Analysis of frequency distribution of ground fault-current magnitude in transmission networks for electrical safety evaluation

Electric Power Systems Research 173 (2019) 100–111

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Analysis of frequency distribution of ground fault-current magnitude in transmission networks for electrical safety evaluation

T

M. Coppoa, F. Bignucoloa, R. Turria, H. Griffithsb, , N. Haridb, A. Haddadc ⁎

a

Department of Industrial Engineering, University of Padova, Padova 35131, Italy Advanced Power and Energy Center, ECE Department, KUST, Abu Dhabi, United Arab Emirates c Advanced High Voltage Engineering Research Centre, Cardiff University, Cardiff, UK b

ARTICLE INFO

ABSTRACT

Keywords: Ground Potential Rise Grounding Power system modeling Power system faults

Ground Potential Rise (GPR) evaluation under fault conditions is important for the evaluation of electrical safety risk to human beings who are present in and around electric installations. Current national and international standards recommend such evaluation is based on a set of ‘worst-case’ conditions that include a simplified fault current calculation procedure based on assuming a maximum value. Accordingly, this approach may lead to over-design of grounding systems in certain cases by overestimating individual risk. To address this and move towards a comprehensive probabilistic assessment of risk, a more detailed model of the electric network is required for fault current magnitude evaluation. This paper describes the application of the multi-conductor method to determine fault current distribution on transmission networks. First, a model of a portion of the UK transmission network is built and the fault current and its distribution among the phases, ground wire and ground is evaluated for three example fault positions. It is found that the position of the fault along the line, substation bus section status and proximity to generation affect greatly the current distribution. Then, a transmission model based on the national network is built, and the effects of system loading and generation on fault current magnitude were also considered. A complete frequency distribution of fault current magnitude is obtained and the results demonstrate the value of such description for the probabilistic assessment of human safety in grounding system design.

1. Introduction The evaluation of Ground Potential Rise (GPR) under fault conditions is necessary for the design of transmission substation grounding systems. National and international standards [1–3] define a set of deterministic conditions to provide safety for human beings in substations and provide tools to calculate GPR and associated touch and step hazard voltages. While this deterministic approach is straightforward and cost effective in most cases, sometimes it results in expensive or impractical safety mitigation measures; e.g. concerning existing installation extensions [4]. An alternative strategy outlined in Refs. [4,5] may employ quantitative risk assessment (QRA) to assess the individual risk of fatality (IR) and to contain it within the As Low As Reasonably Practicable (ALARP) region. It should be emphasized that the QRA approach would be used as the ‘second stage’ of a two-stage design approach and called upon only in such cases where there would be excessive and uneconomic expenditure from proposed safety risk mitigation [4]. A framework for the QRA approach is set out in the

⁎

landmark publications on risks to human safety [6,7] and is adopted in simplified form in standards for the design of high voltage installations [8,9]. Two CIGRE Technical Brochures, 694 [10] and 749 [11] outline how the QRA approach may be applied to the grounding of transmission towers and in the vicinity of HV installations, respectively. In Refs. [4,5], the authors presented a more detailed QRA assessment method for IR at transmission installations, based on detailed analysis of IECpublished safety data [12]. However, the risk assessment strategy in Ref. [4] focused on the case study of a fault condition only at the subject substation; while realistically, faults are more likely to happen outside on the system and particularly along the transmission lines. A study based on fault current recordings at a substation showed that average values of the ground fault current magnitudes tend to be relatively small compared with the maximum values, due to the variation of fault positions along lines [13]. In Ref. [14], the actual fault position along the lines is indicated, together with other aspects (e.g. actual generation scenario), as one of the factors influencing the fault current. These concepts were applied in

Corresponding author. E-mail address: [email protected] (H. Griffiths).

https://doi.org/10.1016/j.epsr.2019.03.024 Received 18 July 2018; Received in revised form 24 March 2019; Accepted 24 March 2019 0378-7796/ © 2019 Published by Elsevier B.V.

