TB_TrEn_Study_Guide_-_CIGRE_C4.307-+R7_edited+in+Grenoble

CIGRE WG C4.307

Transformer Energization in Power Systems: A Study Guide

Revision 7: September 2012

TB coordinator: M. Martínez Duró

Contents

1. To be reviewed

2. Draft TB

To be reviewed by the WG (existing sections not reviewed in past meetings, updated sections after last meeting, brand new sections)

In bold, what should take the most time (big, controversial, difficult...) In the updated sections, the main changes are identified by yellow background.

Status When What WhereUpdate June 2012 Harmonic content of the inrush currents §2.2

New June 2012 Series sympathetic interaction §2.3.2Update June 2012 Advanced modeling: a topology-based model §5.1.2.D

Update June 2012Uncertainty treatment and stochastic simulation (rewritten)

§5.3

New June 2012 When is a simulation study necessary? §5New June 2012 Wind turbine modelling §5.1.3.5New June 2012 Cables and Wind farms issues §1

Update June 2012 Network equivalents §5.1.3.4Update June 2012 116 MVA energization with surge arrester stress (Case #1) Appendix: §8.3

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Members

Type Members Names with country of origin (two letter code)X. XXX, Convenor (XX), A. XXXX, Secretary (XX),

XXXXX (XX) B. XXXXX (XX)

Copyright © 2011

“Ownership of a CIGRE publication, whether in paper form or on electronic support only infers right of use for personal purposes. Are prohibited, except if explicitly agreed by CIGRE, total or partial reproduction of the publication for use other than personal and transfer to a third party; hence circulation on any intranet or other company network is forbidden”.

Disclaimer notice

“CIGRE gives no warranty or assurance about the contents of this publication, nor does it accept any responsibility, as to the accuracy or exhaustiveness of the information. All implied warranties and conditions are excluded to the maximum extent permitted by law”.

ISBN : (To be completed by CIGRE)

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ISBN : (To be completed by CIGRE)

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Transformer Energization in Power Systems: A Study GuideT Y P E S U B T I T L E I F A P P L I C A B L E

No Extra Cover page or “blank pages”

Use CIGRE abbreviations insofar as possible: Study Committee – SC, Technical Brochure – TB, Working Group – WG

Joint Working Group – JWG, Technical Committee – TC

Photos: must be of reasonable definition (preferably 300 dpi); all figures and tables must be titled, legible and numbered with legends provided.

No Company logos…

Table of Contents

EXECUTIVE SUMMARY................................................................................................................61. Introduction........................................................................................................................... 72. Inrush currents at transformer energization..........................................................................7

2.1. Transformer saturation and inrush currents [NC]............................................................72.2. Harmonic content of the inrush currents [JAMV].............................................................82.3. Sympathetic interaction [HB].........................................................................................12

3. Transient voltage distortion due to transformer energization..............................................203.1. RMS-voltage drop [TM and ZE].....................................................................................203.2. Resonant overvoltages due to network parallel resonances [JAMV].............................22

4. Mitigation techniques and best practices [NC + JAMV]......................................................265. Simulating transformer energization transients [MM]..........................................................28

5.1. Electromagnetic modeling of the network......................................................................295.2. Quantification of the overvoltage stress in the transformers and the surge arresters [MM] 655.3. Parameter uncertainty assessment and stochastic simulation [MM].............................69

6. Appendix: Analytical calculation of inrush current [NC]......................................................767. Appendix: Transformer modelling: Calculation of saturation curve from no-load test and B-H curve [NC] 798. Appendix: Case study examples & simulation results vs. field measurements...................81

8.1. RMS-voltage drop [Zia]..................................................................................................818.2. RMS-voltage drop and inrush currents [Terry]...............................................................848.3. Sympathetic interaction [Terry]......................................................................................90

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EXECUTIVE SUMMARYLorem ipsum dolor sit amet, consectetuer adipiscing elit. Maecenas porttitor congue massa. Fusce posuere, magna sed pulvinar ultricies, purus lectus malesuada libero, sit amet commodo magna eros quis urna.

Explain the technical reasons for conducting the study (system/component failures,

industrial/manufacturer needs for technical improvement, inadequateness of present standards, etc...).

Include reference, if any, to previous CIGRE work on the subject. A limited number of technical or numerical

data may be included, only if strictly necessary.

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1. Introduction[Specific issues related to cables and wind farms:]

Why do we perform transformer energization studies? What is the impact on the system? Issues associated with power quality?

Short explanation of what the TB is intending to do.

Transformer energization is not a new problem in transmission or distribution networks. However, it is getting even more relevant nowadays with the increasing use of cable links and the increase of wind generation. As for cable links, their per length capacitance is much higher than that of overhead lines, therefore they reduce the resonance frequencies of the system and increase the risk of resonant overvoltages. Regarding wind generation, compared to higher power rated transformers, wind turbine transformers are small power rated (several MVA), but they have higher inrush currents (in relative values); on another hand, a wind farm may have tens of transformers, the successive energization of which may generate complex sympathetic interactions; finally, wind farm internal networks and the link to the distribution/transmission network are cable-based, so they tend to reduce the resonance frequencies of the system and thus to increase the risk of overvoltages.

2. Inrush currents at transformer energization

2.1. Transformer saturation and inrush currents [NC]

The transformer core may become saturated due to an abrupt change in the voltage applied to it. This may be caused by switching transients, out-of-phase synchronization of a generator, external faults and faults-clearance. When saturated, a transformer absorbs a magnetization current, also known as inrush current, which can reach several times the nominal current of the transformer. The energization of a transformer normally yield to the most severe case of inrush current as the flux in the core can reach a maximum theoretical value of two to three times the rated flux peak [1]. A qualitative and simplified representation of the inrush current phenomenon is illustrated in Fig. X for energization at voltage zero crossing. The flux-linkage is calculated as the time-integral of the voltage applied to the transformer (upper-left part of Fig. X). The initial value of the flux-linkage is determined by the residual flux in the transformer core prior energization. The flux-linkage/current relation is nonlinear and is determined by the saturation curve of the transformer (upper-right part of Fig. X). Therefore, the magnetization current of a transformer contains harmonics. When a transformer is energized, the initial value of the flux may differ from the prospective flux2. This causes a DC offset of the flux-linkage and a higher-than-rated peak value. The result is an inrush current that may be several time the value of the nominal current. Due to the low relative permeability of the ferromagnetic material in saturation, a marginal increase in the peak of the flux linkage results in a magnification of the inrush current (lower-right corner of Fig. X). The inrush current transient is characterized by asymmetrical current waveforms that are damped in some tens of cycles primarily by the series resistances of the systems (transformer winding resistance, transmission line and cable series resistance, generator winding resistance, etc.). A qualitative representation of the inrush current transient is shown in the lower-left corner of Fig. X.

1 L. F. Blume, Transformer engineering : a treatise on the theory, operation, and application of transformers, 2nd ed. New York, N.Y.: John Wiley and Sons, Inc., 1951.

2 The prospective flux is the flux that would circulate in the transformer core under steady state conditions.There is no induced flux before energization, but the source voltage has the prospect to create an induced flux.

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Fig.X: Qualitative representation of the inrush current phenomenon and the effect of the residual flux.

The main factors affecting the inrush current magnitudes can be divided in:

1) transformer design,

2) initial conditions,

3) network factors.

The design of a transformer can affect the magnitude of the inrush current as it can shift the steady state operating point on the saturation curve. A transformer with an operation point closer to the knee area of the saturation curve is easily brought into saturation. Initial conditions affecting the magnitude of inrush current are residual flux and the point-on-wave (POW) energization. These influence the magnitude of inrush current as affect the DC offset of the flux-linkage and the saturation of the transformer. The residual flux is the flux that remains trapped in the core due to a previous de-energization of the transformer and defines the initial DC offset of the flux in the core. Energization at voltage zero crossing results in the most severe inrush current for a transformer as it induces a flux-linkage of theoretically of up to 2 pu (1 pu DC offset); the residual flux adds on top of that. Energizing a transformer at voltage peak results in no DC offset beside the amount caused by the initial residual flux. High network impedance acts as a limiting factor for inrush current the high current causes a voltage drop at the transformer terminal that limits the saturation of the transformer

2.2. Harmonic content of the inrush currents [JAMV & NC]

Transformer saturation is a highly nonlinear phenomenon and hence the inrush current contains harmonics and DC components besides its fundamental component. The variation of the harmonic content of the transformer inrush current with time has been analysed in several references [1, 2]. To obtain the magnitude and phase shift of each harmonic component, reference [2] proposed the application of a Fourier analysis for each cycle of the inrush current separately.

Figure 2 shows the peak values as a function of time of the main harmonic components of a recorded inrush current [3]. Note that the peak value of any harmonic component during one cycle is generally different from its peak during another cycle. For the case shown in the figure, the 2nd harmonic is by far the dominant one [3]. Note also that the peak values of some harmonics decrease to zero and then increase again to a value higher than the initial value at the instant of energization. A phase shift inversion is also common as the magnitude of a harmonic passes through zero [2].

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Figure 2 Harmonic components of inrush current [3]

Figure 3 shows another example of recorded inrush current waveforms along with the variation of the main harmonic components with time during the energization of a 735 kV, 510MVA autotransformer consisting three single-phase units [1]. Individual harmonics vary with time and some attain their maximum value few cycles after energization.

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Figure 3. Transformer inrush current at energization. Magnitudes of the harmonic currents [1].

The evolution of harmonic currents cannot be generalized as it depends highly on the transformer, feeding network and initial energization conditions. For example, the cases given in Figures 2 and 3 show different developments of 2nd and 3rd harmonics. This implies the need for a case-by-case study when a specific network configuration is identified to be susceptible to overvoltages.

The fact that the individual harmonic currents vary as the inrush current decays, explains one of the most typical characteristics of the overvoltages generated as a consequence of transformer energization. The maximum overvoltages often occur during the decay of the inrush current and not immediately after energization (when the individual harmonics attain their maximum values).

References for this subsection

[1] G. Sybille, M.M. Gavrilovic, J. Belanger, and V.Q. Do, “Transformer saturation effects on EHV system overvoltages”, IEEE Trans. on Power Apparatus and Systems, vol. 104, no. 3, pp. 671-680, March 1985.

[2] R. Yacamini and A. Abu-Nasser, “Transformer inrush currents and their associated overvoltages in HVDC schemes”, IEE Proc., vol. 133, Pt. C, no. 6, pp. 353-358, Sept. 1986.

[3] H.S. Bronzeado, P.B. Brogan, and R. Yacamini, “Harmonic analysis of transient currents during sympathetic interactions”, IEEE Trans. on Power Systems, vol. 11, no. 4, pp. 2051-2056, November 1996.

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2.3. Sympathetic interaction [HB]

Sympathetic interaction in transformers can occur when a transformer or shunt reactor is energized onto a system with long transmission lines in the presence of other electrically close and energized transformers or shunt reactors. In the case of shunt reactors this is unlikely as the saturation level is comparatively high. Sympathetic interaction changes significantly the duration and the magnitude of the transient magnetizing currents in the transformers involved [1-3].

In practice, transformers are energized in series or in parallel with other transformers already in service. On systems with appreciable series resistance, this inrush transient may trigger a transient interaction between the transformer being energized and those that are already in operation. This occurs because the transformers already in service saturates. This saturation is produced by the asymmetrical voltage waveforms on the system busbar due to the asymmetrical voltage drop across the series resistance of the system caused by the inrush current.

The occurrence of saturation of transformers already in operation during the inrush transient of a transformer being energized (sympathetic interaction) was first reported by Hayward [4] in 1941 following field tests trying to establish the reason for false operation of transformer differential relays. It is reported that transient magnetic currents of higher magnitude could flow “not only in the transformer being switched but also in other parallel transformers”, and that these transient currents last longer, with the currents decaying at a much slower rate than it would be in transformers being switched-onto a system on which there were no other transformers connected to. In the 1980’s, this phenomenon appeared in Brazil in an SVC transformer connected to a 230kV busbar, which always tripped out by operation of transformer protection due to high level of dc magnetizing current when one of the 100 MVA, 230/69kV transformers of the same substation was switched onto the 230 kV busbar [1]. Similar occurrences were also noticed during the energization of large shunt reactors near SVCs and HVDC converters [5-7].

2.3.1. PARALLEL SYMPATHETIC INTERACTIONThe transient sympathetic interaction between paralleled transformers may be explained by using the circuit shown

in Figure 1. Parallel here means that only the transformer primary windings are connected in parallel to each other. The secondary windings are not. So, when the switch S is closed, the transformer T2 is energized and its primary winding is connected in parallel with that of the unloaded transformer T1 already connected to the system.

FIGURE 1 – CIRCUIT USED TO EXPLAIN THE SYMPATHETIC INTERACTION PHENOMENON

Figure 2 shows the transient currents which are representative of the simulations where T2 is energized in parallel with the already energized unloaded transformer T1. As can be seen in this figure, the sympathetic interaction phenomenon takes place between the two transformers, with the peaks of the offset magnetizing current i1

(sympathetic magnetizing current) and to the inrush magnetizing current i2 occurring in the opposite direction to each other on alternate half cycles. Note that the supply current is (Figure 2c) is the sum of the currents i1 and i2.

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The sympathetic interaction may be explained as follows: before the switch S being closed only the magnetizing current of the unloaded transformer T1 flows through the system. When the transformer T2 is energized a transient inrush current is drained from the generator (e.m.f.), which flows through the system. Due to the almost entirely unidirectional characteristic of this inrush current, the voltage drop across the series resistance Rs makes the voltage at the transformer's busbar (point of common coupling) asymmetrical. As the flux in a transformer is strictly proportional to the area (integral) of the voltage waveform on the transformer terminals, the flux generated in the transformer T1 begins to be asymmetrical by an amount which may be given by:

(1)

where ∆φ1 is the flux change per cycle in the transformer T1 and r1 is the T1 winding resistance. It should be noted that reactances (system and transformer) have not been taken into account in the above equation as they don’t affect the decay of dc flux.

As the transformer T1 is in service (steady-state), the offset flux in it is zero. Thus, the flux change per cycle ∆φ1 will

produce an increasingly offset flux in the transformer T1 driving it into saturation. As a consequence, a sympathetic magnetizing current i1 is produced in the transformer, increasing gradually from the steady-state value to a considerable magnitude when the transformer saturates. It should be noted that the polarity of the transformer saturation is determined by the sign of ∆φ1.

At the same time, the flux change per cycle ∆φ2 is produced in the transformer T2, which may be given by:

(2)

where r2 is the winding resistance of the transformer T2.

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FIGURE 2 - TRANSIENTS CURRENTS DURING SYMPATHETIC INTERACTION

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Since the polarity of ∆φ2 is opposite in sign to the initial flux offset in the incoming transformer T2 caused by its energizing, the effect of ∆φ2 is to reduce this initial flux offset, producing the well known phenomenon of inrush current decay discussed before.

From equations 1 and 2, it can be seen that at the beginning of the inrush transient both flux changes per cycle ∆φ 1

and ∆φ2 will depend mainly on the voltage drop caused by the inrush current i2. Initially, as the transformer T1 is not saturated, its magnetizing current i1 is very small (steady-state magnetizing current) and essentially symmetrical, hence it does not cause any appreciable flux change per cycle. When the transformer T1 becomes saturated, as it saturates with the opposite polarity to transformer T2, the peaks of the sympathetic magnetizing current i1 will occur with the polarity opposite to the peaks of the inrush current i2 (on alternate half cycles). As a consequence, the voltage asymmetry on the busbar caused by the inrush current i2 during one half-cycle is gradually reduced by the voltage drop produced by the sympathetic magnetizing current i1 during the subsequent half-cycle. This will decrease both flux change per cycle ∆φ1 and ∆φ2 reducing the rate of change of the magnitude of both the increasing current i1 and the decaying current i2. A few cycles later the flux change per cycle ∆φ1 reaches zero and hence i1 stops increasing. Within this cycle:

(3)

Thereafter, the polarity of the flux change per cycle ∆φ1 inverts, reducing the flux in the transformer T1. As a result, the sympathetic magnetizing current i1 begins to decay as does the inrush current i2. Under this condition, the

voltage on the busbar (transformers terminals) presents a waveshape nearly symmetrical with the flux change per cycle in each transformer depending basically on the winding resistance of each transformer, i.e.:

(4)

and

(5)

It is interesting to note that, in this case, the system resistance Rs plays the paradoxical role of keeping both transformers T1 and T2 saturated (on alternated half cycles), with the currents i1 and i2 being concomitantly cause and effect of saturation in the transformers. That is: the voltage drop across the system resistance Rs produced by the current i1 during one half cycle reduces the flux offset in transformer T1 and, at the same time, increases the flux offset in the transformer T2. In the subsequent half cycle is the current i2 that will produce the voltage drop across Rs, which increases the flux offset in the transformer T1 and, at the same time, reduces the flux offset in T2. This sequence repeats itself, developing the phenomenon of sympathetic interaction between the transformers, keeping them saturated for a long period of time. The currents i1 and i2 then will keep flowing for a prolonged period of time until the transformers reach their steady-state magnetizing conditions. This may take several seconds or perhaps minutes, depending essentially on the transformer winding resistances.

2.3.2. SERIES SYMPATHETIC INTERACTIONThis section discusses sympathetic interaction when the transformers are connected in series. Series here means that the primary winding of one of the transformers is connected to the supply system with its secondary winding feeding the transformer that is being energized (Fig. 1).

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Fig. 1 – Circuit used to explain the sympathetic interaction phenomenon in series transformers

Figure 2 shows the transient currents measured during laboratory tests with two small transformers, where Ts2 was energised in series with the unloaded transformer Ts1. As can be seen in this figure, the sympathetic interaction phenomenon takes place between the two transformers, with the current is in the transformer Ts1 (is = i1 + i’2) being equal to the inrush current of Ts2 reflected in the primary of Ts1 and the sympathetic magnetising current in the transformer Ts1 itself, i.e. is = i1 + i’2.

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Fig. 2. a) Current in the transformer Ts1 (is = i1 + i’2); b) Current in the transformer Ts2 (i2).

This phenomenon was also shown by simulating two identical three-phase transformers of 180 MVA, 275/66 kV, in series (Fig. 3). The results (Fig. 4) suggest that the interaction between series transformers is more than somewhat similar to the interaction occurring between the transformers in parallel.

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Fig. 3. Electrical system used in the simulation to investigate the sympathetic interaction between series transformers

Fig. 4. Transient currents calculated during a sympathetic interaction between two identical transformers of 180MVA, 275/66 kV connected in series.

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Note that when T2 is energized its transient inrush current i2 flows through the secondary of transformer T1

producing a corresponding “inrush current” i’2 in the primary of T1. This current (i’2) is added to the current i1 to yield the total current isys = i1 + i’2 which flows through the circuit formed by the supply system and the primary winding of T1. Thus:

(1)

It should be noted that transformer T1 “sees” the inrush current i2 as a load current. The flux change per cycle ∆φ1

in transformer T1 can be given by

(2)

where Rsys is the system resistance and rp1 is the resistance of the primary winding of T1, or

(3)

The flux change per cycle ∆φ1 will drive transformer T1 that was initially in the steady state into saturation, with the offset magnetizing current i1 increasing gradually until the flux change per cycle ∆φ1 becomes zero. In this condition,

(4)

From this point onwards, the flux change per cycle ∆φ1 inverts the polarity so that the offset magnetizing current i1

starts to decay, developing the sympathetic interaction between the series transformers T1 and T2. The rate of decay of the inrush current i2 and, consequently, the rate of decay of the ”primary inrush current” i’2 are essentially determined by the flux change ∆φ2 in transformer T2, which can be described by

(5)

or

(6)

where rs1 is the resistance of the secondary winding of T1 and rp2 is the resistance of the primary winding of T2

connected to the secondary of T1. It should be observed that when eqn. 4 is satisfied, the first two terms of eqn. 6 add to zero. In this condition, the flux change ∆φ2 will depend basically on the voltage drop across the total

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Zia.Emin, 10/04/12,

Add a prime on i2

resistance in the circuit formed by the secondary winding of T1 and the primary winding of T2. This indicates, paradoxically, that the total resistance in the primary side of transformer T1, i.e. Rsys + rp1, does not contribute effectively to the decay of the inrush current in transformer T2 during the sympathetic interaction.

This interaction is greatly reduced if the series resistance of the supply system is small and also if the resistance of the circuit between the transformers is relatively large. The latter could represent the case where transformers are separated by an appreciable length of transmission line.

References:

[1] BRONZEADO H. S., “Transfomer Interaction Caused by Inrush Current“, MSc Thesis, University of Aberdeen (Scotland), 1993.

[2] BRONZEADO H. S. and YACAMINI R, “Sympathetic Interaction Between Power Transformers”, Proc. of 29th Universities Power Engineering Conference -UPEC, Vol. 1, pp. 236-239, University College Galway, Ireland, Sept. 1994.

[3] BRONZEADO H. S., BROGAN P.B., YACAMINI R., “Harmonic Analysis of Transient Currents During Sympathetic Interactions”, IEEE Trans. On Power System, Vol. 11, No. 4, Nov. 1996.

[4] HAYWARD C. D., “Prolonged Inrush Currents with Parallel Transformer Affect Differential Relaying“, AIEE Trans., VoL 60, pp1096-1101, Jan. 1941.

[5] PUENT H. R, BURGES M. L, LARSEN E. V. and ELAI-IL H., “Energization of Large Shunt Reactors Near Static Var Compensators and HVDC Converters”, IEEE Trans. on Power Delivery, Vol. 4, No. 1, pp. 629-635, Jan. 1989.

[6] YACAMINI R and ABU-NASSER A., “Transfomer Inrush Currents and their Associated Overvoltages in HVDC Schemes”, IEE Proc., Vol. 133, Pt. C, No. 6, pp. 353-358, Sept. 1986.

[7] POVH D. and SCHULTZ W., “Analysis of Overvoltages Caused by Transformer Magnetizing Inrush Current”, IEE Trans. on Power Apparatus and System, Vol. PAS-97, NO. 4, pp. 1355-1365, Jul./Ago. 1978

[8] BLUME L F., CAMlLLI G., FARNHAM S. B. and PETERSON H. A., “Transformer Magnetizing Inrush Currents and Its Influence on System Operation”, AIEE Trans., Vol. 63, pp. 366-375, 1944.

3. Transient voltage distortion due to transformer energization

3.1. RMS-voltage drop [TM and ZE]

Transformers may be switched on and off for various reasons, such as to conduct equipment maintenance, to temporarily reconfigure the station bus by local or remote supervisory switching, protective relay operation, and sometimes due to protection misoperation or operator error. While network transformers are seldom switched, generator transformers may be switched more frequently depending on dispatch requirements. Hence, transformer energization or re-energization is a normal planned operation in an electric power system. Sometimes, energizing a transformer results in the transformer drawing a relatively large initial inrush current which decays over time to a much smaller steady state magnetizing current. The transient magnetizing current that occurs during transformer energization (the inrush current) is produced by transformer core saturation following switch-on. This current, which must be supplied from the system sources such as generators and large motors, flows through the network impedance to the transformer being energized. Consequently, there is a voltage drop across the network

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impedance and a drop in the line voltages where the effect increases in the direction towards the transformer. The time required for the inrush current to decay depends on losses in the circuit (including the resistive component of the system impedance and the transformer winding resistance), the transformer leakage and magnetizing impedances and the proximity to other significant energized transformers [1]. Since the magnetizing impedance of the transformer is high, the decay of the inrush current can take seconds to minutes.

Short duration abrupt voltage drops that are caused by faults, by large inrush currents produced by transformer energization or large starting currents during motor starting, are called voltage dips or sags, dip being the IEC terminology and sag, though not formally defined, being a common North American term. Typically, a voltage sag or dip is a decrease in system voltage to between 0.1 pu and 0.9 pu at power frequency, lasting from one-half cycle to one minute. Some high power electronic devices, such as adjustable speed drives, programmable logic-based process controls in the mining and pulp and paper industry, and electronic chip manufacturing plants are voltage sensitive. For such equipment, power quality problems associated with voltage sags are an important design issue. Voltage sags as short as 2 or 3 cycles can affect critical equipment, such as used in the electronic chip manufacturing, and adversely impact the production process. Outages due to poor power quality can have as detrimental impact as sustained power interruptions.

It is important to recognize that, for a power system that is operated prudently, voltage sag caused by transformer energization does not fall into the category of voltage fluctuation and should not be characterized by the flicker curve. Flicker is the impression of fluctuating luminance occurring when the supply to an electrically powered lighting source is subjected to voltage fluctuation [2]. The flicker curve is applicable to a frequency of voltage change events ranging from a few events per hour to 20 or more per second.

Unlike faults, transformer energization is a planned operational event for which the associated voltage sags must be limited to the withstand of voltage-sensitive industrial loads. Unfortunately, there is a lack of standards which quantify voltage sag withstand capability for high voltage industrial loads. The ITIC (Information Technology Industries Council) curve, formerly called CBEMA (Computer Business Equipment Manufacturers Association) curve [3] defines the over-voltage and under-voltage susceptibility for a very limited segment of voltage sensitive loads, namely, information technology equipment operating at 120 V nominal voltage. Utilities normally restrict the voltage sag in the network caused by the operation of their customers and, in the absence of recognized standards, have developed their own internal standards. Some examples of the limitations on rapid voltage change in use in a few countries are given below.

In Great Britain, if flicker-producing voltage fluctuation is not relevant, Engineering Recommendation P28 states that a maximum 3 percent voltage change limit applies, with no distinctions made as to the voltage level or the number of events in a given period. The grid code in France addresses limitation on rapid voltage change and states that the RMS voltage change must be less than 5% for the 63kV to 225kV network and less than 3% for the 400kV network. IEC 61000-3-7, Table 6, provides the planning levels for rapid voltage changes on public supply systems as a function of frequency of occurrence, taking into account all installations which may cause rapid voltage change. For HV/EHV, maximum voltage changes of up to 3% to 5% cannot exceed 4 events per 24 hours and voltage changes of up to 3% cannot exceed 2 events per hour. For MV systems, the permissible voltage changes are basically 1% higher than the HV/EHV levels. For events occurring between 2 per hour and 10 per hour, the voltage change for HV/EHV and for MV is stated as 2.5% and 3%, respectively. In Canada, limitations on rapid voltage change vary from province to province. For example, a Canadian utility allows voltage sags up to 3% of the nominal voltage at the point of common coupling with a frequency of occurrence not to exceed once per hour. Voltage sags from 3% to 6% up to once per eight-hour shift are allowed. With prior approval, this utility may allow a voltage sag of up to 9%. The voltage sag is calculated on a per phase basis using a sliding one-cycle RMS window. The most affected phase is used as the basis for quantifying the voltage sag event. Over the past fifteen years, a reasonably common issue of voltage sag due to transformer energization in this Canadian utility system has occurred with Transmission Customers and Independent Power Producers (IPPs), usually run-of-river hydro projects, who propose to connect relatively large projects in a weak part of the network. Reference [4] describes

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such a case where post-test EMT simulations of transformer energization were able to reproduce, with some success, the magnetizing inrush currents recorded during commissioning tests.

