Some Aspects of Sustainable Energy Conversion during Transient Processes in Electric Power Systems Comprising Generator Circuit Breakers

This paper is a sequel to a study by the authors of electric power systems comprising the generator circuit-breakers (GCBs) at power plant generator terminals. A sustainable assessment of the current interruption requirements of a GCB addresses the main stresses on the generator circuit breaker, revealing that the GCB current interruption requirements are significantly higher than for the distribution network circuit breakers. Hence, generator circuit-breakers are subject to unique demanding conditions caused by specific stresses, namely: high asymmetrical fault currents resulting from high d.c. components of the fault current; greater electrical, thermal and mechanical stresses when interrupting longer arcing time faults; and important dielectric stress after the electric arc extinction caused by the transient recovery voltage (TRV). This paper extends other studies of the authors on the energetic and exergetic transformation chain at the interruption current OPEN ACCESS 2 transient process in an electric power system that comprises the generator circuit-breaker, as well as the transient recovery voltage which appears after the interruption of a short-circuit fed by the synchronous generator or by the main step-up transformer. For achieving the TRV equivalent configuration the authors applied the method of operational symmetrical components (o.s.c.), and utilized the operational impedances of the synchronous generator and of the main transformer, depending on the fault location. Modeling the transient recovery voltage of circuits emphasizes aspects with direct implications on commutation equipment. Thus, the o.s.c. method can be applied at the poles of any breaker, for any eliminated fault type, if the network configuration and elements are known. The TRV, which appears after the interruption of a short-circuit fed by the generator, may be considered like an oscillation, where the oscillation factor and the rising rate (RR) of the TRV are established by the electrical machine parameters: resistance, inductance and capacitance. Consequently, modeling of concentrated equivalent parameters of the synchronous generator at perturbations caused by current interruption transient processes is achieved in this study through an approach based on sustainability concepts. These findings allow for simulations of the transient recovery voltage and comparisons with experimental results.


Introduction
Following the notion that life on Earth demonstrates sustainable energy conversion [1][2][3][4], we aim to highlight that an approach to an electrical power system according to the patterns defined within the sustainability framework could provide a synoptic view of electromagnetic interaction phenomena within an electric power network entailing a generator circuit-breaker (GCB) at the terminals of the synchronous generator of an electric power station [5][6][7].
In previous studies [16][17][18][19] we have shown the configuration of the key fault current encountered by the generator circuit breaker, namely: generator-fed faults (which can be insulated or grounded threephase and two-phases short-circuits); transformer-fed faults (which can be insulated or grounded threephase and two-phases short-circuits); and generator-fed faults, on the high voltage side of the main step-up transformer (which can be insulated or grounded three-phase and two-phases short-circuits, as well as single-phase short-circuits).
-Just after the short-circuit current interruption by the generator circuit-breaker (when the GCB has been subjected to a very high temperature plasma arc), between its opened contacts arises the transient recovery voltage, which constitutes the most important dielectric stress after the electric arc extinction [12][13][14][16][17][18][19].The authors utilized the framework of sustainability and addressed the GCB specific stresses through an exergy analysis of the transformation chain for the interruption current transient process in an electric power system that comprises the generator circuit-breaker [16][17].The exergy balance of the interrupted circuit during the arc burning stored out in X X X + = emphasizes that the stored exergy X stored is represented by the energy embedded in the magnetic field of the generator windings, corresponding to different states (sub-transient, transient and steady-state short-circuit), and embodied in the arc circuit [16][17].
After the electric arc extinction in the GCB, the magnitude and shape of the TRV occurring across the generator circuit-breaker are critical parameters in the recovering gap after the current zero and they are related to further transformation of the exergy stored in the interrupted alternating-current circuit that comprises the synchronous generator and the GCB.
In this paper, for the case of the faults fed by the synchronous generator, we determine the TRV equivalent configuration with operational impedances, taking into account the generator parameters, and applying the operational symmetrical components method.

