10.2.1 Step, touch, mesh and transferred potentials

Electric shock may occur when a person touches an earthed structure during a short-circuit fault or walks within the vicinity of an earthing system during a fault or touches two separately earthed structures. Figure 10.1 illustrates a person in the vicinity of an ac substation subject to step, touch, mesh and transferred potentials. Step potential is the surface potential difference between the 2 ft of a person that are assumed to be 1 m apart. Touch potential is the potential difference between the hands and feet of a person when they are standing 1 m away from an earthed object they are touching with their hands. Mesh potential is the maximum touch potential to be found within a mesh of the substation earthing system. Transferred potential is a case of a touch potential in a remote location where the shock voltage may be approaching the full rise of earth potential of an earth electrode.

Figure 10.1 Illustration of persons subject to step, touch, mesh and transferred potentials

10.2.2 Electrical resistance of the human body

The human body presents a resistance (impedance) to the flow of ac power frequency current which decreases non-linearly with the applied touch voltage. Various parts of the human body such as skin, blood, muscles, tissues and joints, tend to oppose current flow. Broadly speaking, the human body presents an impedance that consists of two parts. The internal impedance which is mainly resistive, and the impedance of the skin at the point of contact which consists of a resistance and capacitance in parallel. Above about 200 V touch voltage, the dc resistance and ac impedance of the human body are generally equal.

Figure 10.2 shows a simplified representation of the resistances of the human body where the following equations can be written

Figure 10.2 Representation of the electrical resistances of the human body

     (10.1)

Measurements of the hand-to-hand and hand-to-foot body resistances suggest that the arm and leg resistances are related by R_Leg ≈ 0.6 × R_Arm. Thus,

     (10.2)

     (10.3)

and if both hands and both feet are in parallel

     (10.4)

and

     (10.5)

The above equations indicate that the effective resistance of interest is between extremities, e.g. from one hand to the other, from one hand to both feet or from one foot to the other. The resistance of internal body tissues is around 300 Ω excluding skin resistance. Including skin, the total resistance from hand to hand is typically between 600 and 3300 Ω. The total body resistance reduces with increasing touch voltage because of the considerable reduction in the skin resistance part due to damage or puncture. For example, for R_Hand-Hand = 1725 Ω at a 100 V touch voltage, R_Arm = 862.5 Ω, R_Leg = 517.5 Ω, R_Hand-Foot = 1380 Ω, R_{One hand-Both feet} = 1121 Ω, R_{Both hands-Both feet} = 690 Ω, R_{Both hands-Body centre} = 431 Ω and R_Foot-Foot = 1035 Ω. The most dangerous currents in terms of ventricular fibrillation of the heart (this is discussed in the next section) are those that flow from left hand to one or both feet. For other paths through the body, IEC 60479-1:2005 provide a rough estimation of the current magnitude required to present an equivalent danger of ventricular fibrillation. For example, for a left foot to a right foot path, the magnitude of the current required is 25 times higher. According to American IEEE Standard 80:2000, the total body resistance from one hand to both feet, hand to hand and foot to foot is generally taken equal to 1000 Ω.