The case in appendix provides an example of the good agreement that can be achieved between EMT simulation and actual field recordings when the network and the transformer are modelled in appropriate detail. The example shows the RMS voltage sags caused by the energization of a pair of 400 kV non-identical generator step-up transformers. The EMT simulation of the maximum voltage sag in each phase agreed very closely with the field test. In order to achieve this agreement, a rather detailed system model was required. This model included dynamic voltage support (SVCs) and mechanically switched shunt capacitor banks. The simulated rates of recovery of the voltages were higher than the measurements, indicating that some further refinement in the resistive losses in the network impedance and transformers could be considered. Having verified the adequacy of the EMT model of the system and transformer for the purposes required, additional simulation studies enabled the selection of a functional and cost effective solution to reduce the worst-case voltage sag, considering a variety of network conditions, to acceptable levels.

References

[1] H.S. Bronzeado, P.B. Brogan and R. Yacamini, “Harmonic Analysis of Transient Currents During Sympathetic Interaction,” IEEE Trans. On Power Systems, vol. 11, no. 4, pp 2051-2056, Nov. 1996.

[2] Energy Networks Association, “Engineering Recommendation P28”, Issue 1, 1989, “Planning Limits for Voltage Fluctuations Caused by Industrial, Commercial and Domestic Equipment in the United Kingdon”.

[3] Federal Information Processing Standards Publications (FIPS) no. 94.

[4] M. Nagpal, T.G. Martinich, A. Moshref, K. Morison and P. Kundur, “Assessing and Limiting Impact of Transformer Inrush Current on Power Quality,” IEEE Trans. On Power Delivery, vol. 21, no. 2, pp890-896, April 2006.

3.2. Resonant overvoltages due to network parallel resonances [JAMV]

Description of the phenomenon: Transformer energization creates transient inrush currents when the transformer’s core becomes saturated. Transformer inrush currents can have a high magnitude with a significant harmonic content. The inrush currents interact with the power system, whose resonant frequencies are a function of the series inductance (associated with the short circuit strength of the system) and the shunt capacitances of lines and cables. This may result in long-duration resonant temporary overvoltages (TOV). Higher inductances (relatively weak systems) and higher capacitances (long lines and cables) yield lower resonant frequencies and a higher chance of TOV, which can have hundreds of peak of about the same magnitude if the TOV duration is several seconds [1].

A transmission system will generally be weak during the first steps of a system restoration following a black out. The equivalent system inductances are then relatively high because relatively few generators are on line and the grid tends to be sparse. Therefore, the first system resonant frequency can be much lower than during normal system operation. Large capacitances also contribute to the low resonant frequencies. One of the major concerns during the early stages of a power system restoration is the occurrence of overvoltages as a result of switching procedures [2]. Energizing equipment during black start conditions can result in higher overvoltages that during times of normal operation.

Harmonic resonance voltages are oscillatory undamped or weakly damped temporary overvoltages (TOVs) that originate from switching operations and equipment non-linearities [2], and are a result of network resonance frequencies multiples of the fundamental frequency. They can be excited by harmonic sources, such as saturated transformers or power electronics, and lead to long lasting overvoltages resulting in arrester failures and system faults and prolong system restoration [2].

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During a restoration phase, the capacitive voltage rise due to charging currents can be sufficient to overexcite transformers and generate significant harmonics. If the combination of the system impedance and the line capacitance is adverse, then a harmonic resonance will result. Harmonic distortions produced by transformer saturation will excite these resonances, which can result in damaging overvoltages. Even if transformers are not continuously overexcited, harmonics generated by magnetizing inrush on energization can be sufficient to excite the resonance. They result from several factors that are characteristic of networks during restoration [2]: (i) the natural frequency of the series circuit formed by the source inductance and line charging capacitance may, under normal operating conditions, be a low multiple of the power frequency; (ii) magnetizing inrush caused by energizing a transformer produces many harmonics; (iii) during early stages of restoration the lines are lightly loaded and transients therefore are lightly damped, which means that the resulting resonance voltages may be very high. In other words, resonance overvoltages can result when transformer energization is made from a supply source with a natural frequency equal or close to a harmonic frequency present in the inrush current. If transformers become overexcited due to power frequency overvoltage, harmonic resonance voltage will be sustained or even grow.

Generator step up (GSU) transformer energizing is often part of a black start analysis. When generator circuit breakers are used to connect auxiliary transformers to the medium voltage generator bus, the GSU transformer can be energized directly from the transmission system, which may initiate a significant TOV [1].

Harmonic resonance overvoltages may also develop when transformers are switched in high voltage cable systems and HVDC stations [3]. Unlike overhead-line systems, cable systems generally have a pronounced resonance point, which occurs at a relatively low frequency because of the high cable capacitance; if this resonance happens to coincide with one of the harmonics produced during transformer energization, TOVs will build up. The AC filter circuits connected at the HVDC stations produce several parallel resonance points in the impedance-frequency characteristic of the system [4], so high saturation overvoltages may occur if the system also has a low degree of damping. This is the case where generators feed the HVDC stations direct (i.e., without local AC loads being connected), although field measurements show that not every system configuration of this kind produces high TOVs.

Resonance Overvoltages During Transformer Energization: An harmonic analysis can be carried out by representing an inrush current as a harmonic current source I(h) connected to the transformer bus. The relation between nodal voltages, network impedance matrix and current injections can be then analyzed by means of impedance equations [5]:

V (h )=Z (h) I (h)(1)

where h represents the harmonic frequencies (multiples of the fundamental frequency), and Z(h) is a symmetrical matrix with as many rows and columns as the harmonic currents. V(h) and I(h) are respectively the vectors of harmonic voltages and currents.

The harmonic current components of the same frequency as the system resonance frequencies are amplified in case of parallel resonance, thereby creating high voltages at the transformer terminals, see previous section. This leads to a higher level of saturation resulting in higher harmonic components of the inrush current which again results in increased voltages. This can happen particularly in lightly damped systems, common at the beginning of a restoration procedure when a path from a blackstart source to a large power plant is being established and only a few loads have been restored [2], [6].

The diagram shown in Figure 3 is used to illustrate a condition that can lead to harmonic resonance overvoltages. It is a very simplified representation of a power system at the early stages of a restoration procedure in which the analysis is concentrated on the energization of transformer that is assumed to be unloaded. The plots of Figure 4 show some simulation results. A conclusion from a frequency scan from the point of connection of the transformer is that the impedance seen from this bus shows a parallel resonance peak at the second harmonic. When the transformer is energized, this resonance condition results in the overvoltage depicted shown in Figure 4c.

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TransformerPower system

equivalent

Figure 3 Diagram of the test system.

Frequency ( Hz)

0 100 200 300 400 500

Imp

eda

nce

a) Impedance at transformer bus

b) Transformer current during energization

c) Transformer terminal voltage

Figure 4. Energization of an unloaded transformer.

The effect of the system resonance frequency, network damping and inrush current on the harmonic overvoltages that can occur in high-voltage cable and HVDC systems were examined in [3]. Both systems are likely to have a low resonance frequency. For high-voltage cable systems, this results from the high charging power and the fact

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that the fault level is still low in the initial stage of construction. The frequency response of such a system is of the single-resonance-frequency type with a quality factor of about 15. In HVDC systems, the high charging power of the filter circuits consisting of tuned filters for 5th, 7th, 11th and 13th harmonics produces parallel resonance at a low frequency. In addition, parallel resonance points may also occur between the filter frequencies. Under conditions of resonance, the magnitude of the overvoltage is a function of the system damping. Therefore, losses of transformers, generators, filter circuits and other system loads must be determined and specified as accurately as possible for the frequency range under consideration.

The key factors for analysis of harmonic overvoltages include the resonance frequency of the network, the system damping (including the network losses and the load connected to the network), the voltage level at the transformer terminals, the saturation characteristic and air core inductance, the remnant flux in the core of the transformer, and the closing time of the circuit breaker pole. Factors that contribute to a higher level of resonance overvoltage are [8]: (1) higher rating of the transformer to be energized; (2) lower value of source fault level; (3) longer circuit length; (4) smaller amount of load in the system; (5) higher than normal system operating voltage; (6) higher designed working flux density of the transformer; (7) position of the tap resulting in lower turns ratio.

References for this subsection

[1] D. Durbak, “Temporary overvoltags following transformer energizing”, Siemens PTI Newsletter, Issue 99, pp 1-3, September 2006.

[2] M.M. Adibi, R.W. Alexander, and B. Avramovic, “Overvoltage control during restoration” , IEEE Trans. on Power Systems, vol. 7, no. 4, pp. 1464-1470, November 1992.

[3] D. Povh and W. Schultz, “Analysis of overvoltages caused by transformer magnetizing inrush current”, IEEE Trans. on Power Apparatus and Systems, vol. 97, no. 4, pp. 1355-1365, July/August 1978.

[4] J.P. Bowles, “Overvoltages in HVDC transmission systems caused by transformer magnetizing inrush currents”, IEEE Trans. on Power Apparatus and Systems, vol. 93, no. 1, pp 487-493 January/February 1974

[5] D. Lindenmeyer, H.W. Dommel, A. Moshref, and P. Kundur, “Analyis and control of harmonic overvoltages during system restoration”, IPST’99, June 20-24, 1999, Budapest.

[6] G. Morin, “Service restoration following a major failure on the Hydro-Quebec power system”, IEEE Trans. on Power Delivery, vol. 2, no. 2, pp. 454-463, April 1987.

[7] A. Ketabi, A.M. Ranjbar, and R. Feuillet “Analysis and control of temporary overvoltages for automated restoration planning”, IEEE Trans. on Power Delivery, vol. 17, no. 4, pp. 1121-1127, October 2002.

[8] C.P. Cheng and S. Chen, “Simulation of resonance over-voltage during energization of high voltage power network”, Electric Power Systems Research, vol. 76, pp. 650-654, 2006.

[9] T. Hayashi et al., “Modeling and simulation of black start and restoration of an electric power system. Results of a questionnaire”, Electra, no. 131, pp. 157-169, July 1990.

[10] G. Sybille, M.M. Gavrilovic, J. Belanger, and V.Q. Do, “Transformer saturation effects on EHV system overvoltages”, IEEE Trans. on Power Apparatus and Systems, vol. 104, no. 3, pp. 671-680, March 1985.

[11] O. Bourgault and G. Morin, “Analysis of a harmonic overvoltage due to transformer saturation following load shedding on Hydro-Quebec - NYPA 765 kV interconnection”, IEEE Trans. on Power Delivery, vol. 5, no. 1, pp. 397-405, January 1990.

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4. Mitigation techniques and best practices [NC + JAMV] Mitigation techniques inventory: Controlled switching, shunt reactances, closing resistors, surge arresters,

network modification, local magnetization, loads, transformer taps… What is your experience (simulation and/or field)? What is the effectiveness of each technique? What are

the practical problems?

[MM: references ‘CBXX’: are Nicola’s, whereas references XX are Martínez Velasco’s, in §3.2]

Several technologies exist to mitigate energization transients in transformers. Reduction of mechanical and electrical stresses, as well as mitigation of TOVs can be achieved using mitigation techniques.

The methods that have been proposed to prevent harmonic resonance overvoltages are listed below [2], [5], [7], [10] - [12].

Adding as much load as possible before energizing the transformer

This leads to a decrease in the magnitude of the impedance and, consequently, to a reduced amplification of the injected harmonic currents. To assure that resonance is damped, sufficient load should be connected to the underlying system at both ends of line to damp the resonance. Analysis for a 500kV line has shown that a load of about two megawatts per km is adequate [2].

Selecting a low impedance path for energization of the transformer

A high source impedance can be reduced by bringing additional generators online since a higher number of generators results in a lower overall inductance and, consequently, in a higher resonance frequency. This means that if generators are added, the resonance peak is shifted to higher frequencies and if generators are omitted, it is shifted to lower frequencies.

Reducing transformer flux density

Harmonic overvoltages caused by over-excitation of transformers can be controlled by selecting a transformer tap which equals or exceeds the power frequency voltage applied before energizing. The influence of the energized transformer tap changer on an inrush current is investigated in [BP9]. Higher number of excited turns gives lower flux density in the core. Changing from the 0.9 p.u. to the 1.1 p.u. tap position results in almost a 50% reduction in the inrush current first peak.

Reducing the system voltage before energizing the transformer

Decreasing a generator’s scheduled voltage leads to a proportional decrease of the pre-switching steady-state voltages. An alternative to generator dynamic voltage regulation would be the reduction of voltage after the generator step up transformer by the appropriate tap position to obtain the minimum voltage on the secondary side. Appropriate switching of reactive power compensation equipment could also be considered to reduce the system voltage. However, with the latter, care should be taken to avoid creating harmonic resonances at undesired frequencies.

Installing pre-insertion resistors in parallel to the circuit breaker energizing the transformer

Pre-insertion resistors require relatively large resistors to be installed in parallel to the main circuit breaker and an effective reduction of inrush current is achieved only by an optimal choice of the resistance value and the pre-insertion time. In addition, circuit breakers with pre-insertion resistors requires significant maintenance over their lifetime and there are concerns on the reliability of these CBs.

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Controlling the switching times of the circuit breaker energizing the transformer

Controlling the switching time (the closing point on the voltage waveform and the delay between individual phases) is another effective method, which can reduce or even eliminate inrush currents during energization.

The first commercial controlled breakers were adopted in the 1990’s, with few installation existent in 1995. A larger adoption occurred at the end of 1990’s thanks to the introduction of effective compensation algorithms for the variation of the operation timing. Controlled switching technology is mainly used for insertion of capacitor (66%). Fewer installations are reported for the energization of transformer and reactors, and an insignificant quantity for line switching [BP7]. Controlled switching is used to energize transformers to mitigate the inrush currents and avoid power quality degradation, false relay tripping, and reduction of internal mechanical stresses. The most common controlled switching strategy implies to energize a first phase at its voltage peak and delay the energization of the remaining two phases by a quarter of a period of the fundamental frequency (5 ms in a 50 Hz system). It should be noted that switching at voltage peak might lead to higher slow front transient overvoltages. Commercial system taking into account residual fluxes are becoming available. Controlled de-energization is also proposed as a mean to control residual fluxes.

The state of the art of controlled switching with and without the consideration of residual fluxes is given in [BP1,BP10]. This publication also discusses the practical calculation of the residual fluxes from integration of voltage waveforms and the possible influence of network disturbances.

Three different synchronized switching strategies (rapid closing, delayed closing and simultaneous closing) are extensively examined and discussed in [BP11, BP12]. The work has been followed up in [BP13] and experimental results are presented. Synchronized switching with delayed closing and rapid closing outperform the proposed simultaneous closing strategy. The rapid closing technique is normally preferred as it avoids possible resonance issues thanks to a minimal delay in the closing all three poles of the CB.

The application of controlled switching with measurement of residual fluxes is reported in several publications in the last 10 years [BP1,BP5,BP6,BP14,BP15]. Closing time scatter and residual flux measurement uncertainties reduce the field performance of controlled switching as reported in [BP16].

Other methods

Alternative methods for inrush current mitigation are continuously been proposed. A method for controlling the POW closing instant and defining a value of residual flux using a dc coil is proposed in [BP8].

An inrush mitigation strategy based on a pre-insertion neutral resistor is presented in [BP17]. This technique is however restricted to star grounded transformers.

Prikler et. al. [BP4] present a synchronized switching method for three-pole spring driven circuit breaker with fixed delay between poles (mechanically staggered). Optimum energization is obtained by controlling the residual flux with controlled de-energization.

Chiesa and Høidalen [BP18] propose a method based on simultaneous-closing strategy. The method is valid for all most common transformer coupling. The optimum energization instant is calculated based on the disconnection instant of the circuit breaker.

The inrush currents during IPP transformer energization must be controlled so as to reduce the unacceptable RMS voltage dip and the temporary overvoltage during recovery of the RMS voltages. There are a few possible approaches that can accomplish this:

Transformer energization by segregated-pole breakers equipped with point-on-wave (POW) control can minimize inrush currents and is very effective, particularly in cases where voltage dip can be high [8]. There are special purpose commercially available relays which calculate the residual flux in the core based

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on measurements from the previous de-energization of the transformer. The closing characteristics of the breakers must be stable and repeatable, with a relatively small pole scatter. The POW relay will account for the closing characteristics of the breaker and, together with the estimated residual flux, will close each pole at the optimal closing point on the voltage wave. With the modern generation of POW controllers, the transformer need not be switched off in a particular phase sequence for the residual flux calculation. The cost of three independent-pole breakers is significantly higher than the cost for a single three-pole breaker.

The use of pre-insertion resistors in the closing device will reduce the inrush current. The rating of the resistors and the insertion duration have to be carefully calculated. There are maintenance issues associated with pre-insertion devices that should be considered.

Energizing the transformer using disconnect switches instead of a breaker helps to reduce inrush currents. A disconnect switch is a relatively slow device so its contacts will pre-strike or flash over close to voltage peak. However, with the arc in air, there could be multiple extinctions and re-ignitions of the current. High frequency transients could be imposed on the transformer windings and this could stress the transformer insulation. Also, the disconnect switch contact might weld closed as a result of the arcing process unless the disconnect is rated for this application.

Energizing the transformer at a reduced voltage, while effective when voltage dip is only marginally above acceptable limits, might not be practical. It will probably not be sufficient to correct very severe voltage dip problems.

De-fluxing the transformer core prior to energization will reduce the worst-case inrush current magnitude but this technique, for most applications, is not very practical.

References

[BP1] Cigré Brochure 263, “Controlled switching og HVAC circuit breaker”, WG A3.07, 2004.

[BP2]

[BP3]

[BP4] L. Prikler, G. Banfai, G. Ban, and P. Becker, “Reducing the magnetizing inrush current by means of controlled energization and de-energization of large power transformers,” Electric Power Systems Research, vol. 76, no. 8, pp. 642–649, May 2006.

[BP5] H. S. Bronzeado, J. C. Oliveira, R. Apolonio, and A. B. Vasconcellos, “Transformer inrush mitigation - Part I: modeling and strategy for controlled switching,” in Cigré international technical colloquium, C. SCA3, Ed., Rio de Janeiro, Brazil, Sep. 12–13 2007.

[BP6] H. S. Bronzeado, S. Oliveira Pinto, J. C. Oliveira, M. L. R. Chaves, and P. Jonsson, “Transformer inrush mitigation - Part II: Field test on a 100MVA threephase transformer,” in Cigré international technical colloquium, C. SCA3, Ed., Rio de Janeiro, Brazil, Sep. 12–13 2007.

[BP7] H. Ito, “Controlled switching technologies, state-of-the-art,” in Proc. Transmission and Distribution Conference and Exhibition 2002: Asia Pacific. IEEE/PES, vol. 2, Oct. 6–10, 2002, pp. 1455–1460.

[BP8] P. C. Y. Ling and A. Basak, “Investigation of magnetizing inrush current in a single-phase transformer,” IEEE Trans. Magn., vol. 24, no. 6, pp. 3217–3222, Nov. 1988.

[BP9] H. S. Bronzeado and J. C. de Oliveira, “The influence of tap position on the magnitude of transformer inrush current,” in IPST’99 - International Conference on Power System Transients, Budapest, Hungary, 1999, pp. 457–61.

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Zia.Emin, 10/04/12,

Text that came from TM example and needs to be blended here.

[BP10] Controlled switching of HVAC CBs - Planning, Specifications & Testing, CIGRÉ SC A3 WG A3.07, Jan. 2004.

[BP11] J. H. Brunke and K. J. Frohlich, “Elimination of transformer inrush currents by controlled switching. I. Theoretical considerations,” IEEE Trans. Power Del., vol. 16, no. 2, pp. 276–280, April 2001.

[BP12] J. H. Brunke and K. J. Frohlich, “Elimination of transformer inrush currents by controlled switching. II. Application and performance considerations,” IEEE Trans. Power Del., vol. 16, no. 2, pp. 281–285, April 2001.

[BP13] U. Krusi, K. Frohlich, and J. H. Brunke, “Controlled transformer energization considering residual flux-implementation and experimental results,” in Proceedings of the IASTED International Conference Power and Energy Systems, Anaheim, CA, USA, 2001, pp. 155–60.

[BP14] A. Mercier, E. Portales, Y. Filion, and A. Salibi, “Transformer controlled switching taking into account the core residual flux - a real case study,” in CIGRÉ, Session 2002, no. 13-201, 2002.

[BP15] H. Kohyama, K. Kamei, and H. Ito, “Application of controlled switching system for transformer energization taking into account a residual flux in transformer core,” in CIGRE SC A3&B3 Joint Colloquium in Tokyo, 2005.

[BP16] A. Ebner, M. Bosch, and R. Cortesi, “Controlled switching of transformers - effects of closing time scatter and residual flux uncertainty,” in Proc. 43rd International Universities Power Engineering Conference UPEC 2008, Sep. 2008, pp. 1–5.

[BP17] S. G. Abdulsalam and W. Xu, “A sequential phase energization method for transformer inrush current reductiontransient performance and practical considerations,” IEEE Trans. Power Del., vol. 22, no. 1, pp. 208–216, Jan. 2007.

[BP18] N. Chiesa, H. K. Høidalen, “Novel Approach for Reducing Transformer Inrush Currents: Laboratory Measurements, Analytical Interpretation and Simulation Studies,” IEEE Trans. Power Del., vol.25, no.4, pp.2609-2616, Oct. 2010.

5. Simulating transformer energization transients [MM]Transformer energization simulation studies provide a significant insight into the phenomenon and is quite desirable due to several interrelated reasons. As energization transients are heavily dependent upon initial conditions, field measurements may not always provide enough information to assess the potential consequences of an energization transient. Due to operational restrictions, it may not always be possible to perform measurements for various running arrangements and system contingencies. At the extreme, measurement conditions may actually be such that the resultant transient is higher than equipment dielectric withstand capability and therefore leading to equipment damage under a field test condition. Therefore, in order to acquire as much information as possible for a set of initial condition, system running arrangements as well as system contingencies without the risk of damaging any equipment it is desirable to perform simulation studies. Furthermore, at network planning stage it is not possible to perform field tests and simulation studies are the only available tool to predict the likely effect of transients. They are equally applicable when choices about several mitigation measures are available for implementation so that the best and most cost effective solution can be identified by way of simulation studies which account for the effect of various system conditions and parameters.

However not every possible transformer energization configuration needs to be studied by simulation as this is not practically possible and most of the system configurations are such that the energization of the transformer will not cause any problem. Hence, the practicing engineer needs to decide which system and transformer configurations could be dangerous and require studying. It is difficult to provide a general and reliable rule of thumb for this, and the decision largely relies on the experience of the engineer and his knowledge of the system.

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In general, two cases need to be considered: RMS-voltage drop and resonant overvoltages. For the RMS-voltage drop, a rough estimate of the drop can be computed from the short-circuit impedance of the supply system and the inrush current analytically calculated (see Appendix). The higher the short-circuit impedance and the higher the rated power of the transformer (for a given rated voltage), the higher the RMS-voltage drop. It should be noted that, if other similar transformers are connected nearby, the user might be able to assume that the voltage drop will be similar based on operational experience. The prediction of resonant overvoltages is more complex. As a general approach, the frequency response of the feeding network and especially the resonant frequencies and their impedances should be considered. If any of the resonance frequencies is below 600-800 Hz, then resonant overvoltages may be generated, with higher probability for larger resonance impedances. For instance, the system in §5.3.4 is potentially dangerous, as its first resonance frequency is close to 200 Hz. On the other hand, for a given voltage rating, the higher the rated power of the transformer, the higher the inrush currents and the higher the risk of resonant overvoltages.

5.1. Electromagnetic modeling of the network

General guidelines to be used for representing system components are those recommended for analysis of low-frequency or slow-front transients [CIGRE TB 39][IEC 71-4][IEEE Gole et al.]. The frequency of interest for transformer energization studies ranges from DC up to 1kHz [Cigre TB 39].

5.1.1. STUDY ZONE [MM]A common practice in electromagnetic transient studies when dealing with large systems is to divide the system into a study zone, where transient phenomena occur, and an external system that encompasses the rest of the system.

In the study zone the system must be modeled in detail. Besides the target transformer and the circuit-breaker energizing it, the study zone includes the equipment in the same or neighboring substations that may interact with the inrush harmonic currents and associated voltages: other transformers (sympathetic interaction, see §2.3), reactors and capacitors, synchronous compensators, generators and voltage regulators, loads, etc. and the lines or cables between them [1].

For the rest of the system, a network equivalent may be used in order to reduce the time needed to build the model and the computational burden. The network equivalent must reproduce the frequency response of the network in the range of the inrush currents frequencies [1][2], especially when simulating resonant overvoltages (it is less critical when simulating RMS-voltage drop). For network equivalent modelling, see §5.1.3.4.

[1] Iravani, R. (chair), A.K.S. Chandhury, I.D. Hassan, J.A. Martinez, A.S. Morched, B.A. Mork, M. Parniani, D. Shirmohammadi, R.A. Walling, “Modeling Guidelines for Low Frequency Transients”, in A. Gole, J.A. Martinez-Velasco and A.J.F. Keri (Eds), Modeling and Analysis of System Transients using Digital Systems, IEEE Special Publication, TP-133-0, 1998.

[2] CIGRE WG 33.02, “Guidelines for Representation of Network Elements when Calculating Transients”, CIGRE TB 39, 1990.

