or collectively at once, they are elements of admittance matrix of object to be modelled. This admittance matrix (Eq. (1)) defines the frequency response behaviour of the modelled transformer revealed at its terminals. When building up the network model out of these measured functions by fitting, it is important that these elements of the matrix are fitted together to enhance the coupling between them. According to the properties of linear systems admittance matrices, the elements in the rows and columns elements are interrelated about the leading
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function Yfit Yfit Z
m X iZ1
n X ci ci ci C C C se C d s K ai iZmC1 s K ai s K ai (5)
We can write the admittance of the network above in the form of the fitted function as: Fig. 6. The p-network equivalent model for two winding transformer consisting of RLCG components.
diagonal. For this reason, even though it may be difficult to fit all elements of the matrix at once, we can undertake fitting of related elements by row or column. In this work, the elements of the nodal admittance matrix are approximated with rational functions consisting of real as well as complex conjugate poles and zeros, by spavectfit a matrix element-fitting version of vectfit [20] fitting technique. These numerical approximations are realised in the form of an RLCG network (Fig. 7) that can be used with ATP, as an element of the p-equivalent model network of Fig. 6, which each of its elements comprises of one such RLCG network. The reasons as to why this network is appropriate for high frequency model for transformers are that it comprises of mixed assemblage of inductance and capacitance that represents the general constructional properties of transformer windings. The inductive elements of the transformer network model are important at low frequencies, whereas the capacitive elements dominate at high frequency. Vectfit approximates a frequency response Y(s) with rational functions, expressed in the form of a sum of partial fractions Yfit ðsÞ z
n X iZ1
ci C d C se s K ai
(4)
where the terms d and e are real constants; while ci and ai are the residues and poles, respectively, which may either be real quantities or complex conjugate pairs. The fitted admittance function can be written in the form such that real residue and poles are separated from that of complex conjugate residue and pole function, so that it is easier to realise RLCG network (Fig. 7) from the fitted
Fig. 7. Structure of RLCG network used in this study to represent each element of p-equivalent model.
YðsÞ Z YCG ðsÞ C YRL ðsÞ C YRLCG ðsÞ
(5.1)
For G0 and C0 branches: YCG ðsÞ Z G0 C sC0 Z Y0 ðsÞ
(5.2)
For series Rs and Ls branches: YRL ðsÞ Z
m X iZ1
ki s K pR i
(5.3)
For RmLmCmGm branches: ( ) n X bi bi C YRLCG ðsÞ Z s K pCi s K pCi iZmC1
(5.4)
The combined network is: YðsÞ Z Y0 ðsÞ C
m X iZ1
n X ki C s K pRi iZmC1
(
bi bi C s K pCi s K pCi
)
(5.5) In the above equations, G0, C0, and ki are real constants; PRi are real poles; PCi and bi are complex poles and residues, and pCi and bi , are their respective poles and residue complex conjugates. For our case m is the number of real poles, while (nKm) is the number of complex conjugate poles, in which case m also represent the number of RL series branches and (nKm) is the number of RLCG branches. These rational functions (Eqs. (5.1)–(5.5)) are used to evaluate the R, L, C and G components from fitted admittance functions, required for each RLCG module network. The approximation is achieved by replacing an initial set of poles with an improved set of poles using a pole relocation method based on least squares approximation of linear problems [20], and the order of the approximation is equal to the number of starting poles, i.e. initial number of poles. Vectfit performs the actual fitting in rectangular (complex) function and approximation (fitting) results are displayed in polar function along with the measured admittance functions. Examples of the approximated admittance (fitted) function superimposed on the measured functions are shown in Fig. 8(a) and (b). The measured functions Y21 and Y22 are the same as that shown earlier in Figs. 4 and 5. Fitting had been performed with smaller bandwidth (1 kHz–1.5 MHz) compared to measured frequency response bandwidth (1 kHz–3 MHz) for the purpose of avoiding passivity problems [21], which manifests itself in
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Fig. 8. (a) Fitted magnitudes of measured self and mutual admittance functions. (b) Fitted phase angle of measured self and mutual admittance functions. Deep black is measured function and grey print is approximated (fitted) function.
time domain simulations. Fitting with smaller bandwidth has been opted instead of performing complicated passivity enforcement procedures. However, in the final analysis the fitted function was verified by checking its ability to predict frequency responses of full measured bandwidth. Approximation (fitting) of the admittance functions is of exemplary quality (match) such that in most parts of the function it is difficult to distinguish between the measured and fitted functions, this is evident in the fitting of Y22, where the fitted function exactly overlays that of measured function. Appreciable good fit has been obtained in both magnitude and phase of the measured admittance frequency responses, all major resonances have been correctly approximated. There is small discrepancy at low frequencies; however, this may have been introduced by measurement errors at these frequencies as discussed in previous sections or fitting associated errors. Besides these small
discrepancies the general combined approximation of all admittance functions for the model is adequately good. A supplementary software routine has been compiled in MATLAB within the vectfit-working environment, for the purpose of evaluating the R, L, C and G components, on completion of the fitting procedure for each RLCG network. The derivation of these rational functions and hence the computation functions for RLCG components from fitted admittance function has been undertaken in [23], and a summary is given in Appendix A. Since there is no unique solution on fitting the functions with varying number of poles and number of iterations, a best fit out of a number of good fits may be selected based on the frequency response of the resulting RLCG network. It is important and realistic to check the response of approximated RLCG network, because after all the fitting process is meant for generating RLCG network and hence the model that can effectively be used in time domain simulations. Therefore, a fit that derives an RLCG network, which reproduces adequately the frequency response of measured or fitted function, is expected to work effectively in a combined model network. Fig. 9 shows the comparison of measured admittance function to the frequency response of RLCG network that has been derived from the fitted function. Frequency response of derived RLCG network is comparable to the fitted admittance functions and sufficiently predicts the responses beyond the maximum fitting frequency of 1.5 MHz (compare with measured admittance functions in Figs. 4 and 5). We have compiled a computer program in MATLAB such that the formulation of the distribution transformer model for lightning transient transfer simulation is realised easily, after the necessary admittance frequency response have been measured. The computer flow chart in Fig. 10
Fig. 9. Comparison of fitted admittance function (Fig. 8(a) and (b)) and response of derived RLCG network. Deep black is fitted admittance function and grey print is RLCG network response.
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Step 5. A MATLAB routine is prepared to organise the RLCG network elements in ATP linear network (creation of local nodes) file format. Each RLCG network so created for ATP is given a filename that reflects the element it represents in the transformer model (Fig. 6). Step 6. Steps 2–5 are repeated until all columns of admittance matrix have been fitted and respective RLCG network is computed. Step 7. MATLAB program is prepared to compile RLCG network files into a model file (creation of global nodes and simulation parameters), which will be used for simulation in ATP. Step 8. ATP program uses file created in step 7 to perform the simulations. In some steps interaction of the user is required whenever prompted to enter node numbers, file names, etc.
5. Comparison of measured and simulated results
Fig. 10. Computer flow chart for formulation of transformer model.
shows various stages of the program where in some stages interaction of the user will be required, especially in verification of fitting the functions and checking if the derived RLCG network reproduces the measured admittance functions. Descriptions of the flow chart steps are as follows: Step 1. Short circuit driving point and mutual admittance functions, Y11, Y22, Y21 or Y12 are measured/obtained in frequency domain by Network Analyser, e.g. Figs. 4 and 5. Step 2. Acquired admittance functions are assigned with code names with reference to the element it represents in admittance matrix. The elements are fitted together by arranging them in column using vectfit, which works in MATLAB environment, e.g. Fig. 8(a) and (b). Step 3. Fitted functions are computed with MATLAB compiled routine to equivalent RLCG network. Step 4. The computed RLCG network frequency response is compared with measured admittance responses with help of a compiled MATLAB routine, e.g. Fig. 9.
