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An enhanced tool for fault analysis in multiphase electrical systems
Electrical Power and Energy Systems 75 (2016) 215–225
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
An enhanced tool for fault analysis in multiphase electrical systems Débora Rosana Ribeiro Penido, Leandro Ramos de Araujo ⇑, Márcio de Carvalho Filho Department of Electrical Engineering, Federal University of Juiz de Fora, Juiz de Fora, MG, Brazil
a r t i c l e
i n f o
Article history: Received 10 March 2014 Received in revised form 5 August 2015 Accepted 11 September 2015
Keywords: Component modeling Current-injection method Distribution systems Fault analysis Simultaneous fault Unbalanced operation
a b s t r a c t This paper describes a new methodology for the fault analysis of an n-conductor electrical system, in which the phase imbalances, neutral cables, groundings, and other inherent characteristics of distribution systems are considered. The proposed methodology, which is based on the current-injection method, allows faults to be represented in a simple way and may be used to analyze several fault types, including internal, series, and simultaneous faults. The results of several cases are presented to show the efficiency of the proposed method. Ó 2015 Elsevier Ltd. All rights reserved.
Introduction One of the main objectives in the distribution system analysis is the development of methodologies that are able to represent electrical systems in detail, considering their different topologies and the variety of equipment present in these systems, especially in the context of smart grids [1–7]. The application of advanced methodologies allows engineers to achieve better operating conditions and plan more effectively. Currently, several methodologies for fault analysis consider that electrical networks are symmetrical and balanced, and they model the equipment in a simplified manner, and, through the use of symmetrical components, it is possible represent various asymmetric faults [8–10]. It is noteworthy, however, that the method of utilizing symmetrical components often introduces simplifications in the analysis, making this approach not the most suitable for all cases. For example, the fault analyses in unbalanced systems introduce coupling between the sequence networks, causing the loss of the main advantage of the symmetrical components. This fact can be observed in single-phase, two-phase, or three-phase lines that are not perfectly transposed. Among the works that address studies of faults using symmetrical components, several improvements in the simplifications that are usually adopted have been proposed [9–11]. Several configurations are used in distribution systems, and the three-phase four-wire configuration with multiple neutral ⇑ Corresponding author. Tel./fax: +55 32 2102 3442. E-mail address: [email protected] (L.R. de Araujo). http://dx.doi.org/10.1016/j.ijepes.2015.09.005 0142-0615/Ó 2015 Elsevier Ltd. All rights reserved.
grounding is widely used. This configuration has a low installation cost, provides greater safety for people and equipment, and has a high sensitivity to faults [12]. It is emphasized that methodologies based on symmetrical components do not provide good results when used in this type of system. Also noteworthy is that the correct representation of the neutrals and groundings is essential for fault analysis in distribution systems. In recent years, many researchers have proposed methodologies for fault analysis focusing on unbalanced systems. Previous studies have proposed the representation in phase coordinates, but with some simplifications; for example, the neutral cable is not explicitly represented and its effects in the phases are incorporated by Kron reduction or simply ignored, despite the large use of neutral cables in distribution systems [13–15]. This kind of simplification may lead to incorrect results, especially in unbalanced systems [12]. Some authors have proposed specific methodologies for fault analysis in distribution systems, thereby improving some aspects of the analysis [16–25]. Despite significant advances in the representation of systems with faults [25], many points can be improved. Cited as examples are fault between different voltages points, detailed representation of equipment, representation of magnetic couplings between different circuits, internal faults, improved representation of loads in high impedance faults, and simultaneous faults. These points, if addressed, will allow more accurate analyses and avoid misleading results that may pose risks to both human and equipment safety. This paper proposes a method that allows the analysis of faults in any kind of multiphase electrical system, including internal,
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In Step F.3, if an internal fault is indicated in a transformer, the procedure given in Section ‘Transformers’ is used to represent the equipment in fault. In Step F.4, RLC (resistances, inductances, and/or capacitances) elements are inserted to represent the various types of faults, as shown in Section ‘RLC components’. The right-hand vector (f) and Jacobian matrix (J) are assembled in Steps F.5 and F.7. In Step F.6, a convergence test is performed, e = 106. In Step F.8, increments are calculated and the equation is presented in (1). The contributions to right-hand vector (f) is calculated according to the equations presented in Sectio n ‘Component models’. The contributions to the Jacobian matrix (J) are the first order derivative of f in relation to state variables (VRe and VIm). Derivative equations are not presented for space reasons.
series, and simultaneous faults. When compared with other methodologies of fault analysis, several advantages can be cited for the n-conductor simultaneous fault method (NSFM): i. It allows modeling of any electrical system in phase coordinates, including mutual impedances between phases of the same circuit or mutual impedance between circuits of the same or different voltage level. ii. It allows the representation of any type of transformer connection in direct form. iii. It allows the representation of faults in multiphase systems, e.g., single-phase, two-phase, and three-phase. iv. It allows the representation of any type of electrical system (distribution, transmission, or industrial), radial or meshed, and dispersed generation. v. It allows the explicit representation of groundings, ground wires, and neutral cables. Any type of grounding can be represented (isolated, high-impedance, low-impedance, solid-grounded, or any combination) including even safety groundings. vi. It allows the representation of simultaneous faults in a simple and intuitive form. vii. It allows the simulation of faults between different points in the system, e.g., the contact of a medium-voltage cable with a low-voltage cable in distribution systems. viii. It allows the simulation of internal faults in equipment, e.g., transformers.
ð1Þ
where DVRe and DVIm are the solution, that increment of phase to ground voltage, real and imaginary respectively. DIRe and DIIm are the net current injected (Section ‘Component models’). Jk,m is the partial first order derivative as shown in (2).
It is emphasized that all these points improve the results of fault analyses. It is noteworthy that all the advantages listed can be executed directly in NSFM without the need for additional calculations or simplifications, which are usually required in many current methodologies. This paper is organized as follows: The NSFM is developed in Section ‘n-Conductor simultaneous fault method’, numerical examples are provided in Section ‘Applications’, and conclusions are presented in Section ‘Conclusions’.
2 @ DI 6 Jk;m ¼ 4 @ DI
@ DIRe
3 k
k
@V Re
Imk
@ DIIm
k
k
@V Im
@V Re
k k
7 5
ð2Þ
In Step F.9, state variables are updated. In Step F.10, update load parameter. If |VLD| 6 vLim, a and b is set to 2 (constant impedance – Section ‘Loads’). If |VLD| > vLim, a and b is set to original value. VLD is the voltage in the load terminals and vLim is load minimum voltage.
n-Conductor simultaneous fault method The n-conductor simultaneous fault method (NSFM) was developed to be a general methodology that allows the analysis of any faulted electric power system.
Rek
@V Im
The results are shown in Step F.11.
Methodology
Component models
In the NSFM, the network and the equipment are modeled in phase coordinates and electrical quantities (p.u. quantities are not used). The equations of the electrical system are written as equations of current injection in rectangular coordinates. The proposed method is presented in Fig. 1, where z is the state variable and f are current-injection equations for all the system nodes. The state variables are phase-to-ground voltage and are represented in rectangular coordinates using the real and imaginary parts.
Transmission and distribution lines The transmission and distribution lines are modeled as a coupled n-phase p-equivalent lumped-parameter circuit and the equations presented in (3) and (4) are used to model a distribution line connected between the sets of nodes k and m [8].
2
Ikm;lin In Step F.0 of the proposed algorithm (Fig. 1), all electrical system and equipment data are loaded, including the location and type of fault that will be simulated. In Step F.1, the pre-fault conditions are calculated through a power flow program. In Step F.2, the electric system is updated and additional nodes are created to simulate the faults informed in Step F.1; for example, if a fault is indicated in the middle of a transmission line, one or more additional nodes are created and the line is split.
Z k1;m1
6 6Z 6 k2;m1 6 ¼6 . 6 . 6 . 4 Z kn;m1 2
Z k11
6 6 Z k21 6 þ6 6 .. 6 . 4 Z kn1
Z k1;m2
Z k1;mn
Z k2;m2 .. Z kn;m2 Z k12
7 Z k2;mn 7 7 7 7 7 7 5 Z kn;mn
Z k1n
Z k22 .. Z kn2
.
