On the Testing of Three-phase Equipment Under Voltage Sags

This paper provides insight into the testing of three-phase equipment exposed to voltage sags caused by faults. 
The voltage sag recovers at the fault-current zeros, leading to a ‘discrete’ voltage recovery, that is, the fault is cleared in 
different steps. In the literature, the most widespread classification divides ‘discrete’ sags into 14 types. The authors study shows that it is generally sufficient to consider only five sag types for three-phase equipment, here called ‘time-invariant 
(TI)’ equipment. As the remaining nine sag types cause identical equipment behaviour in Park or Ku variables, the number of laboratory tests (or of extensive simulations) on equipment under sags is reduced by a ratio of 14/5. The study is validated by simulation of a three-phase induction generator and a three-phase inverter, which are ‘TI’, and a threephase 
diode bridge rectifier, which is not ‘TI’. Both analytical study and simulation results are validated by testing a three-phase induction motor and a three-phase diode bridge rectifier.


I. INTRODUCTION
HE voltage in sags caused by faults has a discrete recovery.Sags are commonly classified into fourteen types [1].This paper shows that it is generally sufficient to consider only five discrete sag types for grid-connected equipment (e.g., induction and synchronous machines, power inverters or active rectifiers), called time-invariant (TI) in this paper.TI equipment meets the following three conditions: (I) the pre-fault dynamic three-phase electrical variables (e.g., the three-phase currents and/or fluxes) are constant when expressed in Park or Ku variables in the synchronous reference frame; (II) in controlled equipment, the control strategy is implemented in Park or Ku variables (not in abc phase variables); (III) there is no neutral connection.
The remaining nine sag types lead to identical equipment behavior in transformed variables (regardless of the reference frame).Thus, the number of experimental tests (or of simulations) on TI equipment under sags is reduced by a ratio of 14/5.
In order to validate the study, a squirrel-cage induction This research work has been supported by the Spanish Ministry of Economy and Competitiveness through project DPI2011-28021.
generator, an inverter and a diode bridge rectifier are simulated during voltage sag events.As the diode bridge rectifier does not meet condition I, the grouping of sags is only applicable to the first two devices.The results illustrate that the grouped sags produce identical effects on both.

II. VOLTAGE SAG CHARACTERIZATION AND CLASSIFICATION
A voltage sag is characterized by four parameters [1], namely duration (Δt), depth (h), fault current angle (ψ) and sag type.Sags are mainly caused by faults.Three-phase faults generate symmetrical sags, i.e., type A sags, while one-or two-phase faults generate unsymmetrical sags, i.e., types B, C, D, E, F and G sags.This classification is given in [2] and Appendix I.
Faults are cleared at the fault-current zeros; that is, faultclearing does not occur instantaneously but in different steps, resulting in a discrete voltage recovery.The ways to fully clear the same type of fault are classified into fourteen groups in [1], five of which refer to symmetrical sags (named A 1 , A 2 , A 3 , A 4 and A 5 ) while the other nine refer to unsymmetrical sags (denoted as B, C, D, E 1 , E 2 , F 1 , F 2 , G 1 and G 2 ).Appendix II shows the discrete fault-clearing instants for each sag type and the sag sequence during fault clearance.
The sag classification is simplified if the zero-sequence voltage is removed.Then, -According to Appendix I, sag type B is a particular case of sag type D (a sag type D with h = 1/3…1 has the same positive-and negative-sequence voltages as a sag type B with h = 0…1).
-According to Appendix I, sag types E and G are equivalent as they have identical positive-and negativesequence voltages.
-According to Appendix II, sag types A 3 and A 5 are equivalent if the sag sequence during voltage recovery is considered.
Thus, only ten sag types must be studied assuming no zerosequence voltage: A 1 , A 2 , A 4 , A 5 , C, D, F 1 , F 2 , G 1 and G 2 .

III. SAG COMPARISON
The study of sag effects on grid-connected equipment relies on two approaches: to consider or ignore the transformer connections.