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5.1.2. TRANSFORMERS [NC]

A. Physical aspects to be represented

The transformer being energized has to be represented in sufficient detail so that it generates the inrush currents that are the primary source of current harmonics injection in the system. Other transformers in the study zone have to be modeled accurately but do not require the same level of detail unless sympathetic inrush is foreseen. An exact representation of transformers is more complex than any other network component required for energization studies. According to CIGRE TB 39 [T1], IEC 71-4 [T2], CIGRE WG 33.10 [T3] and IEEE PES [T4], when studying transformer energization and resonance phenomena, the following aspects need to be represented (in order of importance):

leakage impedance and winding resistance nonlinear saturation and core losses air-core inductance magnetic phase coupling residual flux frequency dependent winding losses zero sequence impedance hysteresis and frequency dependent iron losses capacitances.

Leakage impedance and winding resistance

The leakage impedance, also short-circuit impedance, and the winding resistance define the series impedance of the transformer. While the winding resistance is specific for winding, a particular winding does not have a leakage inductance; the leakage inductance is defined from one winding to another.

Nonlinear saturation and core losses

Transformer saturation is an important component of many low frequency electromagnetic transient phenomena, including ferroresonance, temporary overvoltages, and transformer inrush. For most phenomena, the critical transformer saturation parameters are the final slope of the saturation curve (air-core inductance) and the value of the saturation flux (the point where the final slope intercept the zero current axis). Other details of the transformer saturation curve (i.e. the shape in the knee area) are of secondary importance for the study of the worst case scenarios. The location of the core representation in the transformer model topology is also important. Core losses can also be critically important in some low-current phenomena involving saturation, such as ferroresonance and temporary overvoltages.

The transformer core has an intrinsic nonlinear behavior due to the saturation of the ferromagnetic material. This nonlinear characteristic is the prime source of current harmonic injection from a transformer and need to be modeled accurately. Core losses are inherent in the hysteretic behavior of ferromagnetic material.

Air-core inductance

The air-core inductance of a coil is the self-inductance of the coil without the iron-core. It defines the inductance of a winding when the core is completely saturated. It is an important information to extrapolate the nonlinear saturation curve for studies that involve core saturation as energization transients.

Magnetic phase coupling

In three-phase transformers with three-legged or five-legged stacked core and shell core, the coupling between phases is provided through the magnetic core. Therefore, the representation of the core topology is important in three-phase transformers for simulations where phase unbalance, zero sequence, and magnetic phase coupling

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have to be taken into account. Simple, single-phase equivalent models cannot represent the core topology3; an advanced three-phase transformer modeling approach is presented in section D.

Residual flux

The residual flux value is a fundamental parameter during the re-energization of a transformer since it affects the first peak of the inrush current. Due to the flat nature of the saturation curve, a small increase of flux peak (residual flux) can drive the iron core of the transformer into heavy saturation. The residual flux is created when a transformer is disconnected from the power grid. At the end of a de-energization transient both the voltages and currents decrease to zero, however the flux in the core retains a certain value defined as residual flux. A ringdown transient is a natural LC response that appears as the stored energy dissipates whenever a transformer is deenergized4.Due to the ringdown transient the residual flux is always somewhat lower than the theoretical maximum residual flux equal to the interception of the hysteresis curve with the zero current axes (also known as remnant magnetization).

Frequency dependent winding losses

Frequency dependent winding resistance model should be used if the representation of the damping of transients and resonance conditions is important. Series winding losses are frequency dependent due to eddy currents induced in the winding and in other transformer components.

Zero sequence impedance

The zero sequence impedance of a transformer is determined by its delta windings and core configuration. While the delta winding provides linear zero-sequence impedance, the core usually involves nonlinearities. In general, it is not considered accurate to assume that the presence of a delta winding is sufficient to represent inter-winding coupling in a transformer. Both winding coupling and core topology should be represented to obtain an accurate representation of the zero-sequence and unbalanced operation behaviors.

Hysteresis and frequency dependent iron losses

The main limitations of piecewise RL core representation (with linear R and piecewise-nonlinear L) are:

It cannot represent the nonlinear core losses due to the linear R.Core losses are accurate only at rated excitation. Losses are overestimated at excitation lower than rated and underestimated at higher excitation.

It cannot retain residual flux after de-energization; residual flux is always zero. It cannot represent the frequency-dependent and frequency-independent components of the core losses.

Hysteresis and frequency dependent iron losses should be represented to avoid these limitations.

Stray Capacitances

Capacitances are only of marginal importance in the study of low-frequency phenomena which includes transformer energization. However, for energization studies, their representation is important if de-energization

3 Note to advanced users: In order to respect to a certain degree the topology of the core, it is suggested to connect the nonlinear inductance in delta for three-phase core-type transformers, and in wye for three-phase shell transformers and banks of single-phase transformers. The coupling of the core equivalent representation does not need to agree with the coupling of the winding it is attached to. Care should be taken to ensure that the saturation characteristic is rescaled if the winding and core coupling are different.

4N. Chiesa, A. Avendano, H. K. Høidalen, B. A. Mork, D. Ishchenko, and A. P. Kunze, " On the ringdowntransient of transformers", International Conference on Power SystemsTransients (IPST’07) in Lyon, France on June 4-7, 2007

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transients have to be simulated to calculate residual flux. In addition, they play an important role in the study of ferroresonance . They also influence the measurement of transformer excitation characteristic (no-load test).

B. Available models

In its simplest form, a transformer is represented by mean of a single-phase equivalent circuit. Such representation is adequate for single-phase and bank of single-phase transformers. However, since it lacks of a suitable representation of the magnetic coupling of the phases in the core, it can only give approximate results for three-phase transformer with core and shell core types.

This section does not cover all transformer models, but the two most used models as they are implemented in most transients programs. The first model is an extension of the Steinmetz model to multi-phase transformers. The second model uses either a branch impedance or admittance matrix. Both types of models present important limitations for simulating some core designs.

Saturable Transformer Component (STC Model)

Characteristics:

It cannot be used for more than 3 windings. The magnetizing inductance is connected to the star point. Numerical instability can be produced with 3-winding models.

The Saturable Transformer Component (STC)5,6,7is a two- and three-winding single phase transformer model. It is the nonlinear version of the classic Steinmetz model8. This model is based on the star-circuit representation shown in Fig. 2. Saturation and hysteresis effects are modeled by adding an extra nonlinear inductor at the star point. Alternatively, the core representation can be connected to the terminal of the winding closest to the core (normally the LV winding).The primary branch is treated as an uncoupled R-L branch, each of the other windings being handled as a two-winding transformer. It represents the short-circuit impedances between windings, the load and magnetization losses, and the nonlinear inductive magnetization.

5 J. A. Martinez and B. A. Mork, “Transformer modeling for low- and mid-frequency transients - a review,” IEEE Trans. Power Del., vol. 20, no. 2 II, pp. 1625–1632, 2005.

6 M. R. Iravani, A. K. S. Chaudhary, W. J. Giesbrecht, I. E. Hassan, A. J. F. Keri, K. C. Lee, J. A. Martinez, A. S. Morched, B. A. Mork, M. Parniani, A. Sharshar, D. Shirmohammadi, R. A. Walling, and D. A. Woodford, “Modeling and analysis guidelines for slow transients. III. The study of ferroresonance,” IEEE Trans. Power Del., vol. 15, no. 1, pp. 255–265, 2000.

7 H. W. Dommel and et.al., Electromagnetic Transients Program Reference Manual (EMTP Theory Book). Portland, OR: Prepared for BPA, Aug. 1986.

8C. P. Steinmetz and E. J. Berg, Theory and calculation of alternating current phenomena. New York: Electrical World and Engineer, inc., 1897.

Page 32

LM

L2 R2R1

RM

L1

L3 R3

N1:N2

N1:N3

LMRM

Fig. 2: Star-circuit representation of single-phase three-winding transformers. Alternative location of the core representation in gray.

For two winding transformers, the winding resistance and leakage inductance are artificially split between primary and secondary winding. A 50% splitting factor is often used. However, it should be more accurate to put 75%-90% of the leakage inductance on the HV side and the rest on the LV side9,10, as the leakage inductance between the HV winding and the core is normally larger than the one between the LV winding and the core.

The STC model can be extended to three-phase units through the addition of a zero-sequence reluctance parameter, but its usefulness is limited. In addition, when it is used to model three-phase transformers, the inter-phase magnetic coupling is not represented in this model therefore posing a great limitation in studing unbalanced operation.

The input data consist of the R-L values of each star branch, the turn ratios, and the information for the magnetizing branch. This model has some important limitations: it cannot be used for more than three windings, since the star circuit is not valid for N>3, the magnetizing inductance Lm with resistance Rm, in parallel, is connected to the star point, which is not always the correct topological connecting point, and numerical instability has been reported for the three-winding case, although this problem has been identified as the use of a negative value for one short-circuit reactance11,12.

Matrix Representation (BCTRAN Model)

Characteristics:

This model includes all phase-to-phase coupling and terminal characteristics. Only linear models can be represented.

9 Martinez, J.A.; Walling, R.; Mork, B.A.; Martin-Arnedo, J.; Durbak, D.; , "Parameter determination for modeling system transients-Part III: Transformers," Power Delivery, IEEE Transactions on , vol.20, no.3, pp. 2051- 2062, July 2005.

10 The the leakage inductance between the HV winding and the core is normally larger than the one between the LV winding and the core. In energization studies this results in a larger inrush current in pu on the LV side than on the HV side as observed in practice.

11 X. Chen, “Negative inductance and numerical instability of the satura-ble transformer component in EMTP,” IEEE Trans. Power Del., vol. 15, no. 4, pp. 1199–1204, Oct. 2000.

12 T. Henriksen, “How to avoid unstable time domain responses caused by transformer models,” IEEE Trans. Power Del., vol. 17, no. 2, pp. 516–522, Apr. 2002.

Page 33

Excitation may be attached externally at the terminals in the form of non-linear elements. It is reasonably accurate for frequencies below 1 kHz.

The BCTRAN model13 is an n-phase transformer model where inter-winding coupling can be taken into account. The model is linear and assumes phase symmetry. It consists of a coupled RL or RL−1 matrix representing short-circuits impedances between windings, load losses at rated frequency and optionally linear inductive magnetization.

Phase-to-phase couplings, as well as the terminal characteristics, are included with these approaches, but they are linearised and do not consider differences in core or winding topology, since all core designs get the same mathematical treatment.

This model is linear; however, for many transient studies it is necessary to include saturation and hysteresis effects. Exciting current effects can be linearised and left in the matrix description, which can lead to simulation errors when the core saturates. Alternately, excitation may be omitted from the matrix description and attached externally at the models terminals in the form of non-linear elements, see Fig. 1. Such an externally attached core is not always topologically correct, but good enough in many cases.

When the core equivalent representation is connected at the terminal of the innermost winding (normally the LV winding), it has to be connected before the winding resistance, as shown in the figure below. This is required to correctly take into account the damping of inrush current provided only by the winding resistance of the energized winding.

Although these models are theoretically valid only for the frequency at which the nameplate data was obtained, they are reasonably accurate for frequencies below 1 kHz.

C. Practical modeling

Leakage impedance and winding resistance

Leakage impedances and winding resistances can be derived from standard binary short-circuit tests. Information usually available from short-circuit test report are short-circuit voltage in per-cent (vsc%) and active power (Psc) at rated current for each winding pair (binary short circuit test: one winding energized, another short-circuited and all other windings left open). The short-circuit winding resistance and leakage inductance for a winding-pair based on phase quantities are:

13 V. Brandwajn, H. W. Dommel, and I. I. Dommel, “Matrix representation of three-phase n-winding transformers for steady-state and transient studies,” IEEE Trans. Power App. Syst., vol. PAS-101, no. 6, pp. 1369–1378, June 1982.

Page 34

Zia.Emin, 10/04/12,

Nicola has redrawn the figure, please replace.

RSC=k ∙PSC ∙V n

2

Sn2

X SC=k ∙√( vSC %

100∙V n

2

Sn2 )

2

−(PSC ∙V n

2

Sn2 )

2

with k = 1 for single-phase and three-phase wye-connected transformers, and k = 3 for three-phase delta-connected transformers.

Alternatively, the leakage impedance can be analytically estimated from ampere-turn diagram (ATD) based on geometrical winding dimensions [T19].

For a two winding transformer, the inductance representation is rather simple. Different modeling approach are available for the representation of the leakage inductance of multi-winding transformers, with the most common being star-equivalent representation (STC) and fully coupled impedance or admittance formulation (BCTRAN).

Winding losses are modeled with linear resistors to match the measured power frequency short circuit lossess. For two winding transformer a total equivalent resistance (rACsc) is calculated from the short circuit losses that has to be divided between low- and high-voltage windings. It is common practice to equally divide between windings on a pu base. When the DC resistance value is known, it may be preferable to use it in order to achieve higher accuracy based on the following allocation [T14]:

r AC HV=r AC sc ∙r DC HV

rDC HV +r DC LV

r AC LV=r AC sc ∙r DC LV

rDC HV +r DC LV

where r is the resistance expressed in per unit with, r AC HV and r AC LVthe pu AC resistances, r AC scthe pu total short-

circuit resistance and r DC HV and r DC LV the pu DC resistances. The splitting of the AC resistance based on DC

resistance measurements usually results in r AC HV <r AC LV on a per unit base.

For three winding transformer the winding resistance can be obtained by solving the following system of equations:

[Rs c12

Rs c23

Rs c13]=[1 10

0 111 01][R1

R2

R3]

where RSC12 is the binary short circuit resistance between windings 1 and 2 and so on and R1, R2 and R3 are the winding resistances. For more than three-winding transformers the above system of equations become over

defined as the number of binary short-circuit test is nw (nw−1 )

2, larger than the number of windings nw. A subset of

the binary short-circuit test can be used for the calculation of the winding resistance and the remaining values used for verification.

Page 35

An autotransformer is modeled as normal transformer where the primary and secondary winding are connected in series to form the common and series windings. A tertiary delta winding is normally present in three-phase autotransformers. The model should represent the actual series, common, and delta coils as shown in Fig. 3auto, and not the three-winding "black box" equivalent typically assumed. For this reason, autotransformer coil reactances should be calculated with care from short-circuit factory test report14,15.

LT

H

Δ

S

C

Fig. 3auto: Schematic representation of an autotransformer model with series and common connected in series.

Defining the voltages as:

V H=V S+V CV L=V CV T=V D

The series-to-common impedance in pu values is:

ZSC=ZHL ∙( V H

V H−V L)

2

=ZHL ∙(V S+V C

V S)

2

No modification is needed for the common-to-delta impedance in pu values:

ZCD=Z¿

Finally, the series-to-delta impedance in pu values is:

ZSD=ZHL ∙V H V L

(V H−V L)2 +ZHT ∙V H

V H−V L

−Z¿ ∙V L

V H−V L

The autotransformer can therefore be treated as a transformer with three windings S,C,D with short-circuit impedances ZSC, ZCD, and ZSD.

14 H. W. Dommel , S. Bhattacharya , V. Brandwajn , H. K. Lauw and L. Martiacute; Electromagnetic Transients Program Reference Manual (EMTP Theory Book), 1992 :Bonneville Power Admin.

15 B. A. Mork , F. Gonzalez and D. Ishchenko "Leakage inductance model for autotransformer transient simulation", Int. Conf. Power System Transients, 2005. [online] Available: http://www.ipst.org/IPST05Papers.htm.

Page 36

Nonlinear saturation and core losses

One of the most critical steps in the creation of a transformer model for energization transient calculation is the construction of the saturation curves accounting for the nonlinear behavior of the core. Primary source of data available for the calculation of nonlinear saturation and core losses is the standard no-load test. Information usually available from standard no-load test is the no-load current in per-cent and the no-load losses at different excitation voltage. Typical values are limited to 100% and 110% of the rated voltage. It is becoming more common to perform no-load test in an extended voltage range (between 90% and 120%). This is required if higher accuracy in the modeling and calculation of saturation curve is desired.

The simplest approach for modeling saturation and core losses is with a parallel R-L representation as shown in the figure below.

RL

where L is a piecewise non-linear inductance and R is a linear resistance.

The value of the resistance is calculated from the core losses at rated excitation:

R=k ∙V 2

P

with k=1 for single-phase and three-phase wye-connected transformers, and k=3 for three-phase delta-connected transformers.

The value of the inductance is calculated from the magnetization current that is the no-load current from the test report subtracted by its resistive component:

I m=√ I no−load2 −I R

2

Once the magnetization current is calculated for all available no-load test values, the piecewise nonlinear L can be calculated. The process consists in a conversion of the rms values of the magnetization current and voltage to the peak values of current and flux-linkage. Since L is nonlinear the point of the saturation curve cannot be calculated independently from the preceding ones, but sequentially using an analytical [refSATURA,T12] or numerical procedure. This calculation is based on the assumption of a sinusoidal applied voltage. In order to enhance the accuracy of the saturation curve calculation, one should take into account the triplen harmonic correction in transformers tested from the delta winding [T12] and capacitive correction for larger transformer16,17.

Air-core inductance

16Gaudreau, A.; Picher, P.; Bolduc, L.; Coutu, A.; , "No-load losses in transformer under overexcitation/inrush-current conditions: tests and a new model," Power Delivery, IEEE Transactions on , vol.17, no.4, pp. 1009- 1017, Oct 2002 (Figure below from here)

17 Martinez, J.A.; Walling, R.; Mork, B.A.; Martin-Arnedo, J.; Durbak, D.; , "Parameter determination for modeling system transients-Part III: Transformers," Power Delivery, IEEE Transactions on , vol.20, no.3, pp. 2051- 2062, July 2005 (Figures 5,6,9 from here)

Page 37

When the value of the air-core inductance is known, an additional point in the nonlinear saturation curve should be defined beyond the last calculated value from test report data to set the saturated core inductance according to the air-core inductance. The saturated inductance seen from the low-voltage winding is:

Lsat = Lair−core,HV − LHL

with Lair−core,HV the air-core inductance calculated for the HV winding, and LHL the leakage inductance between HV and LV windings and Lsat the final differential magnetization inductance. The inrush current peak value is highly sensitive to the air-core inductance value; therefore, special care should be taken to consider this parameter. The value of air-core inductance can be normally provided by a transformer manufacturer or estimated from design data [T19]. When the core is connected to the star point, the saturated inductance is calculated by removing only a fraction of the total leakage inductance LHL, corresponding to the fraction of leakage placed on the primary side.

Suggested value of the air-core reactance seen from the HV side is 0.3 pu [T3]. 0.3-0.8 pu on self-cooled base for large, high BIL transformers, 0.05-0.15 pu for distribution transformers [T10]. Typical values given by CIGRE TB 39 air-core inductance of the HV winding (outer winding) is 2-2.5 LHL for step-down transformer and 4-4.5.LHL for autotransformers.

Residual flux

A typical value of maximum residual flux is in the range of 0.6-0.8 pu.

In three-phase transformers with a common core (three-legged, five-legged, shell) the residual flux in the three phases (and in the core limbs) are correlated by the magnetic flux equations.

The most common scenario of residual flux analyzed in worst case scenario are:

Single-phase transformers: 0.8/0.6 pu Three-phase transformers:

o one phase at 0.8/0.6 pu, the other two phases at -0.4/0.3 pu (cos2 π3

)

o two phases at 0.7/0.5 pu (sin2π3

), the other phase at zero residual flux.

Frequency dependent winding losses

A simple approach for representing winding resistance frequency dependence is with foster network. It is however critical to obtain the characteristic of the winding resistance as a function of the frequency required as input data for the modeling. The frequency dependency can be estimated analytically but requires a detailed knowledge of the winding configuration and is quite complex18,19.

Taking into account the frequency dependency of the winding resistance will provide higher damping for higher frequency components as well as reduced DC losses. For practical purposes when frequency dependency is neglected the duration if not the amplitude of the resulting voltage is overestimated.

Zero sequence impedance

18 de Leon, F.; Semlyen, A.; , "Detailed modeling of eddy current effects for transformer transients," Power Delivery, IEEE Transactions on , vol.9, no.2, pp.1143-1150, Apr 1994

19 de Leon, F.; Semlyen, A.; , "Time domain modeling of eddy current effects for transformer transients," Power Delivery, IEEE Transactions on , vol.8, no.1, pp.271-280, Jan 1993

Page 38

Hysteresis and frequency dependent iron losses

The core losses can be divided in three components: hysteresis, classic eddy current and excess (or anomalous) eddy current:

P0=Physt+Pcls eddy+P exc eddy

To achieve a more accurate representation, each component should be modeled separately however this is not straightforward. The first difficulty is how to split the total core losses measured at power frequency in the three components. The figure below shows the iron loss in grain oriented material (adapted from20) depends from the steel grade. Indicatively, at 1.7 T each loss component is between 30% and 40% of the total core losses.

Classical and excessive eddy current are approximately proportional to the squared of flux and frequency and can be modeled by a linear resistor:

Peddy∝Bmax2 f 2Peddy ≈

V 2

Reddy

The representation of hysteresis losses is more complex as they are frequency independent and nonlinear. The hysteresis loop of the magnetic material needs to be represented to consider this loss component. Nonlinear

resistors could be used to match more accurately the nonlinear loss behavior; however, their transient performance is somewhat uncertain. A non-linear resistance can be used to represent the excitation dependent hysteresis core

losses, however this representation can represent the average loss per period, but fails to accurately reproduce the exact waveforms and hysteresis loops as shown in the figure below. In addition, this model cannot represent the

rate independent property of the hysteresis losses.

20 R. Girgis, M. Hastenrath, and J. Schoen, “Electrical steel and core performance,” Presentation, Oct. 2009. [Online]. Available: http://www.transformerscommittee.org/info/f09-presentations.htm

Page 39

Advanced models for the representation of hysteresis are Preisach and Jiles-Atherton. While these models may be accurate, they are complex and it is hard to obtain model parameters. Their use is justified only in special simulations.

A common used model that can provide a simple representation of the hysteresis loop is a piecewise nonlinear hysteretic inductor (also known as type 96 hysteretic inductor). While this model received some critics because it cannot represent minor loops and sub-loops cycles accurately, it can be built from standard test report data. One part of the losses is attributed to eddy current losses and is included in a parallel resistor. The other part is assigned to hysteresis losses and is included in the type 96 component. The hysteresis losses PH are further scaled to the maximum excitation with a Steinmetz coefficient of 2 as shown in (11) which seems reasonable according to21.

PH=Physt ∙( λmax

λrated)

2

Where Physt is the hysteresis part of the total core losses at rated excitation (at λrated). λmax is the final point of the hysteresis loop as shown in the figure below.

21 F. De Leon, A. Semlyen: “A simple representation of dynamic hysteresis losses in power transformers”, IEEE TPWRD, Vol. 10, pp. 315-321, Jan. 1995

Page 40

The width of the hysteresis loop is assumed to be constant (but goes to zero at the second highest flux-linkage point) and calculated as22

W =PH

f ∙(λnp−1+λnp−2)[ A ]

where f is the power frequency and the flux-linkages λnp-1 and λnp-2 are given in the piece-wise nonlinear curve in the figure below and their sum multiplied by the width W becomes the area of the hysteresis loop.

In this approach the maximum residual flux is determined by the intersection of the hysteresis loop with the zero-current axis. One should be careful to verify that the resulting maximum residual flux has a reasonable value in the range of 0.6-0.8 pu and that the measured saturation, when plotted on the hysteresis characteristic lies within the loop.

Capacitances

Source of capacitances in a transformer are parasitic winding and bushing capacitances and includes:

Capacitance between a winding to the core or ground including the bushing capacitance: CLG and CHG

Capacitance between two concentric winding of the same phase: CHL

Capacitance between two phases (only for the HV or outermost windings): CPh

For low-frequency transients they can be modeled as lumped parameters with two capacitive networks as the one shown in the figure below connected each to one end of the windings.

22H. K. Høidalen, N. Chiesa, A. Avendaño, B. A. Mork: "Developments in the hybrid transformer model – Core modeling and optimization", IPST2011, International conference of power system transients, 2011.

Page 41

Data for the modeling of capacitance are normally included in test report or can be obtained from a system of simple measurement at the terminals of the transformer. However, CPh is not normally available and can be neglected without any significant impact on the overall results.

D. Advanced modeling: topology-based models

Topology-based models address the limitations of standard models and many approaches have been proposed. [5] Their derivation is performed from the core topology and can represent very accurately any type of core design in low- and mid-frequency transients if parameters are properly determined.

The main characteristics are:

Topological correct core model (triplex, three-legged, five-legged, shell). Effects of saturation in each individual leg of the core. Interphase magnetic coupling.

Typical drawbacks of advanced models are the limited availability of input data and the difficulty in estimating of the model parameters. Many models have been proposed but almost none has become widely adopted. Only two model are known today to have been partially implemented in commercially available tools (UMEC23,24 and Hybrid Transformer25,26,27,28,29 models).

23 W. Enright, O. B. Nayak, G. D. Irwin, J. Arrillaga, An Electromagnetic Transients Model of Multi-limb Transformers Using Normalized Core Concept, IPST '97 Proceedings, Seattle, pp. 93-98, 1997.

24 W. Enright, N. Watson, O. B. Nayak, Three-Phase Five-Limb Unified Magnetic Equivalent Circuit Transformer models for PSCAD V3, IPST '99 Proceedings, Budapest, pp. 462-467, 1999.

25 Bruce A. Mork; Francisco Gonzalez; Dmitry Ishchenko; Don L. Stuehm; JoydeepMitra; , "Hybrid Transformer Model for Transient Simulation—Part I: Development and Parameters," Power Delivery, IEEE Transactions on , vol.22, no.1, pp.248-255, Jan. 2007

26 Bruce A. Mork; Francisco Gonzalez; Dmitry Ishchenko; Don L. Stuehm; JoydeepMitra; , "Hybrid Transformer Model for Transient Simulation—Part II: Laboratory Measurements and Benchmarking," Power Delivery, IEEE Transactions on , vol.22, no.1, pp.256-262, Jan. 2007

27 Mork, B.A.; Ishchenko, D.; Gonzalez, F.; Cho, S.D.; , "Parameter Estimation Methods for Five-Limb Magnetic Core Model," Power Delivery, IEEE Transactions on , vol.23, no.4, pp.2025-2032, Oct. 2008

Page 42

The Hybrid Transformer model is an engineering transformer model based on limited input data.The modeling of the transformer is based on the magnetic circuit transformed to its electric dual. The leakage and main fluxes are then separated into a core model for the main flux and an inverse inductance matrix for the leakage flux. The copper losses and coil capacitances are added at the terminals of the transformer. In the current implementation the model can represent two- and three-windings three-phase transformer. Three-limb, five-limb, shell and three-phase bank (triplex) transformer core constructions are possible configurations in the model where only relative dimensions are required. Input data can be specified as typical values, test report, and design information.

