Abstract
Although of a great extent in time, the research works directed at studying transients in synchronous generators have not yet provided fully sufficient comparative studies in respect to sudden short circuits of the machine. The present paper puts forward novel and comprehensive process models for dynamic simulation of short circuit faults of initially unloaded synchronous generators, using the generalized d-q-0 mathematical model as starting point in derivation. Distinct from the time-domain analysis, the technique proposed here allows an effective comparative overview by employing a specialized procedure to perform repeated time-domain simulations accompanied by peak values recording for the various circumstances. The time consuming matrix numerical inversion at each step of integration, usually performed when selecting currents as state variables, is eliminated by advancing the process models in a convenient split matrix form that allows the symbolic processing. Also, the computational efficiency is being increased by introducing a set of auxiliary variables common to different state equations. The models derivation is carried out without altering the structural equations of the generalized d-q-0 mathematical model of synchronous generators whilst the simulation results are both compared and discussed in detail.
1. Introduction
The essential component of an electric power system is the three-phase a.c. generator. Nowadays almost all power generators are of synchronous type. In normal operation, synchronous generators have two synchronously rotating fields. One field is produced by the rotor, driven at synchronous velocity, and excited by d.c. current. The other field is produced in the stator winding by the a.c. phase currents. Synchronous generators have generally saliency of magnetic poles, what makes their mathematical modeling involved. Although of a great extent in time, the analytical approaches have not yet provided adequate solutions and explanations in respect to transients such as sudden short circuits of the machine, thus leaving the reader with inadequate understanding. The synchronous generator failure is usually caused by external causes such as lightning strikes, heavy rain, strong winds, or contamination of insulators. Long-term average short circuit statistics indicate that 70 percent of synchronous generator faults are of line-to-ground type and result from insulator flashover during electrical storms. Only 5 percent of short circuit occurrences are represented by the balanced three-phase category.
Synchronous generator short circuit studies are an essential tool for the power system designer. The task is to calculate the fault conditions and to provide protective equipment designed to isolate the faulted generator from the remainder of the system in the appropriate time. The interrupting capacity of breakers should be chosen to accommodate the largest of short circuit currents and hence care must be taken not to base the protection decision simply on the results of a balanced three-phase short circuit. The circuit breakers are capable of carrying for a short time the specified short circuit current. However, the possibility of catastrophic failure exists if the short circuit currents are not properly calculated and the breakers are subjected to fault duties that exceed their rating. The stator phase and rotor field currents at short circuit stressing take dangerous values, thermally overloading the installation. The critical value of electromagnetic torque, during short circuit transient, has to be known by the generator designer to appraise the mechanical strength of the structure.
The vast majority of commercial software designed for short circuit transient analysis are based on the empirical calculations encompassed by the accepted standards. References [1–8] describe and highlight the significant differences among the three international approaches to the analysis of short circuit currents: the International Electrotechnical Commission IEC 60909 Standard, the UK Engineering Recommendation ER G7/4, and the American IEEE C37.010 Standard. Dynamic simulation of short circuit faults is always an option, not expressly for validation of the results received from standardized calculations but also for an accurate and effective representation of the transient behavior [9–14]. The present paper describes various novel, comprehensive, and general process models for simulation of synchronous generators short circuit transients. The models derivation was carried out without altering the structural equations of the generalized mathematical model of synchronous generator. The proposed simulation technique offers the advantage of an increased computational efficiency by introducing auxiliary variables, common to different state equations. The time-consuming matrix inversion at each step of numerical integration, performed when currents are selected as state variables and with a view to computing the currents derivatives, is eliminated by advancing the models in a convenient split matrix form that allows symbolic processing. For practical purposes, besides the time-domain analysis, the peak values of short circuit characteristic quantities are depicted as dependencies upon the initial value (at short circuit occurrence) of the rotor lag angle. Such kind of representation considerably facilitates the examination of the differences among the results corresponding to different short circuit types, hence permitting to study the synchronous generator behavior closely.