The nature of transient coupling between the two transformer circuits depends on type of winding connection [23] and the transformer grounding system used. As aforementioned, most of the lightning related perturbations on distribution lines are caused by indirect lightning strikes. Sometimes, these indirect strikes induce transient voltages that are below the protectors threshold value, but has sufficient rise times to excite transformer resonances. Figs. 11 and 12 show the comparison of experimental and simulation results of transferred transients from primary to secondary circuits, with consideration of rise times of lightning surges that are expected in practise.
Fig. 11. Comparison of experimental and simulated results from same signal excitation.
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From measured and simulated results we note that the maximum magnitude (peak) of the transferred transients to LV network from HV terminals increases with an increase in the rate of rise (steepness) of stimulus surges. It is revealed in Fig. 12 that steep preliminary pulses can be transferred through transformer to low voltage installation with less attenuation. Fast rising pulses have more influence on the transferred voltage than the corresponding low rising pulses. Generally, the model reproduces sufficiently the oscillatory frequency of the transferred surges. The model shows that it is capable of reproducing the transient transfer behaviour of distribution transformer due to various lightning transient rise times (wide frequency content) coupling to MV distribution lines.
6. Conclusion
Fig. 12. (a) Experimentally measured transferred transient to LV circuit due to steep surges on HV terminal. (b) Model simulation result of transferred transient to LV circuit due to chopped surges application on HV circuit.
The results of simulation (Fig. 11) show that the model is predicting the experimentally measured transfer voltages. In order to make realistic comparison of the two transferred voltages (experimental and simulation), the input voltage for simulation was obtained by sampling the experimental input waveform, and applying linear interpolation in synchronisation to time intervals used in ATP simulation. The rise time of the excitation signal is about 1.2 ms, and Fig. 11 shows excellent match and reproduction of experimental results in terms of magnitude and time constants. In most cases, distribution transformers have transient protectors, commonly installed on its primary terminals. Operation action of these arresters sometimes results in chopped surges, an action which causes the generation of steep tail similar to steep fronts that represents transients of higher frequencies. In Fig. 12(a) and (b), comparison is made between experimental and model simulation results on the transferred voltage to LV network due to an arbitrary steep surge application on primary terminals.
The simplified formulation of model for estimating transferred lightning transient from the more exposed MV lines to customer installation has been presented. Comparison of admittance functions for model formulation derived from traditional procedure of impulse response to those obtained in frequency domain has been presented. Advantages and disadvantages of both methods have been discussed; however, in final analysis the advantages of measuring admittance function in frequency domain opted in this work outbalances disadvantages. Main consideration is on its simplicity, less time and easiness with which measurement of parameters may be carried out on field. Furthermore, the need for computational convolutions (time to frequency domain) and associated errors on filtering are avoided with realisation of less noise in measured admittance functions. The network analyser which was previously used in measurement of frequency response driving point (self) admittance function has been effectively used in measuring mutual admittances as well in this work. We have shown that even though the network analyser acquires the admittance functions at low voltage signals, the obtained functions are comparable to those derived from impulse response measurements at higher voltage signals. The use of Network Analyser has made possible for improvement and extension of the frequency bandwidth of the resulting distribution transformer model up to 3 MHz. This simple model is a linear network model suitable for use in estimating transients that are in the frequency range where the transformer behaves linearly. The simplified model reproduces satisfactorily the experimental measured transferred transients in all lightning transient frequency spectra expected in practise. Non-linear behaviour cannot be studied with this model and therefore, it is not a stand-alone model, but can be used as add on model to existing ATP low frequency models. Although, various loading conditions of distribution transformer has not been considered in this work, the extreme case of lightly loaded is considered, in view of the worst case the customer network will be exposed to.
M.J. Manyahi, R. Thottappillil / Electrical Power and Energy Systems 27 (2005) 241–253
This work has not dealt with lightning caused differential mode transfer of transients, i.e. transients that appear at HV terminals due to unsynchronised operations of surge protectors, and hence transferred to LV circuits. However, in our work to be submitted soon we are addressing the problem of arrester installation practise and the generated differential mode transients and their transfer through transformer windings. Finally, we have achieved further model formulation simplification by a compiled program in MATLAB, which on entering admittance data carries out fitting, computation of equivalent network and formulation of the model for direct use in time domain simulation (ATP) and hence the simulations. Acknowledgements The authors would like to acknowledge valuable advises of Bjo¨rn Gustavsen and for providing the vectfit program used in this work. We greatly appreciate the financial support of Swedish International Development Agency (SIDA) on this work and support of Mr M.J. Manyahi’s PhD work at Uppsala University.
251
For the computation of component values of admittance function, above equations need to be written in the form of the fitted admittance function: i:e: Yfit Z
n X iZ1
ci C se C d s K ai
(A5)
The resulting fitted function equation shows that there are multiple branches of RL and RLCG branches with a single CG branch, with the terms e and d as constant real terms, while ci and ai are the real or complex residue and poles, respectively. Therefore, the fitted admittance function expression can be expanded such that real residue and poles are separated from that of complex conjugate residue and pole function of Eq. (A5)
m n X X ci ci c Yfit Z C C i C se C d s K ai iZmC1 s K ai s K ai iZ1 (A6) Comparing the last two terms of Eq. (A6) with Eq. (A2) Y0 Z sC0 C G0 hes C d Therefore C0 Z e and G0 Z d
Appendix A
For multi-RL branches (A3) becomes
The description below elaborates on the procedure that have been derived for determining values of the individual components of RLCG module directly from admittance function fitted by vectfit. A MATLAB program is compiled based on these expressions such that at the end of vectfit fitting, all values of individual components of respective admittance function will be available. Total admittance of RLCG network module is the sum of the branch admittances: Y Z Y0 C
m X
YSi C
iZ1
n X
YMi
(A1)
Y0 Z sC0 C G0
(A2)
The low frequency branches of inductance and resistance are represented by 1 sLSi C RSi
sCMi C GMi Z ðsLMi C RMi ÞðsCMi C GMi Þ C 1
m X iZ1
1=LSi s C RSi =LSi
(A8)
where m is the number of RL branches. Comparing (A8) with first term of (A6) YS Z
1=LSi c h i s C RSi =LSi s K ai
such that LSi Z
1 ci
ðci realÞ
(A9)
and RS i Z K
ai ci
ðci and ai are realÞ
(A10)
Similarly, Eq. (A4) can be factorized to produce conjugate complex functions G
YMi Z
s2 C s
s L1M C CM MLiM i i i RMi R G C LM C CM1LM C CMMi LMMi
GMi CM i
i
i
i
i
(A11)
i
(A3)
The medium frequency resonant RLCG branches are represented by: YMi
YS Z
iZmC1
Since RLCG network module may have multiple branches for YS as well as YM. The high frequency branch of capacitance and conductance is represented by:
YSi Z
(A7)
let 2a Z
GMi RMi C CMi LMi
and g2 Z
RM G M 1 C i i CMi LMi CMi LMi
G
(A4)
YMi Z
s L1M C CM MLiM i
i
i
s2 C 2as C g2
(A12)
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let g2 K a2 Z u2
(A13)
It follows that for complex real and residue values (ai and ci)
G
YMi Z
s L1M C CM MLiM i
i
C Mi Z
i
ðs C aÞ2 C u2
ci C ci Kðc aiCci ai Þ Kðci aiCci ai Þ u2 C a2 K 2a K iciCc ciCc i
s Z
1 LMi
G C CM MLiM i i
½s K ðKa K juÞ½s K ðKa C juÞ
(A20) (A14)
rewriting middle part of (A6) in partial fractions form of (A14) such that: bR K jbi bR C jbi C ½s K ðKa K juÞ ½s K ðKa C juÞ
YM Z
(A15)
Hence ai ZKaK ju and ci Z bR K jbi Equating the numerators of Eq. (A14) to that of (A15), since both have the same denominator s
G Mi 1 C hci ðs K ai Þ C ci ðs K ai Þ LMi CMi LMi
hence 1 Z ci C ci LM i LMi Z
1 ci C ci
and finally G Mi Z
GMi CMi CMi
G Mi Z
Kðci ai C ci ai Þ Kðc aiCci ai Þ Kðci aiCci ai Þ u2 C a2 K 2a K iciCc ciCc i
i
(A21) Component values of admittance module can be calculated from the fitted function by Eq. (A7) for capacitance and conductance branch from constant terms, and Eqs. (A9) and (A10) for series inductance and resistance branch from real values of pole and residues. Finally, Eqs. (A16), (A19)–(A21) for RLCG branch computed from complex values of poles and residues.