.
31 2
31 2
7 Z k2n 7 7 7 7 7 5 Z knn
V k1 V m1
3
7 6 6 V k2 V m2 7 7 6 7 6 7 6 . . 7 6 . 5 4 V kn V mn V k1
3
7 6 6 V k2 7 7 6 7 6 6 .. 7; 6. 7 5 4 V kn
ð3Þ
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217
Fig. 1. Algorithm of the NSFM.
2
Imk;lin
Z k1;m1 Z k1;m2 6Z 6 k2;m1 Z k2;m2 ¼6 .. 6 .. 4 . . Z kn;m1 Z kn;m2 2 Z m11 Z m12 6 Z m21 Z m22 6 þ6 . .. 4 .. . Z mn1 Z mn2
31 2 3 Z k1;mn V m1 V k1 7 Z k2;mn 7 6 V m2 V k2 7 7 7 6 7 .. 7 6 5 5 4 . V mn V kn Z kn;mn 3 2 3 Z m1n 1 V m1 6 7 Z m2n 7 7 6 V m2 7 7 6. 7: 5 4 .. 5 Z mnn V mn
ð4Þ
where Zkx,my are the self-impedances and mutual impedances between nodes x and y and phases k and m, and Zkxy and Zmxy are the shunt impedance of node k or m on phases x and y. The proposed methodology can also simulate faults in phases between lines of different circuits, considering the mutual impedances and even different voltage levels. Despite not being usually studied, this information can be useful in distribution systems to analyze the effects of a contact between conductors of medium voltage to low voltage or in studies of faults in transmission lines that share the same right-of-way. Loads In this methodology, all loads are specified as constant impedance. The current injection of a load connected between nodes k and m (Ik and Im) is shown in (5).
Ik ðzÞ ¼
jV km ja Pkm jjV km jb Q km ; V km
ð5Þ
Im ðzÞ ¼ Ik ðzÞ; where Pkm and Qkm are the active and reactive load connected between nodes k and m, respectively. Vkm is the voltage phasor between nodes k and m. a and b are the exponential load model.
Transformers The transformer model used in this work was enhanced to allow the simulation of internal faults. The fault can be between a turn and ground, between turns of the same windings, or between distinct windings, including between different voltage levels (primary and secondary). In the NSFM, the transformer model uses intermediate waypoints (wpis) in the windings to simulate internal faults, and these points are defined according to the type and location of internal fault that will be simulated. Fig. 2 presents an algorithm to construct the faulted transformer nodal admittance matrix to represent an internal fault. Detailed descriptions of the steps indicated in the algorithm in Fig. 2 are presented in the following. The transformer data are loaded in Step P.1. If the system under study is isolated or high-impedance grounded, the shunt and parasitic capacitances of the transformer should also be informed. These capacitances are inserted in the terminal nodes of the transformer after calculation of the transformer nodal admittance matrix. In Step P.2, the internal fault data are read. In Step P.3, the primitive impedance matrix is assembled. The procedure for inserting four wpi nodes for simulation of an internal fault is illustrated in Fig. 3. Fig. 3a presents a delta-wye transformer and Fig. 3b presents a detail of the windings highlighted in bold in Fig. 3a. Three types of faults are shown in Fig. 3b: (i) turn-ground (Rwg); (ii) between turns of the same winding (Rww); and (iii) between turns of different windings (Rwy). It is noteworthy that several other types of faults can be simulated using this model. The impedances of faults are placed externally by equipment-type RLC (discussed in Section ‘RLC components’). The primitive impedance matrix of the windings illustrated in Fig. 3 is shown in (6). Z1+,1a is the impedance between node
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Fig. 2. Algorithm to construct the Ybus of a generic transformer for internal faults.
In the fourth step (P.4), the nodal incidence matrix is built (Ainc). (7) represents the incidence matrix (Ainc) of the faulted winding in Fig. 3b where the rows represent the partial windings (divided by waypoints) and the columns represent the nodes of the electrical system (external and internal) in which the partial windings are connected.
ð7Þ
In Step P.5, the winding-voltage adjustment matrix is built (Wwv). This matrix is used to deal with the partial winding voltage, where DV is the voltage level between the transformer nodes (ex: V1a and V1b) and is a percentage of the nominal winding voltage. The winding voltage adjustment matrix of Fig. 3b is presented in (8).
Fig. 3. Details of the transformer windings.
V1+ and the wpi node V1a, the position of the V1a node is informed in Step P.2 as a percentage of the winding, i.e., Z1+,1a is a percentage of Z1. Z1a,1b; Z1b,1; Z2+,2a; Z2a,2b; and Z2b,2 follow the same assembly procedure. Zmag is the mutual impedance between the windings [26] and must be informed in P.2. Here: V1+, V1, V2+, and V2 are the external connection of the windings of the transformer. V1a, V1b, V2a, and V2b are the waypoint nodes. The number of waypoints created is a function of the type of fault that will be simulated. Z1 and Z2 are the total impedance of the windings. Rwg, Rww, and Rwy are examples of fault resistance.
2 6 6 6 6 Zprimitiv e ¼ 6 6 6 6 4
3
Z 1þ;1a
7 7 7 7 7 þ ½Zmag : 7 7 7 5
Z 1a;1b Z 1b;1 Z 2þ;2a Z 2a;2b Z 2b;2
ð6Þ
ð8Þ In Step P.6, the generic transformer nodal admittance matrix (Ytrf) is calculated as (9).
Ytrf ¼ AtInc: Wwv Z1 primitiv e Wwv AInc: :
ð9Þ
Generators and substations The generators and substations are modeled in NSFM as a voltage source behind the impedance, as shown in Fig. 4. The parameters of the voltage source are calculated for the pre-fault conditions, and then are kept fixed during the fault calculations: this is done because no voltage regulators or transformer tap actions are performed during the initial instants of the fault.
D.R.R. Penido et al. / Electrical Power and Energy Systems 75 (2016) 215–225
V s;i1 V s;i2 ¼ DV s ;
219
ð12Þ
where s = {a, b, c}. Ps,ger and Qs,ger are the active and reactive power of the voltage source, respectively, which are state variables. DVs is the pre-fault internal voltage of the sources. RLC components Many network components can be modeled with RLC. The RLC elements are also used to represent faults in this work. The equations of an RLC connected between two nodes (k and m) are shown in (13) and an RLC connected between node k and ground is indicated by (14).
Fig. 4. Generator or substation model.
Where Vai1, Vai2, Vbi1, Vbi2, Vci1, and Vci2 are the connection nodes of the voltage sources representing phases a, b, and c of the machines or substations. Vae1, Vbe1, and Vce1 are the external connection nodes of the voltage sources of machines or substations. zabc source is the impedance matrix of the machine or short-circuit impedance of the substation. DVa, DVb, and DVc are the internal voltages of the sources, calculated for pre-fault conditions. The current equations of the zabc source between nodes Vse1 and Vsi1 (Ie1 and Ii1) are shown in (10), the current contributions of the voltage sources are presented in (11), and the voltage specification equations in (12).
2
Z aa
6 Ie1 ðzÞ ¼ 4 Z ba
Z ab Z bb
Z ca Z cb Ii1 ðzÞ ¼ Ie1 ðzÞ Is;i1 ðzÞ ¼
Z ac
31 2
Vae1 Vai1
7 6 7 Z bc 5 4 V b e 1 V b i 1 5 ; Z cc Vc e1 Vc i1
jV s;i1 V s;i2 j2 Ps;ger jV s;i1 V s;i2 j2 Q s;ger ðV s;i1 V s;i2 Þ
Is;i2 ðzÞ ¼ Is;i1 ðzÞ
3
;
ð10Þ
ð11Þ
Ik ðzÞ ¼ ykm ðV k V m Þ; Im ðzÞ ¼ ykm ðV m V k Þ; Ik ðzÞ ¼ yk ðV k Þ;
ð13Þ ð14Þ
where Vk = VRe,k + jVIm,k and Vm = VRe,m + jVIm,m are the voltage phasors of nodes k and m. Fig. 5 shows several possibilities of RLC element connections. It is possible to represent many fault types, including simultaneous faults or phase-neutral faults. It is also possible to represent even complicated faults in a simple form (Fig. 6). Some of these faults are impossible to represent in many methodologies. It is also possible to represent all the fault types shown in Fig. 5 simultaneously with no extra computational effort or additional considerations. In addition, it is emphasized that the tool allows the explicit representation of the neutral wire; therefore, it can provide much more precise results in systems that are not solidly grounded. Groundings and parasitic capacitances Any existing grounding can be modeled in the proposed methodology without the need for approximations. RLC elements can be used in conjunction with other equipment, such as
Fig. 5. Some fault representations.