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Fig. 1. Schematic network representation with the multi-conductor approach.

the probabilistic short circuit algorithm through a Monte Carlo approach presented in Ref. [15], whereas fault position along the line and fault type are considered in the probabilistic approaches described in Refs. [16] and [17]. In this context, the present work focuses on the development of a detailed methodology for calculating the relative frequency distribution of ground current for an entire transmission network and accounting for a full annual load cycle with corresponding generation variation. Commercial software is available to determine ground fault current distribution [18]. Generally, the approach is focused on a single loading condition and modeling is oriented to a radially configured system connecting to a central substation [19,20] and not suited to carrying out the multiple studies required for this work. While EMTP software, usually applied to studying transient and dynamic power systems phenomena, may be employed to perform steady-state power frequency solutions of fault conditions in the phase domain, such platform neither lends itself easily to set up a complete national network, nor, perform multiple studies. Previous research has shown the importance of representing in detail the impedances associated with the ground-return path and the mutual couplings between each pair of conductors as discussed in Refs. [21–25] together with detailed representation of tower footing resistances which strongly influence the ground-return current magnitude [26–29]. In this paper, ground current computations are implemented in MATLAB and based on the multi-conductor approach described in Refs. [30–32]. This approach is based on the Carson–Clem equations for which application to the power frequency computations required for this work is sufficiently accurate [33–36]. First, the network modelling approach is described followed by the fault analysis methodology and validating studies with EMTP-RV. Then the method is applied to determine fault current distribution of a small generic multi-phase transmission grid. Finally, the method is applied to a complete national transmission network which enables the calculation of relative frequency distributions of ground fault current at selected nodes, based on a large number of multiple random location fault studies occurring over an annual load cycle. The results show a large skew in the frequency distribution towards lower current magnitudes and that the maximum (worst case) ground return fault current magnitude occurs relatively infrequently.

fault current calculation requirements, may introduce difficulties for the correct evaluation of the ground-return current during faults. To overcome this issue, a multi-phase approach is used in this work to allow the analysis of power systems including a generic number of conductors and uniquely representing the ground return path. This methodology was described in Ref. [30] and then developed and extended for application to transmission networks [31] and a summary of its main features are outlined below. Fig. 1 shows a schematic generic representation of the power system including the network, loads, generators and grounding. The network is represented by self and mutual admittances between system buses, which are represented as n-phase ports and formed from a primitive admittance and incidence matrix. Shunt elements and grounding impedances are then connected to these ports, completing the connections between phases and neutrals to ground. 2.1. Branch elements models Branch elements are represented by an n-phase lumped parameters circuit as depicted in Fig. 1, including the longitudinal impedance Z and transversal admittance Yt components defining the self and mutual couplings between the circuits and between each circuit and ground. These terms are written in matrix form and used to compose the branch element’s admittance matrix as shown in Eq. (1):

Z

1

+

Yt 2

Z

1

YBranch =

Z

1

Z

1

+

Yt 2

(1)

The YBranch matrix defined in (1) represents the relationship between currents (positive if entering) and voltages (with respect to a common zero-voltage reference) for the 2n ports of the branch element. The detailed representation of the power transmission lines is reported in Appendix A. To complete the network model, transformers are represented in the same reference frame as other branch elements, in order to define submatrices YTr for inclusion in the system matrix, as depicted in Fig. 1. Transformer representation in the phase-frame of reference can be found in the published literature for configurations including ‘open wye, open delta’ and ‘open delta, open delta’ configurations [37–39]. Different voltage levels in transmission networks are typically connected through low-impedance autotransformers which can be modeled with a wye–wye connection with grounded neutral on both sides. Power plants supplying the transmission grid are usually connected through a delta–wye step-up transformer with the generators connected to the delta side and the grounded neutral on the grid’s side. A generalization of transformer representation in the phase frame of

2. Network modeling approach Generally, in power system analysis, the network elements are represented by their sequence (positive, negative and zero) equivalent models, assuming the elements to be symmetric or approximated to a symmetric geometry. Such simplifications, while acceptable for many 101

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Fig. 4. Line-to-earth voltage magnitudes along the line.

3. Fault analysis methodology The network component models are combined in the system admittance matrix Y, which can be used to calculate the network’s steady state pre-fault condition from which the fault regime is defined and computed. 3.1. Pre-fault simulation

Fig. 2. Tower layout of the double-circuit line [41].

A pre-fault simulation is used to initialize the power flows in the grid, assuming a particular generation and load scenario. The correction-current-injection methodology mentioned in the previous section is used to compute the power flow in the multi-conductor network model, defining constant admittance terms for the shunt elements and calculating the current injection needed to adjust their power exchange according with the respective voltage dependency. 3.2. Short-circuit analysis Once the steady-state regime is computed as in (1), setting the prefault conditions of the power system, the fault condition is introduced by defining an additional admittance matrix YF in which only the fault admittances (green unfilled boxes in Fig. 1 representing any faulted conditions between phases and to earth) are stored, introducing self and mutual coupling. The fault current solution is obtained by directly solving (3).

Fig. 3. Currents on the faulted phase (1R), earthwire and ground.

reference has been proposed by the authors in Ref. [40] and it is used to model transformers in this work.