10.2.3 Effects of ac current on the human body

The effects of an ac electric current passing through the vital parts of a human body depend on magnitude, frequency and duration of this current. The most common physiological effects of electric current on the human body, in order of increasing current magnitude, are perception, muscular contraction, unconsciousness, ventricular fibrillation, respiratory nerve blockage and burning. Accordingly, a number of touch current thresholds are defined such as perception, reaction or muscular contraction, let-go and ventricular fibrillation. The threshold of reaction is usually taken as 0.5 mA independent of duration. The threshold of let-go current is the maximum touch current at which a person can let-go of the electrodes they are holding. This threshold is assumed in IEC 60479-1:2005 to be equal to 10 mA for adult males and 5 mA for all population of adult males, females and children. Ventricular fibrillation is a condition of incoordinate action of the main chambers (ventricles) of the heart resulting in immediate arrest of blood circulation. Humans are vulnerable to the effects of power frequency currents and values as low as tens of mA can be lethal. In comparison with power frequency current, it is generally accepted that the human body can tolerate a slightly higher current at 25 Hz and approximately five times larger current at zero frequency i.e. dc current. Fortunately, the human body may tolerate hundreds of amperes in the case of lightning surges because of their very short microseconds duration. The threshold of perception is generally taken as 1 mA. Currents from 0.5 to 5 mA, usually termed let-go currents, do not impair the ability of a person holding an energised object to control his muscles and release it. From 5 mA with a duration of 7 s up to 200 mA with a duration at 10 ms, currents are quite painful. They can result in strong involuntary muscular contraction, difficulty in breathing and immobilisation but do not usually cause organic damage and the effects are temporary and would disappear if the current is interrupted. It is generally not until a current magnitude of around 100 mA that ventricular fibrillation, stoppage of the heart or inhibition of respiration might occur and cause injury or death. In such cases, resuscitation does not work and the only known remedy is controlled electric shock. Industrial practice usually attempts to ensure that shock currents are kept below the ventricular fibrillation threshold in order to prevent death or injury. According to American IEEE Standard 80:2000 that provides guidance on safe earthing practices in ac substations, the non-fibrillating 50–60 Hz current is related to the energy absorbed by the human body and is given by

     (10.6)

where I_B is the tolerable body current and t_s is the duration of shock current in seconds. The constant 0.116 is based on a body weight of 50 kg and a probability of ventricular fibrillation of 0.5% in a large population of adult males. Equation (10.6) gives tolerable body currents of 367 and 116 mA for 0.1 and 1 s, respectively. Based on the permissible body current, American IEEE Standard 80:2000 provides criteria for the permissible touch and step voltages for a 50 kg person as follows:

     (10.7)

     (10.8)

where ρ_S is the resistivity of earth surface material in Ωm, I_B is the permissible shock current given in Equation (10.6), the number 1000 is the body resistance in Ω and C_S is a derating factor of the earth’s surface layer given as

     (10.9a)

and

     (10.9b)

and K = (ρ − ρ_S)/(ρ + ρ_S) is the reflection coefficient between the surface layer having a resistivity of ρ_S and the soil beneath having a resistivity of ρ, r, in m, is the equivalent radius of the foot that is assumed a conducting plate on the earth’s surface and usually taken equal to 0.08 m. h_S is the thickness of the surface layer in m.

10.3.1 Functions of substation earth electrode system

Substation earth electrode systems provide a means to carry and dissipate electric currents into earth under both normal and short-circuit conditions. Also, they are designed to provide a degree of safety for persons working or walking in the vicinity of earthed equipment so that they are not exposed to the danger of critical electric shock. In providing an earthing point for various equipment associated with the substation, the substation earth electrode system must fulfil the following requirements:

10.3.2 Equivalent resistance to remote earth

Buried hemisphere

The equivalent resistance to remote earth of a buried hemisphere, shown in Figure 10.3, is given by

Figure 10.3 Buried hemisphere earth electrode

     (10.10a)

where ρ_e is average earth resistivity in Ωm and r is hemisphere radius. The arrows represent the diffusion of current through the earth. For example, for ρ_e = 100 Ωm and r = 1 m, R_G ≈ 15.9 Ω.

If the sphere is buried at a depth of h in m which is large compared to twice its radius, the resistance to earth is given by

     (10.10b)

One driven vertical rod

Driven earth rods are generally solid and circular and are effective in small area substations or where low soil resistivity strata, into which the rod can penetrate, lies beneath a layer of high soil resistivity. The resistance to earth of a single isolated earth rod driven vertically into earth, shown in Figure 10.4, is given by

Figure 10.4 An isolated driven vertical rod earth electrode

     (10.11)

where L and r are the length and radius of the rod in m, respectively. For example, for a driven rod in an 11 kV substation with L = 3 m, r = 0.01 m and ρ_e = 100 Ωm,

Where the vertical electrode is surrounded by an infill material such as semi-conductive concrete or cement, the earth resistance is given approximately by

     (10.12)

where ρ_i is resistivity of infill material in Ωm, r_i is radius of shell of infill material in m and r_rod is the radius of the rod in m. The resistivity of concrete infill material lie in the range of 30–90 Ωm.