2. The Generalized Model of Synchronous Generator
We start with the commonly accepted picture of salient-pole synchronous generator, assuming that, besides the field winding, the damper winding can be fairly represented by two axis equivalent circuits. As it is put forward, the generalized mathematical model of salient-pole synchronous generator encompasses two distinctive sets of structural equations [14–18]. These are the differential equations, that is, voltage and motion equations, and the algebraic correlations between flux linkages and currents (the flux equations).
The voltage equations of synchronous generators are given by means of the following ordinary differential equations:
(i)for the stator winding:wherein denote voltages, currents, and flux linkages, respectively, while subscripts , , are associated with the stator axis components and the zero sequence component of voltages, currents, and flux linkages:
(ii)for the rotor windings:
where symbols , , denote variables and parameters associated with field winding and the axis damper windings, respectively.
The electromagnetic torque is given by
with known as the number of generator pole pairs.
The equation of mechanical motion is
wherein represents the driving turbine torque, and is the equivalent moment of inertia.
The standard flux equations of salient-pole synchronous generators are given by means of the following flux linkage-currents relationships:
wherein index denotes the stator and rotor leakage inductances, while
represent the axis components of magnetizing current space-phasor [19]. Having in view the flux equations (9)–(14) together with (15), the following relations are easily seen:
Notice that rotor quantities are referred to stator. It has also to be emphasized that the lag angle of the rotor is measured between the stator phase axis and the direct () axis, and its decreasing in time corresponds to a positive rotor angular velocity as illustrated in Figure 1. Hence
The direct and converse Park-Gorev transforms are given by
respectively. The substitutions
are to be performed in (18) and (19). The mapping (18) performs the transformation of the stator winding variables to a coordinate system in which the rotor is stationary. We identify equivalent windings in the direct and quadrature axes. The direct axis (axis) winding is the equivalent of one of the phase winding, but aligned directly with the field. The quadrature (axis) winding is situated so that its axis is perpendicular to the axis of rotor field winding.
3. Mathematical Modeling
3.1. Preliminaries
In order to improve legibility in presentation and to increase computational efficiency during numerical integration, the following set of auxiliary variables is introduced:
We proceed now to employ the current-based expressions (12)–(14) with the purpose of eliminating the flux variables from rotor voltage equations (5) and (6). The following processed voltage equations result:
with rotor excitation voltage as input variable. Also, replacing the stator axis flux linkages in generalized expression (7), using correlations (9) and (10), we eventually receive the electromagnetic torque as expression in terms of only axis currents:
We consider the most general case of significant deviations from the rated angular displacements. We also assume that the electric changes involved are much faster than the resulting mechanical changes produced by the generator/driving turbine velocity control. Thus, we assume that the driving turbine power is a constant for the purpose of transient calculations. Additionally, we consider that initially the generator operates under no-load at constant rotor angular velocity. These assumptions in connection with motion equation (8) simply lead to valid not only before but also after the instant of short circuit occurrence. Therefore, the equation of mechanical motion (8) yields
3.2. Structuring Methodology of the Line-to-Ground Short Circuit Model
Let phase A be the faulted phase. The restrictive (boundary) conditions in this case are
The fact that phase is shorted to ground is used. This leads to
The converse Park-Gorev transform (19) yields the phase flux linkage in terms of the components:
As a result of time-related differentiating in (29), considering (17), condition (28) can be put forward employing the stator axis flux linkages, their time-related derivatives, including also the time-related derivative of the zero sequence component:
or by considering the flux linkages as provided by the flux linkage-currents relationships (9)–(11):
Having in view the direct Park-Gorev transform (18), where restrictive conditions (27) are to be considered, the stator axis currents come forth as functions of phase current:
Differentiating in (32) and eventually taking into account (17), one obtains
Relationships (32) and (33) allow us to select the set of state currents in a convenient manner. More precisely, the stator axis currents and their time-related derivatives in (22)–(24) and (31) can now be replaced by employing expressions (32) and (33) that merely incorporate phase current and its time-related derivative. Eventually, the state currents vector will include phase current, field current, and the damper axis currents. It is interesting to be pointed out that as a result of subtracting (23) out of (22) one obtains an equation that does not incorporate the stator axis currents as variables:
Since (34) does not require further processing, in order to facilitate the model derivation, this equation has been used as a replacement for (22). The resulted model of line-to-ground short circuit incorporates six differential equations, namely, the processed form of (23), (24), and (31) coupled with (26), (17), and (34) in their original form. Thus, both angular velocity and rotor lag angle also stand as state variables. With expressions (15) of magnetizing current axis components and expressions (32) of stator axis currents, the electromagnetic torque (25) gets an expression in terms of only selected state currents:
3.3. Structuring Methodology of the Line-to-Line Short Circuit Model
Let phase be unfaulted (short circuit fault between phases and ). The restrictive conditions are
The voltage equations for the generator phases and are
Thus, condition (36) can be rewritten:
or by retaining phase B current as state variable and having in view condition (37):
Applying the converse Park-Gorev transform (19), we get
As a result of time-related differentiating in (41), taking into account (17), condition (36), now expressed by (40), can be advanced using the stator axis flux linkages and their time-related derivatives:
or by considering the stator d-q axis flux linkages as provided by flux linkage-currents relationships (9) and (10):
Having in view the direct Park-Gorev transform (18), wherein restrictive conditions (37) are to be considered, we can establish the stator axis currents as functions of phase B current:
Differentiating in (44) and, eventually, taking into account linking (17), we obtain
Relationships (44) and (45) allow us to remove the stator axis currents along with their time-related derivatives out of equations (23), (24), and (43). Eventually, the state currents vector will encompass phase current, field current, and the damper axis currents. With expressions (15) and correlations (44), the electromagnetic torque (25) gets an expression in terms of only available state currents:
Similar to the case of line-to-ground short circuit, previously examined, the line-to-line short circuit model also incorporates six differential equations, more precisely the processed form of (23), (24), and (43) coupled with (26), (17), and (34) in their original form.
3.4. Structuring Methodology of the Three-Phase Short Circuit Model
The restrictive conditions in this case are
or by applying the direct Park-Gorev transform (18),
By employing the current-based expressions (9) and (10) to eliminate the flux variables from stator voltage equations (1) and (2), wherein condition (48) has to be considered, the following processed voltage equations result:
The short circuit model follows by coupling (49) and (50) with the processed rotor voltage equations (22)–(24) and with (26) and (17) with the mention that, in this case, the electromagnetic torque preserves the expression (25), given in terms of axis currents. Thus, the model incorporates seven differential equations, and the vector of state variables includes the stator axis currents, the rotor (field, damper) currents, angular velocity, and the rotor lag angle.
At each step of numerical integration, the stator phase currents result by means of converse Park-Gorev transform (19) having in view condition (47) that points the annulment of the zero sequence component:
4. Dynamic Simulation
Having employed the developed models, we have obtained different characteristic curves, a novelty with respect to available data in literature. With a view to numerical integration, we have implemented [20] an eight-order Adams predictor-corrector scheme [21], the startup being tackled by the fourth-order Runge-Kutta method [21, 22]. The small truncation error of the integrator, coupled with an extended 10 bytes data representation, ensures high numerical integration accuracy. Both process models describing here the line-to-ground and the line-to-line short circuits possess the following internal structure:
with matrix L of the form
From the six differential equations encompassed by structure (52), the first equation plainly represents (34). The second and the third equations represent (23) and (24), processed by employing relationships (32) and (33) for the case of line-to-ground short circuit and by employing (44) and (45) for the case of line-to-line short circuit, respectively. Also, the fourth differential equation of structure (52) stands for the processed form of (31) and (43), respectively. For the case of line-to-ground short circuit, for which in (53), one identifies:
while for the case of the line-to-line short circuit fault, when
In order to increase the computational efficiency, we have decided to avoid the numerical inversion of (53) at each step of numerical integration by loading a symbolic processor (Appendix). For the case of three-phase short circuit fault, one observes that the expressing of state currents time-related derivatives is straightforward. More precisely, the derivatives of the axis currents (stator axis, field and damper axis currents) have been extracted solely from the system of (49), (22), and (23) while (50) and (24) have yielded the time-related derivatives of q-axis currents (stator axis and damper axis currents).