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percentage of transformer winding relative to the neutral point. Ideal windings have been considered in the simulations. Fault at 50% of high-voltage winding, connection-grounded wye–grounded wye To simulate an internal transformer fault, a wpi node was created at 50% of the high-voltage winding in the transformer located between three-phase busbars 2 and 3 (Fig. 7). The fault was represented by a negligible resistance (108 X).The load on busbar 4 has been neglected. The detailed results of the simulation are presented in Fig. 8. As seen in the figure, although the secondary circuit was open, there was a fault current flow in the high-voltage winding of the transformer, even with the low-voltage winding open. This occurred because voltage is applied to segment 2.a–2.f of the transformer, and this segment is magnetically coupled to segment 2.f–2.n (autotransformer effect). Thus, currents of equal and opposite magnitude were created in both segments.
Fig. 6. Medium to low voltage fault in distribution system.
transformers, to model high impedance grounding (for example, generators grounding transformers or zig–zag transformers). The parasitic capacitances of equipment like CTs, VTs, transformers, surge suppressors, and other equipment are modeled as RLCs. These capacitances should not be neglected for fault studies in ungrounded or high-impedance grounding systems.
Applications The IEEE test feeders [27] have been developed to provide validation of distribution systems analysis software and algorithms. Three of these systems have been chosen to demonstrate the modeling and some simulation capabilities of NSFM.
IEEE4 test system The one line diagram of IEEE4 is shown in Fig. 7. The bus 1 voltages VAN, VBN, and VCN have fixed pre-fault values 7200\0.0° V, 7200\120.0° V, and 7200\120.0° V, respectively. Also shown is a detailed representation of the transformer windings located between the three-phase busbars 2 and 3. The p% value is the
Fault in the high-voltage winding, grounded wye–grounded wye, variation in p% Based on the diagram shown in Fig. 7, the value of p% was varied from 100% (phase to ground fault) to 1% (neutral to ground fault). The load was considered in this case. Fig. 9 shows the voltages at the transformer terminals with variation of p%. It is noted that phases b and c present overvoltage for faults close to node 2.a. The currents of phases a, b, and c of circuit 1–2, the internal fault current (Int. Fault), the current in the grounding of the generator (Grd. Gen), and the current in the grounding of the transformer (Grd. Trf) are shown in Fig. 10. Note that: (i) the current in phase a and the current in the grounding of the generator reduce at the same rate, because the current in the grounding of the generator is the phase a fault current plus the load unbalanced current; (ii) the internal fault current is always greater than the phase current (phase a in this case), because this internal fault current is the sum of the current in phase plus the induced current in the other part of the winding of the transformer; and (iii) the current of the grounding transformer increases as the fault moves toward neutral, which occurs because of the magnetic coupling and the induced current. Fault in the high-voltage winding, delta–grounded wye, variation in p% The transformer connected between busbars 2 and 3 (Fig. 7) was turned into a delta-ground wye transformer, and the p% value was varied from 100% (phase a fault) to 0% (phase b fault). Fig. 11 shows the phase voltages at the transformer terminals. It is noted that phase a and b voltages vary in modulus as expected and phase c presents a small overvoltage close to the fault terminal. The variations in the fault currents are shown in Fig. 12. Note that (i) the internal fault current is the sum of the fault currents
Fig. 7. IEEE 4 test system.
D.R.R. Penido et al. / Electrical Power and Energy Systems 75 (2016) 215–225
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Fig. 8. Transformer internal fault.
Fig. 9. Voltages in grounded wye–grounded wye transformer internal fault.
Fig. 11. Voltages, delta–grounded wye transformer internal fault.
Fig. 10. Currents in grounded wye–grounded wye transformer internal fault. Fig. 12. Currents, delta–grounded wye transformer internal fault.
of phases a and b and (ii) the current in the grounding of the generator is equal to the current internal fault, because this is the only path for current return.
Fault between windings of different voltages, grounded wye–grounded wye Wpi nodes were created in the windings of high and low voltage to simulate an internal fault between windings of different voltages. These nodes were created in the middle (50%) of the windings and the load was neglected. The results of this simulation are presented in Fig. 13. As seen in the figure, fault current circulation occurred at the high-voltage winding of the transformer and the bottom of the
low-voltage winding: both effects are explained by the magnetic coupling between the windings (Section ‘Transformers’). A pure three-phase methodology or representation in sequence components does not include much detail, and resolution of this case becomes very complicated or impossible in this kind of system representation. Three-busbars test system The three-busbars system, shown in Fig. 14, was based on [28]. The area of high-resistance neutral grounding is the generator to the low-voltage winding of the power transformer (160 MVA) and to the high-voltage winding of the unit auxiliary transformer
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Fig. 13. Fault between windings.
Fig. 14. High-resistance neutral grounding.
Table 1 Voltages (p.u.), high-impedance grounding.
Table 3 Voltages (p.u.), Ungrounded.
Node
Case 1
Case 2
Case 3
Case 4
Node
Case 1
Case 2
Case 3
V1.a V1.b V1.c V1.n V2.a V2.b V2.c V3.a V3.b V3.c
0.973 0.973 0.973 0.000 0.965 0.965 0.965 0.961 0.961 0.961
0.059 1.649 1.707 0.965 0.000 1.672 1.672 0.961 0.961 0.961
0.000 1.732 1.732 1.000 0.000 1.732 1.732 1.000 1.000 1.000
0.329 0.000 1.509 0.526 0.000 0.000 1.509 0.000 0.871 0.871
V1.a V1.b V1.c V1.n V2.a V2.b V2.c V3.a V3.b V3.c
0.973 0.973 0.973 0.000 0.965 0.965 0.965 0.961 0.961 0.961
0.059 1.649 1.707 0.965 0.000 1.672 1.672 0.961 0.961 0.961
0.000 1.732 1.732 1.000 0.000 1.732 1.732 1.000 1.000 1.000
Table 2 Currents (A), high-impedance grounding.
Table 4 Currents (A), ungrounded.