Figure "Hybrid" from 28

The UMEC transformer model is based on the concept of a unified magnetic equivalent circuit. A normalized core is used in order to remove the requirement of design data; only relative dimensions are required. The magnetic network is derived from the transformer core topology. Three-limb, five-limb, and three-phase bank (triplex) transformer core constructions are possible configurations in the model. The magnetic network representing the core and leakage inductances is described with a matrix formulation using a permeance matrix (in the current implementation the number of winding is limited to two for three-phase transformers). Both the magnetic coupling between windings of different phases and the coupling between windings of the same phase are taken into account. Winding and core losses are not included in the magnetic circuit and are represented by an equivalent linear resistance at the winding terminals. Load losses are equally divided on a p.u. base and represented by linear series resistances connected at one terminal of each winding. Core losses are assumed linear and equally divided on a p.u. base between primary and secondary windings. The use of a magnetic network does not allow a simple representation of topological core losses, therefore they are represented by linear shunt resistances connected at the terminals of each winding.

28 H. K. Høidalen, B. A. Mork, F. Gonzalez, D. Ishchenko, N. Chiesa, "Implementation and verification of the Hybrid Transformer model in ATPDraw", Electric Power Systems Research, Volume: 79, Issue: 3, Pages: 454-459, 2009

29 H. K. Høidalen, N. Chiesa, A. Avendaño, B. A. Mork "Developments in the hybrid transformer model – Core modeling and optimization", IPST11

Page 43

Figure "UMEC" from 24

These two topology-based models are compared with standard model for the study of a black start energization30. The study concludes that a topologically correct core model produces higher accuracy results when simulating highly nonlinear and unbalanced electromagnetic transients.

30 N. Chiesa, H. K. Høidalen, M. Lambert, M. Martínez Duró, " Calculation of Inrush Currents – Benchmarking of Transformer Models ", International Conference on Power Systems Transients (IPST2011) in Delft, the Netherlands, June 14-17, 2011

Page 44

Table 1: Representation of transformers.

STC(Star-equivalent or T model)

BCTRAN(Pi-equivalent or matrix representation)

UMEC(Topological model)

Hybrid Transformer (Topological model)

Simplified Representation (ideal transformers are omitted)

See Figure "UMEC" See Figure "Hybrid"

Short circuit Requires the artificial splitting of leakage inductance

Matrix formulation based on inverse inductance formulation

Matrix formulation using a permeance matrix

Based on the BCTRAN approach.

# windings 3 (stable with 2, possible instability with 3 due to negative inductance).

No limitation No limitation, limited to two windings in current implementation

No limitation, limited to three windings in current implementation

Core connection At the star point or terminals of the innermost winding

At the terminals of the innermost winding

N/A: No separation between core and leakage

Fictitious winding at the core surface is introduced to create a topological connection point for the core

Core modeling Equivalent single-phase representation. No magnetic coupling

Equivalent single-phase representation. No magnetic coupling

A magnetic network represents both the core and leakage inductances. Doubtful handling of open-circuit data

Topological model representing the core construction and magnetic coupling between phases

References

[T1] CIGRE WG 33.02, “Guidelines for Representation of Network Elements when Calculating Transients”, CIGRE brochure 39, 1990.

[T2] IEC TR 60071-4:2004, “Insulation co-ordination – Part 4: Computational guide to insulation co-ordination and modelling of electrical networks”.

[T3] CIGRE WG 33.10, “Temporary overvoltages. Test case results”, Electra 188, 2000.

Page 45

[T4] A. M. Gole, J. A. Martinez-Velasco, and A. J. F. Keri, Modeling and Analysis of System Transients Using Digital Programs, , 1999, IEEE PES Special Publication

[T5] J. A. Martinez and B. A. Mork, “Transformer modeling for low- and mid-frequency transients - a review,” IEEE Trans. Power Del., vol. 20, no. 2 II, pp. 1625–1632, 2005.

[T6] H. W. Dommel and et.al, Electromagnetic Transients Program Reference Manual (EMTP Theory Book): Prepared for BPA, 1986.

[T7] C. P. Steinmetz and E. J. Berg, Theory and calculation of alternating current phenomena. New York: Electrical World and Engineer, inc., 1897.

[T8] T. Henriksen, “How to avoid unstable time domain responses caused by transformer models,” IEEE Trans. Power Del., vol. 17, no. 2, pp. 516–522, Apr. 2002.

[T9] L. F. Blume, Transformer engineering : a treatise on the theory, operation, and application of transformers, 2nd ed. New York, N.Y.: John Wiley and Sons, Inc., 1951.

[T10] IEEE PES Task Force on Data for Modeling System Transients, “Parameter determination for modeling system transients: Part III. Transformers”, IEEE Transactions on Power Delivery, Vol. 20, No. 3, 2005, p. 2038-2044.

[T11] V. Brandwajn, H. W. Dommel, and I. I. Dommel, “Matrix representation of three-phase n-winding transformers for steady-state and transient studies,” IEEE Trans. Power App. Syst., vol. PAS-101, no. 6, pp. 1369–1378, June 1982.

[T12] N. Chiesa and H. K. Høidalen, “Analytical algorithm for the calculation of magnetization and loss curves of delta connected transformers,” IEEE Trans. Power Del., 2010, accepted for publication, TPWRD-00589-2009.

[T13] N. Chiesa and H. K. Høidalen, “Hysteretic iron-core inductor for transformer inrush current modeling in EMTP,” in PSCC 2008 - 16th Power Systems Computation Conference, Glasgow, Scotland, Jul. 2008.

[T14] N. Chiesa, B. A. Mork, and H. K. Høidalen, “Transformer model for inrush current calculations: Simulations, measurements and sensitivity analysis,” IEEE Trans. Power Del., vol. 25, no. 4, pp. 2599–2608, Oct. 2010.

[T15] B. A. Mork, F. Gonzalez, D. Ishchenko, D. L. Stuehm, and J. Mitra, “Hybrid transformer model for transient simulation: Part I: development and parameters,” IEEE Trans. Power Del., vol. 22, no. 1, pp. 248–255, Jan. 2007.

[T16] H. K. Høidalen, B. A. Mork, F. Gonzalez, D. Ishchenko, and N. Chiesa, “Implementation and verification of the hybrid transformer model in ATPDraw,” Electr. Power Syst. Res., vol. 79, no. 3, pp. 454 – 459, Mar. 2009, special Issue: Papers from the 7th International Conference on Power Systems Transients (IPST).

[T17] H. K. Høidalen, N. Chiesa, A. Avendaño, and B. A. Mork, “Developments in the hybrid transformer model – Core modeling and optimization,” in IPST’2011, 2011.

[T18] Martinez, J.A.; Walling, R.; Mork, B.A.; Martin-Arnedo, J.; Durbak, D.; , "Parameter determination for modeling system transients-Part III: Transformers," Power Delivery, IEEE Transactions on , vol.20, no.3, pp. 2051- 2062, July 2005

[T19] Kulkarni, Kapade, "Transformer Engineering. Design and Practice," Marcel Dekker, Inc., 2004

Page 46

5.1.3. OTHER COMPONENTS OF THE SYSTEM

5.1.3.1. Shunt reactorsReactors can be represented similarly to transformers without the leakage impedance. In a shunt reactor model there is only a single inductance instead of leakage and magnetizing inductances of multi-winding transformers.

Air-gaps are normally used in shunt reactors to define the total inductance. The equivalent per-phase effect of the air-gaps should be represented by a linear inductance in series to the magnetizing inductance as shown in the figure below. The resistance for representing the core losses should be placed in parallel to only the magnetizing inductance.

Lair-gap

RmLm

The rules for modeling saturation effects and frequency dependent losses for transformers apply also to shunt reactors. Consideration should be given to the fact that for three-phase reactors the zero-sequence impedance is generally lower than the positive sequence impedance.

5.1.3.2. Overhead-lines [MM] According to the IEC 71-4 [1] and CIGRE TB 39 [2] recommendations for the modelling of temporary overvoltages and the CIGRE WG 33.10 [3] recommendations for the modelling of transformer energization, the important phenomena to be represented in overhead-line (OHL) models are:

The line asymmetry (especially in untransposed lines) and the coupling with other circuits. The frequency dependency of the ground mode.

Line asymmetry and coupling with other circuits

The line asymmetry is given by the different distances between the conductors and by the different distances between the conductors and the ground. The coupling with other circuits is present when several circuits are in the same corridor.

The modelling of the line asymmetry is important to accurately represent the resonance frequencies and impedances for each phase, which may be excited by the harmonic components of the inrush currents. This is especially important for untransposed lines where both resonance frequencies and impedances may be quite different for different phases. Transposition tends to reduce the asymmetry effect. However, if no resonant harmonic overvoltage is expected (for instance, if the total line length is short), the line asymmetry may be neglected.

Notice that if the line asymmetry must be represented, the sequence impedances of the lines used in steady-state or power-flow studies are not informative enough, as they assume the line is symmetric. Instead, the line model parameters must be calculated from the spatial disposition of the conductors.

Page 47

Frequency dependency

According to [1][2][3][5], in the TOV frequency range the variation of the line parameters in the aerial mode (i.e., direct sequence if line symmetry assumed) is negligible, whereas it may be important in the ground mode (zero sequence). Indeed, direct sequence inductance and resistance are fairly constant up to 1 kHz, but zero sequence values are very dependent on frequency because of the skin effects in the earth return [4][5].

The variation of the line parameters with frequency is important because it will affect the damping of the harmonic overvoltage components. However, if no harmonic overvoltage is expected, frequency dependency may be neglected ([6] shows no difference in the inrush current damping whether the frequency dependency of the line parameters is represented or not).

To illustrate the effect of the line asymmetry and the frequency dependency of the line parameters, let’s consider the harmonic impedance of the network in Figure 1, where an unloaded 3x550 MVA 400/20 kV step-up transformer is energized at the end of an 84 km double-circuit untransposed overhead line fed by the 400 kV network.

OHL, double-circuit, untransposedNetwork equivalent, 400 kV 3x550 MVA (triplex)

+?i 1 2

405/20 +

25Ohm

L1

+RL1

+

20kVRMSLL /_0

?i

+

+

+

+

+

+

+

FD+

abcccc

bbb

aaa

FIGURE 1: TEST CIRCUIT #1: TRANSMISSION NETWORK + OHL + TRANSFORMER

The Figure 2 shows the network impedance for a direct (left) and a zero sequence (right) current injection at the transformer location. On the top figures, the line parameters are frequency dependent; on the bottom ones, they are constant and calculated at the power frequency (50 Hz).

These curves illustrate the effect of the line asymmetry: no matter the figure considered, the resonance frequencies and impedances are different for the three phases. The effect of the frequency dependency is also clearly shown: in this frequency range (below 1 kHz), the impact of the frequency dependency is moderate in the direct sequence (figures on the left) but it is very important in the zero sequence (figures on the right).

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FIGURE 2: NETWORK IMPEDANCE: EXACT-PI (TOP) COMPARED TO CASCADED-PI (BOTTOM) FOR DIRECT (LEFT) AND ZERO (RIGHT) SEQUENCE EXCITATIONS.

Other aspects: Corona effect, Surge arresters

The corona effect involves the ionization of the air surrounding the line conductors. Its consequences are the increase of the losses and capacitance of the line [7][8][2]. The increase in capacitance is important for travelling wave phenomena (switching or lightning studies), as it reduces the steepness of the surges. For temporary overvoltages, the increase of the losses will result in additional damping. The corona effect occurs if the corona inception voltage is reached, which is usually about 30% to 40% higher than the line rated voltage [8] (yet, in systems with high rated voltage, the difference with the corona inception voltage may be smaller). However, as it is very difficult to model, the corona effect is usually neglected (therefore the modeling remains in the pessimistic side) [3][9].

If the overvoltages are high enough, the surge arresters will absorb part of the energy, thus contributing to damping. Temporary overvoltages release large energy, however, the surge arresters energy capability is relatively small and hence the principal concern is whether or not they sustain damage during the energization transient [3] (see §5.2).

Practical modeling

In the frequency domain, a line is modeled by a lumped RLC multiphase representation, i.e., a PI-circuit, and it is easy to represent the line asymmetry, the coupling with other circuits and the frequency dependency of the parameters. The line asymmetry and the coupling with other circuits are taken into account if the model parameters are calculated from the spatial disposition of the conductors. The frequency dependency is modeled by recalculating the PI-circuit parameters at each frequency –the model is then called “Exact-PI” (or “PI-exact”).

For the time domain, the table below shows the standard modelling possibilities depending on the model and the input data. Three standard models are available: cascaded-PI, Bergeron and J. Marti’s. Available input data may be just sequence characteristics of the line, i. e., series impedance and shunt capacitance in positive and zero sequence, or the spatial disposition of the conductors and their resistivity and diameter.

Page 49

TABLEAU 1: STANDARD OHL REPRESENTATIONS

Line asymmetryNo Yes

Frequencydependency

NoPI/sequence

Bergeron/sequencePI/conductor

Bergeron/conductorYes JMarti/conductor

If no frequency dependency is to be modeled, a cascaded-PI representation may be used, where several PI-circuits are connected in series to represent the total line length. These PI-circuits are calculated at a fixed frequency (see below, model frequency)31. The maximum line length (Lmax) to be modeled by a single PI-circuit depends on the maximum frequency (fmax) of the simulated phenomena. Reference [1] recommends this expression relating both parameters:

Lmax=ν

5⋅f max

where is the propagation speed of the electromagnetic wave (300 km/ms). Considering that inrush harmonics are negligible beyond 1 kHz, each PI-circuit can account up to 60 km length.

The “Bergeron model” (also known as “CP model”) takes into account the distributed nature of the line parameters [10]. Therefore it is more suitable than the PI model for the calculation of switching overvoltages due to travelling-waves reflections. The parameters of the model are constant, though, and thus may be used when no frequency dependency is to be represented. However, this model may be quite sensitive to the model frequency specified by the user (see below).

In the widely known J. Marti’s model [11] (also known as “FD model”), parameters are both distributed and frequency dependent. Therefore, this is the only standard model that is able to represent the frequency dependency of the line parameters.

If line symmetry is assumed (for instance, if the line is transposed), the cascaded-PI model and the Bergeron model parameters can be calculated from the sequence characteristics of the line. If the line asymmetry is modeled or if the JMarti model is used, the model parameters are calculated from the spatial disposition of the conductors and their physical characteristics. When computing parameters from conductor’s spatial disposition, it is very important to include the shield wires (if they exist) and to ground them or to set them to zero voltage.

The three models take ground resistivity as an input parameter. The ground resistivity value along the line is uncertain and varies with climatic conditions, but [5] shows that the line impedance sensitivity to this parameter is very low below 5 kHz.

Furthermore, in all the three line models, cascaded-PI, Bergeron and JMarti, a unique model frequency must be specified. In the cascaded-PI model, this is the frequency at which the parameters are calculated. In the Bergeron and JMarti models, it is the frequency at which is calculated the modal transformation matrix Q. As the main harmonic component of the energization voltages and currents is the power frequency one [12], the power frequency seems the most suitable value to be used in the line models.

31 As already mentioned, in the frequency domain EMT programs recalculate the parameters of the PI-circuits at each frequency, so the frequency dependency is taken into account. In the time domain (where the signals contain more than one frequency), however, the parameters of the PI-circuits are kept constant and thus no frequency dependency is modeled. In order to distinguish between these two usages of the same equivalent circuit, one calls “exact-PI” or “PI-exact” the PI-circuits used in the frequency domain and “nominal-PI” the PI-circuits used in the time domain [4].

Page 50

Bergeron and JMarti are distributed-parameter models and as such they are suitable to model both harmonic TOV and possible travelling-wave switching surges at the circuit-breaker closing. However, these models limit the maximum simulation time-step to half the value of the propagation time in the shortest line. Notice also that for low frequencies (below 1 kHz), Bergeron and JMarti models perform well for single-circuit lines, but much worse for double or triple-circuit lines [13].

References

[1] CIGRE WG 33.02, “Guidelines for Representation of Network Elements when Calculating Transients”, CIGRE brochure 63, 1990.

[2] IEC TR 60071-4:2004, “Insulation co-ordination – Part 4: Computational guide to insulation co-ordination and modelling of electrical networks”.

[3] CIGRE WG 33.10, “Temporary overvoltages. Test case results”, Electra 188, 2000.

[4] Dommel, H. W., EMTP Theory Book, Vancouver, Canada, Microtran Power System Analysis Corporation, 1992.

[5] Martinez, J. A., B. Gustavsen, and D. Durbak, "Parameter Determination for Modeling System Transients—Part I: Overhead Lines", Transactions On Power Delivery 20(3): 2038-2044, 2005.

[6] Martínez Duró, M., "Damping Modelling in Transformer Energization Studies for System restoration: Some Standard Models Compared to Field Measurements," in IEEE PowerTech 2009, Bucarest, 2009.

[7] Imece, A.F., D.W. Durbak, H. Elahi, S. Kolluri, A. Lux, D. Mader, T.E. McDermott, A. Morched, A.M. Moussa, R. Natarajan, L. Rugeles, E. Tarasiewicz, “Modeling Guidelines for Fast Front Transients”, in A. Gole, J.A. Martinez-Velasco and A.J.F. Keri (Eds), Modeling and Analysis of System Transients using Digital Systems, IEEE Special Publication, TP-133-0, 1998.

[8] Das, J. C., Transients in Electrical Systems; Analysis, Recognition, and Mitigation, McGraw-Hill, New York, 2010.

[9] CIGRE WG 33.210, “Temporary overvoltages: causes, effects and evaluation”, Proceedings of the 33rd Session. CIGRE, Paris, France, 1990.

[10] Dommel, H. W., “Digital computer solution of Electromagnetic Transients in single and multiphase networks”, IEEE Transactions, Vol. PAS-88, pages 388-399, April 1969.

[11] Marti, J., “Accurate Modeling of Frequency Dependent Transmission Lines in Electromagnetic Transient Simulations”, IEEE Trans. On Power Apparatus and Systems, vol PAS-101, pp 147-157, 1982.

[12] Sybille, G.; Gavrilovic, M.M.; Belanger, J.; Do, V.Q., "Transformer Saturation Effects on EHV System Overvoltages", Power Apparatus and Systems, IEEE Transactions on, vol.PAS-104, no.3, pp.671-680, March 1985.

[13] Martí, J. R., H.W. Dommel, L. Martí, and V. Brandwajn, "Approximate transformation matrices for unbalanced transmission lines", Proceedings PSCC, 9th Power System Computation Conference, Butterworths, London, pp. 416-422, August 1987.

5.1.3.3. Cables [MV]The subject of cable modelling has been extensively covered by C4 WG502 (Power System Technical Performance Issues Related to the Application of Long HVAC Cables). A full review of modelling issues related to underground and submarine HVAC cables is included in Chapter 3 of TB****. Furthermore, recommendations are made for the usage of models for various types of studies. The frequency of interest for transformer energization studies ranges from DC up to 1kHz [Cigre TB 39]. In fact, the phenomena of highest concern in the presence of long HVAC cable circuits or large concentration of short HVAC cables is normally harmonic resonance excited by inrush currents, hence the frequency of interest is normally restricted the low harmonic range.

The major difficulty in modelling underground cables is the fact that they are highly nonlinear in nature; due mainly to the frequency dependency of conductors (skin effect), as well as the ground or earth return path. This frequency dependency is more pronounced than in overhead lines.

Page 51

Because of the complex geometry, the most accurate methods to calculate cable parameters are those based on finite element methods. However, finite element methods are not suitable for the types of EMT programs cable models presently available in commercial or free-ware simulation tools.

Unless the cable is very short in length, a distributed parameter model is recommended for transient studies. Furthermore, for accurate results at various frequencies, the selected model must also be capable of reproducing the frequency dependency behaviour of the cable.

The exact pi model is only recommended for frequency scans. However, this model is not adequate for transient studies, such as transformer energization, where more than one frequency component needs to be considered.

Distributed parameter model at constant frequency (eg Bergeron) is only recommended when the studies involve a single frequency, such as power frequency load flow or very low harmonics.

Frequency dependant modal domain models (eg. JMARTI) are very accurate in most overhead line configurations and simple cable systems because the transformation matrix can be assumed constant with very little error in the time domain solution. When this approximation can be done, these type of models lead to a very efficient time domain simulation.

When complex cable configurations, or a large range of frequencies need to be considered, frequency dependent phase domain models must be used (eg ARMA, zcable model and Universal Line model). These models accurately represent the frequency dependency of the cable parameters and are, therefore, recommended for studies covering various frequencies, such as transients and harmonic resonance studies.

In terms of the required accuracy in the input data, the results of a sensitivity analysis carried out by C4 -502 WG indicate that the conductor radius and the permittivity and thickness of the main insulation layer of the cable are the most critical parameters in the model. Other parameters such as conductor resistivity, sheath thickness and resistivity, inner semi-conductive layer or earth resistivity have a lesser impact on the accuracy of the model for the low frequency range of interest in transformer energization studies.

5.1.3.4. Network equivalents [JAMV, YV & MM]In order to reduce computational burden, a reduced three-phase (system equivalent) model of the power supply (external zone) may be required in some transient studies. The aspects to be considered for obtaining this model in transformer energization studies include the type of transient that can be caused when energizing the transformer, the size of the external system that will be represented by the equivalent power supply model, and the information to be used for developing the supply model.

For RMS-voltage drop studies in networks where no resonant overvoltages are generated (without low frequency resonances), only the short-circuit impedance of the supplying network (the network at the target transformer terminals) needs to be represented. If in addition switching surges due to line energization or reenergization are to be simulated, it is generally sufficient to represent the lines, at least in meshed systems, up to two busbars back from the energizing bus; the remaining part of the system may be represented by power frequency network model. The Power frequency network equivalents are described below.

However, if the supplying network has low frequency impedance resonances, the inrush currents and the temporary overvoltages observed during transformer saturation can be very sensitive to the frequency response of the network. Therefore the modelling of the network has to represent resonances from DC to few kHz (less than 1 or 2 kHz), at least at the nodes of the study zone where transformer saturation is studied. This chapter will discuss about the size of the network and the network equivalent models which are of first importance regarding the correct evaluation of possible resonances during transformer saturation.

It is difficult to give general recommendations in terms of minimal number of nodes or distance to be accurately modeled for transformer energization studies. Reference [9] provides some recommendations for the size of the

Page 52

study zone to calculate impedance on the HV and EHV grids. For EHV networks, this reference suggests that the complete primary transmission network has to be modelled. However, a complete modelling of the network leads to several problems: very time-consuming simulations, high risk of human modelling errors and difficulties to update the data to take into account different network configurations (for example peak and off-peak loads).

If the frequency response of the network is known (from measurements or from a complete network model used for a frequency scan), a network equivalent can be built that reproduces its frequency response. With this technique, the external zone is assumed to be a linear portion of the system that can be reduced to gain calculation speed. Since the behaviour of the external system is assumed linear and the generators in the external system are not close to the study zone, they are represented by power-frequency voltage sources, which avoid electro-mechanical-type low-frequency transients in the model. Two network equivalent techniques are described below: an approximate single resonance network equivalent and a fully frequency dependent network equivalent (FDNE) reproducing the exact response of the network.

If the frequency response of the network is unknown or if the user wants an accurate result without using a sophisticated frequency dependent equivalent, another method consists in modelling the real network (the lines or cables and the substation equipment) up to a certain distance from the target transformer and representing the rest of the system by power frequency equivalents. This technique is described below as Detailed modelling of the network in the neighbourhood of the target transformer.

Power frequency network equivalent

Power frequency network equivalent consist of a voltage source in series with the sequence impedances of the network; sequence impedances are easily obtained by performing short-circuit calculations with the model of the whole network usually available for steady-state power flow analysis. Outside the study zone, mutual coupling between multiple equivalent points could also be implemented using network reduction techniques. Figure 4 shows some of these power frequency network equivalents [3].

The first type a) represents the short circuit impedance (Thevenin equivalent) of the connected system; the X/R ratio is selected to represent the damping (the damping angle is usually in the range 75º-85º).

The second type b) represents the surge impedance of connected lines. This equivalent may be used to reduce connected lines to a simple equivalent surge impedance and where the lines are long enough so that reflections are not of concern in the system under study.

If the connected system consists of a known Thevenin equivalent and additional transmission lines, the two impedances may be combined in parallel as in Figure 2c. It should be noted however, that this approach may yield an incorrect steady-state solution if the equivalent impedance of the parallel connected lines is of comparable magnitude to the source impedance. In such a case it may not be possible to lump the source and the lines into single equivalent impedance.

Page 53

R X

Z

R X

Z

a) Short-circuit impedance

b) Surge impedance

c) Short-circuit impedance + Surge impedance

Fig. 4. Conventional power frequency network equivalents [3].

Single resonance network equivalent

The model proposed in [2] is shown in Figure 5 and can be obtained from information about the short-circuit currents and the frequency response of the power supply system (external zone). This model simplifies the network frequency response to a single resonance frequency. To compute Z0 and Z1, the zero and positive-sequence impedance of the source, one can assume that they are mostly inductive, i.e. Z0 j2ωfL0, and Z1 j2ωfL1, being f the power frequency. L0 and L1 are calculated from the values of single-phase and three-phase short-circuit currents and by the formula:

Z0

Z1

=3I 3 ph− sc

I 1 ph−sc

−2(1)

Resonance capacitors are determined from the zero- and positive-sequence resonance frequencies by:

C1=1

L1 ω12

C0=1

L0 ω02

C ph=1L1ω1

2 Cn=

3L0ω0

2−L1 ω12

(2)

Then the damping resistors are given by:

R1=k √ L1

C1

R0=k √ L0

C0

Rph=kω1 L1 Rn=k3 (ω0 L0−ω1 L1)

(3)

where the factor k defines the level of attenuation.

Page 54

Fig. 5. Equivalent circuit of the power supply for low-frequency transients [2].