[I] = {[Y] + [YF]} [E]

The generators’ voltages and currents resulting from the pre-fault analysis are used to calculate the E1 voltage (as defined in Appendix B), which is assumed constant during the fault study. The sub-transient period is assumed and the sources’ reactances are set according to their sub-transient value xd”.

2.2. Sources and shunt elements models All shunt elements are represented as a parallel connection of a constant admittance and a current source injecting a correction-current term, as can be seen in Fig. 1. Voltage dependency can be included for these elements, according to the well-known ZIP model, with coefficients of a load model comprised of constant impedance Z, constant current I, and constant power P, as detailed in Ref. [30]. It should be noted that components represented as constant admittances (e.g. grounding impedances, fault conditions) may still be represented in the same way, by setting to zero the injected current. Referring to Fig. 1, red unfilled boxes represent load and generator admittances whereas green solid filled boxes represent grounded admittance elements. Once all the admittance terms are included in the system admittance matrix Y, the power flow solution is obtained by solving the wellknown Eq. (2):

[I] = [Y] [E]

(3)

3.3. MATLAB computer routine for fault current calculation validation The developed computer routine is validated using a small case study network including a comparison with results obtained using EMTP-RV [41]. The case study comprises a 400 kV double-circuit line with a single earth-wire equipped with 4-wire bundles having a length of 135.42 km. The tower layout of the line is depicted in Fig. 2, whereas data about the conductors can be found in Ref. [41]. The simulated condition is a phase-to-earth-wire occurring at mid-line. The results of the fault current distribution and line-to-earth voltage along the line are shown in Figs. 3 and 4, and it is shown that they perfectly overlap with the results reported in Ref. [41], based on a multi-conductor cell analysis approach (MCA). Table 1 reports line voltages computed at the fault location compared with the values published in Ref. [41], which include also those calculated using EMTP-RV. The relative error, εr, in the computations

(2)

In order to represent correctly the generators in the phase frame of reference and iteratively solve the power flow problem of Eq. (2), additional steps are required as detailed in Appendix B. 102

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Table 1 Results comparison between the adopted methodology and the one in Ref. [41], including validation with software EMTP-RV. Tower phase number

1 2 3

‘MCA’ [41] EMTP-RV [41] Adopted methodology in this paper εr (between 2 and 3)

1R (faulted) Current (kA)

1S Voltage (kV)

1T

2R

2S

2T

6.680 6.740 6.715 −0.37%

307.700 308.800 306.902 −0.62%

304.600 301.700 302.495 0.26%

140.400 141.400 142.395 0.70%

294.800 295.300 293.854 −0.49%

303.800 301.900 304.445 0.84%

from the model developed in this work with respect to EMTP-RV is also reported in Table 1, showing a good agreement (εr always lower than 1%). 4. Case study of a small network To illustrate the application of the computational technique outlined in Sections 2 and 3, a portion of the UK National Grid transmission network was modeled. This network, shown in Fig. 5, comprises 13 substations with two voltage levels, 400 kV and 275 kV, connected by double circuit overhead lines with a total length of ≈665 km and with two auto-transformers. The tower line geometry adopted in this study is shown in Fig. 6, with circuit phases arranged in the low reactance configuration and with a single ground wire. The lines were modeled exactly according to their tower type and conductors, which vary from line to line and even in sections within the same line. Such a level of detail included in the model can be inferred from the minimum and maximum distance values reported in Fig. 6. The data used to build this network model was obtained from Ref. [42], including information about the layout and line composition, along with historical data regarding the power demand and generation scenarios. Relatively long transmission lines depart from substations S12 and S1 and, in this small network study, the power contribution of the remainder of the grid to this portion of the network has been considered concentrated at substation S1 (considering the presence of a power station at S12). Short circuit analysis was conducted on the case study network using a steady-state study to set the system’s pre-fault conditions. The demand assumed for this case is 2708 MW (corresponding to the maximum load condition occurring in the winter season), giving an average substation demand of ≈160 MW and the highest value at substation S2 (730 MW). Three power plants are connected to substations S7, S9 and S12 as shown in Fig. 5, having a total generation capacity of about 3000 MW. For this study, the power generation associated with each plant is as

Fig. 6. Tower geometry with minimum and maximum distances.

follows: 900 MW for Gen 1500 MW for Gen 2 and 1260 MW for Gen 3. In this model, the remaining portion of the grid is represented as an equivalent voltage source connected to substation S1, with a short circuit power of 1000 MVA. A short circuit analysis has been conducted considering the transmission line between substations S5 and S11, highlighted in Fig. 5. Utilizing one of the flexible features of the developed code, this line has been discretized considering each of the spans in order to analyze the fault current distribution in the ground wire and the ground along the line, as depicted in Fig. 7a. Three studies were carried out, based on a phase-to-ground-wire fault on phase a of Circuit 1 of the line between substations S5 and S11 occurring at three different locations along the transmission line as shown in Fig. 7a: 1 Fault Section #1 (FS#1): located at 9.6 km from S5 (1/3rd of the way along the line); 2 Fault Section #2 (FS#2): located at 25.8 km from S5 (2/3rd of the way along line); 3 Fault Section #3 (FS#3): located at the receiving substation, S11.