Modelling of Transient Recovery Voltage (TRV) during Interruption of Short-circuits Fed by the Synchronous Generator
The dielectric strength within the extinction chamber of the circuit-breaker should increase to a higher value than the TRV; otherwise the electric arc may rekindle [14].The circuit-breaker capability in interrupting the new arc is strongly affected since the gas-filled space between the two electrodes has an important conductivity and this time the electric arc exergy would be entirely used in a destructive process.
Although the mechanical disconnection of the three phases is performed simultaneously, because of the currents angle phase, the arcs extinction on the phases cannot be simultaneous.It first switches the arc on the phase so its current is passing first through zero.Therefore, the TRV on the three phases are different, even in the same three-phase fault interruption.Moreover, the TRV occurring across the generator circuit breaker depends both on the fault location and on the fault nature [14][15][16][17][18][19][20][21][22].
In the sustainability framework of this paper, we determine the TRV when the three-phase shortcircuit fed by the synchronous generator is interrupted, using the operational symmetrical components method [16][17][18][19].Basically, any non-symmetrical three-phase system of currents or voltages can be decomposed into two symmetrical three-phase systems (of different sequences) and a system of single-phase quantities [20,[23][24][25][26].The method application presumes linearity, having been used together with the symbolic representation, in a simplified complex, of the sinusoidal quantities.On basis of this transformation, applied to the phase instantaneous quantities of a three-phase system, we obtain complex quantities (which lose their physical significance), named +,-,0 coordinates or instantaneous symmetrical components.In the paper we follow the same idea of transformation of phase instantaneous quantities at the generator circuit-breaker terminals, in the moments subsequent to its disconnection.
The TRV occurring across the generator circuit breaker is considered only at t > 0, like a damped oscillation by high frequency (f>>50 Hz), starting from zero.Consequently, it is noted that both the TRV and the high frequency currents and voltages achieve the original function conditions.
Applying the operational calculus to the relations determined by the instantaneous symmetrical components method, we obtain the operational symmetrical components (o.s.c.) of the phase voltages and currents [16][17][18][19].Moreover, the o.s.c. of the phase voltages and currents -on the synchronous generator side -are related by operational equations, of the form: where denote the phase operational impedances of the synchronous generator in the symmetrical regimes by positive, negative and zero sequences.Applying the o.s.c.method, the operational image of the TRV occurring across the GB poles after the current interruption can be determined.If the phase current interruption is modelled by the current injection i k -equal but opposite to the eliminated one -the operational image of the TRV occurring across the generator breaker can be expressed in mathematical form: (2) where I k (p) represents the operational image of the switched current and Z e (p) the operational equivalent impedance across the generator circuit breaker.
We take into consideration the fault location in which the short-circuit currents that must be interrupted by GCB are fed by the synchronous generator.The considered generator has the stator three-phase windings connected by configuration Y. Consequently, the GB can be submitted to the insulated or grounded three-phase and two-phases short-circuits.In the paper, for illustration, we analyze the insulated three-phase short-circuit.
The TRV calculation configuration for the moment of the GB first phase opening is represented in Figure 1.
The current interruption has been considered as result of injection on phase A of the current under the restriction:  (5) Equation ( 5) allows the operational representation in Figure 2a.
Applying the method of operational symmetrical components to determine the TRV which appears after the interruption of a short-circuit fed by the generator or the main transformer represents an original contribution of the authors of this study.The method is suitable for application at the poles of any circuit breaker, for any eliminated fault type (fed by the main transformer or by the synchronous generator), if the elements and configuration of the power network are known.Note that, in analytical studies, the calculation of the transient recovery voltage at the circuit breakers situated in the proximity of the generators is made on basis of the equivalent configurations with concentrated parameters.The TRV which appears after the interruption of a short-circuit fed by the generator may be considered like an oscillation with the oscillation factor and the rising rate (RR) of TRV established by the electrical machine parameters: resistance, inductance, and capacitance.Consequently, addressing the analysis of synchronous generator behavior during an electromagnetic perturbation (caused by the current interruption) in an electric power system comprising generator circuit-breakers, and the assessment of the values of the concentrated equivalent parameters of the synchronous generators windings follow in this study.