Multiple driven vertical rods in a hollow square

Figure 10.5 shows multiple driven vertical rods in parallel arranged in a hollow square with adjacent rods being equally spaced. The equivalent resistance to earth of the combined rods is given by

Figure 10.5 Multiple driven vertical rods in a hollow square earth electrode

     (10.13)

where R_G is the resistance of a single isolated rod given in Equation (10.11), N is the number of rods along each side of the square, d is the distance between adjacent rods in m and λ(N) is a factor that depends on the number of rods along each side of the hollow square. British Standard BS EN 7430:1998 Table 3 gives the value of λ as a function of N where X = 2.71 for N = 2, λ = 7.9 for N = 10 and λ = 9.4 for N = 20. For example, using the previous example rod data, N = 20 and d = 10 m, we obtain

Buried horizontal strip or wire

Horizontal strip or round wire conductor electrodes are very useful where high resistivity earth layer underlies shallow surface layers of low resistivity. The resistance to earth of a single strip or round wire conductor, shown in Figure 10.6, is given by

Figure 10.6 Buried horizontal strip or round wire earth electrode

     (10.14)

where k = 1 for a strip conductor and k = 1.3 for a round wire conductor, L is length of strip or wire in m, h is depth in m and w is width of strip or diameter of round wire in m. For example, for a round wire conductor with L = 50 m, h = 0.5 m, w = 0.02 m and ρ_e = 100 Ωm, we obtain

Buried vertical or horizontal flat plate

Vertical or horizontal flat plate electrodes have large surface area and are used in soils where it is difficult to drive rods or where soil resistivity is very high. The resistance to earth of a buried vertical or horizontal flat plate, shown in Figure 10.7, is given by

Figure 10.7 Buried horizontal or vertical flat plate earth electrode

     (10.15a)

where h is depth to plate centre in m, r is the radius of an equivalent circular plate in m and is given by

     (10.15b)

where A is area of plate in m².

For example, for a 1.1 m × 1.1 m plate buried at a depth of 1 m in a soil having

If the plate is placed on the surface of the earth, h = 0 and R_G = 50/(4 × 0.62) = 20.2 Ω.

Buried horizontal ring of wire

The resistance to earth of a buried horizontal ring, shown in Figure 10.8, is given by

Figure 10.8 Buried horizontal ring of wire earth electrode

     (10.16)

where r is radius of ring in m, h is depth in m and r_w is radius of wire in m. For example, for ρ_e = 100 Ωm, r = 0.5 m, h = 0.5 m and r_w = 0.01 m,

Buried horizontal grid or mesh

Figure 10.9 illustrates a horizontal grid or mesh system where R_G represents the equivalent resistance to remote earth and R_m represents the local conductor resistance within the mesh.

Figure 10.9 Buried horizontal grid or mesh earth electrode

The mesh is generally buried at a depth of 0.3−1 m. The resistance of the metal conductors forming the mesh is generally negligible compared to the resistance of the volume of earth in which the mesh is buried. The resistance to earth of a buried grid or mesh, is given by

     (10.17)

where in m and A is area of mesh in m². is sum of lengths of buried conductors. For example, for a large 400 kV/132 kV substation with a 171 m × 171 m square mesh that consists of 20 × 20 conductor rods at 9 m intervals buried in a soil having ρ_e = 100 Ωm, , = (20 × 171) × 2 = 6840 m and

Combined horizontal mesh with driven vertical rods around periphery

Using Figure 10.10, the total equivalent resistance of an earthing system that consists of a combination of horizontal mesh and vertical rods electrodes is given by

Figure 10.10 Combined horizontal mesh and driven vertical rods around periphery earth electrode

     (10.18)

where R_G(mesh) and R_G(N) are as given in Equations (10.17) and (10.13), respectively, R_Mut is a mutual resistance and is given by

     (10.19)

where L is length of driven vertical rod in m and w is width of strap in m. For example, from the previous mesh and multiple vertical rods, R_G(mesh) = 0.274 Ω and R_G(N) = 2.36 Ω. For w = 0.05 m, the mutual resistance is

Therefore, the total equivalent resistance is

In this example, R_Eq is practically equal to R_G(mesh). Thus, the buried horizontal mesh is by far the most effective earth electrode and the additional vertical rods have introduced negligible improvement.