Fig. 5. Case study network with faulted line highlighted with the dashed line. 103

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Fig. 7. Detail of the transmission line between substations S5 and S11 with fault locations (a) and a circuit diagram to illustrate current direction convention at a generic faulted node (b).

Fig. 8. Fault current distribution on the transmission line between substations S5 and S11 in Circuit 1 under the three study fault conditions: ‘1’ (a), ‘2’ (b) and ‘3’ (c).

location are identified with subscripts A-1 and B-1, and where If is the current flowing through the fault impedance (here assumed to be Rf = 0.1 Ω), Iew are the components flowing in the ground-wire and Igr is the ground current. In Fig. 8a–c the fault current distribution results are reported for each of the phases of Circuit 1, for the ground-wire and for the ground-return path.

In these studies, the grounding impedances are assumed purely resistive with values for the substations (RgSS) and towers (RgTW), respectively of 1 Ω and 10 Ω. Where data is available, individual measured tower footing resistance can be represented in the code developed for these simulations. In Fig. 7b, the incoming phasor currents on either side of the fault

Table 2 Phase currents on the two circuits in the sending end (A) and receiving end (B) in the three fault sections (FS#1, FS#2 and FS#3). Busbar circuit breakers at substations 11 and 5 either closed and open (C–O), open and open (O–O) or closed–closed (C–C). Switch status

Fault Section #1 IA

C–O

O–O

C–C

a b c a b c ew a b c a b c ew a b c a b c ew

Fault Section #2 IB

IA

Fault Section #3 IB

IA

mag [kA]

angle [°]

mag [kA]

angle [°]

mag [kA]

angle [°]

mag [kA]

angle [°]

mag [kA]

angle [°]

6.575 0.260 0.555 1.659 0.306 0.757 2.098 5.474 0.482 1.179 0.000 0.000 0.000 2.077 6.018 0.301 0.697 1.119 0.304 0.807 2.103

−112 −152 105 45 −152 97 51 −107 −154 86 0 0 0 55 −112 −133 107 33 −134 103 49

3.923 0.260 0.555 1.659 0.306 0.757 5.411 4.807 0.482 1.179 0.000 0.000 0.000 5.137 4.674 0.301 0.697 1.119 0.304 0.807 5.548

57 −152 105 45 −152 97 −116 50 −154 86 0 0 0 −119 56 −133 107 33 −134 103 −116

5.158 0.199 0.650 1.046 0.242 0.819 1.683 4.619 0.391 1.284 0.000 0.000 0.000 1.705 4.097 0.205 0.826 0.656 0.204 0.850 1.711

−111 −170 99 33 −167 95 44 −107 −167 85 0 0 0 49 −110 −144 101 −63 −146 101 41

5.302 0.199 0.650 1.046 0.242 0.819 5.843 5.866 0.391 1.284 0.000 0.000 0.000 5.686 6.984 0.205 0.826 0.656 0.204 0.850 6.273

58 −170 99 33 −167 95 −116 52 −167 85 0 0 0 −120 58 −144 101 −63 −146 101 −116

4.095 0.169 0.777 0.571 0.197 0.917 1.113 3.993 0.311 1.435 0.000 0.000 0.000 1.169 2.346 0.189 0.993 2.180 0.194 0.897 1.147

−111 150 93 −2 160 91 44 −106 164 82 0 0 0 51 −105 147 89 −104 144 93 38

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Table 3 Gross fault currents (kA) at the three fault points (FS#1, FS#2 and FS#3). Bus switch status S5–S11