A. Considering the Transformer Connections
Let us suppose that a 1-phase-to-ground fault, i.e., a type B The delta-wye (Dy) and wye-delta (Yd) transformers eliminate the zero-sequence voltage and only modify the phase of the positive-and negative-sequence voltages (they do not alter their modulus if the transformation ratio is unity).Thus, the Dy and Yd transformers change the sag type.In the example of Table I, the type B sag is transformed into a type C sag by the first Dy transformer and then into a type D sag by the second Dy transformer.Regarding Dd or Yy transformers, they are equivalent to two cascade Dy transformers.
With regard to the sag initiation and clearance instants, t i and t f1 , respectively, the following comments can be made: -Instant t i can take any value as the fault can be initiated at any arbitrary instant.
-Instant t f1 is defined by the fault clearance process and depends on the faulted phases (type of fault) and the fault current angle, ψ.Thus, this instant is not arbitrary, but can take the discrete values in Appendix II only.
Table I also shows the modulus and angle of the transformed voltage in the complex plane, v f , when applying the Ku transformation [3] (v f is a complex notation for the Park dq components) in the synchronous reference frame.This voltage is calculated as where ω is the grid voltage pulsation and Ψ 0 is the transformation angle at instant t = 0 s (the transformation angle in the synchronous reference frame is Ψ = ωt + Ψ 0 ).It is worth noting that the Ku variables in the synchronous reference frame are used in most equations and examples in the paper only for simplicity reasons, neither the use of Park or Ku variables nor the reference frame affects the results.
As can be seen in ( 1), the transformed voltage v f depends on Ψ 0 , which can be freely chosen.Table I shows that this angle does not influence the time evolution of the modulus of v f but alters the time evolution of the angle of v f in one offset angle.We will come back to this point in Subsection IV-A.
The results of Table I exhibit the following features: (a) During the fault, the angle of the phase a voltage, α a , which depends on the transformer clock number and fault type, varies with the voltage levels (for example, α a = 0° at voltage levels I and III in Table I while α a = -90° at voltage level II).
(b) As evident, the during-fault voltages (the abc phase and transformed voltages) start and end at the same instants t i and t f1 at all levels.
(c) The time evolution of the modulus of v f is identical at all voltage levels.
(d) The time evolution of the angle of v f is identical at all voltage levels by appropriate selection of Ψ 0 (e.g., Ψ 0 = 0° for level I, Ψ 0 = -90° for level II and Ψ 0 = 0° for level III).

B. Ignoring the Transformer Connections
In the technical literature sag types are usually modeled without considering the transformer connections (e.g., the sags in Appendix I).It is further assumed that phasors b and c are symmetrical with respect to phasor a, and α a is null.Table II repeats the sag types of Table I but modeled by ignoring the transformer connections: the type C sag is shifted by 90°, while the type B and D sags are maintained as in Table I.
As α a in sag type C has been forced to be null, its time evolution in Table II exhibits the following changes: The initial and final instants, t i and t f1 , are time-shifted 90° with respect to those of types B and D.
(b) The time evolution of the modulus and angle of v f is time-shifted 90° with respect to that of types B and D.

C. Final Remarks
The aim of the paper is to demonstrate that several pairs of sag types (e.g., C and D) produce identical TI equipment behavior.Just like the transformed voltages of types C and D only differ in one offset angle in Table I (unless Ψ 0 is chosen properly), and one time shift in Table II, considering or ignoring the transformer connections is related to the influence of one offset angle (or the Ψ 0 choice) and of one time shift, respectively, on equipment behavior.

IV. TRANSFORMED VOLTAGE ANALYTICAL EXPRESSION
The expression for the transformed voltage of an unsymmetrical sag in the synchronous reference frame is obtained from (1) as where V p and V n are the rms value of the positive-and negative-sequence voltages, and φ p and φ n are their angles.
The modulus and angle of the transformed voltage (2) are Note that (3) is valid for any unsymmetrical system.As phases b and c are symmetrical with respect to phase a in the sags of Appendix I, angles φ p and φ n are where ξ is a binary variable equal to 0 or 1 depending on the sag type: ξ = 0 for sag types A, C, E and G, and ξ = 1 for sag types B, D and F. Two observations can be made from (3), which agree with the results of Table I and Table II: 1) The transformed during-fault voltage oscillates at a pulsation equal to twice the grid pulsation (2ω).
2) The transformed pre-and post-fault voltage is , where a j e V V α = is the phasor of the phase a pre-fault voltage.This transformed pre-fault voltage can be chosen as the angle reference for Ψ 0 = α a .For example, the transformed voltage at level II in Table I is the angle reference for Ψ 0 = α a = -90°.