Current
Case 1
Case 2
Case 3
Case 4
Current
Case 1
Case 2
Case 3
Iger,a Iger,b Iger,c RIcap,a RIcap,b RIcap,c Isc Igen.n
3080.9 3080.9 3080.9 2.016 2.016 2.016 0.000 0.000
3087.1 3082.3 3079.6 0.123 3.416 3.538 8.137 448.6
8.428 3.588 3.588 0.000 3.589 3.589 8.428 464.6
17,148 17,145 3.126 0.683 0.000 3.127 17,149 244.6
Igen,a Igen,b Igen,c RIcap,a RIcap,b RIcap,c Isc Igen,n
3080.9 3080.9 3080.9 2.016 2.016 2.016 0.000 0.000
3081.4 3081.4 3081.4 0.123 3.416 3.538 6.003 0.000
6.217 3.589 3.589 0.000 3.589 3.589 6.217 0.000
D.R.R. Penido et al. / Electrical Power and Energy Systems 75 (2016) 215–225 Table 5 IEEE13 results. Node
IA (kA)
IB (kA)
IC (kA)
Diff (%)
SUB 692 680 675 671 650 634 633 632
13.70 3.35 2.91 3.07 3.35 8.42 15.27 4.15 4.80
13.70 3.27 2.84 3.04 3.27 8.42 15.13 4.02 4.70
13.70 2.96 2.55 2.75 2.96 8.42 14.72 3.80 4.39
0 13.17 14.11 11.63 13.17 0 3.73 9.21 9.34
(15 MVA). This system is grounded by a step-down transformer and a small value resistor in the generator neutral. The load connected at busbar 3 represents an equivalent system load. The parasitic capacitances of the equipment are given in Fig. 14 and were connected at busbar 1 to simplify presentation of the results. Two types of grounding were simulated in this case: (i) high-resistance neutral grounding and (ii) an ungrounded system. High-resistance grounding with neutral resistor Four cases were simulated. In the first case, faults were not inserted; in the second case, a single-phase fault was applied in
223
node 2.a; in the third case, a single-phase fault was applied in node 2.a but the load was neglected; and in the fourth case, two simultaneous single-phase faults were applied, one in node 1.b and the other in node 2.a. The results are shown in Table 1. The generator terminal voltages are balanced in the first case and a neutral shift and overvoltage of 165% in phase b and 170% in phase c of Cases 2 and 3 can be noted, as expected in a singlephase fault in a high-resistance ground [28]. The currents in some of the equipment are shown in Table 2. The generator currents presented small magnitude variation between Cases 1 and 2, and in this case, the power delivery are not interrupted in this kind of fault [28].The current and voltage across the generator neutral resistor are usually used to identify faults in this system. The proposed tool allows simulating simultaneous faults in different locations, as done in Case 4. This case has a behavior similar to a phase-phase-ground fault. Ungrounded system The grounding of the generator of Fig. 14 has been removed in this analysis. Three cases were simulated in this modified system: In the first case, faults were not inserted; in the second case, a single-phase fault was applied in node 2.a; and in the third case,
Fig. 15. NEV test system.
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D.R.R. Penido et al. / Electrical Power and Energy Systems 75 (2016) 215–225 Table 6 NEV test system results. Cases – currents (A)
1.a 1.b 1.c 2.a 2.b 2.c 3.a 3.b 3.c 4.a 4.b 4.c Neutral Telecom1 Telecom2 Telecom3 Telecom4
Differences (%)
1
2
3
4
5
d1
d2
59 211 224 141 376 358 89 281 345 6816 486 651 2657 762 774 795 833
81 218 207 185 388 331 158 288 304 5741 489 563 1544 426 428 435 451
82 213 214 198 371 346 161 271 324 4968 534 628 1062 330 345 363 389
94 216 199 227 385 318 191 285 290 5059 482 539 1030 274 271 272 279
92 212 206 223 371 333 186 272 309 4593 531 587 714 222 232 244 262
37.2 3.3 7.6 31.2 0.2 7.5 77.5 0.3 11.8 15.8 0.6 13.5 41.9 44.0 44.7 45.3 45.9
13.5 2.8 7.5 22.7 4.6 8.8 21.0 6.3 11.7 25.0 10.7 16.5 117.6 91.9 84.2 77.8 72.2
Table 7 NEV test system – fault to neutral comparisons. Fig. 16. Pole 16 details.
Currents (A)
1.a 1.b 1.c 2.a 2.b 2.c 3.a 3.b 3.c 4.a 4.b 4.c
Fig. 17. NEV – fault current modules at node 16.4.a.
a single-phase fault was applied in node 2.a, but the load was neglected. The results are shown in Tables 3 and 4. The generator terminal voltages are balanced in the first case and a neutral shift and overvoltage of 173% in phases b and c of Cases 2 and 3 can be noted, as expected in a single-phase fault in an ungrounded system. It may be noted the fault currents circulate in the parasitic capacitances. IEEE13 test case Three-phase faults were simulated in all busbars of the IEEE13 test system. The loads are neglected. In Table 5, the proposed method was compared with a traditional symmetrical components application. It can be noticed that in some busbars there is a difference of up to 14% between the phases. This can be an unacceptable error depending on the objectives of the study and this is a very small system. This result shows the need for multiphase tools for the analysis of unbalanced systems. The proposed method was compared with references [11,13,15,24] and the results was the same in all methods. IEEE NEV test case The NEV (Neutral to Earth Voltages) test system (Fig. 15) was simulated to demonstrate the efficiency of the proposed methodol-
Proposed (F–N)
[11,13,15]
Differences (%)
59 211 224 141 376 358 89 281 345 6816 486 651
71 204 218 175 359 348 157 257 313 5878 513 670
20.3 3.1 2.6 24.1 4.4 2.8 76.4 8.4 9.3 13.7 5.5 2.9
ogy in the representation of complex systems [27]. The main objective of this section is to show the effects of simplifications commonly used in fault analysis. Five test cases based in the NEV were chosen to evaluate the effects of the simplifications and node 16.4.a (pole 16, circuit 4, phase a) was chosen for application of the phase-ground fault. The value of all fault resistances in this case was 108 X. Case 1: No simplification was made and the fault was applied between node 16.4.a and the neutral node (16.n) through a resistor (Ran), as shown in Fig. 16. Case 2: No simplification was made and the fault was applied between node 16.4.a and the ground through a resistor (Rag), as shown in Fig. 16. Case 3: The mutual impedances between circuits were not considered, but the mutual impedances between phases of the same circuit were considered. All poles were considered. Fault: node 16.4.a to ground, same as Case 2. Case 4: All mutual impedances were considered (phases and circuits). Only notable poles were represented (circled numbers). Fault: node 16.4.a to ground, same as Case 2. Case 5: Just the mutual impedances between phases of the same circuit were considered and only notable poles were represented. Fault: node 16.4.a to ground, same as Case 2. The fault current modules at node 16.4.a are presented in Fig. 17. Note a considerable difference in the fault currents
D.R.R. Penido et al. / Electrical Power and Energy Systems 75 (2016) 215–225 Table 8 NEV test system – fault to earth comparisons. Currents (A)
1.a 1.b 1.c 2.a 2.b 2.c 3.a 3.b 3.c 4.a 4.b 4.c
Proposed (F–E)
[11,13,15]
Differences (%)
81 218 207 185 388 331 158 288 304 5741 489 563
71 204 218 175 359 348 157 257 313 5878 513 670
12.3 6.4 5.3 5.4 7.4 5.1 0.6 10.7 2.9 2.4 4.9 19.0
between Case 1 and Case 2, of approximately 23%. This result shows that the Kron reduction (used to incorporate the effects of the neutral cables in the phase cables) can result in great errors in electrical systems that are not solid grounded and should be avoided. The resistance of 100 X is the value of the pole ground grid. It can be noted that result errors vary with the function of the used simplification, with the fault current differences between Cases 2 and 5 being approximately 26% (all faults are phaseground). This result again shows the importance of correct modeling of the network. The values of currents in the 13.2 kV feeder are shown in Table 6. The d1 column shows the percentage differences between Case 1 and Case 2 and presents many values greater than 40% in telecommunications and neutral cables and a difference of approximately 15% in phase a of circuit 4; these results can greatly impact studies of equipment sizing, electrical protection, and personal safety. The percentage differences between the highest and lowest values of Cases 2, 3, 4, and 5 are presented in column d2. The high values of discrepancies between simulations shows that simplifications can provide erroneous results and this is not a good approximation. In Table 7 is presented a comparison between the proposed method and some recent fault analysis methods. In this case was considered a fault between node 16.4. and the neutral node (16n). The [11,13,15] methods showed similar results among them. Although, can be noted considerable differences in the results when compared with proposed method, especially in the currents in branches 3.a and 4.a. In Table 8 is simulated a fault between phase and earth (node 16), the differences between the methodologies have decreased, but still notable. It should be noted that were made simplifications in the NEV system to tests in the methodologies [11,13,15] (Kron reduction and no mutual impedances between circuits). Conclusions This paper presents a methodology for fault analysis in multiphase systems with which it is possible represent any kind of fault, including simultaneous and internal faults. The methodology performs calculations directly in phase coordinates. The NSFM allows a detailed representation of the electrical systems in fault conditions, allowing the representation of various elements, e.g., neutral cables and grounding, which can significantly impact the results of the analysis of a fault, especially in unbalanced systems.
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Because of the great flexibility of the NSFM, it can be used for analyses from balanced electrical systems to unbalanced multiphase systems without modification of the proposed methodology. Therefore, the potential uses for NSFM are very broad.