(A16) References
and GMi Z Kðci ai C ci ai Þ CMi LMi
(A17)
GMi Kðci ai C ci ai Þ Z ci C ci CMi
(A18)
but 2a Z
GMi RMi C Z Re½aj CMi LMi
therefore GM RMi Z 2a K i LMi CMi RM i
i
Kðci ai C ci ai Þ 1 Z 2a K ci C ci ci C ci
(A19)
from Eq. (A13) g 2 Z u2 C a 2 Z
RM GM 1 C i i CMi LMi CMi LMi
1 Z u2 C a 2 CMi LMi Kðci ai C ci ai Þ Kðci ai C ci ai Þ K 2a K ci C ci ci C ci
[1] Task Force Report (Low voltage surges requirement for distribution transformers). Secondary (Low-side) surges in distribution transformer. IEEE Trans Power Deliv 1992;7(2):746–56. [2] Holmberg P. Modeling the transient response of windings, laminated steel cores and electromagnetic power devices by means of lumped circuits, Doctoral Thesis, Uppsala University, Sweden; 2001. [3] de Leon F, Semlyen A. Complete transformer model for electromagnetic transients. IEEE Trans Power Deliv 1994;9(1):231–9. [4] Vaessen PTM. Transformer model for high frequencies. IEEE Trans power Deliv 1988;3(4):1761–8. [5] Morched A, Marti L, Ottevangers J. High-frequency transformer model for the EMTP. IEEE Trans Power Deliv 1993;8(3):1615–26. [6] Chimklai S, Marti JR. Simplified three-phase transformer model for electromagnetic transient studies. IEEE Trans Power Deliv 1995; 10(3):1316–25. [7] Woivre V, Arthaud JP, Ahmad A, Burais N. Transient overvoltage study and model for shell-type power transformers. IEEE Trans Power Deliv 1993;8(1):212–20. [8] Smith DR, Puri JL. A simplified lumped parameter model for finding distribution transformer and secondary system responses to lightning. IEEE Trans Power Deliv 1989;4(3):1927–36. [9] Burrage LM, Veverka EF, McConnell BW. Steep front short duration low impulse performance of distribution transformers. IEEE Trans Power Deliv 1987;PWRD-2(4):1152–6. [10] Caldecott R, Liu Y, Sebo S, Kasten DG, Wright SE. Measurement of the frequency dependent impedance of major station equipment. IEEE Trans Power Deliv 1990;5(1):474–80. [11] Eriksson AJ, Stringfellow MF, Meal DV. Lightning-induced overvoltages on overhead distribution lines: experiment. IEEE Trans Power Apparatus Syst 1982;PAS-101:960–8. [12] Rachidi F, Nucci CA, Ianoz M, Mazzetti C. Response of the multiconductor lines to nearby lightning return stroke electromagnetic fields. IEEE Trans Power Deliv 1997;12(3):1404–11.
M.J. Manyahi, R. Thottappillil / Electrical Power and Energy Systems 27 (2005) 241–253 [13] Task Force 15.09 (IEEE Lightning and Insulator Subcommittee). Review of research on non-standard lightning voltage waves. IEEE Trans Power Deliv 1994;9(4):1972–82. [14] Rubinstein M, Uman MA, Medelius PJ, Thomson EM. Measurements of the voltage induced on an overhead power line 20 m from triggered lightning. IEEE Trans Electromagn Compat 1994;36(2):134–40. [15] Georgiadis N, Rubinstein M, Uman MA, Medelius PJ, Thomson EM. Lightning-induced voltages at both ends of a 448-m power-distribution line. IEEE Trans Electromagn Compat 1992;34(4):451–60. [16] Barker PP, Mancao RT, Kvaltine DJ, Parrish DE. Characteristic of lightning surges measured at metal oxide distribution arrester. IEEE Trans Power Deliv 1993;8(1):1404–11. [17] Fernandez MI, Rambo KJ, Rakov VA, Uman MA. Performance of MOV arresters during very close, direct lightning strikes to a power distribution system. IEEE Trans Power Deliv 1999;14(2):411–8.
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[18] IEEE Recommended Practice for Routine Impulse Test of Distribution Transformers, IEEE Std c57.138; 1998. [19] Bellaschi PL. Characteristics of surge generators for transformer testing. AIEE Trans 1932;June:975–83. [20] Gustavsen B, Semlyen A. Rational Approximation of Frequency Domain Responses by Vector Fitting. IEEE paper PE-194-PWRD-011-1997. [21] Gustavsen B, Semlyen A. Enforcing passivity for admittance matrices approximated by rational runctions. IEEE Trans Power Syst 2001; 16(1):97–104. [22] Dugan RC, Kershaw SS, Smith SD. Protecting transformers from low side current surges. IEEE Trans Power Deliv 1990;5(4):1892–901. [23] Manyahi MJ. Response of transformers to lightning transients and their transfer between the transformer circuits. Licentiate Thesis, Uppsala University; 2000.