10.4.1 Overhead line earth wire and towers earthing network

Most high voltage lines and generally all extra high voltage lines have one or more earth wires. The earth wire is usually bonded to the tower top and connected to the earth meshes of substations where the line terminates. Terminal towers of lattice steel structures and poles of metallic construction are usually bonded to the substation earth electrode system. The line’s earth wire, as seen from the substation earthing system, usefully serves to extend this earthing system beyond the substation due to the tower connections to earth via the tower footing resistances. Figure 10.11 illustrates a single-circuit overhead line with a single earth wire where Z_S is the self-series impedance of the earth wire per span with earth return. A span is the distance between two adjacent towers and Z_S can be calculated using Equation (3.19a). R_T is the tower earthing or footing resistance with the impedance of the tower itself being negligible at power frequency. For homogeneous earth resistivity ρ_e and with only earth wires present, i.e. no counterpoise conductor, the tower earthing or footing resistance R_T is given by

Figure 10.11 Illustration of overhead line earth wire and tower footing resistance network of a single-circuit overhead line

     (10.20)

where r is the radius of an equivalent hemispherical electrode having the same earth resistance as the tower footing. This radius is in the order of 1–2 m for smaller towers and may be 6–10 m for large towers with multiple legs. For example, for a typical 275 kV tower used in England and Wales having an equivalent radius of 2.8 m, the tower footing resistance is equal to 5.7 Ω, for a 100 Ωm earth resistivity.

10.4.2 Equivalent earthing network impedance of an infinite overhead line

In order to calculate the equivalent impedance of the combination of the earth wire and tower footing resistances, the line or number of spans is assumed infinite. Using Figure 10.11, the earth wire per span and tower footing resistance can be represented as a ladder network that consists of an infinite number of series–parallel elements as shown in Figure 10.12(a).

Figure 10.12 Equivalent earthing impedance of the infinitely long line shown in Figure 10.11: (a) ladder network representation of the earth wire and tower footing resistances and (b) derivation of equivalent impedance of a ladder network

Along the entire line length, the span lengths are assumed identical and the footing resistances of each tower are assumed equal. The tower at the fault location is not included in the equivalent circuit of the earth wire/tower footing. Assume that the driving point impedance of this ladder network as seen from the fault location is Z_∞. Since the line is assumed infinitely long, the addition of one more series–parallel element does not change Z_∞. This is illustrated in Figure 10.12(b) where

Solving this equation for Z_∞, the practical solution is given by

     (10.21)

It is interesting to note that if Z_S = R_T then Z_∞/R_T = 1.6180 a number known since medieval times and related to the Fibonacci numbers which were first published in the Liber Abaci in 1202 AD. Equation (10.21) shows that the driving point impedance of this ladder network depends on the earth wire characteristics, e.g. material and size, tower footing resistance and earth resistivity along the line route. For example, consider a 132 kV single-circuit overhead line with a single aluminium conductor steel reinforced (ACSR) earth wire that has three spans per km and a self-impedance of (0.18 + j0.72)Ω/km. The tower footing resistance R_T = 10 Ω. The driving point or ladder equivalent impedance is equal to

In practice, the line length beyond which the driving point impedance remains nearly constant is not infinite. This is discussed in the next section.

10.10.1 General

The current flowing in a single-core cable or the flow of unbalanced currents in the cores of a three-core cable induces voltages in the metallic sheath/armour of the cable. Under an earthed fault condition, and if the sheath/armour is earthed at each end, a current will circulate in the sheath/armour earth loop. This current is part of the earth fault current and the remaining part is the earth return current that returns through earth. The cable’s screening factor is dependent on the cable’s layout, earthing method, sheath/armour material, terminal earthing resistances and earth resistivity.