Frequency-Dependent Network Equivalents (FDNE)

Complex equivalents which properly represent the frequency response characteristic can be used to reduce the study to model and the computational time of system.

The aim is to obtain a three-phase equivalent circuit which has the same frequency response as the detailed network from the nodes connected to the portion of the system represented in detail (i.e., the study zone), see Figure 6.

Fig. 6. Linear network equivalent [2].

The linear modelling of the external system is normally based on an admittance formulation which defines the relation between voltages V and currents I on the terminals of the equivalent [8]:

YV=I (4)

The development of a frequency-dependent network equivalent (FDNE) usually involves the following procedure [11]: (i) simulation of the system to obtain the frequency response (either impedance or admittance) to be modelled by the equivalent; (ii) fitting of model parameters (identification process); and (iii) implementation of the FDNE in the simulation tool.

The known frequency response admittance characteristic of the external system can be estimated by fitting it to a function of the appropriate order:

Page 55

f fit (s )=p0+ p1s+ p2 s2+⋯+ pN sN

1+q1s+q2s2+⋯+qN sN

(5)

Or the equivalent form:

f fit (s )=p0+∑k=1

N ck

s−ak (6)

Since the values of f(jωp) are known at an arbitrarily large number of frequency points, (6) can be expressed as an overdetermined fitting problem in the 2n + 1 variables a1, a2, …, aN, and c0, c1, c2, …, cN. However, this is a nonlinear problem that cannot be solved by linear regression methods.

Early work reported in literature used frequency domain computed data to fit parameters to the model in (5) [12]. References [13], [14] overcome ill-conditioning problems of FDNE, by dividing the frequency response into sections. Other techniques like column scaling, adaptive weighting, and iterations step adjustment are also utilized in these references. A time domain approach to obtain the fitted function (6) using Prony analysis was presented in [15]. Time domain approaches have also been applied to identify the external system as a digital filter in [16], [17]. In [18], [19], and [20], the external system is modelled using lumped parameters. Vector Fitting is another option that converts the problem in (5) into a linear problem as described below [21] - [25].

Many practical applications involve several busses. In such multi-port cases, the same modelling procedure is applicable as vector fitting can be applied to several elements simultaneously. In practice, one stacks the elements of Y into a single vector and subjects it to vector fitting which produces a rational model with a common pole set, which after rearrangement of fitting parameters gives the following pole-residue model

Y=∑k=1

N Rk

s−ak

+R0

(10)

A symmetrical model is obtained by fitting only the upper (or lower) triangle of Y

When the terminals of the equivalent include more than a single three-phase bus, the modelling becomes more challenging as error magnification problems may arise. When applying a voltage source to one bus, the model is required to produce large short-circuit currents with a short circuit applied to the other bus, and small currents if the second bus is open (charging currents). This behaviour is reflected in the admittance matrix Y by large and small eigenvalues, respectively. Direct fitting of the elements of Y may easily result in corruption of its small eigenvalues, which may lead to error magnification with certain terminal conditions. Some approaches such as modal vector fitting overcome this problem by assigning high weights to the small eigenvalues of Y in the least-squares fitting process [26].

More details on the determination of frequency-dependent network equivalents can be consulted in [8] and [27].

To derive such network equivalent some expertise is required and very rarely the practical engineer will be able to carry out the whole procedure. In such cases it is advisable to use the model once it is clear the detail with which it must be represented.

Detailed modelling of the network in the neighbourhood of the target transformer

As previously mentioned, the supplying network can be represented by modelling the real network (the lines or cables and the substation equipment) up to a certain distance from the target transformer and representing the rest of the system by power frequency equivalents. In order to decide up to which distance the system is to be

Page 56

represented in detail, either at the same voltage level (horizontal distance) or at different voltage levels (vertical distance), the following rule is suggested: progressively increase the distance (horizontal and vertical) of the detailed modelling until the results do not change in a significant manner. As a general principle, the model should be extended in priority to the electrical nodes containing the higher capacitive components (including cables), as these will affect the most the frequency response. Figures 1, 2 and 3 illustrate this technique. They provide the network impedance seen from the target transformer depending on the horizontal and vertical distances modelled in detail (before the power frequency equivalents). Error: Reference source not found shows that, for this particular case, representing the 400 kV network up to three nodes away from the target transformer is enough to represent approximately the resonance (the response when representing up to four nodes is close). Note that in mesh networks a distance of three nodes away from the target transformer involves more than three nodes. In this particular case, about ten 400 kV nodes were at a three nodes distance from the target transformer. Error:Reference source not foundFig 2 and 3 shows that, for this particular case, representing 225 kV up to three nodes is enough (representing up to four, five and six nodes provides a similar response).

0 100 200 300 400 500 600 700 800 900 10000

100

200

300

400

500

600

700

800

frequency (Hz)

Impe

danc

e (O

hms)

Impact of 400 kV network reduction on the frequency response at Sub A

400kV network up to 1 node400kV network up to 2 nodes400kV network up to 3 nodes400kV network up to 4 nodes400kV network complete

Fig. 1. Network modelling: frequency response as a function of the horizontal distance (same voltage level) between the target transformer and the power frequency equivalents.

Page 57

0 100 200 300 400 500 600 700 800 900 10000

100

200

300

400

500

600

frequency (Hz)

Impe

danc

e (O

hms)

Impact of 225 kV network reduction on the frequency response at Sub A

400kV network complete400kV network complete + 225 kV network up to 1 node400kV network complete + 225 kV network up to 2 nodes400kV network complete + 225 kV network up to 3 nodes

Fig. 2. Network modelling: frequency response as a function of the vertical distance (lower voltage levels) between the target transformer and the power frequency equivalents.

0 100 200 300 400 500 600 700 800 900 10000

50

100

150

200

250

300

350

400

450

frequency (Hz)

Impe

danc

e (O

hm)

Impact of 225 kV network reduction on the frequency response at Sub A (2)

400kV network complete + 225 kV network up to 3 nodes

400kV network complete + 225 kV network up to 4 nodes400kV network complete + 225 kV network up to 5 nodes

400kV network complete + 225 kV network up to 6 nodes

Fig. 3. Network modelling: frequency response as a function of the vertical distance (lower voltage levels) between the target transformer and the power frequency equivalents (next).

Summary

For RMS-voltage drop studies in networks where no resonant overvoltages are generated, a simple power frequency network model can be used to represent the network outside the study zone.

When the frequency response of the network is of concern, a more detailed model of the system is required. If the user has the frequency response of the network, a network equivalent can be built. A single resonance network equivalent is easy to build but it simplifies the frequency response. For better accuracy, a FDNE should be used, but then this requires a good expertise. Another option, if the user is not familiar with network equivalents or if he

Page 58

has no information on the frequency response of the network, is to build a detailed model of the network in the neighbourhood of the target transformer.

References

[1] CIGRE WG 33-10, “Temporary overvoltages: Causes, effects and evaluation”, CIGRE Session, Report 33-210, Paris, 1990.

[2] IEC TR 60071-4, “Insulation Co-ordination - Part 4: Computational Guide to Insulation Co-ordination and Modeling of Electrical Networks”, 2004.

[3] D.W. Durbak, A.M. Gole, E.H. Camm, M. Marz, R.C. Degeneff, R.P. O’Leary, R. Natarajan, J.A. Martinez-Velasco, K.C. Lee, A. Morched, R. Shanahan, E.R. Pratico, G.C. Thomann, B. Shperling, A.J.F. Keri, D.A. Woodford, L. Rugeles, V. Rashkes, and A. Sharshar, “Modeling guidelines for switching transients,” Chapter 4 of Modeling and Analysis of System Transients using Digital Systems, A. Gole, J.A. Martinez-Velasco and A.J.F. Keri (Eds), IEEE Special Publication, TP-133-0, 1998.

[4] CIGRE WG 05 of Study Committee 13, “The calculation of switching surges – III. Transmission line representation for energization and re-energization studies with complex feeding networks”, Electra, no. 62, pp. 45-78, 1979.

[5] CIGRE WG 33-02, “Guidelines for representation of network elements when calculating transients”, CIGRE Brochure 39, 1991.

[6] A.S. Morched and V. Brandwajn, “Transmission network equivalents for electromagnetic transients studies,” IEEE Trans. on Power Apparatus and Systems, vol. 102, no. 9, pp. 2984-2994, September 1983.

[7] A.S. Morched, J.H. Ottevangers, and L. Marti, “Multiport frequency dependent network equivalents for the EMTP,” IEEE Trans. on Power Delivery, vol. 8, no. 3, pp. 1402-1412, July 1993.

[8] U.D. Annakkage, N.K.C. Nair, Y. Liang, A.M. Gole, V. Dinavahi, B. Gustavsen, T. Noda, H. Ghasemi, A. Monti, M. Matar, R. Iravani, and J.A. Martinez, “Dynamic system equivalents: A survey of available techniques”, IEEE Trans. on Power Delivery, vol. 27, no. 1, pp. 411-420, January 2012.

[9] A. Robert and T. Deflandre, CIGRE WGCC02, “Guide for assessing the network harmonic impedance”, Electra, vol. 167, August 1996.

[10] J. Arrillaga et al, CIGRE Joint TF 36.05.02/14.03.03, “AC system modelling for AC filter design - An overview of impedance modelling”, Electra, vol. 164, February 1996.

[11] N. Watson and J. Arrillaga, Power Systems Electromagnetic Transients Simulation, IET Power and Energy Series, no. 39, 2003.

[12] L. Marti, “Low-Order approximation of transmission line parameters for frequency-dependent models” IEEE Trans. on Power Apparatus and Systems, vol. 102, no. 11, pp. 3582-3589, November 1983.

[13] T. Noda, “Identification of a multiphase network equivalent for electromagnetic transient calculations using partitioned frequency response”, IEEE Trans. on Power Delivery, vol. 20, no. 2, pp. 1134-1142, April 2005.

[14] T. Noda, “A binary frequency-region partitioning algorithm for the identification of a multiphase network equivalent for EMT studies”, IEEE Trans. on Power Delivery, vol. 22, no. 2, pp. 1257-1258, April 2007.

[15] J. Hong and J. Park, “A time-domain approach to transmission network equivalents via Prony analysis for electromagnetic transients analysis”, IEEE Trans. on Power Systems, vol. 10, no. 4, pp. 1789-1797, November 1995.

[16] H. Singh and A. Abur, “Multiport equivalencing of external systems for simulation of switching transients”, IEEE Trans. on Power Delivery, vol. 10, no. 1, pp. 374-382, January 1995.

[17] A. Abur and H. Singh, “Time domain modelling of external systems for electromagnetic transients programs”, IEEE Trans. on Power Systems, vol. 8, no. 2, pp. 671-679, May 1993.

[18] A.S. Morched, J.H. Ottevangers, and L. Marti, “Multi-port frequency dependent network equivalents for the EMTP”, IEEE Trans. on Power Delivery, vol. 8, no. 3, pp. 1402-1412, July 1993.

[19] V.Q. Do and M.M. Gavrilovic, “An iterative pole removal method for synthesis of power system equivalent networks”, IEEE Trans. on Power Apparatus and Systems, vol. 103, no. 8, pp. 2065-2070, August 1984.

[20] V.Q. Do and M.M. Gavrilovic, “A synthesis method for one-port and multi-port equivalent networks for analysis of power system transients”, IEEE Trans. on Power Systems, vol. 1, no. 2, pp. 103-113, April 1986.

[21] B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by vector fitting”, IEEE Trans. on Power Delivery, vol. 14, no. 3, pp. 1052-1061, July 1999.

[22] B. Gustavsen, “Computer code for rational approximation of frequency dependent admittance matrices”, IEEE Trans. on Power Delivery, vol. 17, no. 4, pp. 1093-1098, October 2002.

[23] B. Gustavsen and A. Semlyen, “Application of vector fitting to state equation representation of transformers for simulation of electromagnetic transients”, IEEE Trans. on Power Delivery, vol. 13, no. 3, pp. 834-842, July 1998.

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[24] B. Gustavsen and A. Semlyen, “Enforcing passivity for admittance matrices approximated by rational functions”, IEEE Trans. on Power Systems, vol. 16, no. 1, pp. 97-104, February 2001.

[25] D. Deschrijver, B. Gustavsen, T. Dhaene, “Advancements in iterative methods for rational approximation in the frequency domain”, IEEE Trans. on Power Delivery, vol. 22, no. 3, pp. 1633-1642, July 2007.

[26] B. Gustavsen and C. Heitz, “Fast realization of the modal vector fitting method for rational modeling with accurate representation of small eigenvalues”, IEEE Trans. on Power Delivery, vol. 24, no. 3, pp. 1396-1405, July 2009.

[27] B. Gustavsen and T. Noda, “Techniques for the Identification of a Linear System from its Frequency Response Data”, Appendix A of Power System Transients. Parameter Determination, J.A. Martinez-Velasco (ed.), CRC Press, 2009.

5.1.3.5. Generators [MM]Two cases must be distinguished: synchronous and asynchronous generators. According to IEC TR 60071-4 and CIGRE TB 39 guidelines for low-frequency transients modelling [1][2], the synchronous machines must be modelled by the generalized Park’s model based on the complete equations in the direct and quadrature axes, representing saturation, excitation and mechanical torque, as well as voltage and speed controls. The effect of the capacitances is negligible.

However, in transformer energization studies the speed control may be neglected as transformer energization does not generate any significant active power variation. As for the voltage control, the modelling must be aware of the behavior of the regulator in presence of the voltage harmonics, a fact not taken into account in power-flow models.

If no significant voltage distortion is generated at the energization, the generator may be represented by an ideal sinusoidal voltage source with appropriate phase angles (from load flow) in series with the subtransient direct reactance and the armature resistance, thus neglecting generator dynamics and voltage control [3][4].

As mentioned in §5.1.1, if the synchronous machines are far enough from the transformer being energized, they can be included in a network equivalent.

The case of wind turbine generators (WT) using asynchronous machines is different. The behaviour of wind turbine generators during a voltage change is complex and is mainly determined by the electronic converter controls designed to comply with the Fault Ride Through (FRT) requirements of Distribution and Grid Codes. These requirements are different from country to country (for instance, some codes require maximise Mvar injection during voltage dip and other codes require MW injection proportional to the retained voltage), therefore it is not possible to use a generic model. The user should ask for a detailed model from the WT manufacturer applicable for the case under investigation.

References

[1] CIGRE WG 33.02, “Guidelines for Representation of Network Elements when Calculating Transients”, CIGRE brochure 63, 1990.


[3] CIGRE WG 33.10, “Temporary overvoltages. Test case results”, Electra 188, 2000.

[4] Martínez Duró, M., "Damping Modelling in Transformer Energization Studies for System restoration: Some Standard Models Compared to Field Measurements," in IEEE PowerTech 2009, Bucarest, 2009.

5.1.3.6. Loads [TM]Loads comprise a fundamental component of the real power system, no less important than the generating stations which provide the electrical power and the transmission and distribution systems which convey the power to the system loads. In simulations investigating the effects of transformer energization on the power system, the appropriate representation of loads in the network model is important for the following reasons:

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1. Loads must be included in the network model to achieve correct initial conditions for an EMT case, prior to any time domain simulation. The EMT model and resulting three-phase load flow should be based on and agree with a load flow calculation from a load flow program, as would be generated by Transmission Planning people.

2. Normal aggregate loads provide the main damping of transient and temporary overvoltages and may affect resonance conditions, particularly at higher frequencies [1], [2], [3].

3. Loads can transiently and dynamically interact with the inrush current, particularly for loads that are of significant size.

System loads can have a significant effect on the response of the network to transformer energization. Loads provide important damping of transient overvoltages as could be caused by excitation of low order harmonic parallel resonances in the network from the harmonic components in the magnetizing inrush currents. Customer’s synchronous motors contribute to the system fault levels and may help to mitigate voltage sag due to the flow of inrush current through the network. On the other hand, the response of large induction motors when terminal voltages sag, due to their voltage versus slip characteristic, will be to absorb increasing reactive power, which may aggravate the voltage sag after a few cycles. For transformer energization studies it is particularly important to include, in the system model, large loads that are located close to the transformer to be energized.

If the subject of the EMT simulation study is transient overvoltages, then the worst case would be a light load case where the damping of overvoltages provided by the loads is minimum. The presence of shunt capacitor banks in the network and in customer premises can contribute significantly to transient overvoltages. Industrial customers having significant power factor correction shunt capacitor banks or distribution stations having switchable distribution level shunt capacitor banks should be modelled at the minimum load for which the capacitor banks would still be in service. On the other hand, if voltage sag is the concern, a worst case would be obtained by modelling distribution system loads as being high but industrial customers should be modelled with a minimum number of large rotating machines in service. This, obviously, requires some knowledge of the customer’s various operating modes if a realistic case is to be created. If the rate of the recovery of the voltages is expected to be a concern, then large induction motors and their dynamics should be explicitly included in the system model.

Loads can be categorized as industrial, commercial, and residential/distribution, as will be described below. The modelling of different types of loads in EMT programs is described as follows.

Industrial Loads

These loads, which are generally relatively large, are supplied directly from the transmission system via isolating breaker(s), disconnect switches and intertie transformer(s). Pulp and paper mills, mining loads where ore is crushed and processed, compressor loads to liquefy natural gas, steel mills, ship loading and off-loading facilities, and the petrochemical industry are examples of industrial loads. Figure 1 shows a simplified electrical single-line diagram of a 30 MW industrial load, in this case a copper-gold mine.

The supply voltage at the point of interconnection with the utility can range from 60 kV to 345 kV (perhaps higher). The size of the load can range from a few MW to hundreds of MW, depending on the utility supply voltage. This type of load typically consists of a mixture of large synchronous and induction motors supplied at 2 kV to 25 kV according to power rating, a number of smaller induction motors supplied from lower voltage utilization busses and, finally, a possibly large number of quite small induction motors supplied from utilization busses where voltage can be as low as a few hundred Volts. In general, there could also be motors that are supplied from motor drives based on high power electronic converters, where the horsepower ratings of the motors could range from a few tens of kW to many thousands of kW. Usually, large harmonic-producing motor drives, such as LCI, cyclo-converters and variable speed drives, will have some associated harmonic filter banks and/or shunt capacitor banks for power factor correction. Industrial loads may also contain a relatively small component dedicated to providing the power requirements for lighting, heating and ancillary services.

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Figure 1 Simplified Single-Line Diagram of a Typical Industrial Load Having Synchronous and Induction Motors supplied from two Independent Busses, and Power Factor Correction Capacitor Bank.

Large industrial loads that are located electrically close to the transformer to be energized or along the path of the inrush current, should preferably be modelled in detail. This will require that the model developer have an up-to-date electrical single-line diagram of the customer’s facilities. The single-line should show the ratings and locations of the intertie and low voltage transformers, the disposition of the LV utilization busses, the larger motors, motor drives, power factor correction capacitor banks and harmonic filter banks. Ideally, large rotating machines, such as generators, synchronous motors and induction motors should be represented by electro-dynamic machine models.

The utility intertie and lower voltage transformers should be modelled according to Section 5.1.1. Saturation effects should be included when the customer is located electrically close to the transformer being energized, due to concerns of sympathetic interaction [4].

Synchronous motors (or generators if the customer has self-generation) should be modelled as per Section 5.1.3.5 by the generalized Park’s model, for example, EMT programs Synchronous Machine Model. Modelling should include the excitation system and mechanical torque. Large induction motors should be modelled by using, for example, the Universal Machine Model. Reference [5] lists typical data for induction motor models over a large range of kW ratings. The behaviour of the mechanical load with change in speed should be accounted for by the model.

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In many cases, however, this approach of detailed modelling of rotating machinery becomes impractical, due to lack of data or because of overwhelming complexity. Some simplifications to the modelling can be made that provide conservative yet usually reasonable results for transformer energization studies. Multiple synchronous machines supplied from the same bus can be combined into a single equivalent machine and the model can then be simplified as ideal sinusoidal voltage sources behind the d-axis subtransient reactance with a resistor in series per phase to account for the armature winding resistance (see §5.1.3.5). The subtransient reactance is normally around 0.2 pu based on machine ratings. In a similar way, induction motors can be simplified as ideal voltage sources behind the operational impedance of the motor which, for low frequency transients, is the locked-rotor impedance. The locked rotor impedance of squirrel-cage induction motors is generally around 0.2 pu based on ratings.

Large converter drives, such as LCI or adjustable speed, should be modelled explicitly but, if this is not possible, they could be approximated and treated as a DC converter terminal as seen from the system side. As a crude approximation, the converter can be modelled as a constant impedance load. Any associated filtering and shunt capacitor banks should be explicitly represented.

Many small low voltage (typically 600V and below) motors supplied from the same bus, where they are separated from the utility supply by several layers of transformers, can be treated as an aggregate load and approximated by a constant impedance equivalent R in parallel with jωL. This should account for the real and reactive power consumption of the load. Adjustable speed drives to small motors can be aggregated with the small motors that are directly connected to the same low voltage AC bus and included in the constant impedance model.

Shunt capacitor banks used for power factor correction and harmonic filter banks, where the voltage rating is more than a few hundred volts, should be explicitly modelled. These capacitor banks may affect the lower frequency series and parallel resonances as seen from the transformer being energized and can influence the transient response of the network.

Smaller Commercial Loads

These loads, generally smaller than typical industrial loads, are supplied from feeder circuits originating from a distribution substation at voltages usually at or below about 35 kV. Examples of this type of load would be sawmills, commercial refrigeration plants, and the induction motor driven pumps that are found in an oil and gas field for extraction of fluids.

The induction motors will be smaller than the large ones found in most industrial customers, and ratings could be from a few kW to possibly a few hundred kW. Some motors might be fed from adjustable speed drives.

The modelling of individual commercial loads, for transformer energization studies, should follow the guidelines given for industrial loads where the motors are not expected to have a large impact on the simulation. The largest motors can be aggregated, where appropriate, and modelled as ideal sinusoidal voltage sources behind locked-rotor impedance. The remainder of the load, including all of the small motors, can be combined and modelled as constant impedance equivalent to reflect the real and reactive power consumption. Shunt capacitor banks should be included in the model, if the kvar rating is significant.

Distribution Substation for Residential/Light Commercial Loads

These loads are generally supplied from a system of feeder circuits emanating from a distribution substation. While the individual loads are relatively small (perhaps only a few kW), there could be very many such loads involved. Residential loads comprise the usual domestic small low voltage single-phase induction motors (for refrigerators, washing machines, clothes dryers, pumps etc.), consumer electronics, resistive heating, and so on.

The modelling of a distribution system, where the total load is the aggregate of many small low voltage loads, presents some difficulties; the system exhibits some frequency-dependence and the nature of the load varies with

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time of day. Ideally, this complicated system would be represented by a frequency-dependent equivalent circuit based on field measurements. In practice, however, seldom are such field measurements available for the model developer. The literature provides ideas for modeling such loads for harmonic studies [1], [2] and [3]. For such studies loads are often represented by simple models consisting of series and/or parallel Rs and Ls derived from the active and reactive power flow parameters. These Rs and Ls can be given harmonic weighting factors to reflect frequency dependency. For transformer energization studies, it might sometimes be impractical to develop a frequency-dependent model for a low voltage distribution-type load, so a simple constant impedance equivalent comprised of R in parallel with ωL to ground should be sufficient. This equivalent should account for the steady state real and reactive power consumption of the load being represented.

It is important to model shunt capacitor banks (switchable or fixed) within or electrically close to the distribution substation. Also, if there are distribution cables emanating from the station that are of significant length (few hundred metres) then the effective shunt capacitance of these should be calculated and combined into an equivalent shunt capacitance at the station LV bus.

Modeling of a Transformer Which is Not Unloaded

Sometimes the transformer to be energized, for simulation purposes, cannot be considered to be unloaded. Firstly, there might be some load connected to the transformer LV bus so that, when the transformer is picked up, so is the load. An example would be a distribution station transformer where the feeder breaker or recloser remains closed even though the transformer and distribution system are isolated from the grid and are de-energized. In this case the inrush current, when the transformer is energized, includes the effects of a “cold-load pickup”. Secondly, there could be one or more secondary transformers and some low voltage auxiliary load. An example could be a generator transformer with an LV station service transformer to auxiliary loads comprised of small motors, heating, lighting, and so on. The ratings of these secondary transformers will normally be significantly less than the rating of the main transformer. The presence of a secondary transformer and/or connected load can affect the overall inrush currents and the response of the external network compared to the results obtained with only the main transformer.

If the MVA rating is significant compared to the main transformer, a station service transformer (or other secondary transformer) should be included in the system model. The transformer will have a nonlinear magnetizing characteristic and residual flux in the core as a result of being switched off. The model developer should be aware that a secondary transformer might be comprised of single-phase units rather than being a three-phase unit. Ideally, it should be modeled in accordance with the guidelines discussed elsewhere in this Guide for modeling of the main transformer. Three-phase motors will probably not be a consideration since the motor contactors will have been opened by motor protection and they should remain disconnected for the duration of the transformer energization. Small low voltage single-phase motors, on the other hand, will likely still be connected to their low voltage system and should be included in the modeling. They can be aggregated where appropriate and represented by a locked-rotor equivalent impedance. Heating, lighting and small electronic loads can be modelled as an equivalent constant impedance load, as discussed in the previous subsection.

References

[1] “Guide for Assessing the Network Harmonic Impedance”, A.Robert, T. Deflandre, Working Group: CC02, Electra No 167, pp. 96-131, Aug. 1996.

[2] “AC System Modelling for AC Filter Design – An Overview of Impedance Modelling”, J. Arrillaga, L. Juhlin, M. Lahtinen, P. Ribeiro, A.R. Saavedra, Joint Task Force: 36.05.02/14.03.03, Electra No 164, pp. 132-151, Feb. 1996.

[3] “Harmonics, characteristic parameters, methods of study, estimates of existing values in the network”, Working Group 36-05, Electra No 77, pp. 35-59.

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[4] H.S. Bronzeado, P.B. Brogan and R. Yacamini, “Harmonic Analysis of Transient Currents During Sympathetic Interaction,” IEEE Trans. On Power Systems, Vol. 11, No 4, pp. 2051-2056, Nov. 1996.

[5] EPRI Report TR-107332 “Evaluation of Distribution Capacitor Switching Concerns,” prepared by Electrotek Concepts, Inc. Oct. 1997.