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Contents lists available at ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
An enhanced tool for fault analysis in multiphase electrical systems Débora Rosana Ribeiro Penido, Leandro Ramos de Araujo ⇑, Márcio de Carvalho Filho Department of Electrical Engineering, Federal University of Juiz de Fora, Juiz de Fora, MG, Brazil
a r t i c l e
i n f o
Article history: Received 10 March 2014 Received in revised form 5 August 2015 Accepted 11 September 2015
Keywords: Component modeling Current-injection method Distribution systems Fault analysis Simultaneous fault Unbalanced operation
a b s t r a c t This paper describes a new methodology for the fault analysis of an n-conductor electrical system, in which the phase imbalances, neutral cables, groundings, and other inherent characteristics of distribution systems are considered. The proposed methodology, which is based on the current-injection method, allows faults to be represented in a simple way and may be used to analyze several fault types, including internal, series, and simultaneous faults. The results of several cases are presented to show the efficiency of the proposed method. Ó 2015 Elsevier Ltd. All rights reserved.
Introduction One of the main objectives in the distribution system analysis is the development of methodologies that are able to represent electrical systems in detail, considering their different topologies and the variety of equipment present in these systems, especially in the context of smart grids [1–7]. The application of advanced methodologies allows engineers to achieve better operating conditions and plan more effectively. Currently, several methodologies for fault analysis consider that electrical networks are symmetrical and balanced, and they model the equipment in a simplified manner, and, through the use of symmetrical components, it is possible represent various asymmetric faults [8–10]. It is noteworthy, however, that the method of utilizing symmetrical components often introduces simplifications in the analysis, making this approach not the most suitable for all cases. For example, the fault analyses in unbalanced systems introduce coupling between the sequence networks, causing the loss of the main advantage of the symmetrical components. This fact can be observed in single-phase, two-phase, or three-phase lines that are not perfectly transposed. Among the works that address studies of faults using symmetrical components, several improvements in the simplifications that are usually adopted have been proposed [9–11]. Several configurations are used in distribution systems, and the three-phase four-wire configuration with multiple neutral ⇑ Corresponding author. Tel./fax: +55 32 2102 3442. E-mail address: [email protected] (L.R. de Araujo). http://dx.doi.org/10.1016/j.ijepes.2015.09.005 0142-0615/Ó 2015 Elsevier Ltd. All rights reserved.
grounding is widely used. This configuration has a low installation cost, provides greater safety for people and equipment, and has a high sensitivity to faults [12]. It is emphasized that methodologies based on symmetrical components do not provide good results when used in this type of system. Also noteworthy is that the correct representation of the neutrals and groundings is essential for fault analysis in distribution systems. In recent years, many researchers have proposed methodologies for fault analysis focusing on unbalanced systems. Previous studies have proposed the representation in phase coordinates, but with some simplifications; for example, the neutral cable is not explicitly represented and its effects in the phases are incorporated by Kron reduction or simply ignored, despite the large use of neutral cables in distribution systems [13–15]. This kind of simplification may lead to incorrect results, especially in unbalanced systems [12]. Some authors have proposed specific methodologies for fault analysis in distribution systems, thereby improving some aspects of the analysis [16–25]. Despite significant advances in the representation of systems with faults [25], many points can be improved. Cited as examples are fault between different voltages points, detailed representation of equipment, representation of magnetic couplings between different circuits, internal faults, improved representation of loads in high impedance faults, and simultaneous faults. These points, if addressed, will allow more accurate analyses and avoid misleading results that may pose risks to both human and equipment safety. This paper proposes a method that allows the analysis of faults in any kind of multiphase electrical system, including internal,
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In Step F.3, if an internal fault is indicated in a transformer, the procedure given in Section ‘Transformers’ is used to represent the equipment in fault. In Step F.4, RLC (resistances, inductances, and/or capacitances) elements are inserted to represent the various types of faults, as shown in Section ‘RLC components’. The right-hand vector (f) and Jacobian matrix (J) are assembled in Steps F.5 and F.7. In Step F.6, a convergence test is performed, e = 106. In Step F.8, increments are calculated and the equation is presented in (1). The contributions to right-hand vector (f) is calculated according to the equations presented in Sectio n ‘Component models’. The contributions to the Jacobian matrix (J) are the first order derivative of f in relation to state variables (VRe and VIm). Derivative equations are not presented for space reasons.
series, and simultaneous faults. When compared with other methodologies of fault analysis, several advantages can be cited for the n-conductor simultaneous fault method (NSFM): i. It allows modeling of any electrical system in phase coordinates, including mutual impedances between phases of the same circuit or mutual impedance between circuits of the same or different voltage level. ii. It allows the representation of any type of transformer connection in direct form. iii. It allows the representation of faults in multiphase systems, e.g., single-phase, two-phase, and three-phase. iv. It allows the representation of any type of electrical system (distribution, transmission, or industrial), radial or meshed, and dispersed generation. v. It allows the explicit representation of groundings, ground wires, and neutral cables. Any type of grounding can be represented (isolated, high-impedance, low-impedance, solid-grounded, or any combination) including even safety groundings. vi. It allows the representation of simultaneous faults in a simple and intuitive form. vii. It allows the simulation of faults between different points in the system, e.g., the contact of a medium-voltage cable with a low-voltage cable in distribution systems. viii. It allows the simulation of internal faults in equipment, e.g., transformers.
ð1Þ
where DVRe and DVIm are the solution, that increment of phase to ground voltage, real and imaginary respectively. DIRe and DIIm are the net current injected (Section ‘Component models’). Jk,m is the partial first order derivative as shown in (2).
It is emphasized that all these points improve the results of fault analyses. It is noteworthy that all the advantages listed can be executed directly in NSFM without the need for additional calculations or simplifications, which are usually required in many current methodologies. This paper is organized as follows: The NSFM is developed in Section ‘n-Conductor simultaneous fault method’, numerical examples are provided in Section ‘Applications’, and conclusions are presented in Section ‘Conclusions’.
2 @ DI 6 Jk;m ¼ 4 @ DI
@ DIRe
3 k
k
@V Re
Imk
@ DIIm
k
k
@V Im
@V Re
k k
7 5
ð2Þ
In Step F.9, state variables are updated. In Step F.10, update load parameter. If |VLD| 6 vLim, a and b is set to 2 (constant impedance – Section ‘Loads’). If |VLD| > vLim, a and b is set to original value. VLD is the voltage in the load terminals and vLim is load minimum voltage.
n-Conductor simultaneous fault method The n-conductor simultaneous fault method (NSFM) was developed to be a general methodology that allows the analysis of any faulted electric power system.
Rek
@V Im
The results are shown in Step F.11.
Methodology
Component models
In the NSFM, the network and the equipment are modeled in phase coordinates and electrical quantities (p.u. quantities are not used). The equations of the electrical system are written as equations of current injection in rectangular coordinates. The proposed method is presented in Fig. 1, where z is the state variable and f are current-injection equations for all the system nodes. The state variables are phase-to-ground voltage and are represented in rectangular coordinates using the real and imaginary parts.
Transmission and distribution lines The transmission and distribution lines are modeled as a coupled n-phase p-equivalent lumped-parameter circuit and the equations presented in (3) and (4) are used to model a distribution line connected between the sets of nodes k and m [8].
2
Ikm;lin In Step F.0 of the proposed algorithm (Fig. 1), all electrical system and equipment data are loaded, including the location and type of fault that will be simulated. In Step F.1, the pre-fault conditions are calculated through a power flow program. In Step F.2, the electric system is updated and additional nodes are created to simulate the faults informed in Step F.1; for example, if a fault is indicated in the middle of a transmission line, one or more additional nodes are created and the line is split.
Z k1;m1
6 6Z 6 k2;m1 6 ¼6 . 6 . 6 . 4 Z kn;m1 2
Z k11
6 6 Z k21 6 þ6 6 .. 6 . 4 Z kn1
Z k1;m2
Z k1;mn
Z k2;m2 .. Z kn;m2 Z k12
7 Z k2;mn 7 7 7 7 7 7 5 Z kn;mn
Z k1n
Z k22 .. Z kn2
.