10.10.2 Single-phase cable with metallic sheath

Consider a single-phase cable with a metallic sheath but no armour having a length L in km. The cable is fed from a single-phase source at end 1 and subjected to a short-circuit fault at end 2 as shown in Figure 10.16(a). The cable core is denoted C and the sheath is denoted S and the latter is earthed at each end. The sheath terminal earthing resistances are denoted R₁ and R₂. Figure 10.16(b) shows the cable sheath earthing equivalent circuit. I_F, I_S and I_ER are the conductor fault current, sheath current and earth return current, respectively.

Figure 10.16 Calculation of screening factor for a single-phase cable with metallic sheath earthed at each end: (a) current distribution during a one-phase to earth short-circuit fault and (b) equivalent cable sheath circuit

The series voltage drop across the length of the conductor and sheath is given by

     (10.42)

From Equation (10.42) and Figure 10.16, we have

Therefore, the cable’s screening factor is given by

     (10.43)

The earth return current and sheath current are given by

     (10.44)

The sheath current can be expressed as follows:

     (10.45)

As in the case of an overhead line, the current returning through the cable sheath consists of a conductive component and an inductive component given by

     (10.46a)

     (10.46b)

I_i is the inductive component of the sheath current due to inductive coupling between the core conductor and the sheath through the mutual impedance Z_CS. I_C is the conductive component of the sheath current due to conduction between the cable’s earthing terminals 2 and 1.

Equation (10.43) of the cable’s screening factor k includes the effect of the terminal earthing resistances. However, it is sometimes required to represent the effect of the cable parameters only. Therefore, with the terminal resistances set to zero, the cable’s screening factor is given by

     (10.47)

Equation (10.47) can be rewritten using Equations (3.101a) and (3.103) as follows:

     (10.48)

where R_S(ac) is the sheath’s ac resistance and Z_SS is given in Equation (3.101a).

It is interesting to note that the self-impedance of the faulted core conductor Z_CC does not have any effect on the cable’s screening factor and hence on the current that returns through the sheath and also on the earth return current. Using I_F = 3I^Z, the earth return current and sheath current are given by

     (10.49a)

     (10.49b)

10.10.3 Three-phase cable with metallic sheaths

We presented in Chapter 3, a wide variety of three-phase cable circuit layouts and earthing arrangements. We will now illustrate the technique of calculating the cable’s screening factor for a general three-phase cable layout since the technique can be extended and applied to other layouts including armoured cables. Figure 10.17 illustrates a three-phase single-core cable having a length L in km and metallic sheaths, but no armour, and laid out in a flat symmetrical arrangement. At both ends of the cable, the sheaths are solidly bonded and connected to earth electrode where the terminal earthing resistances are denoted R₁ and R₂.

Figure 10.17 Calculation of screening factor for a three-phase single-core cable having a symmetrical flat arrangement and solidly bonded earthed sheaths: (a) current distribution during a one-phase to earth short circuit and (b) equivalent circuit of faulted cable conductor and sheaths

The cable is fed by a three-phase voltage source at end 1 and is open-circuited at end 2, the cable cores are denoted C1, C2 and C3, and the sheaths are denoted S1, S2 and S3. Under a single-phase to earth short-circuit fault on phase R, or conductor C1, at end 2, Figure 10.17(a) shows the resultant currents returning through the three sheaths and the current returning through earth. Figure 10.17(b) shows an equivalent circuit representation of the cable under this short-circuit fault condition. Using Equation (3.86), the series voltage drop across the cable, is given by

     (10.50a)

and

     (10.50b)

From Equation (10.50) and the symmetrical cable layout shown in Figure 10.17(a), we have Z_S1S1 = Z_S2S2 = Z_S3S3 and Z_S1S2 = Z_S2S3. Thus, we can write

     (10.51a)

and

     (10.51b)

After some algebra, it can be shown that the cable’s screening factor is given by

     (10.52)

Z_S1S1 is the self-impedance of sheath S1 with earth return, Z_S1S2 is the mutual impedance between sheaths S1 and S2 with earth return, and Z_S1S3 is the mutual impedance between sheaths S1 and S3 with earth return. Z_C1S1, Z_C1S2 and Z_C1S3 are the mutual impedances between the faulted phase conductor R and sheaths S1, S2 and S3, with earth return, respectively. All impedances are in Ω/km. The self-impedance Z_S1S1 can be calculated using Equation (3.101a) and all mutual impedances can be calculated using Equation (3.103).