Simplified model for estimation of lightning induced transient transfer through distribution transformer M.J. Manyahia,b,*, R. Thottappillilb a Faculty of Electrical and Computer Systems Engineering, University of Dar es Salaam, P.O. Box 35131, Dar es Salaam, Tanzania Division for Electricity and Lightning Research, The Angstrom Laboratory, Uppsala University, Box 539, SE-751 21 Uppsala, Sweden
b
Accepted 25 February 2003
Abstract In this work a simplified procedure for the formulation of distribution transformer model for studying its response to lightning caused transients is presented. Simplification is achieved by the way with which the model formulation is realised. That is, by consolidating various steps for model formulation that is based on terminal measurements of driving point and transfer short circuit admittance parameters. Sequence of steps in the model formulation procedure begins with the determination of nodal admittance matrix of the transformer by network analyser measurements at the transformer terminals. Thereafter, the elements of nodal admittance matrix are simultaneously approximated in the form of rational functions consisting of real as well as complex conjugate poles and zeros, for realisation of admittance functions in the form of RLCG networks. Finally, the equivalent terminal model of the transformer is created as a p-network consisting of the above RLCG networks for each of its branches. The model can be used in electromagnetic transient or circuit simulation programs in either time or frequency domain for estimating the transfer of common mode transients, such as that caused by lightning, across distribution class transformer. The validity of the model is verified by comparing the model predictions with experimentally measured outputs for different types of common-mode surge waveform as inputs, including a chopped waveform that simulate the operation of surge arresters. Besides it has been verified that the directly measured admittance functions by the network analyser closely matches the derived admittance functions from the time domain impulse measurements up to 3 MHz, higher than achieved in previous models, which improves the resulting model capability of simulating fast transients. The model can be used in power quality studies, to estimate the transient voltages appearing at the low voltage customer installation due to the induced lightning surges on the high voltage side of the transformer. The procedure is general enough to be adapted for any two-port devices that behaves linearly in the frequency range of interest. q 2004 Elsevier Ltd. All rights reserved. Keywords: Distribution transformer model; Lightning induced transients; Admittance functions
1. Introduction Despite improvements in transient over-voltage protection over the years [1], lightning has continued to be a major cause of outage and equipment damage on low voltage distribution networks. Most modern electronics equipment/ systems found in consumer’s installation today are sensitive to the effects of transient over-voltages. Although, it is well known that there are various ways by which lightning as * Corresponding author. Address: Faculty of Electrical and Computer Systems Engineering, University of Dar es Salaam, P.O. Box 35131, Dar es Salaam, Tanzania. E-mail addresses: [email protected] (M.J. Manyahi), [email protected] (R. Thottappillil). 0142-0615/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2003.02.001
a source of transient can couple to electrical networks and induce transient over-voltages, the growing concern is how these voltage surges and transients are reaching consumer installations. Power distribution transformer is an important component of the distribution network by providing ultimate connection of low voltage (LV) customer installations to the medium voltage (MV) distribution network. Besides its role of providing isolation between the two circuits operating at different voltage levels, it also acts as a gateway for conveying the unwanted transient over-voltages in the MV network to the LV customers and vice versa. Consumer equipment that is connected to low voltage power network may have no direct exposure to lightning, however, lightning induced transients can be transferred to this
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equipment as a result of strikes on exposed lines on the primary side of the transformer. Even though transient protectors are incorporated on the primary side of the transformer, the transferred residual transients to secondary side may be a threat to connected equipment, a fact that prompts for a need of advanced understanding of transients propagation through distribution transformers. Even though efforts to understand the transfer of transients across the distribution transformer have been carried out for many years, it is widely accepted that, a comprehensive and generally acceptable transformer model for transient simulation does not exist. The reason behind this is that transformers come in different design and configurations, and with different parameters as well as different installation practices. There are two different approaches for modelling transformers. The first one, based on the detailed electromagnetic modelling of the windings and the core [2,3] are required for understanding the influence of transformer design parameters on the transient response of the transformer. However, such detailed models are difficult to use if the primary aim is to predict the transfer of lightning transients across transformer windings in an existing distribution network. The second approach based on modelling the transformer using the voltage and current measurements at its terminals [4–6] is ideally suited for predicting the transients arriving at the consumer installation in the existing distribution network, even though it may not give as much insight into the influence of the design parameters on the transient response. Various transformer terminal models for high frequency or transient studies have been proposed in recent years [4–7], most of them for power transformers. Formulation of these models has been based on building up of equivalent linear network, from approximations of impedance or admittance function frequency responses. The principle differences between these models are on how the admittance functions are obtained and the specific methods used to approximate these frequency response functions to form the equivalent network of linear components. Different methods are used for obtaining the external terminal admittance functions, namely, sequence admittance measurement [5], impulse response measurement [4], and measurements at low frequencies with subsequent extrapolation to higher frequencies [6]. In spite of their differences in their formulation procedure, majority of these terminal models has been devoted for studying transients in power transformers, not in distribution transformers. Also, the models are bandwidth limited (upper frequency of several kilohertz to maximum 1 MHz), a limitation that makes them unsuitable for faithful simulation of the transient response to steep fronted or chopped pulses, which are crucial in studying response to lightning caused transients. On other hand, when distribution transformers have been covered, most of their models and transient response studies have been directed towards the main objective of protecting transformers from
damage [7–9]. Recently, studies are carried out on distribution transformer transient responses in view of their failures due to lightning induced transients in secondary (LV installations) [1,22]. Apparently for models derived by fitting admittance functions, accurate approximations (fitting) over a frequency range of interest and the choice of network representation of the model plays an important criterion for better numerical simulation and stability of the model. Fitting technique employed in some of these models [5] for approximation of rational functions used in realisation of the elements of model networks, involves many steps of mathematical optimisation performed by portioning the admittance function over various frequency ranges. Piecewise fitting have major influences in the portions being fitted, and is one of the reasons for reduction of the fitting accuracy and hence the accuracy of the resulting model. Nearby lightning strikes in transmission system may not be of concern to components operating at that level of voltage (200 kV and above), in contrast to switching transients (differential mode) generated in GIS (Gas insulated systems) in the same system. Meanwhile in distribution networks, both direct and indirect lightning strikes are of much concern to network components, and respective connected consumer’s installations. The problem of lightning caused transients becomes more pronounced in distribution network when overhead lines that are very long are used throughout the urban and rural areas or in both MV and LV network due to increased exposure to lightning strikes. In this work a simplified procedure for the formulation of distribution transformer model is presented. The paper is organised as follows. Section 2 describes the assumptions, along with its justification, used in the modelling. The model that has been implemented in this study is based on admittance network representation proposed in [5]. Section 3 describes the measurements performed on the transformer. Frequency domain measurements are undertaken with the use of Network Analyser method adopted by Caldecott et al. [10]. In their setup they made use of Network Analyser HP3577A along with a shunt resistor for measurement of current, which is essential for the computation of power transformer impedance. However, in this work the admittance functions are directly measured with a configuration consisting of Network Analyser HP4195A, Pearson probe 2877 for current measurement, and computer interlinked to Network Analyser through GPIB card for measured parameter storage. Acquired admittance frequency responses were thereafter, verified by comparing with corresponding responses that are derived from time domain (impulse response) measurements. Section 4 describes approximation (fitting) of the measured admittance function, and computation of equivalent RLCG network used in model formulation. Computed RLCG network responses are verified by comparing measured admittance function. An outline of computer program, which is used in this work for
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performing various steps of model formulation, is presented. Section 5 provides comparisons of model predictions with experimentally measured responses, for different stimulus surge applications. Chopped pulses and standard lightning impulse voltage have been used to verify the model response to transients that are encountered in practise. Section 6 summarises the work pointing out the benefits of the model and drawbacks on the use of the model.