FS #1

FS #2

FS #3

O–C O–O C–C

10.447 10.078 10.644

10.414 10.298 11.026

11.288 11.258 13.128

As can be seen from the results of the three studies shown in Fig. 8, different fault locations along the line result in significantly different values of currents flowing in the phase on which the fault is simulated, although the gross value of current at the point of fault is similar in magnitude for each of the three cases (around 11 kA). Of relevance to grounding analysis, it can be understood that the impedances along the circuit behind the fault point, which are different according to each study, affect greatly the ground wire and ground current magnitudes. In this particular network, the double circuit transmission line under consideration is connected between two substations, one of which (S11) is operated with a split busbar. This arrangement results in a particularly unbalanced condition, because the two circuits of the same line carry significantly different currents in terms of amplitude and phase displacement. Such an arrangement is not easily replicated with simplified standard approaches to fault current distribution analysis. With reference to Fig. 8a–c, it can be seen that the value of Igr at substation S5 varies between 1.2 kA and 2.5 kA, while at substation S11, it is between 1.4 kA and 4.6 kA. Hence, it is clear that the fault position will influence greatly the magnitude of Ground Potential Rise at a substation and this indicates that accounting for statistical variation in fault location would reduce significantly the typical GPR in comparison with the worst case scenario. An interesting result arising from this split busbar arrangement can be observed in Fig. 8b, where the ground-wire current reaches a higher value in the section beyond the fault location towards S11. Table 2 reports the current magnitudes and phase angles at the three fault sections FS#1, FS#2 and FS#3 while, in Table 3, the gross fault current in each instance is reported. Three scenarios are considered changing the status of the busbar switches at S5 and S11 to either Open (O) or Closed (C) positions. For example, in the table, C–O means that S5 bus-section switch is closed while S11 bus-section switch is open. The results in Table 2 are provided for each phase of the two circuits on the sending end (A) and receiving end (B) sections. As expected, the currents in the two adjacent sections of phase a of Circuit 1, either side of the fault point, tend to be opposite and the same goes for the ground wire currents. Since other substations in the vicinity of the fault are operated with a closed busbar switch, a proportion of the current carried by phase a of Circuit 1 circulates in phase a of Circuit 2. Considering fault position #2, this proportion is around 20%, as can be seen from Table 2. When both switches are open (O–O), the phase currents in the healthy circuit are zero. Comparing the results of the first scenario (S5-Closed, S11Open) with those in the third scenario (both switches closed), it can be deduced that the distribution of currents on the phases, and therefore on the ground-wire is significantly different, due to the different circuit topology. From the results reported in Table 2 and according to the direction conventions depicted in Fig. 7b, it can be seen that the phase current phasors at the fault section tend to have opposite directions at the two ends A and B only in one of the circuits, resulting in different phasor combinations on the two sides of the fault. This is a consequence of different impedances between the fault location and each source, influencing the magnitude of the current flowing through the groundwire on each side of the fault point. This variation in current magnitude either side of the fault according to fault position is clearly visible comparing the results of fault conditions 1 and 2 in the O–C configuration. As could be seen in Fig. 8a, as the fault in position FS#1 occurs closer to substation S5 compared with substation S11, there is a lower series impedance in the section towards S5 compared with the other section (towards S11), resulting in a relatively higher current flowing

Fig. 9. Case study network of national supply system at 400 kV and 275 kV.

from the S5 section of the ground wire. With the third bus-connection scenario (C–C), there is a lower impedance between the fault section and the sources, leading to higher gross fault currents for all fault scenarios, as reported in Table 3. Similar observations were made about current distribution in the case of a simulated phase-to-ground fault; i.e. the ground return current assumes different values from one section to the other as a function of the ground return path impedance with respect to each source providing the fault current. It is evident that only a detailed representation of the network around a fault position can determine fault current path distribution accurately. 5. Case study of a full transmission network The developed computational code was applied to a model of the complete UK England and Wales transmission system. Modeling a National Grid system allows (i) a much higher level of detail in the ground-return path of the fault current, (ii) a more realistic network layout, (iii) an examination of random faults on transmission lines, (iv) modeling a full range of generation scenarios according to demand variation, and most importantly, (v) the calculation of the ground return current at every point of the system due to all considered faults. From such a comprehensive evaluation, the construction of the frequency distribution of the ground return current at any grounding point of the system is obtained. 5.1. System description and model assumptions Fig. 9 shows the layout of the England and Wales transmission network, which comprises 400 kV and 275 kV double circuit transmission lines (indicated in red and green respectively). The network is not intended to be fully accurate but rather an exemplar network of realistic size, topology and appropriate parameter values. In the figure, triangle markers indicate 62 generating station positions. In order to determine 105