For three-core cables and three single-core cables in a touching trefoil or equilateral layouts, the screening factor can be derived from Equation (10.52) by setting Thus,

     (10.53)

Substituting Equations (3.101a) and (3.103) into Equation (10.53), it can be shown that the cable’s screening factor is given by

     (10.54)

where R_S(ac) is the sheath ac power frequency resistance, f is the nominal power frequency, d is the distance between the centres of cable phases, r_is and r_os are sheath inner and outer radii, and D_erc is the depth of equivalent earth return conductor given in Equation (3.15).

Equations (10.52) and (10.53) of the cable’s screening factor k include the effect of the terminal earthing resistances R₁ and R₂, but if it is required to represent the effect of the cable parameters only, these resistances can be set to zero. However, the effect of the terminating resistances is important since they act to reduce the screening factor and hence the earth return current and increase the current that returns through the cable’s sheaths. Using I_F = 3I^Z, the earth return current and total sheath current are given by

     (10.55a)

     (10.55b)

As in the case of a single-phase cable, the sheath current I_S consists of an inductive and a conductive component but we leave this simple derivation for the reader.

It is to note that in the case of three single-core cables of symmetrical flat formation, two different screening factors can be obtained. One would correspond to a short-circuit fault on the cable conductor laid in the central position and the other for a short-circuit fault on one of the cable conductors occupying an outer position. In our analysis here, we have considered the latter case. The derivation of the screening factor for the former case is straightforward using the method we have presented. Further, the method can be used to derive screening factors for cables having other physical layouts as well as armoured cables.

In most practical cable installations in medium, high and extra high voltage systems, and unlike overhead lines, the majority of the earth fault current for the fault conditions presented in this section returns through the cable sheaths and the rest through the earth.

10.15.1 Background

The proximity of overhead power lines, underground cables or traction lines to adjacent structures that have metallic parts such as communication cables, fences, surface or underground pipelines can produce harmful voltages in these structures. Our focus in this book is on the voltages produced in metal pipelines by overhead power lines. Metal pipelines are usually formed of steel tubes that are welded together and used for the transportation of various substances such as crude oil, natural gases, water, liquefied petroleum gases and sewage. The length of pipelines may range from several kilometres to hundreds or even thousands of kilometres. Most pipelines are usually buried at low depth although some may be installed above ground. Since the soil is electrolytic, ac corrosion of buried pipelines may occur when ac current is exchanged between the pipe and the earth. Therefore, buried pipelines have a few millimetre thick coating that insulates the metal from the surrounding earth and provides the primary protection against corrosion. New coatings include polyethylene and epoxy but for old pipelines, bitumen and glass cloth were used. Cathodic protection systems provide additional protection against ac corrosion. The technique consists of applying a low dc voltage along the pipe that negatively polarizes the metal with respect to earth thereby helping to minimise electrochemical corrosion of the metal by the soil. The risk of ac corrosion of pipelines begins at a much lower ac pipeline voltage than that which endangers safety. Generally, the ac voltage between the pipeline and a reference electrode above the pipeline should be less than 10 V if earth resistivity is greater than 25 Ωm but should not exceed 4 V if earth resistivity is less than or equal to 25 Ωm.

Metal pipelines are conductors that are generally insulated from earth. In proximity to overhead power lines, pipelines may be exposed to electrical interference for part of their length and this causes voltages to appear on the pipeline. Many countries specify maximum permissible touch voltages to protect pipeline workers. Under permanent or steady state conditions, maximum permissible touch voltages tend to range from 15 to 65 V. Under short duration fault conditions, the range is generally from 300 to 1500 V depending on short-circuit fault clearance times. In some countries, the limits are reduced if the public has access to the pipelines.