Equations of Synchronous Generator at Electric Perturbation
In this study we have been adopted the usual hypotheses to define the ideal synchronous machine [20,[23][24][25][26], neglecting the iron saturation and hysteresis phenomena and ferromagnetic core losses.The ideal synchronous machine will have the same self and mutual leakage reactances of the stator as the substituted real machine.In terms of the effects that depend on the rotor position, the idealization goes far to considering each stator winding as having a sinusoidal distribution.The positive directions (of voltage and current) are taken into account according to the generator convention.
In Figure 3 there are specified the axes of the phases A, B and C of the ideal synchronous machine, as well as the phase axes (L, M) and (d, q), respectively, of the equivalent synchronous machines; the rotor rotating sense being also specified in Figure 3.The magnetization directions of currents i A , i B , i C , and i L , i M , respectively, are fixed in the reference frame related to stator, whereas the magnetization directions i d , i q are fixed in the reference frame related to rotor.Since through a synchronous generator perturbation a sudden transformation of the operation conditions is understood [23,[26][27][28], we can assume that the first phase current interruption by the generator circuit-breaker constitutes a synchronous generator perturbation.Because of the extremely small perturbation duration, the voltage regulator action is imperceptible.For the same reason it can as well be considered that the rotor position (specified by the angle N 0 , measured between the axis of phase A and the longitudinal axis d) remains unchanged over the whole perturbation duration.
The sudden variation of the current on one phase (for instance, on phase A) will determine the sudden variations Oi d and Oi q of the equivalent machine currents in the frame coordinates (d,q), as follows: sin cos (7) That means the synchronous generator will be submitted to a transient regime.Consequently, at the first moment, assuming the superconducting rotor circuits, the variations of the fluxes linked to the equivalent machine windings in coordinates d,q (caused by the sudden variations of the currents) will be: Considering the recurrence relations between the coordinates (d, q) and (L, M) [3][4][5]: it can be hypothesized that these sudden variations Oi d and Oi q would be determined by some correspondent variations Oi L and Oi M (unknown, for the time being) of the currents flowing through the stator windings (L, M) of the equivalent machine.Similarly, the fluxes' variations OQ L and OQ M are determined: If relations (9) are substituted in (8), with the further results in (10), we obtain the dependencies among the fluxes' variations OQ L , OQ M and the corresponding currents' variations Oi L , Oi M : cos sin cos sin cos sin sin cos (11) With the supplementary notations: (12) equation (11) takes the form: 8 The fluxes' variations OQ L and OQ M implicitly determine further voltage variations.These will result from the synchronous generator voltage equations in coordinates L, M: or after substituting the expressions of OQ L and OQ M : sin cos (15) Writing with capitals the operational Laplace images of voltages and currents, the system of equations ( 15) becomes: cos sin sin cos (16) Equations ( 16) ascertain, in a general case, the link between the current variations and voltage variations at the terminals of the windings L,M of the synchronous generator.In the context of some local restrictions, these equations can be simplified, both for short-circuit and load disconnection.