A. Influence of the Initial Angle of the Transformation (Ψ 0 )
Fig. 1a illustrates the Ψ 0 influence on the transformation of the abc phase voltages of a type D sag with depth h = 0.4, duration ∆t = 2.5T and angle α a = 0°.It is observed that Ψ 0 only causes an offset in the angle of v f , as said in Subsection III-A, and does not affect its modulus.The cases with Ψ 0 = -90°, 0° and 90° in Fig. 1a correspond to the examples of Table I and Table II.
Fig. 1b shows the Ψ 0 influence on the anti-transformation of a given voltage v f .It is observed that the same voltage v f can be related to a type D sag (for Ψ 0 = 0° or 180°), type C sag (for Ψ 0 = 90° or -90°) or other non-defined sags in Appendix I.As a consequence of this similarity, it is expected that the dynamic behavior of the studied equipment owing to type C or D sags will be identical in transformed variables.This is studied in detail in Section V.In practice, the most common values for α a and Ψ 0 are  Angle of vf α a = 0°, which implies that V a is the angle reference for all phasors.-Ψ 0 = α a = 0°, which implies that the pre-fault voltage v f is the angle reference for all transformed variables.

B. Grouping of Sag Types
This subsection analyzes the characteristics of two sags which are different in abc phase variables but identical in transformed variables.Let us assume two unsymmetrical sags 1 and 2, whose positive-and negative-sequence voltages are (1) (1) (1) (1) According to (3), both sags have identical transformed voltages v f (1) and v f (2) (i.e., they have identical moduli and angles with identical time evolution) for ( 1 ) ( 2 ) If only the first three relations in (6) are satisfied, v f (1) and v f (2) differ in one offset angle.This is the case when considering the transformer connections in sag modeling (Subsection III-A).If the first two relations in (6) are satisfied but the third is not, the time evolution of v f (1) and v f (modulus and angle) is the same apart from a time shift ∆φ as As can be seen, the time evolution of i sf and ω m corresponding to the grouped sag types (A 1 -A 2 , D-C and G 1 -F 1 ) is identical but time-shifted 90° in all cases.Then, the relations in (10) are satisfied.Note that this is valid not only for the machine dynamic variables (currents and speed), but also for the electromagnetic torque, which is expressed in function of the transformed variables.

B. Three-Phase Inverter
A 0.1 MW grid-connected three-phase inverter whose parameters are given in Table IV is simulated.The generic structure of the synchronous reference frame control is considered [5].Although the control is carried out in Park variables (dq components), the simulation results are shown in Ku variables (forward component) for clarity purposes.Fig. 3 shows the time evolution of the transformed current (modulus and angle), i f , injected to the grid.Apart from the inverter switching commutations, it is apparent that the time evolution of the transformed current i f is the same but time-shifted when the inverter is under the grouped types (A 1 -A 2 , D-C and G 1 -F 1 ).Then, the relations in (10) are again satisfied.
It is worth noting that the use of an averaged model (neglecting the inverter switching harmonics) would provide an identical time evolution in the transformed currents of the grouped sag types (apart from the well-known time shift).
Finally, although the simple control structure of [5] does not contain independent controls for the positive-and negativesequence currents occurring during the unsymmetrical sags, it is good enough to illustrate the similarities between the grouped sag types.As the results of this paper are valid for   any reference, even for the synchronous reference frame of the negative-sequence voltage, the inverters with independent controls for the positive-and negative-sequence currents [6]- [8] exhibit identical behavior under the two grouped sag types.