.
31 2
31 2
7 Z k2n 7 7 7 7 7 5 Z knn
V k1 V m1
3
7 6 6 V k2 V m2 7 7 6 7 6 7 6 . . 7 6 . 5 4 V kn V mn V k1
3
7 6 6 V k2 7 7 6 7 6 6 .. 7; 6. 7 5 4 V kn
ð3Þ
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217
Fig. 1. Algorithm of the NSFM.
2
Imk;lin
Z k1;m1 Z k1;m2 6Z 6 k2;m1 Z k2;m2 ¼6 .. 6 .. 4 . . Z kn;m1 Z kn;m2 2 Z m11 Z m12 6 Z m21 Z m22 6 þ6 . .. 4 .. . Z mn1 Z mn2
31 2 3 Z k1;mn V m1 V k1 7 Z k2;mn 7 6 V m2 V k2 7 7 7 6 7 .. 7 6 5 5 4 . V mn V kn Z kn;mn 3 2 3 Z m1n 1 V m1 6 7 Z m2n 7 7 6 V m2 7 7 6. 7: 5 4 .. 5 Z mnn V mn
ð4Þ
where Zkx,my are the self-impedances and mutual impedances between nodes x and y and phases k and m, and Zkxy and Zmxy are the shunt impedance of node k or m on phases x and y. The proposed methodology can also simulate faults in phases between lines of different circuits, considering the mutual impedances and even different voltage levels. Despite not being usually studied, this information can be useful in distribution systems to analyze the effects of a contact between conductors of medium voltage to low voltage or in studies of faults in transmission lines that share the same right-of-way. Loads In this methodology, all loads are specified as constant impedance. The current injection of a load connected between nodes k and m (Ik and Im) is shown in (5).
Ik ðzÞ ¼
jV km ja Pkm jjV km jb Q km ; V km
ð5Þ
Im ðzÞ ¼ Ik ðzÞ; where Pkm and Qkm are the active and reactive load connected between nodes k and m, respectively. Vkm is the voltage phasor between nodes k and m. a and b are the exponential load model.
Transformers The transformer model used in this work was enhanced to allow the simulation of internal faults. The fault can be between a turn and ground, between turns of the same windings, or between distinct windings, including between different voltage levels (primary and secondary). In the NSFM, the transformer model uses intermediate waypoints (wpis) in the windings to simulate internal faults, and these points are defined according to the type and location of internal fault that will be simulated. Fig. 2 presents an algorithm to construct the faulted transformer nodal admittance matrix to represent an internal fault. Detailed descriptions of the steps indicated in the algorithm in Fig. 2 are presented in the following. The transformer data are loaded in Step P.1. If the system under study is isolated or high-impedance grounded, the shunt and parasitic capacitances of the transformer should also be informed. These capacitances are inserted in the terminal nodes of the transformer after calculation of the transformer nodal admittance matrix. In Step P.2, the internal fault data are read. In Step P.3, the primitive impedance matrix is assembled. The procedure for inserting four wpi nodes for simulation of an internal fault is illustrated in Fig. 3. Fig. 3a presents a delta-wye transformer and Fig. 3b presents a detail of the windings highlighted in bold in Fig. 3a. Three types of faults are shown in Fig. 3b: (i) turn-ground (Rwg); (ii) between turns of the same winding (Rww); and (iii) between turns of different windings (Rwy). It is noteworthy that several other types of faults can be simulated using this model. The impedances of faults are placed externally by equipment-type RLC (discussed in Section ‘RLC components’). The primitive impedance matrix of the windings illustrated in Fig. 3 is shown in (6). Z1+,1a is the impedance between node
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Fig. 2. Algorithm to construct the Ybus of a generic transformer for internal faults.
In the fourth step (P.4), the nodal incidence matrix is built (Ainc). (7) represents the incidence matrix (Ainc) of the faulted winding in Fig. 3b where the rows represent the partial windings (divided by waypoints) and the columns represent the nodes of the electrical system (external and internal) in which the partial windings are connected.
ð7Þ
In Step P.5, the winding-voltage adjustment matrix is built (Wwv). This matrix is used to deal with the partial winding voltage, where DV is the voltage level between the transformer nodes (ex: V1a and V1b) and is a percentage of the nominal winding voltage. The winding voltage adjustment matrix of Fig. 3b is presented in (8).
Fig. 3. Details of the transformer windings.
V1+ and the wpi node V1a, the position of the V1a node is informed in Step P.2 as a percentage of the winding, i.e., Z1+,1a is a percentage of Z1. Z1a,1b; Z1b,1; Z2+,2a; Z2a,2b; and Z2b,2 follow the same assembly procedure. Zmag is the mutual impedance between the windings [26] and must be informed in P.2. Here: V1+, V1, V2+, and V2 are the external connection of the windings of the transformer. V1a, V1b, V2a, and V2b are the waypoint nodes. The number of waypoints created is a function of the type of fault that will be simulated. Z1 and Z2 are the total impedance of the windings. Rwg, Rww, and Rwy are examples of fault resistance.
2 6 6 6 6 Zprimitiv e ¼ 6 6 6 6 4
3
Z 1þ;1a
7 7 7 7 7 þ ½Zmag : 7 7 7 5
Z 1a;1b Z 1b;1 Z 2þ;2a Z 2a;2b Z 2b;2
ð6Þ
ð8Þ In Step P.6, the generic transformer nodal admittance matrix (Ytrf) is calculated as (9).
Ytrf ¼ AtInc: Wwv Z1 primitiv e Wwv AInc: :
ð9Þ
Generators and substations The generators and substations are modeled in NSFM as a voltage source behind the impedance, as shown in Fig. 4. The parameters of the voltage source are calculated for the pre-fault conditions, and then are kept fixed during the fault calculations: this is done because no voltage regulators or transformer tap actions are performed during the initial instants of the fault.
D.R.R. Penido et al. / Electrical Power and Energy Systems 75 (2016) 215–225
V s;i1 V s;i2 ¼ DV s ;
219
ð12Þ
where s = {a, b, c}. Ps,ger and Qs,ger are the active and reactive power of the voltage source, respectively, which are state variables. DVs is the pre-fault internal voltage of the sources. RLC components Many network components can be modeled with RLC. The RLC elements are also used to represent faults in this work. The equations of an RLC connected between two nodes (k and m) are shown in (13) and an RLC connected between node k and ground is indicated by (14).
Fig. 4. Generator or substation model.
Where Vai1, Vai2, Vbi1, Vbi2, Vci1, and Vci2 are the connection nodes of the voltage sources representing phases a, b, and c of the machines or substations. Vae1, Vbe1, and Vce1 are the external connection nodes of the voltage sources of machines or substations. zabc source is the impedance matrix of the machine or short-circuit impedance of the substation. DVa, DVb, and DVc are the internal voltages of the sources, calculated for pre-fault conditions. The current equations of the zabc source between nodes Vse1 and Vsi1 (Ie1 and Ii1) are shown in (10), the current contributions of the voltage sources are presented in (11), and the voltage specification equations in (12).
2
Z aa
6 Ie1 ðzÞ ¼ 4 Z ba
Z ab Z bb
Z ca Z cb Ii1 ðzÞ ¼ Ie1 ðzÞ Is;i1 ðzÞ ¼
Z ac
31 2
Vae1 Vai1
7 6 7 Z bc 5 4 V b e 1 V b i 1 5 ; Z cc Vc e1 Vc i1
jV s;i1 V s;i2 j2 Ps;ger jV s;i1 V s;i2 j2 Q s;ger ðV s;i1 V s;i2 Þ
Is;i2 ðzÞ ¼ Is;i1 ðzÞ
3
;
ð10Þ
ð11Þ
Ik ðzÞ ¼ ykm ðV k V m Þ; Im ðzÞ ¼ ykm ðV m V k Þ; Ik ðzÞ ¼ yk ðV k Þ;
ð13Þ ð14Þ
where Vk = VRe,k + jVIm,k and Vm = VRe,m + jVIm,m are the voltage phasors of nodes k and m. Fig. 5 shows several possibilities of RLC element connections. It is possible to represent many fault types, including simultaneous faults or phase-neutral faults. It is also possible to represent even complicated faults in a simple form (Fig. 6). Some of these faults are impossible to represent in many methodologies. It is also possible to represent all the fault types shown in Fig. 5 simultaneously with no extra computational effort or additional considerations. In addition, it is emphasized that the tool allows the explicit representation of the neutral wire; therefore, it can provide much more precise results in systems that are not solidly grounded. Groundings and parasitic capacitances Any existing grounding can be modeled in the proposed methodology without the need for approximations. RLC elements can be used in conjunction with other equipment, such as
Fig. 5. Some fault representations.