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For the current study, the range was discretized into 10 bins, as shown in Fig. 11. A finer bin size could be used if required. According to the demand level associated with each bin and considering the above mentioned ranking order, the blue color map in Fig. 9 is used to indicate the order of appearance of each generator. 5.2. Accuracy and availability of data The accuracy and availability of data used in probabilistic-based or quantitative risk analysis is important for the correct prediction of, in this case, individual risk. However, the issue about required accuracy also applies to the estimation of the worst case condition which, in itself, is one condition in the set of probabilistic scenarios. In all areas of electrical engineering computer simulation, judgement must be applied about the required degree of detail in the circuit model and also the accuracy of the data used. However, it is acknowledged that assessing the probability distribution function of touch voltage is subject to uncertainty of data (knowledge uncertainty) in some of the key parameters (e.g. presence probability and activity statistics of workers at electrical installations). Where such uncertainty exists in probabilistic description, the so-called precautionary principle should be applied [5]. The precautionary principle helps to mitigate against complacency about low risk events where the ‘absence of evidence of risk’ is sometimes taken as ‘evidence of absence of risk’. For the case study of the full transmission network used in this paper, some uncertainty in generation ranking order or generation availability over the annual load cycle may be introduced. For example, if the subject substation is close to a generator whose presence is very uncertain, the precautionary principle could be applied by considering the condition that would result in higher fault current levels. A detailed sensitivity analysis taking into account parameter uncertainty and its effect on overall individual risk is the subject of a separate investigation.

Fig. 10. Half-hourly time series of system demand.

Fig. 11. Probability distribution of system demand (10 bins).

a particular system demand level which generators are connected to the system, the generators were assigned an approximate ‘ranking order’ based on energy source and then, of those with the same energy source, by year of commissioning, with the lighter shade of blue indicating a higher ranking order. The assumed ranking list is: wind, nuclear, biomass, combined heat and power (CHP), combined cycle gas turbine (CCGT), coal, coal/oil, open cycle gas turbine (OCGT), and pumped storage. For the current studies, the stochastic nature of wind generation is not accounted for. For a given system demand, the total generation amount is adjusted accordingly. The probability distribution of system demand was obtained from a typical annual data set of half-hourly demand readings, with total demand varying from 21 GW to 61 GW, as shown in Fig. 10.

5.3. Computations 10,000 steady state system computations were carried out with (i) random fault location at substations and line sections and (ii) a random fault time of occurrence in the annual load cycle, taking into account the range of system loading, associated generation deployment and network configuration. The fault occurrence rate is assumed to be uniform per km and the fault rate for each is weighted according to its line length. Three example substation locations, indicated as s/s #1, s/s #2 and s/s #3 and shown in Fig. 6 have been selected to demonstrate

Fig. 12. Ground return current distribution associated to faults in substations #1, #2 and #3. For each case, the gross fault and ground return currents are reported for the respective substation. 106

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Fig. 14. Fault cases in the vicinity of s/s #1.

constructed for all transmission substations or indeed any fault point on the system including any transmission tower. Fig. 13a–c reports the relative frequency distribution plots for the three example substations. From Fig. 13, it can be seen that, for the majority of fault occurrences, the ground return current at the substation is much lower than the maximum value. These distributions are heavily skewed towards low current magnitudes that are unlikely to cause electrocution hazards. Compared to maximum values, the average ground return current is lower at least by one order of magnitude and, in some cases e.g. in the case of s/s #1, up to three orders of magnitude. It can be seen that the shape of the distribution varies according to substation location, and this reinforces the need to use a full network model for accurate assessment of ground return current distribution. An example, taking data from Fig. 13a for Substation #1 can serve to illustrate the link between the relative frequency distribution and fault rate. With reference to Fig. 10a, if the four event clusters above 1000 A are taken as significant GPR contributors, the cumulative relative frequency distribution of these events is 4 × 0.0002 = 0.0008. It can be assumed that the overall fault rate for the UK transmission network is approximately 200 faults per annum [43]. Then, the cumulative probability of significant GPR events affecting Substation #1 would be 4 × 0.0002 × 200 = 0.16 faults per annum. Using the National Grid in-house empirical formula [44] based on average fault rate data from the system which includes an assumption about the extent of the network producing a significant GPR and assuming 3 lines and 6 switchgear bays, the 'substation fault rate' would be 0.135 faults per annum which is similar to that calculated from Fig. 13a. It is also interesting to note how the ground return current may be lower in the substations directly connected to the faulted one rather than in others, depending on where generators are connected (see for instance Fig. 12a). To investigate further the effects of fault location and different demand scenarios, a detailed analysis has been performed considering faults around s/s #1 as depicted in Fig. 14. Three double-circuit lines connect s/s #1 to the rest of the grid and, for each line, different fault locations are assumed to occur half-way along and very close to each end. In addition and for each fault location, three demand scenarios are considered (1, 5 and 10). Table 4 presents the results of gross fault

Fig. 13. Relative frequency distribution of ground return current at selected substations.