5.1.3.7. Circuit-breakers [NC] How do you model the circuit-breakers? Ideal switch & grading capacitors

When representing low-frequency phenomena (below 5 kHz), a simple ideal breaker can be used for representing closing operation [CB1]. It is very important to represent mechanical pole spread: see §Error: Reference source notfound.

If a circuit breaker if equipped with a closing resistor it has to be simulated. Grading capacitors are also to be included in the CB model as the residual voltage on the secondary side of the breaker may interact with the residual flux in the transformer. The effect of prestrike phenomena is negligible.

The accurate representation of opening operation in the low frequency range is important if the employed transformer model is capable to represent de-energization transient and retain the residual flux. In this case, a more advanced CB model capable to represent the arc condition has to be used. The model should at leas represent the high current interruption and current chopping capability of the CB; restrike characteristic and high frequency current chopping can be neglected [CB1]. Black box ark models can be used for this scope [CB2,CB3].

References

[CB1] Cigré Brochure 39, “Guidelines for representation of network elements when calculating transients”, WG 02, SC 33, 1990

[CB2] Electra 117, “Practical application of arc physics in circuit breakers. Survey of calculation methods and application guide.”, WG 13.01, 1988

[CB3]Electra 194, “” Application of black box modelling to circuit breakers”, WG 13.01, 1993

5.2. Quantification of the overvoltage stress in the transformers and the surge arresters [MM]

The previous sections of this document have shown how to simulate the transient voltages and currents generated during the energization of transformers. Due to resonance phenomena already described, harmonic resonant overvoltages may appear during the energization that could damage the equipment insulation. This section deals with the assessment of these potential damaging effects of the overvoltages.

In the IEC typology, resonant overvoltages belong to the class of temporary overvoltages (TOV) [9]. In order to prevent any damage to the substation equipment, the amplitude and duration of the calculated overvoltages must be compared to the TOV withstand capability of the most vulnerable equipment. For that, phase-to-ground and phase-to-phase voltages must be distinguished.

Most vulnerable equipment to phase-to-ground and phase-to-phase overvoltages

The TOV withstand capability depends on the equipment considered. In general, for phase-to-ground TOV lasting a few seconds or less, the surge arresters connected between phase and ground to protect the transformers from switching and lightning surges are the most vulnerable equipment; for longer durations, the most vulnerable

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equipment to phase-to-ground TOV generally are the power transformers [1][2][3]. As arresters between phases are seldom used, three-phase power transformers are then the most vulnerable equipment to phase-to-phase TOV.

Note that per unit phase-to-phase withstand of three-phase power transformers is usually smaller than the per unit phase-to-ground withstand capability of the surge arresters (see figure below); this means that a given overvoltage may be below arresters’ (phase-to-ground) withstand capability but above transformers’ phase-to-phase withstand capability.

It follows from these considerations that the calculated phase-to-ground voltages must be below the withstand capability of the surge arresters and transformers, and that the phase-to-phase voltages must be below the three-phase transformers phase-to-phase withstand capability.

Evaluation of the surge arresters stress

According to the standards [4][6], the manufacturers provide information on the surge arrester TOV withstand capability. This information is given in the form of a voltage/duration curve. These curves provide de maximum allowable duration of a power frequency voltage of given constant amplitude, as illustrated in Figure 3.

FIGURE 3: TYPICAL V-T WITHSTAND CAPABILITY CURVES FOR A SURGE ARRESTER AND A TRANSFORMER

Usually, two curves are provided with and without prior energy absorption (taking into account a possible prior surge). To be conservative, it is recommended to use prior energy curves. In the IEC standard, the rated voltage of the arrester corresponds to the TOV withstand for 10 s with prior energy. Then, the V/t curve takes this general form [4][7]:

V TOV ,max=(10T TOV ,max )

B 1

⋅V rated

Calculated phase-to-ground TOV must be compared to the TOV capability provided by the surge arrester manufacturer [5][6][7]. Thus, if the calculated overvoltage amplitude and duration are V and t, it must be verified that

V < VTOV,max(t)

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Or, alternatively,

t < TTOV,max(V)

However, transient overvoltages due to transformer energization have time-varying amplitude. In order to take this into account, [3] suggests dividing the voltage signal in several constant-amplitude regions (see Figure 4) and cumulating the effects:

t1/TTOV,max(V1) + t2/TTOV,max(V2) + … < 1

FIGURE 4: VARIABLE AMPLITUDE OVERVOLTAGE (SINGLE FREQUENCY) (SOURCE: [3])

Moreover, if the voltage contains harmonics and therefore it is not sinusoidal, it should be treated as purely sinusoidal at power frequency having amplitude equal to the maximum instantaneous voltage. This is equivalent to assume a power frequency envelope of the voltage.

An alternative method to assess the arrester stress is to calculate its energy duty due to the overvoltage and compare it to its energy absorption capability (given in kJ/kV). This method has the advantage that it takes harmonics into account (whereas TOV withstand capability provided by manufacturers is for power-frequency voltages) [6]. However, the energy absorption capability depends on the form of the overvoltage [8], and the manufacturer kJ/kV values are in general for switching surges [7]; furthermore, the energy absorption capability is not covered by the standards.

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Evaluation of the transformers stress

As already mentioned, for TOVs with short duration, the transformers’ phase-to-ground withstand capability is generally larger than that of the surge arresters. Therefore, it can be concluded that if the surge arresters are capable of withstanding the simulated overvoltages, the transformer phase-to-ground insulation should also be capable. However, it must be emphasized that this will not be true for longer duration TOVs. The case of phase-to-phase overvoltages is different: they do not affect the surge arresters (surge arresters between phases are uncommon) but they do stress the inter-phase insulation of three-phase power transformers.

a) Using the short-duration power frequency test level

The overvoltage withstand information usually available for transformers are the standard test levels for power frequency (short and long duration), switching impulse and lightning impulse32. According to IEC 71 on insulation coordination [9], the standard withstand voltage test for temporary overvoltages is the short-duration power frequency test. Thus, the calculated overvoltages should be compared to the voltage applied during the short-duration power frequency test.

Moreover, when comparing expected overvoltages to equipment standard test levels, IEC 71 suggests using a safety factor in order to “compensate for the differences in the equipment assembly, the dispersion in the product quality, the quality of installation, the ageing of the insulation during the expected lifetime, other unknown influences.” [10]. For internal insulation, IEC suggests using a security factor of 1.15, which means a 13% reduction of the initial insulation strength. As for the ageing effect alone, [11] suggests that the reduction in transformer strength is between 5 to 10%.

In conclusion, following the IEC 71, the simulated overvoltages should be compared to the voltage applied during the short-duration power frequency test reduced by a safety margin varying from 5% to 13%. Notice that this method of evaluating the stress pays no attention to the duration of the overvoltage: there is only a voltage threshold beyond which the transformer withstand capability is exceeded.

The user must also consider the overfluxing capabilities of the transformer [2]; for instance, for transformers connected to generators, IEC 76 asks for 1.4 times the rated voltage during 5 s.

b) Building V-t curves

However, the threshold method may be too conservative and lead to unrealistic and too restrictive results. This is due to the fact that the transformer overvoltage withstand is a function of the duration, as for the surge arresters (see Figure 3), and the duration of the short-duration power frequency test is much larger than that of the harmonic TOV generated by transformer energization. The power frequency test duration is usually between 30 and 60 seconds, whereas the duration of the energization TOV is seldom larger than several seconds. Thus, the transformer’s actual withstand to energization TOV is much larger than the short-duration power frequency test level.

A more realistic and less conservative evaluation of the transformer stress is obtained by using voltage/duration curves as those presented for the surge arresters [1][2][12], both phase-to-ground and phase-to-phase. These V/t curves, however, aren’t covered by the standards and must be individually requested to the manufacturers [13]. If no curve can be obtained from the manufacturer, a tentative V/t curve may be built by interpolating between voltage levels induced during the standard tests, as shown in [11], to which a safety factor should also be applied. This is illustrated in the Figure 3 above, where phase-to-ground and phase-to-phase TOV capability curves have been built for a transformer by semi-logarithmic interpolation between the phase-to-ground and phase-to-phase voltages induced during the switching and power frequency tests, reduced by a safety factor of 1.3 suggested by the manufacturer [14].

32 Not all the tests are always performed, but standards provide test conversion factors in order to calculate equivalent tests levels.

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References

[1] WG 33.10, “Temporary overvoltage withstand characteristics of extra high voltage equipment”, Electra 179, 1998.

[2] Adibi, M.M., R.W. Alexander and B. Avramovic, “Overvoltage Control During Restoration”, IEEE Transactions on Power Systems, Vol. 7, No. 4, pp. 1464-1470, Nov 1992.

[3] WG 33.210, “Temporary overvoltages: causes, effects and evaluation”, Proceedings of the 33rd Session. CIGRE, Paris, France, 1990.

[4] IEC 60099-4:2009, “Surge arresters – Part 4: Metal-oxide surge arresters without gaps for a.c. systems”.

[5] IEC 60099-5:2000, “Surge arresters – Part 5: Selection and application recommendations”.

[6] IEEE C62.22-2009, “IEEE Guide for the Application of Metal-Oxide Surge Arresters for Alternating-Current Systems”.

[7] Hileman, A. R., Insulation Coordination for Power Systems, CRC Press, Boca Raton (Fl), 1999.

[8] Martinez-Velasco, J. A. (ed.), Power system transients: parameter determination, CRC, Boca Raton (Fl), 2010.

[9] IEC 60071-1:2006, Insulation co-ordination – Part 1: Definitions, principles and rules.

[10] IEC 60071-2:1996, Insulation co-ordination – Part 2: Application guide.

[11]Balma P. M. , R. C. Degeneff, H. R. Moore, L. B. Wagenaar, “The Effects of Long Term Operation and System Conditions on the Dielectric Capability and Insulation Coordination of Large Power Transformers”, IEEE Transactions on Power Delivery, Vol. 14, No. 3, July 1999.

[12]Antonova, N.P, et al., “Temporary overvoltages and their influence upon the insulation level of the equipment”, CIGRE Session, Paris, 1990.

[13]Manuel Martínez Duró, Alain Tanguy, "Lack of Information about Transformer Dielectric Withstand to Harmonic Temporary Overvoltages", Contribution in the Proceedings of the Cigré 43th Session, Paris, 2010.

[14]Martínez Duró, M., R. Denis, “Parameter uncertainty assessment in the calculation of the overvoltages due to transformer energization in resonant networks”, paper C4-204, CIGRE Session, Paris, 2012.

5.3. Parameter uncertainty assessment and stochastic simulation [MM]

The previous sections have shown how to model the system in order to simulate the energizing transient, i. e., the inrush currents, the RMS-voltage step change and the possible resonant overvoltages. Yet, when modeling the system, the value of a number of parameters is not precisely known, that is, they are affected by some uncertainty. The variation of many of these parameters in their uncertainty range does not affect the results (i. e., the computed currents and voltages); thus, these uncertainties can be neglected. However, this will not be the case for a small set of parameters: those for which the results are highly dependent on the particular parameter value within its uncertainty range. Therefore, the uncertainty on these parameters cannot be neglected but rather has to be taken into account in the study.

Due to their dependency on uncertain parameters, the currents and voltages are not deterministic but rather stochastic variables, their values depending on the particular range taken by the uncertain parameters. Therefore, the output of the simulation study is not a single figure but a probability distribution. Often, however, the user is not interested in the whole distribution of the output but in the risk of exceeding a particular threshold limit, for instance a given RMS-voltage drop limit or the equipment overvoltage withstand limit. This is achieved by computing the threshold-exceeding probability, pthr.

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In order to compute the probability distributions of the stochastic variables (currents, voltages, stresses…) and the threshold-exceeding probability, a number of simulations must be performed to assess the effect of the uncertainties associated with each input parameter. For this, the important uncertain parameters must be identified and characterized, and a suitable stochastic simulation technique must be used. The following sections treat each of these aspects. For a general framework on treatment of uncertainty the reader is referred to [1].

5.3.1. IMPORTANT UNCERTAIN PARAMETERSMany input parameters may be affected by uncertainties; however, only those whose variation may have an important influence on the output variable need to be considered. It is possible to identify potentially important parameters thanks to the knowledge of the physical phenomenon under study. Regarding the inrush currents at transformer energization, it is well established that they are highly dependent on two random parameters: the residual flux in the core before the energization and the circuit-breaker point-on-wave switching times. Indeed, it is widely accepted that the residual flux within the transformer core does not change over time unless there are external influences33. As for the circuit-breaker switching times, they can occur any time in the sine wave and in general they are not simultaneous because of the mechanical pole spread. A third uncertain parameter affecting the inrush currents may be taken into account: the air-core inductance of the transformer, as its value cannot be measured and is analytically calculated by the manufacturer with some estimated accuracy. Notice that the residual flux and switching-times values change at each energization due to the intrinsic random behavior of the physical phenomena, whereas the air-core inductance has a constant value that is not precisely known due to a lack of knowledge. These two kinds of uncertainties are called, respectively, aleatory and epistemic uncertainty [1].

If the system resonance frequencies are low (say below 800 Hz), then the (epistemic) uncertainties on the capacitive and inductive components of the system must be taken into account, as very small variations of these parameters can tune the system resonances to the inrush current harmonic frequencies and thus lead to resonant overvoltages [2]. As an example, in the power restoration configuration presented in §5.3.4, two parameters influencing the network frequency response are considered as affected by some uncertainty: the generator reactance and the phase-to-ground capacitance of the overhead lines.

5.3.2. UNCERTAINTY QUANTIFICATION AND MODELINGThe uncertainty of each uncertain parameter must be quantified and modeled by a probability distribution. For most of the parameters, a uniform distribution can be used as there is no reason to favor one or another value in their uncertainty range, i. e., all the possible values are equally probable. (A triangular distribution could also be used if the central value is considered the most probable.)

Circuit-breaker closing times

The circuit-breaker closing times uncertainty can be modeled by four random parameters: a common order time, the same for the three poles of the CB, and the random offset time of each pole due to the mechanical pole spread: tclose,i=torder+toffset,i i=A, B, C. The common order time may take any value in the sine wave, thus following an uniform probability distribution over the power frequency period T (equal to 20 ms if f=50 Hz). The offset time of each pole is considered to follow a Gaussian law whose mean is zero (the three poles tend to close simultaneously) and whose standard deviation () is calculated from the maximum pole span, MPS (i. e., the maximum delay between poles). The three toffset,i follow the same Gaussian law N(0, ). As the interval ±3. has 99.7% probability in a Gaussian distribution, the standard deviation may be calculated from this interval width as =MPS/6. A 5 ms maximum delay between poles can be considered if no specific information on the CB is available [3][4]. However,

33 Grading capacitor together with the stray capacitance of the bus bars and transformer induce a power frequency voltage on the transformer after the opening of the CB. [RF1] suggest that the presence of grading capacitor has the tendency to decrease the residual flux in the transformer after de-energization.

[RF1] Cigré Brochure 263, “Controlled switching og HVAC circuit breaker”, WG A3.07, 2004.

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delays up to 20-40 ms have been measured in special circumstances. For MPS=5 ms, torder=U[100, 120] ms, toffset,i=N(0, 0.83) ms.

Residual flux

The residual flux can be modeled by two parameters: the residual flux amplitude and a parameter identifying the flux pattern. For the residual flux amplitude (0), we consider that it is uniformly distributed between zero and a maximum value of 80% of the rated flux [5][6][7]. For the residual flux distribution in the core, little knowledge is available (see however [6][7][8], among others). Reference [2] suggests considering twelve equally probable residual flux patterns: 0, -0/2, -0/2, 0, -0, 0 and all the possible phase and sign permutations. The actual pattern is selected by a uniformly distributed random parameter N=1..12.

Air-core inductance

The air-core reactance can be modeled by a uniform distribution centered at the manufacturer’s value and covering the accuracy range (20% accuracy is often suggested by the manufacturers).

Supplying network parameters

The uncertainty on the inductive and capacitive parameters of the system must be considered when the resonance frequencies of the system are low (generator inductances, overhead line inductances and capacitances, etc.). The level of uncertainty depends on the degree of confidence in the data used to build the system model. For instance, overhead-line sequence parameters may have been calculated for an average tower and conductor disposition, not the actual ones. In the example below, §5.3.4, some other uncertainties are considered in a power restoration configuration.

5.3.3. STOCHASTIC SIMULATION: THE MONTE CARLO METHODStochastic simulation techniques provide a way to compute the output variables probability distributions given the probability distributions of the input parameters. The best known technique is the Monte Carlo simulation, the only one that will be treated here. As Monte Carlo simulation can be very slow, a number of alternative techniques have been suggested to accelerate the process [10][11]. Some of them have been compared for transformer energization studies in [2].

The Monte Carlo method

The Monte Carlo method [9][10][11] estimates the output variable probability distribution by generating a sufficient number of random samples of this variable according to the random probability distributions of the uncertain parameters. For example, if the output variable is the RMS-voltage drop and the uncertain parameters are the residual flux and the CB closing times, then the RMS-voltage drop probability distribution is estimated by simulation, the residual flux and CB closing times values used in each simulation being randomly generated according to their respective probability distributions. The estimate of the cumulative distribution function (CDF) of the output variable (in this case the RMS-voltage drop), Y, F(y)=Prob(Y<y), is then obtained as:

F ( y )= 1N ∑

i

1 yi< y (1)

where N is the number of simulations performed, yi (i=1,2..N) are the output variable outcomes and 1yi<y equals 1 if yi<y and 0 otherwise.

Usually, the user is mainly interested in the probability of exceeding a particular threshold limit of the output variable, y0, pthr=Prob(Y>y0), for instance the probability of exceeding a given RMS-voltage drop limit. Then, only one point of F(y) needs to be calculated, as the threshold-exceeding probability corresponds to pthr=1-F(Y<y0). This is given by:

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pthr=1N ∑

i

1yi> y0 (2)

i.e. the threshold-exceeding probability equals the number of the output variable simulation outcomes that exceed the threshold divided by the total number of simulations.

For the estimate to be good (unbiased) in the Monte-Carlo method, the only condition that must be met is that the outcomes of the output variable are randomly generated according to the probability distributions of the uncertain parameters.

The second essential point of the Monte-Carlo method is that the statistical error of the output variable estimate can be assessed by means of a (asymptotic) confidence interval of the estimate. In the case of a threshold-exceeding probability, the 95% confidence interval is easily calculated as:

pthr∈ pthr±(1 .96⋅√ p thr⋅(1− pthr)N ) (3)

If the empirical threshold-exceeding probability is zero, i. e., if no simulation exceeds the threshold limit, then the so-called rule of 3 provides the following confidence interval for pthr [14]:

pthr=0 ⇒ p thr∈[0 ,3N ] (4)

Explicit Monte Carlo algorithm

According to the previous exposition, the list below presents the explicit steps for a Monte Carlo study wanting to calculate a threshold-exceeding probability, pthr :

0) Choose a stop criterion; for instance, the desired maximal confidence interval width pthr,MAX

1) Generate a random sample xi of the uncertain parameters2) Simulate de output yi for xi

3) Compute the confidence interval width pthr (equation 3 or 4)4) Plot the convergence graph (i. e., the confidence interval as a function of the number of simulations: see

Figure 7)5) Test convergence of the algorithm: if pthr ≤ pthr,MAX continue to step 5; else go to step 1

Plotting the convergence graph (step 4) is not a necessary step but it is highly recommended in order to check the whole process.

Minimal number of simulations

The confidence interval is said asymptotic because it is valid only if the error is Gaussian distributed, which, according to the Central Limit theorem, is true only as N tends to infinity. In practice, this means that N must be large enough. How large N needs to be depends on the problem. 30 is usually suggested as a minimal N. Another rule of thumb states that a binomial distribution approximates a Gaussian distribution when p.N>5 and (1-p).N>5, where p is pthr or F(y) (estimated by their empirical estimates).

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Computer implementation

EMT simulation packages rarely implement stochastic simulation capabilities (usually they can only generate random switching times for circuit breakers). Thus, the random parameter generation and confidence interval calculations need some form of programming either in the simulation package itself (if it has scripting capability), or through coupling with a numerical computational software.

5.3.4. EXAMPLE OF APPLICATION TO TRANSFORMER ENERGIZATION STUDIES

As an example, consider we are interested in computing the stress due to the overvoltages generated when energizing the transformer of the following network [2] (for an application to the RMS-voltage drop assessment, see [15]):

FIGURE 5: STUDY CASE

The figure below presents the feeding system frequency response to direct excitation, showing two resonance points, the first close to 200 Hz and the second close to 700 Hz.

FIGURE 6: FEEDING SYSTEM FREQUENCY RESPONSE TO POSITIVE EXCITATION (ALL THREE PHASES SHOWN)

A stress rate function has been built according to the indications in §5.2, so each simulation is associated to a single figure, called the global stress rate, GSR. If the GSR exceeds 1, then the energization is considered as leading to failure. The purpose of the study is to compute the threshold-exceeding probability GSR>1, i. e., the risk of failure at the energization, pf.

All the uncertain parameters we have mentioned in §5.3.1 are included in the study: As for the parameters influencing the inrush currents, the uncertainty on the switching times of the circuit-breaker poles (t0, toffset,A, toffset,B, toffset,C), the residual flux at the energized transformer (0,N) and the air-core inductance of the transformer (Lair-core) are taken into account. As the first system resonance frequency is low, 200 Hz, the uncertainty on two parameters influencing the system frequency response is considered: the generator reactance (L’’

d,gen), whose value is provided by the manufacturer with a given accuracy; and the conductors height of the overhead lines (HOHL), affected by some uncertainty due to the averaging of the tower heights, the varying sag of the conductors and the terrain

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topography. Reference [2] suggests using 15% accuracy for both parameters. Notice that 15% uncertainty on the conductors’ height is equivalent to 3% uncertainty on the phase-to-ground capacitance. Both uncertainties are modeled by uniform distributions centered at their nominal values.

Figure 6 shows the confidence interval for the risk of failure, pf, provided by the application of the Monte Carlo algorithm depending on the number of simulations performed. This figure shows that the confidence interval becomes shorter as the number of simulations increases (although the reduction slows down as the number of simulations increases, a known drawback of the Monte Carlo method).

FIGURE 7: 95% MONTE CARLO CONFIDENCE INTERVAL FOR PF AGAINST THE NUMBER OF SIMULATIONS, COMPARED TO THE CONFIDENCE INTERVAL FOR N=10000

After a sufficient number of simulations, the confidence interval will be small enough and the user will have a fairly good idea of the risk of failure, pf, which depends only on the system being modeled and the considered uncertainties on the parameters.

If the final value of pf is too high to be accepted, the user can try to reduce epistemic uncertainties, for instance by performing specific measurements on selected uncertain parameters, in order to compute more realistic risk estimation [16]. For that, the user should focus on the uncertain parameters that affect the result most. For the above example case, Figure 6 shows that the final calculated risk is around 3-3.5%. This risk is associated with the considered aleatory (residual flux, CB switching times) and epistemic (the rest of the uncertain parameters) uncertainties. If this risk is unacceptable (i. e., the user cannot accept 3-3.5% chance of the transformer being damaged at the energization), then the user may consider reducing the epistemic uncertainties to perform a more realistic risk calculation. For that, the generator reactance will be one of the target parameters for uncertainty reduction, as it has a high influence on the results: for instance, if the generator reactance is taken at its lowest value (L’’

d,gen=0.35±15%), 0.3 pu, the risk of failure (for 1000 simulations) is pf=0%, whereas if the reactance is taken at its highest value, 0.4 pu, the risk of failure is pf=56%.

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References

[1] Rocquigny E. de, N. Devictor, S. Tarantola (eds.), Uncertainty in Industrial Practice: A Guide to Quantitative Uncertainty Management, Wiley & Sons Publishers, 2008.

[2] Martínez Duró, M., R. Denis, “Parameter uncertainty assessment in the calculation of the overvoltages due to transformer energization in resonant networks”, paper C4-204, Set of papers of the 44th CIGRE Session, Paris, 2012.

[3] CIGRE WG 33.02, Guidelines for representation of network elements when calculating transients: CIGRE Technical Brochure 39, 1990.


[5] Povh, D., W. Schultz, 1978, “Analysis of Overvoltages caused by Transformer Magnetizing Inrush Currents”, IEEE Trans. on Power Apparatus and Systems vol. PAS-97 no. 4, p. 1355-1365.

[6] Brunke, J.H., “Elimination of Transient Inrush Currents When Energizing Unloaded Power Transformers”, Doctoral Dissertation, Swiss Federal Institute of Technology, Zurich, 1998.

[7] Holmgren, R., Jenkins, R.S., Riubrugent, J., “Transformer Inrush Current,” CIGRE paper 12-03, CIGRE Paris, pp. 1-13, 1968.

[8] Chiesa, N., A. Avendaño, H. K. Høidalen, B. A. Mork, D. Ishchenko, and A. P. Kunze, “On the ringdown transient of transformers,” in IPST’07 - International Conference on Power System Transients, no. IPST-229, Lyon, France, Jun. 2007.

[9] Robert, C.P., G. Casella. Monte Carlo Statistical Methods, New York: Springer-Verlag, 2004.

[10]Kroese, D. P., Taimre, T., Botev, Z.I., Handbook of Monte Carlo Methods. New York: John Wiley & Sons, 2011.

[11]Glasserman, P., Monte Carlo Methods in Financial Engineering, Springer, 2004.

[12]M.H. Kalos and P.A. Whitlock, Monte Carlo Methods, John Wiley & Sons, 1986.

[13]Rubinstein, R. Y., Simulation and the Monte Carlo Method, John Wiley and Sons, New York, 1981.

[14]Hanley JA, Lippman-Hand A., “If nothing goes wrong, is everything all right? Interpreting zero numerators”, JAMA 1983; 249: 1743-1745. 35.

[15]Martinez Velasco, J., J. Martin-Arnedo, “Voltage Sag Stochastic Prediction Using an Electromagnetic Transients Program”, IEEE Transactions on Power Delivery, Vol. 19, 2004, pp. 596- 602.

[16]Martínez Duró, M., “Cost-effective uncertainty treatment”, Contribution in the Proceedings of the Cigre 44th Session, Paris, 2012.