2. Transformers in power distribution network 2.1. Dominance of indirect strikes In most cases the three-phase MV overhead line conductors are either spaced in horizontal or triangular construction, with or without a grounded shielding wire placed at the pole-top. The clearance distances of line conductors to ground and phase conductor spacing, besides other influencing factors, may depend mainly on the level of operating voltage. Due to their low height and presence of other tall objects within their proximity, distribution lines do not attract many direct lightning strikes. On the other hand, indirect strikes are considered to be more frequent and are thought to be capable of inducing transient over-voltages of up to 300 kV [11]. 2.2. Common-mode versus differential mode Generally, the lateral distance (for strikes in proximity of line) is large compared to the distance of separation between the phase conductors. Therefore, the phase conductors can be assumed to be at equidistant from the point of strike, without any appreciable error. This assumption is realistic for overhead distribution lines, which are of concern/interest to this work, because the ratio of phase conductor separation to the height above ground level is smaller compared to that of HV transmission lines. In a study by Ericksson et al. [11] on 9.9-km experimental MV overhead line, it was found that waveshapes of induced surges are identical on all the three phases. In their recent work, Rachidi et al. [12] have confirmed that the induced voltages are practically proportional to the conductors height above ground level. In particular reference to horizontally spaced conductors, they have shown that almost same voltage is induced on each of the three-phase conductors, and the effect of overhead grounded wire is to reduce the induced voltages on phase conductors. With above review, it is reasonable to assume that the induced surges in primary distribution lines have the same magnitude and waveshape for all the three-phase conductors (Common Mode, CM). This generalized assumption holds, because even if the conductors are not arranged in horizontal configuration, i.e. in equilateral triangle or vertical configuration, their vertical separation is very small compared to the pole height. In spite of the generalized assumption of common mode induced voltages, it is possible also that during unbalanced operation of
243
protective devices (arresters), a transient voltage between phase conductors may be produced (Differential Mode, DM). Therefore, the lightning caused transients are predominantly CM even though small DM transients may appear at the transformer terminals. 2.3. Frequency content of lightning induced transient Lightning caused transient voltages are rarely having standard waveshape [13], they can have very fast components depending on the ground conductivity, connected loads, and reflections from terminations, etc. [14,15], and hence the need for consideration of higher frequencies in modelling network components. This has been confirmed by measurements at distribution arresters performed by Barker et al. [16] and Fernandez et al. [17]. They recorded lightning caused surges that were quite steep, with rise times of first stroke caused surges in the order of few microseconds, while, for subsequent strokes the surges had rise times that are less than 0.5 ms. From the rise times of lightning caused transient overvoltages [14–17], it can be safely assumed that the upper frequency content of these transients are several megahertz. In transfer of transients across transformers, high frequency range is very important because they are less affected by the nominal turns ratio of the transformer. The transformer model developed in this paper is tested for its accuracy up to 3 MHz and hence well suited to evaluate conducted transients arriving at consumer installations in a distribution network. 2.4. Linearity of transformer at high frequencies Distribution transformers are characterised by high turns ratio between primary and secondary winding, and have high ratio between the basic insulation level and the rated voltage of the low voltage winding [18]. The impedance of high voltage winding is higher than that of corresponding low voltage winding. It has been shown experimentally [4] that transformers behave linearly above 1 kHz and iron core does not play any significant role in the voltage and current transfer at high frequencies. Woivre et al. [7] found that transformer windings behave as linear system (without saturation effects) for lightning caused over-voltages. Exploiting the linearity property of transformers at high frequencies, we can write general two-port nodal admittance equations that relates the voltages and currents at the accessible transformer terminals as " # " #" # I1 V1 Y11 Y12 Z (1) I2 Y21 Y22 V2 2
Iia
3
6 7 ½Ii Z 4 Iib 5 Iic
(2.1)
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2
Via
3
6 7 ½Vi Z 4 Vib 5
(2.2)
Vic 2
3
Yij;aa
Yij;ab
Yij;ac
6 ½Yij Z 4 Yij;ba
Yij;bb
Yij;bc 7 5
Yij;ca
Yij;cb
Yij;cc
(2.3)
where [Ii], [Vi] and [Yij] are sub-matrix 3!1 and 3!3, respectively, relating to three-phase transformer terminals.
3. Experiment for parameter measurement The elements of the nodal admittance matrix given in Section 2.4 may be obtained from short-circuit tests at the terminals of distribution transformer. All terminals except one that is connected to test signal source are grounded. Currents are measured in all terminals along with the stimulus voltage in turns or all measurements are done at once depending on channels available for oscilloscopes or data acquisition system. These terminal measurements of admittance parameter may be performed in either frequency or time domain. In the time domain method, impulse voltage and current waveforms at the transformer terminals are measured, and the corresponding frequency domain admittance function is computed using Fourier transform. The frequency domain measurements of admittance function is realised by utilising a tracking generator, which sweeps the defined frequency range, and at sampled points within the range, complex admittance is computed. Since, all the admittance function measurements are carried out at reduced voltage levels (nondestructive voltage levels); the corresponding measured currents are comparatively low. It is important, therefore, that true currents are measured rather than the true voltages, by inserting current probe on the load side and voltage probe on the source side. This setup is of special importance in eliminating recording equipment loading effects, especially when performing the measurements of driving point impedance. Accurate and easy determination of admittance functions, whether it is obtained from time domain derivation or direct frequency domain measurements, are important factors for accurate and simple model formulation. The distribution transformer used in this work is a three-phase, 11/0.4 kV, 50 kVA, with Dzn10 winding connections.
condition for impulse response (time domain) measurement is that the impulse source should produce impulses that are steep enough to contain sufficient high frequencies that will excite the transformer resonances adequately in relevant frequency range. It is essential to have a well-designed impulse source that is capable of providing same test impulse voltages and waveshape within small tolerances for various transformer winding connection and different admittance parameter measurements. Moreover, due to wide difference in primary and secondary circuit impedances, it is sometimes necessary to design impulse source circuits for tests carried on each transformer circuit (i.e. primary and secondary). However, if the measurements have to be carried on a number of distribution transformers of different sizes and rating, it will be difficult to modify impulse source circuit for each measuring case by changing component values. Impulse response measurements have been preferred from the early days [19] because it has been easier to construct impulse sources for variety of voltage levels, and more importantly the generated impulses replicates the transient nature of surges encountered in practice. Furthermore, an impulse under consideration contains all frequency spectra, with the highest frequency content determined by its rise time or steepness. The accuracy of impulse waveform recordings will depend on the settings and bandwidth of the equipment used, i.e. storage oscilloscopes, voltage and current probes. Fig. 1 shows the setup for impulse response measurement of distribution transformer voltage and currents used for deriving admittance functions. The impulse generator used in this setup was designed inhouse and is capable of providing a non-standard open circuit impulse of 0.4/20 ms with highest value of peak voltage up to 2.5 kV. Lecroy oscilloscope LC540, with corresponding high voltage probe and Pearson current probes 2877, were the equipment used for the setup. MATLAB routines were compiled for the purpose of performing FFT function of acquired waveforms and hence compute the respective admittance function. The need for impulses that have higher rise time (for high frequency responses), which have substantial long tail (decay to zero), would require higher sampling rate on the recording oscilloscope, with a requirement of long
3.1. Time domain parameter measurement As mentioned earlier, time domain measurements has been performed in this work for the purpose of obtaining parameters for computation of admittance functions, as a comparison reference to corresponding frequency domain measured functions, to be described later. The necessary
Fig. 1. Experimental setup for measuring impulse response voltage and current values, for computation of driving point and mutual admittance functions.