First Phase Current Interruption
The initial conditions corresponding to the three-phase short-circuit at the terminals of the synchronous generator are specified by u A (t)=u B (t) = u C (t).After the current interruption in phase A, the synchronous generator is passing for a short time in a two-phases short-circuit, with u B (t)= u C (t). Calculation of voltage at the terminals of winding M: (17) emphasizes that on whole perturbation duration u M =0, and consequently OU M =0 as well.Therefore, the current variations OI L and OI M are linked by the relation: By substituting further the relation above in expression ( 16) we obtain: Equation ( 19) constitutes precisely the operational equation of the synchronous generator in the coordinates L,M at the current interruption in phase A.
In the case of the load disconnections, after the first phase current interruption, the synchronous generator moves to operate in an unsymmetrical regime.
Noting that the line voltage u BC =u B -u C remains actually unchanged, and also that u M = u BC / 3, we determine that in this case as well OU M =0 over the whole perturbation duration.Consequently, in the framework of load disconnection, the synchronous generator behavior at the first phase current interruption will be depicted as well by equation (19).
Based on the notations (12) for equation (19) we obtain the form: cos sin (20) or: (21) Since the coordinates L of voltages and currents are determined with the following relations: (22) and the considered perturbation solely affects the quantities of phase A, we directly find in operationa that: Further equation ( 21) takes the form: (24) Equation ( 24) allows settling the simplified operational circuit, without considering the capacitance of the synchronous generator at the first phase interruption, as depicted in Figure 4.This is used to assess the equivalent phase circuit structure of the synchronous generator during the current interruption process, with the aim to determine the transient recovery voltage across the generator circuit breaker.

Synchronous Generator Operational Circuit at First Phase Current Interruption
Studies on synchronous generator windings over the interruption processes [27][28][29][30] have emphasized the finding that the windings could be seen as long straight conductors.In such situation the capacitance is mainly distributed between each winding and the ferromagnetic stator core, because both the capacitive coupling among the phase windings and the longitudinal capacitance (among winding turns) are very small and can be neglected.
Analyzing the first phase current interruption process when it is considered the equivalent concentrated capacitances C eg (see Figure 5), it is ascertained that through the stator winding A: Still, one could note that the capacitance presence will determine further the forming of an oscillatory circuit in which, because of the perturbation, high frequency currents (10-30 kHz) will appear.The magnetic field created in the synchronous generator by these high frequency currents has a spectrum completely different from the useful magnetic fields, rather it having been approached to the leakage field spectrum [23,26].Consequently, this field does not depend on the rotor position or on iron saturation or excitation degree, being influenced mainly by the constructive machine elements.In this field, each phase winding will have the same inductance L g (see Equation ( 25)), which is completely different from L d " or L q ", and the same resistance R g > R (because of the specific phenomena at the oscillation frequency).
If in equation ( 24) the inductances L d " and L q " are replaced by L g , the resistance R is replaced by R g , and we take into consideration relation (25), in which: (26) the following results: Since the equivalent phase resistance R g is relatively small (namely, maximum (8-10)R), one could assume that the strong damping of the oscillations of the transient recovery voltage can be justified solely on basis of a conductance of losses of the winding insulation at high frequency.This means we admit a parallel circuit for the capacitor [27][28][29][30], as depicted in Figure 6, where: (28) where G 0 denotes the d.c.conductance; C (max) the apparent capacitance over the loading cycle (equal to Q max /U max ); and U h the hysteresis losses' angle (from the cycle area).

Figure 6. Equivalent parallel circuit of the capacitor C eg
The split of the total losses into conduction losses and hysteresis losses has no relevance in applications.Therefore the equivalent circuits are utilized with elements determined directly through the measurement of losses.
Thus, if e 1 is the total losses' angle and C / G = tg g g 2 is the losses' factor, the following result: (29) Without a large error, in applications it may be determined that: Consequently, the operational circuit (27) becomes: With the notation OI Ab = 0 -I A for the current variation in the busbar corresponding to phase A over the current interruption, equation ( 31) can be rewritten in the form: In Figure 7 we depict the operational circuit corresponding to the equation above.This circuit is actually the phase operational representation of the synchronous generator at the disconnection processes' frequency.