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percentage of transformer winding relative to the neutral point. Ideal windings have been considered in the simulations. Fault at 50% of high-voltage winding, connection-grounded wye–grounded wye To simulate an internal transformer fault, a wpi node was created at 50% of the high-voltage winding in the transformer located between three-phase busbars 2 and 3 (Fig. 7). The fault was represented by a negligible resistance (108 X).The load on busbar 4 has been neglected. The detailed results of the simulation are presented in Fig. 8. As seen in the figure, although the secondary circuit was open, there was a fault current flow in the high-voltage winding of the transformer, even with the low-voltage winding open. This occurred because voltage is applied to segment 2.a–2.f of the transformer, and this segment is magnetically coupled to segment 2.f–2.n (autotransformer effect). Thus, currents of equal and opposite magnitude were created in both segments.
Fig. 6. Medium to low voltage fault in distribution system.
transformers, to model high impedance grounding (for example, generators grounding transformers or zig–zag transformers). The parasitic capacitances of equipment like CTs, VTs, transformers, surge suppressors, and other equipment are modeled as RLCs. These capacitances should not be neglected for fault studies in ungrounded or high-impedance grounding systems.
Applications The IEEE test feeders [27] have been developed to provide validation of distribution systems analysis software and algorithms. Three of these systems have been chosen to demonstrate the modeling and some simulation capabilities of NSFM.
IEEE4 test system The one line diagram of IEEE4 is shown in Fig. 7. The bus 1 voltages VAN, VBN, and VCN have fixed pre-fault values 7200\0.0° V, 7200\120.0° V, and 7200\120.0° V, respectively. Also shown is a detailed representation of the transformer windings located between the three-phase busbars 2 and 3. The p% value is the
Fault in the high-voltage winding, grounded wye–grounded wye, variation in p% Based on the diagram shown in Fig. 7, the value of p% was varied from 100% (phase to ground fault) to 1% (neutral to ground fault). The load was considered in this case. Fig. 9 shows the voltages at the transformer terminals with variation of p%. It is noted that phases b and c present overvoltage for faults close to node 2.a. The currents of phases a, b, and c of circuit 1–2, the internal fault current (Int. Fault), the current in the grounding of the generator (Grd. Gen), and the current in the grounding of the transformer (Grd. Trf) are shown in Fig. 10. Note that: (i) the current in phase a and the current in the grounding of the generator reduce at the same rate, because the current in the grounding of the generator is the phase a fault current plus the load unbalanced current; (ii) the internal fault current is always greater than the phase current (phase a in this case), because this internal fault current is the sum of the current in phase plus the induced current in the other part of the winding of the transformer; and (iii) the current of the grounding transformer increases as the fault moves toward neutral, which occurs because of the magnetic coupling and the induced current. Fault in the high-voltage winding, delta–grounded wye, variation in p% The transformer connected between busbars 2 and 3 (Fig. 7) was turned into a delta-ground wye transformer, and the p% value was varied from 100% (phase a fault) to 0% (phase b fault). Fig. 11 shows the phase voltages at the transformer terminals. It is noted that phase a and b voltages vary in modulus as expected and phase c presents a small overvoltage close to the fault terminal. The variations in the fault currents are shown in Fig. 12. Note that (i) the internal fault current is the sum of the fault currents
Fig. 7. IEEE 4 test system.
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221
Fig. 8. Transformer internal fault.
Fig. 9. Voltages in grounded wye–grounded wye transformer internal fault.
Fig. 11. Voltages, delta–grounded wye transformer internal fault.
Fig. 10. Currents in grounded wye–grounded wye transformer internal fault. Fig. 12. Currents, delta–grounded wye transformer internal fault.
of phases a and b and (ii) the current in the grounding of the generator is equal to the current internal fault, because this is the only path for current return.
Fault between windings of different voltages, grounded wye–grounded wye Wpi nodes were created in the windings of high and low voltage to simulate an internal fault between windings of different voltages. These nodes were created in the middle (50%) of the windings and the load was neglected. The results of this simulation are presented in Fig. 13. As seen in the figure, fault current circulation occurred at the high-voltage winding of the transformer and the bottom of the
low-voltage winding: both effects are explained by the magnetic coupling between the windings (Section ‘Transformers’). A pure three-phase methodology or representation in sequence components does not include much detail, and resolution of this case becomes very complicated or impossible in this kind of system representation. Three-busbars test system The three-busbars system, shown in Fig. 14, was based on [28]. The area of high-resistance neutral grounding is the generator to the low-voltage winding of the power transformer (160 MVA) and to the high-voltage winding of the unit auxiliary transformer
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Fig. 13. Fault between windings.
Fig. 14. High-resistance neutral grounding.
Table 1 Voltages (p.u.), high-impedance grounding.
Table 3 Voltages (p.u.), Ungrounded.
Node
Case 1
Case 2
Case 3
Case 4
Node
Case 1
Case 2
Case 3
V1.a V1.b V1.c V1.n V2.a V2.b V2.c V3.a V3.b V3.c
0.973 0.973 0.973 0.000 0.965 0.965 0.965 0.961 0.961 0.961
0.059 1.649 1.707 0.965 0.000 1.672 1.672 0.961 0.961 0.961
0.000 1.732 1.732 1.000 0.000 1.732 1.732 1.000 1.000 1.000
0.329 0.000 1.509 0.526 0.000 0.000 1.509 0.000 0.871 0.871
V1.a V1.b V1.c V1.n V2.a V2.b V2.c V3.a V3.b V3.c
0.973 0.973 0.973 0.000 0.965 0.965 0.965 0.961 0.961 0.961
0.059 1.649 1.707 0.965 0.000 1.672 1.672 0.961 0.961 0.961
0.000 1.732 1.732 1.000 0.000 1.732 1.732 1.000 1.000 1.000
Table 2 Currents (A), high-impedance grounding.
Table 4 Currents (A), ungrounded.