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6. Appendix: Analytical calculation of inrush current [NC]Standard analytical formulas for the calculation of inrush peak and rate of decay are derived from single-phase transformer theory. The inrush current on a three-phase transformer can be calculated analytically based on the analytical formulas for a single-phase transformer and an empirical scaling factor. This factor accounts for the number of phases, core construction and coupling of the transformer34,35,36.

A relatively simple equation based on a sustained exponential decay of the inrush current is given by Bertagnolli37:

(1)

(2)

with:U = voltage across the winding phaseRW = winding resistnaceLair core = air-core winding inductanceBN =rated flux density in the core (peak value)BR = remanent flux density in the core BS = saturation flux density (approx. 2.0-2.05 T)tn = time variable =time constant of exponential decay

This equation is useful for rapid hand calculations due to its simplicity.

The analytical formula proposed by Specht38 is somewhat more accurate as the decay of the dc component of the flux (BR) is considered only during saturation (B > BS):

(3)

34 L. F. Blume, Transformer engineering : a treatise on the theory, operation, and application of transformers, 2nd ed. New York, N.Y.: John Wiley and Sons, Inc., 1951.

35S. V. Kulkarni and S. A. Khaparde, Transformer engineering: design and practice, ser. Power engineering. New York, N.Y.: Marcel Dekker, Inc., 2004, vol. 25.

36W. Schmidt, “Ueber den einschaltstrombeidrehstromtransformatoren,” ElektrotechnischeZeitschrift – Edition A, vol. 82, no. 15, pp. 471–474, Jul. 1961, Switch-on current in 3-phase a-c transformers.

37 G. Bertagnolli, Short-Circuit Duty of Power Transformers, Second Revised Edition. ABB, 1996.

38 T. R. Specht, “Transformer magnetizing inrush current,” AIEE Trans, vol. 70, pp. 323–328, 1951.

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(4)

with the saturation angle: θ=cos−1(BS−BN−BR

BN

)

An improved analytical equation is recommended by Holcomb39:

(5)

(6)

Where ts is the time when the core begins to saturate (B(t) >Bs). tS is calculated for the first period from:

B ( t )=BR+BN ∙(1−cosωt )

thereafter for the following period from:

B ( t )=BS+BN ∙(cos ωt 0−cos ωt)

with t0 the time when the inrush current reaches zero at each cycle calculated from (5) for i(t0)=0. It is assumed that the inrush current is different from zero only between ts and t0.

Equation (1) and (3) calculate only the envelope of the inrush current peaks, not the actual waveform. Equation (5) can be used to calculate analytically an approximate waveform of the inrush current. These three approaches are compared in Fig. 4 using a common set of parameters. The air-core inductance Lair-core of a winding can be calculated as:

(7)

With heq_HV being the equivalent height of the winding including fringing effects. The equivalent height is obtained by dividing the winding height by the Rogowski factor KR (< 1.0)40. This factor is usually determined empirically and is a function of the height, mean diameter, and radial width of a winding. More advanced equations for the calculation of

39 J. F. Holcomb, “Distribution transformer magnetizing inrush current,” Transactions of the American Institute of Electrical Engineers, Part III (Power Apparatus and Systems), vol. 80, no. 57, pp. 697–702, Dec. 1961.

40S. V. Kulkarni and S. A. Khaparde, Transformer engineering: design and practice, ser. Power engineering. New York, N.Y.: Marcel Dekker, Inc., 2004, vol. 25.

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the air-core inductance takes into account the thickness and diameter of the winding in addition to the height for the calculation of the equivalent height41,42.

Fig. 4: Analytical estimation of inrush current.

These type of formulas are commonly used by manufacturers for the estimation of the inrush-current first peak and rate of decay. In-house developed experience factors are used to fine tune these analytical formulas for the design and materials of each specific transformer. In particular, those factors are used to tune the calculation of the angle of saturation, the maximum current, and the rate of decay. In addition, the choice of the empirical formula for the calculation of the transformer reactance ωLair-core affects the final result. It is reported in43 that an accuracy of 40% should be expected from analytical formulas for the calculation of inrush current.

41 T. H. Fawzi and P. E. Burke, “The accurate computation of self and mutual inductances of circular coils,” IEEE Trans Power App. Syst., vol.-97, no. 2, 1978, pp. 464–468.

42 Wheeler, H.A.; , "Inductance formulas for circular and square coils," Proceedings of the IEEE , vol.70, no.12, pp. 1449- 1450, Dec. 1982

43 T. R. Specht, “Transformer magnetizing inrush current,” AIEE Trans, vol. 70, pp. 323–328, 1951.

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7. Appendix: Transformer modelling: Calculation of saturation curve from no-load test and B-H curve [NC]

Transformer data: 66MVA 238/13.8kV star-delta transformer three-leg stacked core (60Hz)

Leakage inductance: 10% (at 66MVA)

Air-core inductance: 26% from HV winding (at 66 MVA)

No-load Test Report

Voltage (%) Current (%) Losses (kW)

90 0.049 28.01

95 0.055 32.02

100 0.067 36.73

105 0.101 43.61

110 0.204 53.06

The no-load test is performed on the LV side (delta-connected)

The core equivalent resistance at 100% rated excitation is:

R=k ∙V 2

P=3 ∙

138002

36730=15 554 Ω

The calculation of nonlinear saturarion curve is more complex. First the loss current is calculated for each excitation point:

I loss=Losses(kW )

k ∙V∙

100Voltage (% )

=Losses(kW )∙ 103

3 ∙13.8 ∙ 103 ∙100

Voltage (% )

The phase no-load current for the delta-connected winding is:

I no−load=Current (%)

100∙

SkV

=Current (%)

100∙

66 ∙106

3 ∙13.8 ∙ 103

and is calculated for each excitation point. Then the loss current is subtracted from the total no-load current to obtain the magnetization current. Note that the loss current is resistive and the magnetization current is inductive, therefore 90 degree shifted:

I mag=√I no−lo ad2 −I loss

2

Voltage (%) Current (%) Losses (kW)No-load current, rms (A)

Loss current, rms (A)

Magnetization current, rms (A)

90 0.049 28.01 0.787 0.752 0.235

95 0.055 32.02 0.878 0.814 0.331

100 0.067 36.73 1.061 0.887 0.583

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105 0.101 43.61 1.610 1.003 1.260

110 0.204 53.06 3.264 1.165 3.049

The flux-linkage is calculated from the test voltages:

φ=√2 Vω

=√2 ∙ 13.8∙ 103

2 ∙ π ∙ 60∙

Voltage(% )100

The rms magnetization current is then converted to the peak flux-current nonlinear characteristic [T12].

The final slope of the saturation curve from the LV side (for core model connected to the LV terminals) is:

Lsat LV=Lair−core−LHL=26−10

100∙

3∙ (13.8 ∙ 103 )2

66 ∙106 ∙1

2 π 60=3.67 mH

Flux-linkage (Wb-t)

Peak Current (A)

Peak Current* (A)

46.5914 0.3318 0.3318

49.1798 0.8983 0.9628

51.7682 1.3082 1.4579

54.3566 3.2919 3.8583

56.9450 8.1679 9.8087

56.9778** 16.3359 18.7346

* with triplen-harmonic elimination

** additional point to set final slope

Since the transformer is energized from the delta-connected winding, the triplen harmonics circulate in the delta winding and do not contribute to the magnetization of the core. This effect should be corrected to obtain a more accurate saturation curve.

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Sub A

Sub B

Sub C

Sub K

Sub F

Sub E

Sub N

Sub M

Sub LSub

D

Sub G

8. Appendix: Case study examples & simulation results vs. field measurements

8.1. RMS-voltage drop [Zia]

The example given below relates to energization of two banked generator transformers at Substation A shown in Figure 1 and in a more detailed single line electrical diagram in Figure 2. The depression in voltage took place upon closure of circuit breaker X190 while circuit breaker X290 was open and the bus coupler circuit breaker was closed.

FIGURE 1 SCHEMATIC DIAGRAM OF THE AREA

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to Sub G

to Sub G

to Sub B

to Sub B

400/19 kV 345MVAYnd1



X190

X290

Sub A

FIGURE 2 SINGLE LINE DIAGRAM OF SUBSTATION A

Figure 3 shows the captured rms voltage traces for the three phases at Substation A following the energization of the two banked transformers. Pre-energization rms voltages were recorded to be 237.1, 236.8 and 235.1 kV for phases A, B and C respectively and those rms voltages at 30ms following energization were measured to be 231.7, 221.2 and 227.8 kV. The latter figures represent a voltage depression of around 2.28%, 6.59% and 3.11% respectively. It is of particular interest to note the effect of nearby SVC (at Substations F and L) and rSVC (at substation B) following the dip at voltage. It is important to note that rms voltages have also been depressed in the surrounding substations following the energization but not to the same extend as in Substation A. The corresponding combined inrush current is shown in Figure 4 where the peak value is 2115.9 A on phase B (notice that at approximately eight periods after the energization the saturation of current transformers (CT) is apparent in phases B and C44). It should be noted that the combined peak current is not double the amount expected had each transformer was energized individually (1300 to 1600 A) due to the interaction and magnetic core characteristic.

44 The CT saturation can be visually detected due to the sudden variation of the DC offset in the current waveforms and the atypical shape of the half-period that in the first periods is close to zero. CT saturation is caused by the asymmetrical current waveforms and exhibits in a distortion of the measured currents. If the CT saturation is evident after few cycles, the value of the current measured in the first cycle can be assumed quite accurate. When reproducing these waveforms in an EMT program, it is of no meaning to reproduce the current waveforms beyond the first few cycles without also modeling the CTs. This behaviour is common for general purpose substation CTs. Rogowski coils are also affected by similar issues: even though they have a linear behaviour and do not saturate, they cannot measure DC. Only special current transducers with a bandwidth from DC to some kHz can be used to accurately measure inrush current transients without introducing measurements errors. Laboratory measurements performed with such current transducers shows that half inrush current cycle remains nearly zero for the entire duration of the energization transient.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7220

222

224

226

228

230

232

234

236

238

X: 0.11Y: 236.8

X: 0.11Y: 237.1

X: 0.11Y: 235.1

X: 0.15Y: 221.2

X: 0.14Y: 227.8

Vol

tage

rm

s (k

V)

Time (s)

X: 0.14Y: 231.7

FIGURE 3 MEASURED RMS VOLTAGE TRACES AT SUBSTATION A DURING ENERGIZATION

As the measurements can only be representative of a particular system condition, the phenomenon was modelled using an EMT program software to check presence of other possible worst case conditions as well as the effect of point on wave switching, transformer residual flux and possible system contingencies in the area as the area is not very strong on the available short-circuit level. The model was extended to include the area shown in Figure 1. Transformer at substations B, G and F had all their saturation characteristics included as well as the SVC and mechanically switched capacitor bank connected on the tertiary of some transformers. Various loads present in the given area were modelled based on a parallel R and L to represent the MW and Mvar loading at the time. All line models were based on the geometrical data give distributed parameters. The result of the simulated rms voltages at Substation A is given in Figure 5. Considerable effort was made to include the voltage control logic of SVCs to get results as close to measurements as possible with extremely good agreement. It should be noted that pre-energization rms voltages on each phase are somewhat higher than the nominal system line voltage of 400kV as it is part of the operational strategy for the area.

Sensitivity analysis for various system topologies were also checked to find the best conditions for subsequent energization in the short term. PoW switching was ruled out as it would take significant time to install. The effect of sympathetic inrush was found to be not a major contributor to the observed voltage dip in simulation. Best results were obtained by setting transformer taps at their maximum which gives considerably less onerous voltage dip and this along with the use of voltage control plant in the area was recommended for subsequent energizations.

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FIGURE 4 COMBINED TRANSFORMER INRUSH CURRENT

0.1 0.2 0.3 0.4 0.5 0.6 0.7220

222

224

226

228

230

232

234

236

238

240

X: 0.09Y: 237.8

X: 0.09Y: 237.7

X: 0.09Y: 236.7

X: 0.13Y: 223.2

X: 0.13Y: 229.2

X: 0.13Y: 232.3

Time (s)

Vol

tage

RM

S (

kV)

Voltage RMS at Langage when X190 was closed

FIGURE 5 SIMULATED RMS VOLTAGE TRACES AT SUBSTATION A DURING ENERGIZATION.

8.2. RMS-voltage drop and inrush currents [Terry]

8.2.1. Background

A new independent power producer (IPP) with a 250 MW gas turbine generator connected to the BC Hydro 138 kV network on Vancouver Island in 2000. Figure XXX shows a simplified single-line diagram of the 138 kV network in the vicinity of the IPP. Dunsmuir Substation (DMR) is a major switching station on Vancouver Island that has a strong connection to the main BC Hydro grid on the mainland through two 500 kV AC submarine cables. Three 89 km long 138 V transmission lines connect DMR and the 150 MW John Hart (JHT) hydroelectric generating station. A 200 MW pulp and paper mill (PPM) with power quality sensitive loads is supplied from two 3.7 km lines from JHT. The IPP is located between JHT and the mill and the IPP transformer is tapped into each of the two

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138 kV circuits supplying the mill. The IPP step-up transformer is rated 315 MVA 138/21 kV, 14.9% leakage impedance, HV grounded-star LV delta windings. Since the IPP does not have black-start capability, the transformer must be energized from the grid to supply auxiliary load so that the unit can be started.

Proximity of the IPP to a very large customer, where power quality is important, necessitated the assessment of voltage dip due to inrush current during energization of the IPP step-up transformer to determine if, and how, it needed to be limited. The short circuit level of the BC Hydro system at the point of interconnection (POI) is approximately 1850 MVA at 132 kV. Using a “back-of-the-envelope” calculation [1], voltage dip was estimated as 29% as a worst case. This indicated that energization of the IPP transformer could pose power quality problems for voltage sensitive loads at the nearby mill. BC Hydro limits voltage dip caused by the operation of IPPs to RMS voltage dip at the POI of between 3% and 6%, on the most affected phase, with a frequency of not more than once per eight hours. Subject to prior approval by BC Hydro, the IPP may be permitted to impose a voltage sag of up to 9%. It therefore became apparent that a more accurate and rigorous assessment of voltage dip due to inrush would be necessary if the IPP was to be required to make possibly costly changes to his plant in order to comply with BC Hydro power quality requirements.

FIGURE 1 SINGLE-LINE DIAGRAM OF PART OF THE BC HYDRO VANCOUVER ISLAND 500/230/138KV SYSTEM

8.2.2. Model validation

In June, 2000, as part of the IPP commissioning tests, the 315 MVA generator transformer was energized several times during a time when the mill was shut down for annual maintenance. The inrush currents and the 138 kV phase-to-ground voltages at the transformer bushings were recorded on one working day during four energizations

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of the unloaded transformer. These were recorded at 64 samples per cycle using a metering device with remote access. Prior to the first energization, the transformer had been in a de-energized state for eight months since leaving the factors and very little residual flux should have been present in the core. The highest inrush current was recorded during the third energization, when residual flux would have been present. These field recordings provided a useful benchmark with which to test EMT transformer models.

An EMT model of the relevant part of the BC Hydro Vancouver Island system and the 315 MVA transformer were developed. A simplified single-line diagram of the network model is shown in Figure 2. Positive and zero sequence Thèvenin impedances of the BC Hydro system at DMR were calculated to represent the remaining Vancouver Island network and the 500 kV connection to the main grid, not explicitly modelled. Transmission lines were modelled as 60 Hz constant parameter lines. The JHT generating units were modelled as ideal 60 Hz sinusoidal voltage sources behind subtransient impedance.

FIGURE 2 SIMPLIFIED SINGLE-LINE DIAGRAM OF THE SYSTEM MODELLED IN EMT PROGRAM

The IPP step-up transformer is physically a two-winding three-phase transformer having a three-leg core. The EMT representation assumed three single-phase transformers. This representation, using inter-connected single-phase ideal transformers, is the traditional “T”-equivalent model that has been commonly used for low frequency transient studies. This model is not adequate for transformers that do not have a delta winding to produce inter-phase coupling, which is not the case for the IPP generator transformer of concern here. In the EMT model, the single-phase transformers were connected appropriately, as grounded-star on the 132 kV side and delta on the 21 kV side. The transformer short-circuit test report from the manufacturer was used for calculating the series resistances and leakage reactances (Rp, Xp, Rs, and Xs) shown in Figure 3. The transformer leakage impedance was assumed equally divided between primary and secondary sides of the ideal transformers. From the transformer manufacturer’s open-circuit test of applied voltage versus exciting current, an EMT program auxiliary program was used to calculate the instantaneous flux versus instantaneous exciting current. This was input into a Type “98” pseudo nonlinear reactor model, initially used to represent the saturable iron core (i.e. residual flux was ignored since the saturation characteristic passes through the origin).

The EMT program was used to simulate the energizations of the 315 MVA generator transformer. EMT program voltage sources and the closing point-on-wave were adjusted in the simulations to reflect the same initial conditions at the IPP as during the field tests prior to energization. Comparison of simulation to measurement showed a reasonable agreement for the inrush currents for the first energization but unacceptable match for subsequent energizations.

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FIGURE 3 EQUIVALENT CIRCUIT OF A SINGLE-PHASE T-EQUIVALENT TRANSFORMER (PU VALUES REFER TO THE TRANSFORMER RATING)

Lack of agreement between simulation and field tests, except for the initial energization, indicated that the effects of residual flux on the inrush currents could not be ignored in the modelling. For the first energization, residual flux would not have been important, therefore using a transformer model which ignored it but accounted for saturation effects would have produced acceptable results. For the subsequent energizations, however, residual flux was present and its effects had to be included in the modelling. The polarity and magnitude depended on the transformer condition following the preceding de-energization

The transformer model was then modified to include the hysteresis characteristics of the core. The Type 98 reactor models in the T-equivalent transformers were replaced by EMT Type “96” hysteretic pseudo nonlinear reactors. Upon request, the transformer manufacturer supplied a section of the lower part of the major hysteresis loop, shown in the box on Figure 4. Based on this data, the remaining section of the lower hysteresis loop was estimated and input to the Type 96 model. By considerations of symmetry, the upper hysteresis characteristic is known once the lower curve is defined. The residual flux at zero current was assumed to be the maximum flux that the transformer would retain after de-energization. Residual flux in each hysteretic reactor was selected based on the requirement that fluxes in the three legs must sum to zero. For the third energization, Phase A was modelled with maximum residual flux with negative polarity and the other two phases were modelled with half of this flux, of positive polarity.

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FIGURE 4 OPEN CIRCUIT CHARACTERISTIC OF THE IPP TRANSFORMER. THE LOWER PART OF THE HYSTERESIS CHARACTERISTIC (SHOWN IN THE BOX) WAS PROVIDED BY THE TRANSFORMER MANUFACTURER

8.2.3. Comparison between simulation and field test for the third energization

Figures 5, 6, and 7 compare the transformer inrush currents simulated by EMT program for the third energization with the recordings made during the field tests for the same event. Recorded and simulated data are represented by dotted and solid curves, respectively. This case represented the highest inrush current measured for all four energizations. The crest value of Phase A current is approximately 2500 A, or 1.34 pu, for both simulation and field measurement. There is also good agreement with Phase B inrush current. However, the Phase C simulated current is significantly smaller than the measured current, likely due to differences in the closing point-on-wave for this phase and residual flux assumed for Phases B and C. Figure 8 compares the Phase A-to-ground voltages.

8.2.4. Conclusions

The T-equivalent model for the 315 MVA two-winding three-phase generator transformer, employing Type 96 hysteretic pseudo nonlinear reactors is adequate for predicting the worst case inrush current and voltage dip due to it.

The effects of residual flux in the core plays an important role in the magnitudes of the individual inrush currents and must be accounted for in the transformer model.

Subsequent to the field testing, the IPP replaced his original three-pole 138 kV breaker with an independent pole breaker and point-on-wave switching controlled by a commercially available relay. This relay estimates the residual flux based on voltage measurements made during the previous de-energization and also accounts for the closing characteristics of the energizing breaker to arrive at optimal closing times.

FIGURE 5 PHASE A INRUSH CURRENTS FROM FIELD TEST AND EMT SIMULATION (DASHED LINE), FOR THE THIRD ENERGIZATION.

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Zia.Emin, 10/04/12,

TM to change the curves around

Figure 6 Phase B inrush currents from field test and EMT simulation (solid line), for the third energization.

Figure 7 Phase C inrush currents from field test and EMT simulation (solid line), for the third energization.

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Figure 8 Phase A-to-ground voltages from field test and EMT simulation (solid line), for the third energization.

References

[1] M. Nagpal, T.G. Martinich, A. Moshref, K. Morison and P. Kundur, “Assessing and limiting transformer inrush current on power quality,” IEEE Trans. Power Del., vol. 16, no.2, pp. 276-280, Apr. 2001.

8.3. Sympathetic interaction [Terry]

8.3.1. Background

A new 375 km 287 kV transmission system is being planned to integrate a proposed new run-of-river 300 MW independent power producer (IPP) into the utility’s grid. The first 335 km circuit from the grid to the point-of-interconnection (and future switching station) will be constructed, owned and operated by the utility. The remaining 40 km circuit to the IPP main generating station will be owned and operated by the IPP. The main interconnecting station would be a 500/287 kV substation at the end of a 450 km radial 500 kV shunt-compensated transmission system to the “backbone” 500 kV system. An existing radial 287 kV intertie to a customer-owned 800 MW hydroelectric generating station is terminated at the 500/287 kV interconnecting substation. This self generation supplies the electrical power requirements to a large aluminum plant with surplus power being exported to the grid. It also incidentally helps to support the fault levels in the area.

The new IPP will consist of a main 200 MW generating station with four 13.8 kV hydro units connected to the 287 kV line via two 116 MVA generator transformers. In the future, there will be an additional 100 MW of generation from two other generating plants interconnected from a 69 kV system. As is sometimes the case with run-of-river IPPs, this one will not have black-start capability. This means, of course, that the main generator step-up transformers have to be picked up from the grid. The electrical utility imposes restrictions on the disturbances caused by the operation of transmission level customers and IPPs, including limits on voltage dip, voltage flicker, and harmonic injections. As for limits on voltage dip, IPPs are limited to RMS voltage dips of between 3% and 6%, on the most affected phase, with frequency of occurrence not to exceed once every eight hours. Subject to prior

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approval by the utility, the IPP may be permitted a voltage dip of up to 9%. Voltage dip limits, applied at the point of interconnection (POI), do not address the duration of the dip, similar to the ITIC-CBEMA [1] undervoltage characteristic. Preliminary “back-of-the-envelope” calculations indicated a significant probability of voltage dip problems should the IPP pick up a 116 MVA transformer by randomly closing the 287 kV breaker. It therefore became apparent that a more rigorous assessment of voltage dip due to inrush using EMT simulations would be necessary should the IPP be required to make possibly costly unplanned investment in new equipment in order to comply with the utility’s interconnection requirements.

This report summarizes the EMT modelling and simulations of the worst case transformer inrush currents, where the traditional “T” equivalent model of the transformer has been used. The hysteresis characteristic of the transformer core and the air core inductance assumed for the modelling were based on actual data provided by a major transformer manufacturer for another generator transformer. Simulations demonstrated that unacceptable voltage dip and high temporary overvoltages could be produced by uncontrolled closing of the energizing breaker when a large intertie transformer is energized from the grid, with or without the presence of an already energized companion transformer.

8.3.2. System data

Figure 1 shows a simplified electrical single-line diagram of the proposed new 287 kV transmission system from the existing utility 500/287 kV substation to the 300 MW IPP. Not shown are the configuration of the breakers at the line terminals and the very small 287 kV station service transformer at the POI switching station. Station “S” denotes the existing 500/287 kV interconnecting substation having a three-phase symmetrical fault level of 2520 MVA at the 287 kV bus. Station “B” denotes the 287 kV point of interconnection (POI), located 40 km from the IPP. Generating station “F” is the IPP 287 kV intertie station having two generator transformers, four generators and a future interconnection to a 69 kV system containing two smaller generating plants.

Figure 1 Simplified equivalent single-line diagram of 287 kV system with new IPP

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The 335 km utility-owned circuit will comprise a single flat configuration transposed line from the 500/287 kV substation to the POI. The phase conductors will comprise double-bundle 477 MCM ACSR hawk. A 65 Mvar 287 kV reactor at the interconnecting substation and a 30 Mvar line reactor at the remote end will compensate for most of the line-charging capacitance. Since this circuit will be operated in single-pole trip and reclose mode for single-line-to-ground faults, the shunt reactors will be grounded through neutral reactors. A 50 Ohm series capacitor bank will be located at Station B between the POI and the utility end of the line. The bank will provide 40 percent series compensation for the 335 km 287 kV transmission line. For the purposes of this study, the series capacitor bank is assumed to be bypassed during conditions of low line current. The 40 km circuit from the POI to the generating station F, of similar construction to the main circuit, will have two switchable 25 Mvar line reactors. Both of these reactors would be required to be in service when the IPP generation is out of service. These reactors will have solidly grounded neutrals. Station F will comprise both a generating facility and a transformation station. There will be two 116 Mva 287/13.8 kV HV grounded-star LV delta generator transformers (one transformer connects two 71 Mva generators and the second unit connects one 71 Mva and one 25 Mva generator). A 287/69 kV autotransformer to interconnect a future 69 kV system was not included in the scope of the system model. This study is focussed on the expected worst-case energizations of the two large transformers at Station F and the expected RMS voltage dip and recovery as well as temporary overvoltages.

Table I contains the positive sequence and zero sequence parameters for the 335 km transmission line from Station S to the series capacitor bank as well as the 40 km line to the IPP. Impedances are given in Ohms at 60 Hz and admittances are given in micro-Siemens (µS).

Table I287 kV Transmission Line 60 Hz Parameters for IPP Intertie

Positive Sequence Zero Sequence

FromBus *

ToBus *

Length(km)

R1

OhmsX1

OhmsY1

µSR0

OhmsX0

OhmsY0

µS

S B 335 20.59 123.3 1495. 77.69 475.3 783.

B K 40 2.46 14.7 178. 9.28 56.8 93.5

* Refer to Figure 1

Table II provides the details of the 287kV line reactors located at Stations S, B, and K. The reactors on the 335 km line are fixed whilst the two 25 Mvar reactors at the IPP are switchable, depending upon the number of generating units on line.