The simulations have been carried out for the initial condition of generator no-load operation, with synchronous velocity and rated phase voltage. It has to be emphasized that the per unit (p.u.) system has been adopted. For the purpose of transient calculations, the base quantities are as follows:
(i)voltage base: (ii)current base: (iii)flux linkage base: (iv)resistance base: (v)inductance base: ,(vi)torque base: (vii)angular velocity base: (viii)time base:where subscript denotes the rated values.
4.1. Synchronous Generator Data
The outcomes presented in this paper are all provided in the per unit (p.u.) system and have been obtained for a salient-pole synchronous generator with the following data:
(i)stator winding parameters (in per unit):
(ii)rotor (field and damper) windings parameters (in per unit):
(iii) axis magnetizing inductances (in per unit):
(iv)mechanical time constant:
4.2. Line-to-Ground and Line-to-Line Short Circuits. Results Interpretation
Both line-to-ground and line-to-line short circuits determine the synchronous generator operation to pass into single-phase operation what causes a completely different behavior of the machine compared to the typical case of three-phase short circuit fault.
Two characteristic positions of the rotor can be distinguished at the instant of short circuit occurrence:
(i)the position corresponding to the annulment of the unidirectional (d.c.) component of the stator current (the favorable circumstance);(ii)the position corresponding to the maximum value of the unidirectional (d.c.) component of the stator current (the most unfavorable circumstance).The curves in Figure 2 indicate the dependencies of the peak values of short circuit characteristic quantities upon the initial value (at short circuit occurrence) of the rotor lag angle. Since the dependencies are periodical, with a period of , the curves have been plotted for the initial value of rotor lag angle within the interval []. As mentioned, the lag angle of the rotor, , is measured between the stator phase axis and the direct () axis. Practically, the curves of Figure 2 typify the data returned by a specialized procedure to which sufficient skill has been given to execute progressive modifications of the initial value of rotor lag angle and repeated time-domain simulations, accompanied by the peak values recording. Concerning the time consuming, it has to be noticed that the possibility of detecting the peak values just within the first cycle (2 rad.) leads to a fast enough execution of the “repeated time-domain simulations.”
(a)
(b)
In Figure 3 the time-related evolution curves of the currents occurring at line-to-ground short circuit in the favorable circumstance are presented. This case corresponds to the annulment of the stator current unidirectional component at the instant of short circuit occurrence and implicitly during the entire short circuit process. For more characteristic quantities and, also, for the line-to-ground short circuit, the time-related evolution curves for the most unfavorable circumstance, when the reached peak values come to be the maximum ones, are plotted in Figure 4. It has to be emphasized that the evolution curves characteristic of line-to-line short circuit keep the shapes of the time-related evolution curves obtained for line-to-ground short circuit.
(a)
(b)
(a)
(b)
(c)
(d)
Examining the plots of the stator and field currents, one observes that, whatever the considered interval, the number of field current pulses is double compared to the number of stator current pulses. This is because of the generator single-phase operation; in the favorable circumstance, corresponding to a zero unidirectional component of stator current, the pulsating stator field resolves, referred to the stator, into forward and backward rotating fields moving at rotor velocity as well as at 3, 5, 7, times its velocity. At the same time, the pulsating rotor field resolves, referred to the rotor, into forward and backward rotating fields moving at 2, 4, 6, times the rotor velocity. The influence of stator current unidirectional component upon the pulsating stator field consists in the appearance of components rotating at 2, 4, 6, times the rotor velocity, referred to stator, accompanied by rotor field components rotating at rotor velocity as well as at 3, 5, 7, times this value, referred to rotor. Inspecting the field current plot in Figure 4(b), one observes the presence, in the first moments of short circuiting, of a few high level pulses adjoined to lower level pulses. It is quite natural for the levels of the adjoined pulses to tend to equalization and for the shape of curve in Figure 4(b) to tend to the shape of curve in Figure 3(b) along with the gradual damping of the stator current unidirectional component.