Current
Case 1
Case 2
Case 3
Case 4
Current
Case 1
Case 2
Case 3
Iger,a Iger,b Iger,c RIcap,a RIcap,b RIcap,c Isc Igen.n
3080.9 3080.9 3080.9 2.016 2.016 2.016 0.000 0.000
3087.1 3082.3 3079.6 0.123 3.416 3.538 8.137 448.6
8.428 3.588 3.588 0.000 3.589 3.589 8.428 464.6
17,148 17,145 3.126 0.683 0.000 3.127 17,149 244.6
Igen,a Igen,b Igen,c RIcap,a RIcap,b RIcap,c Isc Igen,n
3080.9 3080.9 3080.9 2.016 2.016 2.016 0.000 0.000
3081.4 3081.4 3081.4 0.123 3.416 3.538 6.003 0.000
6.217 3.589 3.589 0.000 3.589 3.589 6.217 0.000
D.R.R. Penido et al. / Electrical Power and Energy Systems 75 (2016) 215–225 Table 5 IEEE13 results. Node
IA (kA)
IB (kA)
IC (kA)
Diff (%)
SUB 692 680 675 671 650 634 633 632
13.70 3.35 2.91 3.07 3.35 8.42 15.27 4.15 4.80
13.70 3.27 2.84 3.04 3.27 8.42 15.13 4.02 4.70
13.70 2.96 2.55 2.75 2.96 8.42 14.72 3.80 4.39
0 13.17 14.11 11.63 13.17 0 3.73 9.21 9.34
(15 MVA). This system is grounded by a step-down transformer and a small value resistor in the generator neutral. The load connected at busbar 3 represents an equivalent system load. The parasitic capacitances of the equipment are given in Fig. 14 and were connected at busbar 1 to simplify presentation of the results. Two types of grounding were simulated in this case: (i) high-resistance neutral grounding and (ii) an ungrounded system. High-resistance grounding with neutral resistor Four cases were simulated. In the first case, faults were not inserted; in the second case, a single-phase fault was applied in
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node 2.a; in the third case, a single-phase fault was applied in node 2.a but the load was neglected; and in the fourth case, two simultaneous single-phase faults were applied, one in node 1.b and the other in node 2.a. The results are shown in Table 1. The generator terminal voltages are balanced in the first case and a neutral shift and overvoltage of 165% in phase b and 170% in phase c of Cases 2 and 3 can be noted, as expected in a singlephase fault in a high-resistance ground [28]. The currents in some of the equipment are shown in Table 2. The generator currents presented small magnitude variation between Cases 1 and 2, and in this case, the power delivery are not interrupted in this kind of fault [28].The current and voltage across the generator neutral resistor are usually used to identify faults in this system. The proposed tool allows simulating simultaneous faults in different locations, as done in Case 4. This case has a behavior similar to a phase-phase-ground fault. Ungrounded system The grounding of the generator of Fig. 14 has been removed in this analysis. Three cases were simulated in this modified system: In the first case, faults were not inserted; in the second case, a single-phase fault was applied in node 2.a; and in the third case,
Fig. 15. NEV test system.
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D.R.R. Penido et al. / Electrical Power and Energy Systems 75 (2016) 215–225 Table 6 NEV test system results. Cases – currents (A)
1.a 1.b 1.c 2.a 2.b 2.c 3.a 3.b 3.c 4.a 4.b 4.c Neutral Telecom1 Telecom2 Telecom3 Telecom4
Differences (%)
1
2
3
4
5
d1
d2
59 211 224 141 376 358 89 281 345 6816 486 651 2657 762 774 795 833
81 218 207 185 388 331 158 288 304 5741 489 563 1544 426 428 435 451
82 213 214 198 371 346 161 271 324 4968 534 628 1062 330 345 363 389
94 216 199 227 385 318 191 285 290 5059 482 539 1030 274 271 272 279
92 212 206 223 371 333 186 272 309 4593 531 587 714 222 232 244 262
37.2 3.3 7.6 31.2 0.2 7.5 77.5 0.3 11.8 15.8 0.6 13.5 41.9 44.0 44.7 45.3 45.9
13.5 2.8 7.5 22.7 4.6 8.8 21.0 6.3 11.7 25.0 10.7 16.5 117.6 91.9 84.2 77.8 72.2
Table 7 NEV test system – fault to neutral comparisons. Fig. 16. Pole 16 details.
Currents (A)
1.a 1.b 1.c 2.a 2.b 2.c 3.a 3.b 3.c 4.a 4.b 4.c
Fig. 17. NEV – fault current modules at node 16.4.a.
a single-phase fault was applied in node 2.a, but the load was neglected. The results are shown in Tables 3 and 4. The generator terminal voltages are balanced in the first case and a neutral shift and overvoltage of 173% in phases b and c of Cases 2 and 3 can be noted, as expected in a single-phase fault in an ungrounded system. It may be noted the fault currents circulate in the parasitic capacitances. IEEE13 test case Three-phase faults were simulated in all busbars of the IEEE13 test system. The loads are neglected. In Table 5, the proposed method was compared with a traditional symmetrical components application. It can be noticed that in some busbars there is a difference of up to 14% between the phases. This can be an unacceptable error depending on the objectives of the study and this is a very small system. This result shows the need for multiphase tools for the analysis of unbalanced systems. The proposed method was compared with references [11,13,15,24] and the results was the same in all methods. IEEE NEV test case The NEV (Neutral to Earth Voltages) test system (Fig. 15) was simulated to demonstrate the efficiency of the proposed methodol-
Proposed (F–N)
[11,13,15]
Differences (%)
59 211 224 141 376 358 89 281 345 6816 486 651
71 204 218 175 359 348 157 257 313 5878 513 670
20.3 3.1 2.6 24.1 4.4 2.8 76.4 8.4 9.3 13.7 5.5 2.9
ogy in the representation of complex systems [27]. The main objective of this section is to show the effects of simplifications commonly used in fault analysis. Five test cases based in the NEV were chosen to evaluate the effects of the simplifications and node 16.4.a (pole 16, circuit 4, phase a) was chosen for application of the phase-ground fault. The value of all fault resistances in this case was 108 X. Case 1: No simplification was made and the fault was applied between node 16.4.a and the neutral node (16.n) through a resistor (Ran), as shown in Fig. 16. Case 2: No simplification was made and the fault was applied between node 16.4.a and the ground through a resistor (Rag), as shown in Fig. 16. Case 3: The mutual impedances between circuits were not considered, but the mutual impedances between phases of the same circuit were considered. All poles were considered. Fault: node 16.4.a to ground, same as Case 2. Case 4: All mutual impedances were considered (phases and circuits). Only notable poles were represented (circled numbers). Fault: node 16.4.a to ground, same as Case 2. Case 5: Just the mutual impedances between phases of the same circuit were considered and only notable poles were represented. Fault: node 16.4.a to ground, same as Case 2. The fault current modules at node 16.4.a are presented in Fig. 17. Note a considerable difference in the fault currents
D.R.R. Penido et al. / Electrical Power and Energy Systems 75 (2016) 215–225 Table 8 NEV test system – fault to earth comparisons. Currents (A)
1.a 1.b 1.c 2.a 2.b 2.c 3.a 3.b 3.c 4.a 4.b 4.c
Proposed (F–E)
[11,13,15]
Differences (%)
81 218 207 185 388 331 158 288 304 5741 489 563
71 204 218 175 359 348 157 257 313 5878 513 670
12.3 6.4 5.3 5.4 7.4 5.1 0.6 10.7 2.9 2.4 4.9 19.0
between Case 1 and Case 2, of approximately 23%. This result shows that the Kron reduction (used to incorporate the effects of the neutral cables in the phase cables) can result in great errors in electrical systems that are not solid grounded and should be avoided. The resistance of 100 X is the value of the pole ground grid. It can be noted that result errors vary with the function of the used simplification, with the fault current differences between Cases 2 and 5 being approximately 26% (all faults are phaseground). This result again shows the importance of correct modeling of the network. The values of currents in the 13.2 kV feeder are shown in Table 6. The d1 column shows the percentage differences between Case 1 and Case 2 and presents many values greater than 40% in telecommunications and neutral cables and a difference of approximately 15% in phase a of circuit 4; these results can greatly impact studies of equipment sizing, electrical protection, and personal safety. The percentage differences between the highest and lowest values of Cases 2, 3, 4, and 5 are presented in column d2. The high values of discrepancies between simulations shows that simplifications can provide erroneous results and this is not a good approximation. In Table 7 is presented a comparison between the proposed method and some recent fault analysis methods. In this case was considered a fault between node 16.4. and the neutral node (16n). The [11,13,15] methods showed similar results among them. Although, can be noted considerable differences in the results when compared with proposed method, especially in the currents in branches 3.a and 4.a. In Table 8 is simulated a fault between phase and earth (node 16), the differences between the methodologies have decreased, but still notable. It should be noted that were made simplifications in the NEV system to tests in the methodologies [11,13,15] (Kron reduction and no mutual impedances between circuits). Conclusions This paper presents a methodology for fault analysis in multiphase systems with which it is possible represent any kind of fault, including simultaneous and internal faults. The methodology performs calculations directly in phase coordinates. The NSFM allows a detailed representation of the electrical systems in fault conditions, allowing the representation of various elements, e.g., neutral cables and grounding, which can significantly impact the results of the analysis of a fault, especially in unbalanced systems.
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Because of the great flexibility of the NSFM, it can be used for analyses from balanced electrical systems to unbalanced multiphase systems without modification of the proposed methodology. Therefore, the potential uses for NSFM are very broad.
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