the frequency distribution of the ground return current arising from the 10,000 simulations. However, before presenting the frequency distributions of fault current at the three substations, it is important to note the spatial distribution of fault current across the system for any given single fault location. Accordingly, Fig. 12a–c shows a ‘snapshot’ of such distribution of ground return current in the vicinity of a single fault at s/s#1–3, respectively. From this figure, it can be seen that the ground return current magnitude is still significantly high at substations in the vicinity of the fault point. For example, in Fig. 12c, the ground return current is 4 kA at the fault substation, while two substations away the ground return current magnitude is 1.7 kA. Collecting this spatial distribution data for all 10,000 fault simulations enables the relative frequency distribution plots of ground return current to be 107

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Table 4 Detail of the gross fault current and ground-return current for faults occurring along the lines with growing distance from s/s#1. Demand scenario

Distance from s/s #1

1 1 1 5 5 5 10 10 10

0% 50% 100% 0% 50% 100% 0% 50% 100%

Gross fault current [kA]

Ground return current [kA]

L1

L2

L3

L1

L2

L3

16.215 10.812 13.103 22.878 14.298 17.194 34.343 18.688 22.476

16.215 16.249 20.401 22.878 22.258 27.796 34.343 35.562 55.757

16.215 12.201 11.775 22.878 17.856 18.750 34.343 23.756 24.046

5.862 0.101 0.048 8.246 0.132 0.085 12.407 0.172 0.072

5.862 0.589 0.058 8.246 0.847 0.117 12.407 1.268 0.071

5.862 0.316 0.229 8.246 0.398 0.274 12.407 0.567 0.378

current and ground return current for the selected scenarios and noting that s/s #1 is surrounded by generators with different ranking orders. Considering faults occurring at s/s #1 (i.e. at the beginning of each line) for the different scenarios, it could be seen that there is a significant difference in gross fault current between loading Scenarios 1 and 5 (40%) and between Scenarios 5 and 10 (50%). The effect of loading scenario (influencing generation connection) is much less in the case of faults occurring towards the end of each line, as for these faults, there is a relatively higher network impedance in the fault path. The highest gross fault current, 55.8 kA, is obtained in the case of a fault at the end of Line L2 in Scenario 10 (i.e. maximum demand). Table 4 illustrates not only how the gross fault current may significantly vary depending on the fault location, but also how ground return currents at a specific substation may not be proportionally related to the gross fault current. For example, in the case of a fault at the end of line L2, the ground return current at s/s #1 is of similar magnitude as that obtained in the case of a fault occurring at the end of line L1, even when the gross fault currents differ by 250%. From these selected samples of results, it can be seen that not only system loading (and corresponding generation) has a large influence on current magnitude but also the position of the fault is highly influential on the ground return current.

to model easily split substation busbars, an aspect of system operation that influences considerably fault current distribution. In addition, by using a probability distribution of system demand, based on an annual time series, generation and system topology can be described statistically and their effect on fault current can be also captured in detail. From the presented case study on the reduced network, it is demonstrated that a more detailed fault current distribution calculation is required to account for the effect of fault location along the transmission line and the proximity of the selected substations to other substations and power stations. Application of the computation procedure to a full transmission network demonstrated that the calculated relative frequency distributions of ground return current are highly skewed towards low current magnitudes. This means that, in the majority of fault scenarios, there will be a very low probability of high GPR and high associated touch voltages. The studies also confirm that large areas around the fault point are affected and that a complete system model is required in order to describe accurately the probability distribution. Furthermore, relative differences in fault position in the vicinity of a selected substation give large differences in ground return current and this supports the need for a large number of random studies also to describe accurately the probability distribution. The evaluation of GPR in the current standards based on maximum fault current level may, therefore, be said to describe a highly unlikely event, and this work makes a contribution towards the more accurate calculation of individual risk of electrocution to workers and members of the public in and around electricity installations. The probabilistic approach may then be employed as a tool to quantify individual risk in cases where there would be excessive and uneconomic expenditure from proposed safety risk mitigation. Future work will be focused on studying, through sensitivity analysis of parameters such as generation ranking order, grounding impedance values etc., the impact of different ground current frequency distributions on the overall assessment of individual risk of electrocution (of which fault current is one component).