4.3. Three-Phase Short Circuit. Comparative Overview
The sudden three-phase short circuit model is of the following structure:
wherein is the vector of the axis currents, which are here state variables, whilst the electromagnetic torque preserves the expression (25), as being given just in terms of axis currents. It has to be emphasized that the elements of matrix and of column vector do not depend on rotor lag angle , which also stands as state variable.
Since the rotor lag angle, as state variable, only interferes in the last equation of structure (61), it follows that the time-related evolutions of the other state variables of (61) are independent on the initial value (at short circuit occurrence) of rotor lag angle. In other words, the linking equation (17) is not necessary if the purpose is merely the assessment of axis currents and of electromagnetic torque (25), given exactly in terms of axis currents. This obviously is in contrast both with the case of line-to-ground short circuit and with the case of line-to-line fault, where the evolutions of all state variables do depend on the initial value of rotor lag angle. However, in the present case, the lag angle of the rotor decides the values of generator phase currents by means of expressions encompassed by (51). This is the reason why (17) had to be incorporated within structure (61) and, implicitly, the reason why the evolutions of generator phase currents do depend on the initial value of rotor lag angle.
The characteristic quantities necessary for assessing the transient response at sudden three-phase short circuit are displayed in Figure 5, wherein the phase current curve of Figure 5(a) corresponds to the most unfavorable circumstance, for which the reached peak value is the maximum one.
(a)
(b)
(c)
(d)
Figure 6 illustrates a comparative overview on the recorded maximum peak values of short circuit characteristic quantities. Conspicuously, the critical stator phase current occurs at the line-to-ground sudden short circuit, for the initial rotor lag angle of , whilst the critical field current is received at the three-phase fault, independent on the initial circumstance. On the other hand, the electromagnetic torque comes to be critical at the line-to-line short circuit fault, for the initial rotor lag angle of .
5. Conclusion
Synchronous generators are often stressed by short circuit faults having external causes. Long-term average statistics point out that synchronous generators are stressed mostly by line-to-ground and line-to-line short circuit faults. In spite of synchronous generators long history of operation, the research works presented hitherto have not put forward fully sufficient comparative studies with regard to sudden short circuits of the machine. This can be due to the fact that short circuit faults occurring during operation cannot offer sufficient information with a view to fault correct assessment, simply because of the various circumstances in which these faults do practically occur.
This paper presents novel process models for simulation of line-to-ground, line-to-line, and three-phase sudden short circuits in synchronous generators using the generalized model as starting point in derivation. The simulation study is carried out for the initial condition of generator no-load operation at synchronous angular velocity. The simulation technique here makes feasible the survey on the differences among the results corresponding to short circuit faults by means of a specialized procedure that executes repeated time-domain simulations accompanied by peak values recording. The comparative study is performed having in view the most unfavorable circumstance, for which the reached peak values of the characteristic quantities are the maximum ones. A comparative overview plainly reveals that the critical (stator) phase current occurs at line-to-ground short circuit, the field current comes to be critical at the three-phase short circuit whilst the critical torque is reached at the line-to-line short circuit fault.
Appendix
The inverted matrix gets the form
List of Symbols
| : | Instantaneous voltage, current and flux linkage, respectively |
| : | Resistance |
| : | Inductance |
| : | (Rotor) angular velocity |
| : | Time variable |
| : | Torque |
| : | Equivalent moment of inertia |
| : | Number of generator pole pairs |
| : | Lag angle of the rotor |
Subscripts
| : | Denotes the axis (“direct” axis) components |
| : | Denotes the axis (“quadrature” axis) components |
| : | Denotes zero sequence components |
| : | Variables and parameters associated with (rotor) field winding |
| : | Variables and parameters associated with (rotor) d-axis damper winding |
| : | Variables and parameters associated with (rotor) axis damper winding |
| : | Variables and parameters associated with magnetizing circuit |
| : | Suffix to denote leakage inductances |
| : | The synchronous generator phases |
| : | Index for rated (nominal) parameters |
Superscript
| : | Transposed matrix (vector) |