Assessment of Concentrated Equivalent Parameters of Synchronous Generator Windings at Current Interruption Transient Processes
The method of operational symmetrical components has been applied here to determine the operational equations and configurations of the transient recovery voltage which appears at the generator circuitbreaker terminals after the interruption of a short-circuit fed by the synchronous generator.The TRV which appears after the interruption of a short-circuit fed by the synchronous generator may be considered as an oscillation with the oscillation factor and the rising rate (RR) of the TRV established by the electric parameters of synchronous generator: resistance, inductance and capacitance.This assessment step is necessary but not sufficient for the modeling of TRV.That is, the structure of the operational impedances of the generator (p) Z + g and (p) Z - g must be assessed taking into consideration the concentrated equivalent parameters of the synchronous generator.This issue is not simple in a sustainable study, since during the interruption current transient processes the electromagnetic phenomena in the generator are complex, and interactions of the generator with the power electric network occur.Modeling the concentrated equivalent parameters of the generator at perturbations caused by current interruption transient processes is imposed.Studies on the synchronous generators stator windings during the transient processes caused by current interruptions have led to the hypothesis that they can be seen like long straight conductors [27][28][29][30].Thus, the generator capacitance is mainly distributed between each winding and the ferromagnetic core, because both the capacitive coupling among the phase windings and the longitudinal capacitance (among winding turns) are very small and can be neglected.On the other hand, both the mutual inductance among the coils of proximate slots, and the inductance of the frontal terminals zone, can be neglected.Consequently, any phase winding can be seen as a chain of inductances without mutual coupling, having solely capacitances which are uniformly distributed to ground [29][30].
In order to enhance this knowledge, we consider an elementary section (with the length "dx") from the phase winding of a synchronous generator with the parameters (electric resistance R', inductance L', electric capacitance C' and electric conductance G') uniformly distributed.
If the conductance G' = 0 (assuming a winding with a perfect insulation) the section "dx" corresponds to the representation in Figure 8.

Figure 8. Elementary section of a synchronous generator phase winding
On the entire length l, the synchronous generator phase winding (with perfect insulation) is made [29] from identical cells chain of inductances without mutual coupling, having only capacitances which are uniformly distributed to the ground, as in Figure 9.The winding can be seen as a quadrupole, with the operational equation: where: which is the operational constant of propagation, and which represents the wave operational impedance (or the characteristic impedance).By developing in power series the radical and under the restraint of the first two terms, the expressions as before can be approximated by , where the damping coefficient is and the phase constant On the other hand, the concentrated equivalent parameters (and, mainly, the longitudinal capacitance C g of a phase winding) are not defined in an absolute mode and they are dependent both on the stator windings connection and on the situation of the neutral point to the ground.
Further, in this sense, we consider the case of star (Y) connection of windings, with grounded neutral (U 1 =0).We determine, as an original contribution, the concentrated equivalent parameters L g , R g and C g of the equivalent configuration represented in Figure 10, based on certain conditions, linked to the analytical methods of conservation of the impedances, of the self frequencies and of the electrostatic energies, respectively.

Method of Equal Impedances (or Equal Admittances)
This method is based on the identity of the impedances Z A'B' (at the terminals of the concentrated parameters equivalent scheme) and Z AB (at the terminals of the line from the Figure 5, having a shortcircuited terminal).
The long lines' equations, equivalent to the synchronous generator phase winding for U 1 =0, are described by: (39) and: But, the equivalent admittance at the terminals A'B' of the representation in Figure 10 is: From the equality of the expressions of the two admittances, Y A'B' and Y AB , we obtain the relation for C g : Since both the resistance and the inductance interfere with the limitation of the short-circuit currents, their values are adopted as invariant in the two configurations, meaning that . Therefore, equation (42) can be rewritten as: or, taking into account equation (34), the function thx admits the development [7]: At the limit, when 0 l , the capacitance C g has the tendency to attain the maximum, and the long lines' equations, equivalent to the synchronous generator phase winding for U 1 =0, are described by:

Method of Equal Self-Frequencies
The method is based on the identity of the self-frequencies corresponding to the two circuits at the resonance [20,[23][24][25][26][27].
With the complex variable p = j D 0 , in the absence of the losses (R' = 0), the circuits admittances, at the resonance, will be zero.Thus, in the case of the line with distributed parameters, from equation ( 14) (with p = j D 0 ), we obtain: or: , from equation (49) we obtain: The self-pulsation of the concentrated parameters equivalent circuit is: From the equality of the two self-pulsations: Under the assumption l L = Lg , we obtain the equivalent capacitance value: where l C = Ceg it is the total capacitance distributed along the synchronous generator phase winding.