6. Conclusions In this paper, a previously developed multi-conductor based network analysis procedure and code has been adapted and developed for application to a transmission network and accounting for the detailed calculation of ground return current components. This methodology now allows to represent a complete national supply system in detail, without any limitation in the number of nodes or conductors being considered. Furthermore, the procedure is able to perform multiple studies allowing the calculation of the frequency distribution of ground return fault current at any substation or fault point in the system. A particularly useful feature of the developed code is that it is able Appendix A. Transmission line model

The lumped parameter elements of the Z and Yt sub-matrices are determined using the Carson–Clem formulae, which are a widely accepted truncated approximation deduced from the general Carson’s [34] and Pollaczek’s formulas [35], for the calculation of self and mutual impedances when the conductors’ separation is small, throughout the range of power and harmonic frequencies of interest in problems of interference [33]. It should be noted that although for this work the lumped-parameters model is considered sufficiently accurate (line lengths are usually lower than 100 km) a higher precision, if required, could be reached by using a distributed-parameters model as done, for instance, in EMTP-RV. According to the Carson–Clem formulation, the resistive and inductive effects of the ground-return path (i.e. the path followed by a current flowing back through the ground) can be included in the model by adding a resistive term (Re, depending on frequency) and self and mutualinductances (depending on frequency and the soil resistivity and required in the evaluation of the distance De). The self and mutual impedances shown in Fig. A1a are calculated using (A1) and (A2), given two circuits i and j:

Zii = Ri + R e + j 2 10

4

ln

De ri

(A1)

km

108

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Fig. A1. Lumped parameter representation of a double circuit transmission line with a single ground wire.

Zij = Re + j 2 10

4

ln

De dij

km

(A2)

where, Ri: DC resistance [Ω/km]; ri: phase conductor radius [m]; dij: mutual distance between conductors i and j [m] De: depth of the ground return path [m] calculated as:

De = 659

f

[m]

(A3)

As indicated in Ref. [33], Carson–Clem’s formulae can be considered a good approximation until the mutual distance between the conductors is below 13.5% of De. The capacitive couplings between the conductors in the n-phase system may be evaluated by inverting the matrix built by the well-known Maxwell potential coefficients (A4) and (A5):

Pii =

Pij =

1 2

ln 0

1 2

ln 0

2hi ri

km F

(A4)

Dij

km F

(A5)

dij

As highlighted in Fig. A1b, the multi-conductor approach is advantageous especially when dealing with double circuit lines, because it accounts for the mutual couplings between conductors of the same circuit and also between those of adjacent circuits. This feature is particularly useful when the two circuits of the same line are connected to different busbars, e.g. substations operated with normally-opened bus sections. Furthermore, the modeling approach can accommodate diverse tower and line geometry (e.g. lines with multiple ground-wires). Appendix B. Source model Since power plants connected to transmission grids are generally controlled by automatic voltage regulators, the generators busbars are modeled as constant power, constant voltage magnitude (PV) buses from the power flow standpoint (i.e. the pre-fault condition). In this work, since the methodology is based on a correction-current-injection approach, the adjustment of active and reactive power exchanged at PV buses is obtained through a suitable variation of the currents injected by generators. The source model in the phase frame of reference is depicted in Fig. A2 as three

Fig. A2. Source model in the phase frame of reference. 109

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ideal voltage sources, each in series with an impedance equal to the synchronous reactance. A generator operating in PV mode will maintain a fixed magnitude of voltage E2 by varying voltage E1. The power flow problem (1) may be written for generators and loads using (A6):

IG YGG YGL = YLG YLL IL

EG EL

(A6)

where subscript ‘G’ stands for the generator nodes (including the slack bus) and ‘L’ for load nodes. The controlled voltages E2 form the array EG, while EL can be derived by rewriting (A6) as:

EL =

(A7)

Y LL1YLGEG + Y LL1IL

leading to: (A8)

YLGY LL1YLG) EG + YGLY LL1 IL

IG = (YGG

Eq. (A8) may be rewritten separating slack (SL) and PV buses, allowing evaluation of the current injection at PV buses with IPV as a function of the corresponding voltage EPV:

ISL IPV

=

A B C D

E SL EPV

+ [YGLY LL1] [IL]

(A9)

from which the current injection at PV buses is obtained as: (A10)

IPV = C ESL + D EPV + Y GLY LL1 IL

Using (A10), it is possible to set individually the current injection at each port of generator busbars (i.e. each phase of the system). In this work, however, since the sources represent power plants connected to a transmission grid, each generator imposes a balanced voltage triplet. Once the IPV currents are computed, the resulting active powers need to be adjusted to comply with the controlled value Pobj. The active power variation ΔPPV introduced by the IPV currents can be defined as:

PPV = Pobj

Ek IPVk , with k = a, b, c

R

(A11)

k

where Ek is the phase voltage. The active power constraint can be fulfilled by adding suitable correction current terms calculated using (A12):

IP =

PPV E*

(A12)

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