Table II287 kV Line Reactor Details (at 60 Hz)

Reactor Location Line Reactor Neutral Reactor

Line End Mvar Ohms/Phase Ohms

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S 65 1267. 800

B 30 2746. 1600

K 2 x 25 3295. (each) Grounded Neutral

8.3.3. 116 MVA transformers data

Table III shows the basic transformer data that was provided by the IPP in his application for transmission connection to the grid. Since the actual transformer had not been constructed, a manufacturer’s test report providing information required for modelling the unit was not available at the time of the studies. As the core material, dimensions of the core, and other data on the transformer were not known, the hysteresis characteristic, saturation curve and air-core inductance were estimated, based on known data for a similar (reference) generator transformer.

Table III116 MVA Transformer Data

Natural Cooled (ONAN) Rating 116.5 Mva

Voltage of HV Winding 287.0 kV

Voltage of LV Winding 13.8 kV

Connection of HV Winding Grounded Star

Connection of LV Winding Delta

Pos. Seq. Leakage Impedance 14.5 %

Air-Core Inductance* 43.5%

* Based on Manufacturer’s data for another generator transformer

8.3.4. Transformer and network modelling

The 287 kV transmission system to the IPP, shown in Figure 1, together with the 450 km 500 kV single-circuit transmission from the main grid to Station “S” and the 287 kV system to the 800 MW generating station and aluminum plant (details not shown in the Figure) were included in the overall EMT system model. Transmission lines were represented as frequency-dependent lines. Phase and neutral reactors, including the two 25 Mvar reactors at Station F, were modelled as linear inductors (the saturation characteristics of shunt reactors typically have a knee point around 1.5 pu and the air-core inductance is quite high – about one-third of the unsaturated value). The 287 kV 50 Ohm series capacitor bank was assumed to be bypassed since the transmission system is lightly loaded for these studies. Surge arresters were not modelled for these simulations considering the levels of the overvoltages compared to the arrester conduction characteristics. The IPP generators were modelled as Park’s

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machines rather than the simpler representation of three-phase ideal sinusoidal voltage sources behind d-axis subtransient reactance.

The two-winding three-phase transformers under study have three-limb cores, as is common for generator transformers. The flux in each leg of a multi-leg core is not independent but is magnetically coupled to the other legs. For transformer energization studies it is important to account for dynamic core flux, the effects of residual flux in the core from the previous de-energization, and for core flux saturation. The EMT model for the IPP step-up transformer assumed an interconnection of three single-phase transformers. Each single-phase transformer was modelled by the traditional “T”-equivalent, employing ideal transformers where the primary and secondary windings were connected to achieve the grounded-star winding configuration on the 287 kV side and delta configuration on the 13.8 kV side. Hysteretic pseudononlinear reactors were used to model the magnetizing characteristics of each leg of the core. This model also enables the residual fluxes in each leg of the core to be predefined up to the limit of the specified hysteresis characteristic, prior to switching. These nonlinear models were connected across the grounded primary windings of the ideal transformers. The transformer leakage impedance was divided between the primary and secondary sides of the ideal transformers. From zero sequence considerations, the component of leakage impedance on the LV side was brought inside the delta winding.

Favourable comparison in the past between simulation and field recordings of inrush currents provides some justification for use of the T-equivalent transformer model to predict worst-case inrush current, voltage dip and temporary overvoltages due to harmonic resonance with the network. EMT simulations of a field test, when several energizations of a 315 MVA 138/13.8 kV generator transformer were recorded, demonstrated that the inrush currents derived from the T-equivalent model with reasonable assumptions produced acceptable agreement with measured worst case inrush currents [4].

No data was available for the instantaneous flux versus instantaneous exciting current hysteresis characteristic for input to the hysteretic pseudononlinear inductor model. However, this data was available for another two-winding three-phase generator transformer: a 66 MVA 238/13.8 kV unit having a three-limb core. The peak magnetic flux density from which the hysteresis characteristic was derived was 1.9 Tesla. Scaling of these fluxes and the exciting currents to account for a higher nominal primary voltage and Mva rating of the transformer being studied, the major hysteresis characteristic for the hysteretic inductor model was calculated. Figure 2 shows part of the major hysteresis characteristic over a reduced range of exciting current than was actually input to the hysteresis model. This is done to show more detail of the “loop”. The air-core inductance, which is the slope of the flux versus exciting current over the linear part of the characteristic when the core is fully saturated, actually occurs for much higher exciting currents than shown in the Figure. As indicated on Figure 2, this characteristic permits a maximum residual flux in any one leg of 72% of nominal flux. The minimum value of the transformer air-core inductance was assumed to be three times the leakage inductance, or 43.5%. This is in accordance with the manufacturer’s data for the 238 kV reference transformer. Figure 3 shows a simplified single-line diagram of the two-winding transformer model for the generator transformer. The leakage impedance was apportioned between the primary and secondary sides of each ideal transformer. 70% was assumed on the primary side and 30% on the secondary side. By consideration of the actual construction of two-winding three-limb stacked core transformers, an argument can be made that it is reasonable to place most of the transformer leakage impedance on the primary side.

The parameters for the model are given in Table IV.

Table IV

Transformer Model Parameters

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R1 = 0.50 Ohms

X1 = 72.07 Ohms

R2 = 0.0036 Ohms

X2 = 0.2142 Ohms

Rm > 1. MΩ

N = 12.00

IPP 116 MVA Transformer Hysteresis LoopInstantaneous Flux-Current Referred to 287kV Side

at Peak Magnetic Flux Density of 1.9 Tesla

-800

-600

-400

-200

0

200

400

600

800

-6 -4 -2 0 2 4 6

Current (Amp)

Flu

x (V

-s)

Peak FluxMax. Residual Flux72% of Nominal

Air-Core Inductance0.816 H (43.5%)

Figure 2 Hysteresis Characteristic assumed for the 116 Mva transformer at peak magnetic flux density of 1.9 Tesla. Air Core inductance is 0.816 H (43.5%)

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Figure 3 Model (Single-Phase Equivalent) of the 116 Mva 287/13.8 kV Two-Winding Transformer with Hysteresis and Saturation Effects Included.

8.3.5. Simulation of transformer energizations with high inrush current

8.3.5.1. Case #1: Energize the First Transformer T1

Figures 4 to 7 show a simulation of the energization of the first 116 MVA transformer, T1, from the 287 kV bus when no other transformer is connected to the bus. The residual flux assumed in the core and the closing point on the voltage wave were selected to produce a high inrush current in one phase. Since the fluxes must sum to zero in a three-legged core [5], the residual flux distribution in the core was assumed to be +0.72 pu, -0.36 pu, and -0.36 pu in the Phase A, Phase B, and Phase C core legs, respectively. The +0.72 pu residual flux corresponds to the flux at zero current of the major hysteresis characteristic (Figure 2) and was assumed to be the maximum flux that the transformer could retain after de-energization. For these residual fluxes, the closing point-on-wave was selected so that phase A closed at the zero crossing of the voltage wave when the prospective flux would add to the residual flux (i.e. no flux cancellation) to produce a high resultant flux and inrush current in Phase A. All three poles of the energizing switches were closed simultaneously.

Figure 4 shows that the highest inrush current (902 A) occurs in the phase having the highest residual flux. There is considerable distortion in the currents and the DC components appear to decay very slowly following the initial few cycles. Figure 5 shows the three instantaneous phase-to-ground voltages at the 287 kV energizing bus at Station F. As the Figure shows, there is a 1.55 pu temporary overvoltage (TOV) in Phase C within 7 cycles of energization and this overvoltage appears to be sustained throughout the remaining 800 ms of the simulation. Figure 6 shows a one-cycle sample of the Phase C voltage waveform at approximately 0.24 s after energization. The distortion in the waveform is quite pronounced. FFT analysis shows that this waveform, which repeats for many cycles, contains 53% second harmonic, 29% third harmonic and 20% fourth harmonic plus diminishing fifth, seventh and higher harmonics. The high instantaneous overvoltage on Phase C, the most affected phase, occurs for positive half cycles only. Finally, the individual RMS voltages calculated from the previous instantaneous phase voltages, using a one-cycle sliding window, appear in Figure 7. The initial voltage dip on the most affected phase (Phase A) is 35% whereas the voltage dip on Phases B and C are slightly less than one-half of Phase A. The behaviour of the recovery of the RMS voltages indicated on this Figure is interesting . Phase A stays below 90% of the nominal RMS voltage throughout the entire post-energization period of the simulation whereas Phases B and C recover to predisturbance levels within 100 to 200 ms, and then proceed into a modest TOV.

Some discussion of the effects of the TOV on the surge arresters protecting the IPP transformers and entrance circuit breakers is in order. Surge arresters at Station F were not modelled because ones that are rated for a grounded neutral 287 kV application would not have affected the magnitudes of the TOV to any noticeable degree over the duration of the simulations. However, in reality, even though the currents conducted through the phase C arresters during the overvoltage peaks might be very small, the cumulative effects of the energy accumulation might not be small. Even a small energy accumulated over each cycle during a TOV persisting for many seconds

Page 96

could cause the total energy to reach the energy rating of the arrester. There could then be a concern of arrester failure. Figure 6 shows that the TOV waveform is not a fundamental frequency sinusoid, and that the overvoltage occurs in only one polarity. The energy accumulated in the Phase C surge arrester at the Station F 287 kV bus due to the non-sinusoidal overvoltage can be estimated from the conventional TOV capability curve (the TOV strength factor versus time characteristic) provided by all arrester manufacturers. To apply it to this particular case, the TOV should be assumed to be purely sinusoidal at power frequency having an amplitude equal to the maximum instantaneous Phase C to ground voltage. If the maximum instantaneous phase-to-ground voltage per cycle is assumed to be a constant 367 kVpeak then, for a 240 kVrated 2.5kJ/kVr surge arrester, the TOV strength factor Tr is conservatively calculated as 1.08. When no pre-stress is assumed, the TOV withstand curve indicates that the surge arrester would absorb rated energy in about 90 seconds. If the arrester had a pre-stress energy of 2.5 kJ/kVr, rated energy would be reached in about 50 seconds. If the decay of the TOV is to be accounted for, a more refined approach would be to employ a series of sections of sinusoidal waveforms, with each section being of successively reduced amplitude in order to characterize the TOV.

Energize 116 mva 287/13.8 kV Grounded-Star Delta Transformer T1 From the 287 kV BusNo Other Transformers are Connected

-6.00E-01

-5.00E-01

-4.00E-01

-3.00E-01

-2.00E-01

-1.00E-01

0.00E+00

1.00E-01

2.00E-01

3.00E-01

4.00E-01

5.00E-01

6.00E-01

7.00E-01

8.00E-01

9.00E-01

1.00E+00

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time (s)

Tra

nsf

orm

er

T1

Inru

sh

Cu

rren

t (k

Am

p)

at

287

kV

Bu

s

T1-Ia

T1-Ib

T1-Ic

902 Amp

Figure 4: Simulation of the Transformer Inrush Currents

Page 97

Energize 116 mva 287/13.8kV Grounded-Star Delta Transformer T1 From the 287 kV BusNo Other Transformers are Connected

-5.00E+02

-4.00E+02

-3.00E+02

-2.00E+02

-1.00E+02

0.00E+00

1.00E+02

2.00E+02

3.00E+02

4.00E+02

5.00E+02

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time (s)

287

kV

En

erg

izin

g B

us

Ph

.-G

nd

Vo

ltag

es

(kV

)Va

Vb

Vc

364 kVp (1.55 pu) T.O.V.

Distortion Voltages:53% 2nd Harmonic29% 3rd Harmonic20% 4th Harmonic

Figure 5: 287 kV Bus Phase-to-Ground Voltages at the Energizing Bus

Energize 116 mva 287/13.8kV Transformer T1 From the 287 kV Bus0.24 S After Energization - One Cyle of Phase C-Gnd Voltage During the T.O.V.

-4.00E+02

-3.00E+02

-2.00E+02

-1.00E+02

0.00E+00

1.00E+02

2.00E+02

3.00E+02

4.00E+02

0.338 0.34 0.342 0.344 0.346 0.348 0.35 0.352 0.354 0.356 0.358

Time (s)

Ph

ase-

to-G

rou

nd

Vo

ltag

e (k

V)

Vc

Distortion Voltages:53% 2nd Harmonic29% 3rd Harmonic20% 4th Harmonic

367 kVp (1.56 pu)

Figure 6: One Cycle of the 287 kV Bus Phase C Voltages During the TOV

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Energize 116 mva 287/13.8 kV Grounded-Star Delta Transformer T1 From the 287 kV BusNo Other Transfomers are Connected

-40.0

-35.0

-30.0

-25.0

-20.0

-15.0

-10.0

-5.0

0.0

5.0

10.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time (s)

RM

S V

olt

age

Dro

p (

%)

at

28

7 kV

En

erg

izin

g B

us

Phase A Phase B Phase C

35% Voltage Drop on Phase A

One Phase is Still Below 1 pu After 1 Second

Two Phases are Above 1 pu (Temporary Overvoltage)

Figure 7: RMS Voltage Drop at the 287 kV Energizing Bus

8.3.5.2. Case #2: Sympathetic Interaction #1 - Energize Transformer T2 When T1 is Already Energized but Unloaded

An EMT program was used to simulate the energization of 116 MVA transformer T2 against the already energized identical and unloaded T1. The model for transformer T2 was identical to the previous case for T1, including the same residual flux distribution in the core, hysteresis characteristics and air-core inductance. However, the model for already energized T1 was simplified by modelling the saturation characteristics, and extending this to include the fully saturated region rather than using the hysteresis characteristic. Experimentation demonstrated that this modelling approach was completely satisfactory. The same value of air-core inductance (three times leakage inductance) was used in both transformer models. The closing point on the voltage wave was selected to produce a high inrush current in Phase A.

Figures 8 to 12 show the results of the simulation of sympathetic interaction. In Figure 8 the inrush current into transformer T2 is shown and indicates a maximum Phase A inrush current of 959 Ap, which is actually higher than the inrush current when T1 was energized alone (902 Ap). The T1 magnetizing current during the initial 6 cycles of steady state and the subsequent inrush current resulting from energizing T2 appear in Figure 9. As can be seen, the highest T1 inrush current reaches 363 Ap after a delay of about 12 cycles after energizaton occurs and is of opposite polarity to the Phase A inrush current in T2. Figure 10 provides the sum of the T1 and T2 inrush currents which are being supplied from the 287 kV line. The maximum of this total current is 895 Ap, which is almost the same as the peak inrush current obtained just be energizing T1 by itself. The instantaneous phase-to-ground voltages at the energizing bus appear in Figure 11. There is a TOV of 1.46 pu on Phase C which appears 100 ms after energization and is slightly lower than 1.55 pu TOV seen in Case 1. Finally, Figure 12 provides the three RMS voltages at the energizing bus before and during the inrush. The voltage dip in the Phase A voltage is 35%, and is the same magnitude as the dip when T1 was energized by itself. The Figure shows that all three RMS voltages are below 1 pu during the entire post energization period (900 ms) with little indication of recovery. In fact, the Phase A RMS voltage exhibits a declining trend after reaching a recovery to 85% of nominal voltage 150 ms

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after energization. All of these voltage sags occur coincidently with a significant temporary overvoltage in two of the instantaneous voltage waveforms.

Energize 116 mva 287/13.8 kV Grounded-Star Delta Transformer T2 From 287 kV BusSecond 116 mva Transformer T1 is Energized But Not Carrying Load

-8.00E-01

-6.00E-01

-4.00E-01

-2.00E-01

0.00E+00

2.00E-01

4.00E-01

6.00E-01

8.00E-01

1.00E+00

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time (s)

Tra

nsf

orm

er

T2

Inru

sh

Cu

rren

t (k

Am

p)

T2-Ia

T2-Ib

T2-Ic

959 Amp

Figure 8: Simulation of Transformer T2 Inrush Currents


-4.00E-01

-3.00E-01

-2.00E-01

-1.00E-01

0.00E+00

1.00E-01

2.00E-01

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time (s)

Tra

nsf

orm

er

T1

Cu

rre

nt

(kA

mp

)

T1-Ia

T1-Ib

T1-IcT1 Magnetizing

Current

363 Amp

Figure 9: Inrush Currents at the Already Energized but Unloaded Transformer T1

Page 100


-8.00E-01

-6.00E-01

-4.00E-01

-2.00E-01

0.00E+00

2.00E-01

4.00E-01

6.00E-01

8.00E-01

1.00E+00

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time (s)

To

tal S

up

ply

C

urr

en

t (k

Am

p)

to T

1 &

T2

Total_Ia

Total_Ib

Total_Ic

895 Amp

T1 magn.Current

Figure 10: Total Current into T1 and T2 at the 287 kV Energizing Bus

Energize 116 mva 287/13.8kV Grounded-Star Delta Transformer T2 From 287 kV BusSecond 116 mva Transformer T1 is Energized But Not Carrying Load

-4.00E+02

-3.00E+02

-2.00E+02

-1.00E+02

0.00E+00

1.00E+02

2.00E+02

3.00E+02

4.00E+02

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time (s)

287

kV

En

erg

izin

g B

us

Ph

.-G

nd

Vo

ltag

es

Va

Vb

Vc

343 kVp (1.46 pu)

Figure 11: Instantaneous Phase-to-Ground Voltages at the Energizing Bus

Page 101


-40.0

-35.0

-30.0

-25.0

-20.0

-15.0

-10.0

-5.0

0.0

5.0

10.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time (s)

RM

S V

olt

age

Dro

p (

%)

at

28

7 kV

En

erg

izin

g B

us



All Three Phases Are Still Below 1 pu After 1 Second

Figure 12: The RMS Voltages at the 287 kV Energizing Bus

8.3.5.3. Case #3: Sympathetic Interaction #2 - Energize T2 When T1 is Already Energized and a 25 Mva Generator is Connected

This simulation of transformer energization considers another example of sympathetic interaction. Unlike the previous case, the already energized transformer was assumed to be carrying load when the second transformer is energized. An simulation considered the scenario where T1 is energized and carries power from a 25 Mva generator connected to the 13.8 kV secondary windings. Transformer T2, modelled with high residual flux in one core leg, is energized against T1 at the most unfavourable point on the voltage wave. The models for T1 and T2 were exactly the same as for Case 2, including hysteresis characteristics and residual flux distributions in the core of T2 and the saturation characteristics of T1. For the purposes of this study, the generator was modelled as a Park’s machine but machine dynamics and the effects of governor and AVR/Exciter actions were not represented (i.e. constant mechanical torque and constant excitation). Experimentation demonstrated that replacing the Park’s model with a model comprising 60 Hz sinusoidal voltage sources behind d-axis subtransient reactance (X”d), would produce incorrect results during the voltage recovery period but acceptable results during the first few cycles of the inrush.

Figures 13 to 17 show the results of a simulation of this second example of transformer sympathetic interaction. In Figure 13 the inrush currents into transformer T2 are shown and indicate a maximum instantaneous current (in Phase A) of 1049 A. This is a significantly higher peak current than those seen in the previous cases. Figure 14 shows the currents at the 287 kV side of T1, including the initial 6 cycles of load current followed by the effects of the inrush currents when T2 is energized. The total currents (the sum of T1 and T2 inrush currents) appear in Figure 15. Due to the presence of the generator, the maximum current is 792 Ap, which is less than the 895 Ap of Case 2. The 287 kV instantaneous phase-to-ground voltages (in Figure 16) exhibits a T.O.V. of 1.37 pu, which is also less than for Case 2 (1.46 pu) and for Case 1 (1.55 pu). Finally, Figure 17 shows the three RMS voltages at the energizing bus. Voltage dip of 32% and recovery of the RMS voltages are little improved over the case when T1 was unloaded.

Page 102

Some comment should also be made on the three RMS voltages of Figure 17. All three voltages are from 5% to 12% below the nominal voltage and one can expect that the terminal voltages of the 25 Mva generator would be similarly depressed. While generator AVR/exciter was not modelled in the simulation, it is not unreasonable to expect that the effect of including AVR action would have been to boost the machine terminal voltages and therefore the 287 kV voltages, thereby aggravating the TOV. The AVR would not have reduced the TOV.

Energize 116 mva 287/13.8 kV Grounded-Star Delta Transformer T2 AgainstIdentical Energized Transformer T1 Connected to 25 mva Generator

-8.00E+02

-6.00E+02

-4.00E+02

-2.00E+02

0.00E+00

2.00E+02

4.00E+02

6.00E+02

8.00E+02

1.00E+03

1.20E+03

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time (s)

Tra

nsf

orm

er

T2

In

rus

h C

urr

en

t (A

mp

)

T2-Ia

T2-Ib

T2-Ic

1049 Amp

Figure 13: Simulation of Transformer T2 Inrush Current

Page 103


-4.00E+02

-3.00E+02

-2.00E+02

-1.00E+02

0.00E+00

1.00E+02

2.00E+02

3.00E+02

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time (s)

Tra

ns

form

er T

1 I

nru

sh

Cu

rren

t (A

mp

)T1-Ia

T1-Ib

T1-Ic

LoadCurrent

Figure 14: Inrush Currents at the Already Energized Transformer T1


-6.00E+02

-4.00E+02

-2.00E+02

0.00E+00

2.00E+02

4.00E+02

6.00E+02

8.00E+02

1.00E+03

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time (s)

To

tal S

up

ply

Cu

rre

nt

at

287

kV

Bu

s t

o T

1 a

nd

T2

(Am

p)

Total_Ia

Total_Ib

Total_Ic

LoadCurrent

792 Amp

Figure 15: The Total Current Into T1 and T2 (Supplied From the Grid)

Page 104

Energize 116 mva 287/13.8kV Grounded-Star Delta Transformer T2 AgainstIdentical Energized Transformer T1 Connected to 25 mva Generator

-4.00E+05

-3.00E+05

-2.00E+05

-1.00E+05

0.00E+00

1.00E+05

2.00E+05

3.00E+05

4.00E+05

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time (s)

28

7 kV

En

erg

izin

g B

us

Ph

.-G

nd

Vo

ltag

es

(V

)Va

Vb

Vc

321 kVp (1.37 pu)

Figure 16: Phase-to-Ground Voltages at the Energizing Bus


-40.0

-35.0

-30.0

-25.0

-20.0

-15.0

-10.0

-5.0

0.0

5.0

10.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time (s)

RM

S V

olt

age

Dro

p (

%)

at

287

kV

En

erg

izin

g B

us



All Three Phases Are Still Below 1 pu After 1 Second

Figure 17: RMS Voltages at the 287 kV Energizing Bus

8.3.6. Conclusions The worst case voltage dip due to uncontrolled closing of the energizing breaker to pick up a 116 Mva

transformer can be expected to exceed the limits that are acceptable to the utility.

Page 105

A severe voltage dip, as seen in the RMS phase voltages, results when there is a high residual flux in one of the core legs combined with a closing point on the corresponding voltage wave such that the polarity of the residual flux is in the direction of flux buildup after switching.

The voltage dip is aggravated by the high inrush current which has to be supplied through a long radial transmission system.

A temporary overvoltage results when one of the harmonic components of the inrush current excites a parallel resonance in the supply network, as seen from the energizing bus. The resonance can cause the waveforms to be highly distorted and this condition can be sustained for seconds.

For the simulation of transformer energization reported here, it is unlikely that the TOV excited by the transformer inrush current will result in the surge arrester at Station F absorbing rated energy unless it had already been highly pre-stressed prior to transformer energization.

As demonstrated by Case 1, it is possible for the RMS voltages (one or two phases) to recover from the initial voltage dip and go into a temporary overvoltage condition.

Also demonstrated by this study is the possibility of obtaining depressed RMS voltages in the supply at the same time as a temporary overvoltage condition on an instantaneous basis due to the effects of high magnitude and highly distorted inrush currents having large and slowly decaying dc offsets.

Energizing a transformer at the most unfavourable point on the voltage wave against an already energized but unloaded transformer of similar Mva rating may result in a voltage dip as large as would be obtained by energizing the transformer by itself. The post energization recovery of the voltages following a back-to-back energization are also significantly slower.

Energizing a transformer against an already energized transformer of similar Mva rating having generation connected on its secondary side can result in worst case voltage dips of about the same severity as obtained for energizing the transformer alone. Recovery of the RMS voltages back to pre-energization levels may be even more prolonged.

The use of Park’s model is required in simulations to represent synchronous generators that are connected to an already energized companion transformer in order to obtain the correct voltage recovery following transformer energization. If only the first few cycles of the voltage dip are of interest, then representing the generator as three phase ideal sinusoidal voltage sources behind X”d will produce reasonable results.

It is not inconceivable that, for Case 3 which shows a prolonged undervoltage of the three RMS voltages following transformer energization, the real generator’s AVR/exciter, responding to the average of the three RMS terminal voltages, could boost the voltages and thereby aggravate the TOV appearing in the external network.

In simulations of transformer energization, for the modelling of transformers already energized, it is not necessary to model the hysteresis characteristics of the cores of these transformers. It is certainly sufficient to model their saturation characteristics provided that the characteristic includes the fully saturated region (the air-core inductance).

REFERENCES

[1] Federal Information Processing Standards Publications (FIPS) no. 94.

[2] J.J. Brunke, “Elimination of transient inrush currents when energizing unloaded power transformers,” Ph.D. Dissertation ETH No. 12791, Swiss Federal Institute of Technology, Zurich, 1998.

[3] Juan A. Martinez-Velasco, “Power System Transients Parameter Determination,” CRC Press, 2010.

[4] M. Nagpal, T.G. Martinich, A. Moshref, K. Morison and P. Kundur, “Assessing and limiting transformer inrush current on power quality,” IEEE Trans. Power Del., vol. 16, no.2, pp. 276-280, April 2001.

Page 106

[5] J.J. Brunke and K.J. Fröhlich, “Elimination of transformer inrush currents by controlled switching,” IEEE Trans. Power Del., vol. 21, no.2, pp. 890-896, Apr. 2006.

[6] H.S. Bronzeado and R. Yacamini, “Sympathetic Interaction Between Power Transformers,” Proc. Of 29th Universities Power Engineering Conference - UPEC, Vol. 1, pp. 236-239, Sept. 1994

[7] . H.S. Bronzeado, P.B. Brogan, and R. Yacamini, “Harmonic Analysis of Transient Currents During Sympathetic Interaction,” IEEE Trans. on Power Systems, Vol. 11, No. 4, November 1996.

Page 107

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