C. Three-Phase Diode Bridge Rectifier
A three-phase diode bridge rectifier with three line inductors L on the AC-side is simulated [9].The DC-link consists of a capacitor C connected in parallel to a constant current source I dc .Its parameters and operating point are given in Table V. Fig. 4 illustrates the time evolution of the modulus and angle of the transformed current, i f , consumed from the network.Note that this device is not TI equipment as the transformed steady-state pre-fault current i f is not constant despite having used the synchronous reference frame.
As can be observed in the detail of Fig. 4a, the steady-state current i f takes different values at the initial instants t i of the type A 1 and A 2 sags, leading to different dynamic behaviors.As a summary, the time evolution of current i f is different for the grouped sag types (A 1 -A 2 , D-C and G 1 -F 1 ).VII.GROUPING FOR OTHER SAG MODELING APPROACHES Fig. 5 shows three sag modeling approaches which differ in the choice of the voltage recovery instants, t f1 , t f2 and t f3 .Fig. 5a illustrates the approach assumed in this paper, which is the most realistic as it takes into account the fault-clearing process (the fault is cleared at instants t f1 , t f2 and t f3 given in Appendix II [1]).Fig. 5c shows the most usual approach in the literature which, unfortunately, is the least realistic because it considers that the sag ends at any arbitrary instant (i.e., t f1 can take any value) and that the fault clearance is abrupt (i.e., the fault is cleared instantaneously in all affected phases).As many current laboratory sag generators used for equipment   testing are not able to emulate the approach in Fig. 5a, the authors propose the intermediate one in Fig. 5b.This approach considers that t f1 is discrete (see Appendix II) but the fault clearance is abrupt (i.e., the fault is cleared instantaneously in all affected phases).The sag grouping proposed for the approach of Fig. 5a is also valid for that of Fig. 5b.Regarding the approach in Fig. 5c, the grouping is reduced to sag types A, D-C and G-F, as there are no subtypes for the abrupt sag types A, F and G.

VIII. CONCLUSION
This study has shown that, among the fourteen discrete sag types in the literature, it is generally sufficient to consider only five types for the study of the effects of such disturbances on grid-connected equipment.This is because the following sag types cause identical behavior in Park or Ku transformed variables: A 1 -A 2 , A 5 -A 4 , D-C, G 1 -F 1 and F 2 -G 2 .This means that when analyzing equipment behavior under voltage sags the number of simulations (or of laboratory tests) is reduced by an approximate ratio of three.This simplification is valid regardless of the reference frame and the use of Park or Ku variables.
The grouping is valid for time-invariant (TI) equipment, which is easily recognized because: (I) the pre-fault dynamic three-phase electrical variables (i.e., the three-phase currents and/or fluxes) are constant when expressed in the synchronous reference frame (although the grouping is also valid for any other reference frames), (II) in controlled equipment, the control is carried out in transformed variables, not in abc phase variables, and (III) the equipment has no neutral connection.Moreover, the grouping is applicable regardless of whether sags are modeled abrupt or discrete.
The grouping is valid not only for the transformed electrical variables (voltages, currents, fluxes, etc.), but also for the rotor speed, electromagnetic torque and rotor angle in the case of electrical machines, and for any other magnitudes which can be expressed in function of these variables, such as the instantaneous active and reactive powers.
-The angles correspond to voltages expressed by a cosine function.If a sine function is used, the angles must be increased by 90°.

Modulus of v f Angle of v fFig. 1 .
Fig. 1.Initial transformation angle, Ψ0, influence: (a) on the transformation of a type D sag, (b) on the anti-transformation of a given voltage vf.Sag characteristics: Δt = 2.5T, h = 0.4 and ψ = 80°.

Fig. 5 .
Fig. 5. Sag modeling approaches.The graphs correspond to the rms voltage during the event (the fault occurs at ti and is cleared at tf1).(a) Fault clearance is discrete and voltage recovery occurs at different instants tf1, tf2 and tf3 (see Appendix II), (b) fault clearance is abrupt in all affected phases and instant tf1 can only take a specific value (see Appendix II), and (c) fault clearance is abrupt in all affected phases and instant tf1 can take any value.APPENDIX II DISCRETE FAULT-CLEARING INSTANTS OF SAGS (FROM [1]) Type 1st recovery (ωtf1) 2nd recovery (ωtf2) 3rd recovery (ωtf3) Sag sequence

TABLE I TYPE
B, C AND D SAGS: LOCATION AT ALL VOLTAGE LEVELS, abc PHASORS AND TIME EVOLUTION OF THE TRANSFORMED VOLTAGES (WITH INITIAL TRANSFORMATION ANGLE, Ψ0, INFLUENCE).SAG CHARACTERISTICS: Δt = 2.5T, h = 0.1 OR 0.4 AND ψ = 80°

TABLE II TYPE
B, C AND D SAGS WHEN ANGLE SHIFTS DUE TO TRANSFORMER CONNECTIONS ARE NOT CONSIDERED: abc PHASORS AND TIME EVOLUTION OF THE TRANSFORMED VOLTAGES (WITH INITIAL TRANSFORMATION ANGLE, Ψ0, INFLUENCE).SAG CHARACTERISTICS: Δt = 2.5T, h = 0.1 OR 0.