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recording length for a given sweep, in order for the waveforms to be correctly acquired. The requisites for higher sampling rate with long records have a drawback of forfeiting the waveform smoothness due to oversampling noise on the impulse tail. Opting for short record length would result to uncompleted recording of voltage and current waveforms, i.e. they will not have sufficiently decayed to zero, which will consequently cause false interpretation on performing FFT. Accordingly, windowing function need to be applied to last part of the tail or acquired waveforms to ascertain smooth transition of the waveform tail to zero at record end. Therefore, the requirements of broad bandwidth impulses, accurate parameter recordings, efficient waveform processing and computation of admittance functions are the essential features in time domain measurements. Care should be taken to avoid noise contamination to the measuring signal caused by the use of unshielded cables, which will inevitably lead to errors in magnitude and phase of the resulting admittance function. This, however, will depend very much on the level or magnitude of test voltage and the impedance of the system. 3.2. Frequency domain parameter measurement Frequency domain parameter measurement is achieved basically by using a variable frequency signal source (which replaces an impulse source in time domain), accompanied with the use of voltage and current probes. The frequency source generates signals at various frequencies within the bandwidth of interest, and at each frequency, admittance value is computed from measured respective values of voltage and current. In this work, Network analyser has been used in transmission parameter measurement mode. The parameter measured is a complex transfer coefficient T, which is defined as the ratio of the transmitted voltage to the incident voltage expressed as: TZ
Vtransmitted Z t:f Vincident
(3)
The signal source terminal S is connected directly with the reference terminal R (incident terminal) and thereafter connected to ungrounded transformer terminal. Transmission terminal T on the Analyser is connected to current probe, which has an output ratio of 1.0 V/A, which is a mirror conversion of current to transmitted voltage. Fig. 2 illustrates the use of Network Analyser in direct measurement of distribution transformer admittance functions. With its internal operating software the network analyser computes the complex transfer coefficient, which is equivalent to admittance function derived from measured transmitted voltage that is proportional to current, and the incident voltage, which is equal to the source voltage. The analyser has a signal source (tracking generator), that sweeps a broadband of frequencies, and at sampled frequencies measurement is performed with subsequent
245
Fig. 2. Experimental setup for measuring admittance frequency responses in transmission parameter mode of the network analyser.
display of complete amplitude and phase. Coaxial cables are essential for connecting the device under test to Network analysers, which is a 50 U terminated equipment at all its terminals. However, due to the nature of connection to transformer terminals, with a need of placing in a measuring current probe, short length of unshielded connecting leads cannot be avoided. These leads may have a significant influence at high frequencies. In order to reduce the cable influences on all measured parameters, same set of cables of less than 1 m in length were calibrated with the instrument before commencing measurements, and used in various connection setup of equipment under test. Major drawback with the use of Network Analyser (in particular one used in this work) is in short record length of measured parameter. The maximum number of samples over frequency bandwidth of interest is 401 samples, with an option of spreading the sampled points linearly or logarithmically. Therefore, broad bandwidth measurement of parameter will result to poor sampling and hence inaccurate measurement of admittance function frequency response. Additional drawback on measuring admittance function is that its source gives an output test signal with a maximum value of 15 dbmV at 50 U, which limits some measurement of high impedances, although, this problem has not been experienced in measurements of this work. 3.3. Comparison of time and frequency domain parameter measurement In both measuring schemes, Pearson current probes 2877 have been used. For the purpose of substantiating that the current probe has the same influence (identical attenuation) over all frequencies within the bandwidth of interest, probe frequency response was tested to verify their constant amplitude and linear phase behaviour over the bandwidth. Fig. 3 shows the measured current probe amplitude and phase frequency response over a frequency range of 200 Hz– 10 MHz. The current probe frequency responses in Fig. 3 are significantly flat over the measuring bandwidth of interest in this work, i.e. 1 kHz–3 MHz. Conversion factor of the probe is unity over the bandwidth and there is no phase angle shift,
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Fig. 3. Magnitude and phase frequency response of Pearson current probe 2877.
i.e. phase angle is zero over the range. Even at 200 Hz the magnitude response drops by 6% only. Since both admittance-measuring schemes have some advantage and disadvantages, their comparison on suitability of use depends on practical aspect of using the method on field, namely, accuracy and easiness with which the scheme can be applied for complete measurement of parameters. Figs. 4 and 5 show the comparison of mutual and driving point (self) admittance functions obtained from frequency domain measurements and that computed from time domain acquisitions. There is a good correlation in the admittance functions obtained from two measuring schemes, in respect of magnitude and phase angle, with only small discrepancies
Fig. 4. Comparison of frequency domain and time domain derived frequency response of mutual admittance function Y21 between high and low voltage winding.
Fig. 5. Comparison of frequency domain and time domain derived frequency response of low voltage winding self-admittance function Y22.
in phase angles at higher frequencies. However, due to limitation of our signal source for time domain measurements we could not realise impulses that have broad frequency bandwidth considered in this work, i.e. 1 kHz– 3 MHz, for parameter measurement on low voltage winding. That can be noted from Fig. 5, where the bandwidth of computed admittance functions is 40 kHz–4.5 MHz. As mentioned earlier, the signal source for time domain measurements need to be of very good design, so that it is capable of providing same test impulse voltages and waveshape for various transformer winding (high and low voltage) terminal measurements.
4. Model setup Each admittance function is measured or determined independently of other functions in the case of network analyser method because only one channel can be used at one time for that purpose. However, with time domain derived admittance functions a number of these can be obtained in single test depending on number of available channels on the recording devices, e.g. oscilloscopes, and also on the number of admittance function that need to be determined by particular terminal source injection. Regardless of whether these admittance functions are measured/obtained separately or collectively at once, they are elements of admittance matrix of object to be modelled. This admittance matrix (Eq. (1)) defines the frequency response behaviour of the modelled transformer revealed at its terminals. When building up the network model out of these measured functions by fitting, it is important that these elements of the matrix are fitted together to enhance the coupling between them. According to the properties of linear systems admittance matrices, the elements in the rows and columns elements are interrelated about the leading
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247
function Yfit Yfit Z
m X iZ1
n X ci ci ci C C C se C d s K ai iZmC1 s K ai s K ai (5)
We can write the admittance of the network above in the form of the fitted function as: Fig. 6. The p-network equivalent model for two winding transformer consisting of RLCG components.
diagonal. For this reason, even though it may be difficult to fit all elements of the matrix at once, we can undertake fitting of related elements by row or column. In this work, the elements of the nodal admittance matrix are approximated with rational functions consisting of real as well as complex conjugate poles and zeros, by spavectfit a matrix element-fitting version of vectfit [20] fitting technique. These numerical approximations are realised in the form of an RLCG network (Fig. 7) that can be used with ATP, as an element of the p-equivalent model network of Fig. 6, which each of its elements comprises of one such RLCG network. The reasons as to why this network is appropriate for high frequency model for transformers are that it comprises of mixed assemblage of inductance and capacitance that represents the general constructional properties of transformer windings. The inductive elements of the transformer network model are important at low frequencies, whereas the capacitive elements dominate at high frequency. Vectfit approximates a frequency response Y(s) with rational functions, expressed in the form of a sum of partial fractions Yfit ðsÞ z
n X iZ1
ci C d C se s K ai
(4)
where the terms d and e are real constants; while ci and ai are the residues and poles, respectively, which may either be real quantities or complex conjugate pairs. The fitted admittance function can be written in the form such that real residue and poles are separated from that of complex conjugate residue and pole function, so that it is easier to realise RLCG network (Fig. 7) from the fitted
Fig. 7. Structure of RLCG network used in this study to represent each element of p-equivalent model.