Method of Equal Electrostatic Energies
The starting point of the method is based on the statement that two equivalent circuits, for the same voltage amplitude, equally store electrostatic energies [20,[23][24][25][26][27].Thus, admitting a linear variation of the voltage along the winding (see Figure 11), with: , situated at the distance x from the winding terminal, the stored electrostatic energy is , where: It is emphasized that the electrostatic energy stored in the concentrated capacitance C g at the voltage U 2 , and by identification with equation (57) we obtain: In the case of the grounded windings, . Consequently, from the expression (58) we find again that l) C ( 3 1 = C g .

Conclusions
This paper extends other studies of the authors which have examined the exergetic transformation chain at the interruption current transient process in an electric power system that comprises the generator circuit-breaker.
The method of operational symmetrical components has been applied to assess the operational equations and configurations of the transient recovery voltage which appears at the generator circuitbreaker terminals after the interruption of a three-phase short-circuit fed by a synchronous generator.
Applying the method of operational symmetrical components to determine the TRV which appears after the interruption of a short-circuit fed by the synchronous generator represents an original contribution of the authors of this study.Modeling transient recovery voltage of circuits emphasizes aspects with direct implication on commutation equipment.Thus, the o.s.c.method can be applied at the poles of any breaker, for any eliminated fault type, if the network configuration and elements are known.
Further, these findings have been followed by the assessment of the structure of the operational impedances of the generator (p) Z + g and (p) Z - g taking into consideration the concentrated electric parameters of the synchronous generator.During the interruption current transient processes, the electromagnetic phenomena in the synchronous generator are complex, and interactions of the generator with the electric network occur.Modeling of concentrated equivalent parameters of the synchronous generator at perturbations caused by current interruption transient processes is achieved in this study through an approach based on the model of an industrial ecosystem.Thus, adopting the pattern of ideal synchronous machine and applying the overlapping principle and the coordinate transformation, we obtained the operational equations of synchronous generator at sudden load variations.
As another main conclusion the importance is highlighted of the components (L, M).They provide a useful physical point of view concerning the phase current interruption, highlighting the dualist view of industrial ecology, according to which the behavioral analysis of an industrial ecosystem can be performed in two distinct reference frames.
These findings represent a necessary but not sufficient step for TRV modeling.Further modeling the concentrated equivalent parameters of the generator at perturbations caused by current interruption transient processes is achieved in this study through an approach based on sustainability concepts.
Thus, using the equal impedance and equal electrostatic energy methods we obtain for the synchronous generator phase winding equivalent capacitance C g the value C 330 0 eg .
. Applying the equal self-frequency method results in the value C 406 0 eg .
for C g .We conclude that the value of the equivalent capacitance C g of a synchronous generator phase winding (depending on the applied method) varies in the range of , and these findings should be taken into consideration for modeling the TRV on the basis of operational impedances of the synchronous generator during the interruption processes transient processes.

Figure 3 .
Figure 3. Representation of windings and frame systems (L,M) and (d,q) of the synchronous generator

Figure 4 .
Figure 4. Simplified operational circuit of synchronous generator at first phase current interruption

Figure 5 .
Figure 5. Equivalent concentrated capacitances at first phase current interruption

Figure 9 .
Figure 9.Long straight conductors corresponding to the generator phase winding account solely the first two terms, equation (44) becomes:

Figure 11 .
Figure 11.Voltage variation along a phase winding