YðsÞ Z YCG ðsÞ C YRL ðsÞ C YRLCG ðsÞ
(5.1)
For G0 and C0 branches: YCG ðsÞ Z G0 C sC0 Z Y0 ðsÞ
(5.2)
For series Rs and Ls branches: YRL ðsÞ Z
m X iZ1
ki s K pR i
(5.3)
For RmLmCmGm branches: ( ) n X bi bi C YRLCG ðsÞ Z s K pCi s K pCi iZmC1
(5.4)
The combined network is: YðsÞ Z Y0 ðsÞ C
m X iZ1
n X ki C s K pRi iZmC1
(
bi bi C s K pCi s K pCi
)
(5.5) In the above equations, G0, C0, and ki are real constants; PRi are real poles; PCi and bi are complex poles and residues, and pCi and bi , are their respective poles and residue complex conjugates. For our case m is the number of real poles, while (nKm) is the number of complex conjugate poles, in which case m also represent the number of RL series branches and (nKm) is the number of RLCG branches. These rational functions (Eqs. (5.1)–(5.5)) are used to evaluate the R, L, C and G components from fitted admittance functions, required for each RLCG module network. The approximation is achieved by replacing an initial set of poles with an improved set of poles using a pole relocation method based on least squares approximation of linear problems [20], and the order of the approximation is equal to the number of starting poles, i.e. initial number of poles. Vectfit performs the actual fitting in rectangular (complex) function and approximation (fitting) results are displayed in polar function along with the measured admittance functions. Examples of the approximated admittance (fitted) function superimposed on the measured functions are shown in Fig. 8(a) and (b). The measured functions Y21 and Y22 are the same as that shown earlier in Figs. 4 and 5. Fitting had been performed with smaller bandwidth (1 kHz–1.5 MHz) compared to measured frequency response bandwidth (1 kHz–3 MHz) for the purpose of avoiding passivity problems [21], which manifests itself in
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Fig. 8. (a) Fitted magnitudes of measured self and mutual admittance functions. (b) Fitted phase angle of measured self and mutual admittance functions. Deep black is measured function and grey print is approximated (fitted) function.
time domain simulations. Fitting with smaller bandwidth has been opted instead of performing complicated passivity enforcement procedures. However, in the final analysis the fitted function was verified by checking its ability to predict frequency responses of full measured bandwidth. Approximation (fitting) of the admittance functions is of exemplary quality (match) such that in most parts of the function it is difficult to distinguish between the measured and fitted functions, this is evident in the fitting of Y22, where the fitted function exactly overlays that of measured function. Appreciable good fit has been obtained in both magnitude and phase of the measured admittance frequency responses, all major resonances have been correctly approximated. There is small discrepancy at low frequencies; however, this may have been introduced by measurement errors at these frequencies as discussed in previous sections or fitting associated errors. Besides these small
discrepancies the general combined approximation of all admittance functions for the model is adequately good. A supplementary software routine has been compiled in MATLAB within the vectfit-working environment, for the purpose of evaluating the R, L, C and G components, on completion of the fitting procedure for each RLCG network. The derivation of these rational functions and hence the computation functions for RLCG components from fitted admittance function has been undertaken in [23], and a summary is given in Appendix A. Since there is no unique solution on fitting the functions with varying number of poles and number of iterations, a best fit out of a number of good fits may be selected based on the frequency response of the resulting RLCG network. It is important and realistic to check the response of approximated RLCG network, because after all the fitting process is meant for generating RLCG network and hence the model that can effectively be used in time domain simulations. Therefore, a fit that derives an RLCG network, which reproduces adequately the frequency response of measured or fitted function, is expected to work effectively in a combined model network. Fig. 9 shows the comparison of measured admittance function to the frequency response of RLCG network that has been derived from the fitted function. Frequency response of derived RLCG network is comparable to the fitted admittance functions and sufficiently predicts the responses beyond the maximum fitting frequency of 1.5 MHz (compare with measured admittance functions in Figs. 4 and 5). We have compiled a computer program in MATLAB such that the formulation of the distribution transformer model for lightning transient transfer simulation is realised easily, after the necessary admittance frequency response have been measured. The computer flow chart in Fig. 10
Fig. 9. Comparison of fitted admittance function (Fig. 8(a) and (b)) and response of derived RLCG network. Deep black is fitted admittance function and grey print is RLCG network response.
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Step 5. A MATLAB routine is prepared to organise the RLCG network elements in ATP linear network (creation of local nodes) file format. Each RLCG network so created for ATP is given a filename that reflects the element it represents in the transformer model (Fig. 6). Step 6. Steps 2–5 are repeated until all columns of admittance matrix have been fitted and respective RLCG network is computed. Step 7. MATLAB program is prepared to compile RLCG network files into a model file (creation of global nodes and simulation parameters), which will be used for simulation in ATP. Step 8. ATP program uses file created in step 7 to perform the simulations. In some steps interaction of the user is required whenever prompted to enter node numbers, file names, etc.
5. Comparison of measured and simulated results
Fig. 10. Computer flow chart for formulation of transformer model.
shows various stages of the program where in some stages interaction of the user will be required, especially in verification of fitting the functions and checking if the derived RLCG network reproduces the measured admittance functions. Descriptions of the flow chart steps are as follows: Step 1. Short circuit driving point and mutual admittance functions, Y11, Y22, Y21 or Y12 are measured/obtained in frequency domain by Network Analyser, e.g. Figs. 4 and 5. Step 2. Acquired admittance functions are assigned with code names with reference to the element it represents in admittance matrix. The elements are fitted together by arranging them in column using vectfit, which works in MATLAB environment, e.g. Fig. 8(a) and (b). Step 3. Fitted functions are computed with MATLAB compiled routine to equivalent RLCG network. Step 4. The computed RLCG network frequency response is compared with measured admittance responses with help of a compiled MATLAB routine, e.g. Fig. 9.
The nature of transient coupling between the two transformer circuits depends on type of winding connection [23] and the transformer grounding system used. As aforementioned, most of the lightning related perturbations on distribution lines are caused by indirect lightning strikes. Sometimes, these indirect strikes induce transient voltages that are below the protectors threshold value, but has sufficient rise times to excite transformer resonances. Figs. 11 and 12 show the comparison of experimental and simulation results of transferred transients from primary to secondary circuits, with consideration of rise times of lightning surges that are expected in practise.
Fig. 11. Comparison of experimental and simulated results from same signal excitation.
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From measured and simulated results we note that the maximum magnitude (peak) of the transferred transients to LV network from HV terminals increases with an increase in the rate of rise (steepness) of stimulus surges. It is revealed in Fig. 12 that steep preliminary pulses can be transferred through transformer to low voltage installation with less attenuation. Fast rising pulses have more influence on the transferred voltage than the corresponding low rising pulses. Generally, the model reproduces sufficiently the oscillatory frequency of the transferred surges. The model shows that it is capable of reproducing the transient transfer behaviour of distribution transformer due to various lightning transient rise times (wide frequency content) coupling to MV distribution lines.
6. Conclusion
Fig. 12. (a) Experimentally measured transferred transient to LV circuit due to steep surges on HV terminal. (b) Model simulation result of transferred transient to LV circuit due to chopped surges application on HV circuit.
The results of simulation (Fig. 11) show that the model is predicting the experimentally measured transfer voltages. In order to make realistic comparison of the two transferred voltages (experimental and simulation), the input voltage for simulation was obtained by sampling the experimental input waveform, and applying linear interpolation in synchronisation to time intervals used in ATP simulation. The rise time of the excitation signal is about 1.2 ms, and Fig. 11 shows excellent match and reproduction of experimental results in terms of magnitude and time constants. In most cases, distribution transformers have transient protectors, commonly installed on its primary terminals. Operation action of these arresters sometimes results in chopped surges, an action which causes the generation of steep tail similar to steep fronts that represents transients of higher frequencies. In Fig. 12(a) and (b), comparison is made between experimental and model simulation results on the transferred voltage to LV network due to an arbitrary steep surge application on primary terminals.
The simplified formulation of model for estimating transferred lightning transient from the more exposed MV lines to customer installation has been presented. Comparison of admittance functions for model formulation derived from traditional procedure of impulse response to those obtained in frequency domain has been presented. Advantages and disadvantages of both methods have been discussed; however, in final analysis the advantages of measuring admittance function in frequency domain opted in this work outbalances disadvantages. Main consideration is on its simplicity, less time and easiness with which measurement of parameters may be carried out on field. Furthermore, the need for computational convolutions (time to frequency domain) and associated errors on filtering are avoided with realisation of less noise in measured admittance functions. The network analyser which was previously used in measurement of frequency response driving point (self) admittance function has been effectively used in measuring mutual admittances as well in this work. We have shown that even though the network analyser acquires the admittance functions at low voltage signals, the obtained functions are comparable to those derived from impulse response measurements at higher voltage signals. The use of Network Analyser has made possible for improvement and extension of the frequency bandwidth of the resulting distribution transformer model up to 3 MHz. This simple model is a linear network model suitable for use in estimating transients that are in the frequency range where the transformer behaves linearly. The simplified model reproduces satisfactorily the experimental measured transferred transients in all lightning transient frequency spectra expected in practise. Non-linear behaviour cannot be studied with this model and therefore, it is not a stand-alone model, but can be used as add on model to existing ATP low frequency models. Although, various loading conditions of distribution transformer has not been considered in this work, the extreme case of lightly loaded is considered, in view of the worst case the customer network will be exposed to.
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This work has not dealt with lightning caused differential mode transfer of transients, i.e. transients that appear at HV terminals due to unsynchronised operations of surge protectors, and hence transferred to LV circuits. However, in our work to be submitted soon we are addressing the problem of arrester installation practise and the generated differential mode transients and their transfer through transformer windings. Finally, we have achieved further model formulation simplification by a compiled program in MATLAB, which on entering admittance data carries out fitting, computation of equivalent network and formulation of the model for direct use in time domain simulation (ATP) and hence the simulations. Acknowledgements The authors would like to acknowledge valuable advises of Bjo¨rn Gustavsen and for providing the vectfit program used in this work. We greatly appreciate the financial support of Swedish International Development Agency (SIDA) on this work and support of Mr M.J. Manyahi’s PhD work at Uppsala University.
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For the computation of component values of admittance function, above equations need to be written in the form of the fitted admittance function: i:e: Yfit Z
n X iZ1
ci C se C d s K ai
(A5)
The resulting fitted function equation shows that there are multiple branches of RL and RLCG branches with a single CG branch, with the terms e and d as constant real terms, while ci and ai are the real or complex residue and poles, respectively. Therefore, the fitted admittance function expression can be expanded such that real residue and poles are separated from that of complex conjugate residue and pole function of Eq. (A5)
m n X X ci ci c Yfit Z C C i C se C d s K ai iZmC1 s K ai s K ai iZ1 (A6) Comparing the last two terms of Eq. (A6) with Eq. (A2) Y0 Z sC0 C G0 hes C d Therefore C0 Z e and G0 Z d
Appendix A
For multi-RL branches (A3) becomes
The description below elaborates on the procedure that have been derived for determining values of the individual components of RLCG module directly from admittance function fitted by vectfit. A MATLAB program is compiled based on these expressions such that at the end of vectfit fitting, all values of individual components of respective admittance function will be available. Total admittance of RLCG network module is the sum of the branch admittances: Y Z Y0 C
m X
YSi C
iZ1
n X
YMi
(A1)
Y0 Z sC0 C G0
(A2)
The low frequency branches of inductance and resistance are represented by 1 sLSi C RSi
sCMi C GMi Z ðsLMi C RMi ÞðsCMi C GMi Þ C 1
m X iZ1
1=LSi s C RSi =LSi
(A8)
where m is the number of RL branches. Comparing (A8) with first term of (A6) YS Z
1=LSi c h i s C RSi =LSi s K ai
such that LSi Z
1 ci
ðci realÞ
(A9)
and RS i Z K
ai ci
ðci and ai are realÞ
(A10)
Similarly, Eq. (A4) can be factorized to produce conjugate complex functions G
YMi Z
s2 C s
s L1M C CM MLiM i i i RMi R G C LM C CM1LM C CMMi LMMi
GMi CM i
i
i
i
i
(A11)
i
(A3)
The medium frequency resonant RLCG branches are represented by: YMi
YS Z
iZmC1
Since RLCG network module may have multiple branches for YS as well as YM. The high frequency branch of capacitance and conductance is represented by:
YSi Z
(A7)
let 2a Z
GMi RMi C CMi LMi
and g2 Z
RM G M 1 C i i CMi LMi CMi LMi
G
(A4)
YMi Z
s L1M C CM MLiM i
i
i
s2 C 2as C g2
(A12)
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let g2 K a2 Z u2
(A13)
It follows that for complex real and residue values (ai and ci)
G
YMi Z
s L1M C CM MLiM i
i
C Mi Z
i
ðs C aÞ2 C u2
ci C ci Kðc aiCci ai Þ Kðci aiCci ai Þ u2 C a2 K 2a K iciCc ciCc i
s Z
1 LMi
G C CM MLiM i i
½s K ðKa K juÞ½s K ðKa C juÞ
(A20) (A14)
rewriting middle part of (A6) in partial fractions form of (A14) such that: bR K jbi bR C jbi C ½s K ðKa K juÞ ½s K ðKa C juÞ
YM Z
(A15)
Hence ai ZKaK ju and ci Z bR K jbi Equating the numerators of Eq. (A14) to that of (A15), since both have the same denominator s
G Mi 1 C hci ðs K ai Þ C ci ðs K ai Þ LMi CMi LMi
hence 1 Z ci C ci LM i LMi Z
1 ci C ci
and finally G Mi Z
GMi CMi CMi
G Mi Z
Kðci ai C ci ai Þ Kðc aiCci ai Þ Kðci aiCci ai Þ u2 C a2 K 2a K iciCc ciCc i
i
(A21) Component values of admittance module can be calculated from the fitted function by Eq. (A7) for capacitance and conductance branch from constant terms, and Eqs. (A9) and (A10) for series inductance and resistance branch from real values of pole and residues. Finally, Eqs. (A16), (A19)–(A21) for RLCG branch computed from complex values of poles and residues.
(A16) References
and GMi Z Kðci ai C ci ai Þ CMi LMi
(A17)
GMi Kðci ai C ci ai Þ Z ci C ci CMi
(A18)
but 2a Z
GMi RMi C Z Re½aj CMi LMi
therefore GM RMi Z 2a K i LMi CMi RM i
i
Kðci ai C ci ai Þ 1 Z 2a K ci C ci ci C ci
(A19)
from Eq. (A13) g 2 Z u2 C a 2 Z
RM GM 1 C i i CMi LMi CMi LMi
1 Z u2 C a 2 CMi LMi Kðci ai C ci ai Þ Kðci ai C ci ai Þ K 2a K ci C ci ci C ci
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