There are three types of electrical interference from power lines to pipelines. These are generally termed electrostatic or capacitive, electromagnetic or inductive, and resistive or conductive couplings.

10.15.2 Electrostatic or capacitive coupling from power lines to pipelines

Buried pipelines in proximity to overhead lines are not exposed to capacitive coupling from the power line because the earth acts as an electrostatic shield. Only pipelines installed above earth are subject to capacitive coupling from the conductors of overhead lines as illustrated in Figure 10.21(a). Where the pipeline runs physically in parallel with the conductors of the power line, the parallel exposure of the pipeline and power line is termed a parallelism. This is illustrated in Figure 10.21(b).

Figure 10.21 Illustration of electrostatic or capacitive coupling interference from a power line to a pipeline: (a) capacitive coupling and (b) parallel exposure

The coupling occurs under both normal and faulted power system conditions. The coupling causes voltages to appear on the insulated pipeline metal with respect to earth or currents to circulate in an earthed pipeline through the earthing connection. In general, voltage problems caused by capacitive coupling can be easily solved by earthing the pipeline. The pipeline voltages for a given pipeline exposure with the power line can be calculated using matrix analysis techniques as presented in Chapter 3. The self and mutual potential coefficients of the power line conductors are given in Equations (3.2a). Equation (3.2b) can also be used to calculate the mutual potential coefficients between the pipeline and the power line’s conductors where these form a parallelism. The self-potential coefficient of a pipeline close to the earth is given by

     (10.67)

where h_p is the pipeline’s height above ground measured from the pipe’s centre and r_p is the pipeline’s radius, both are in m. Recalling Equation (3.1) and writing it in partitioned matrix form for a multi-conductor system that consists of power lines and pipelines, we can write

     (10.68)

where C, p and E represent the power lines’ phase conductors, pipelines and power lines’ earth wires, respectively. The equation is general and allows for the presence of more than one power line with more than one earth wire and more than one pipeline. All potential coefficients in Equation (10.68) are matrices. The earthed earth wires are now eliminated by substituting V_E = 0 in Equation (10.68), giving

     (10.69a)

where

     (10.69b)

The next step is to apply the pipelines’ earthing constraint to Equation (10.69a). For an insulated pipeline, Q_p = 0 and, from Equation (10.69a), the pipelines’ voltages to earth due to capacitive coupling with the power lines are given by

     (10.70a)

where V_C are the known phase voltages to earth of the power lines. If a person touches pipeline i whose voltage is V_p(i), the current that would flow through the person’s body is determined by the series combination of his contact resistance to earth and the pipeline’s capacitive reactance. In practice, the latter is much greater than the person’s resistance and therefore the discharge current is given by

     (10.70b)

where

     (10.70c)

and L_i is length of pipeline exposed to capacitive coupling in km. If the pipelines are solidly earthed or earthed through a very low impedance, V_p = 0 and, from Equation (10.69a), we obtain

     (10.71a)

Since the phasor equivalent of the current i = dq/dt is I = jωQ, the pipelines’ charging currents are given by

     (10.71b)

The discharge current through the body of a person that touches pipeline i is given by

     (10.71c)

If the pipeline or some of its sections are not parallel to the power line, the distance between the pipeline, or a section of the pipeline, and the various power line conductors is no longer constant. Two such situations are illustrated in Figure 10.22. In both cases, the non-parallel pipeline exposure can be converted to a parallelism where the pipeline is parallel to the power line and is at an equivalent distance from the power line given by

Figure 10.22 Conversion of non-parallel exposures to parallel exposures between a power line and a pipeline: (a) oblique exposure near power line and (b) oblique exposure crossing power line

     (10.72a)

and

     (10.72b)

where x_Eq is the geometric mean distance to the power line and x₁ and x₂ are the minimum and maximum distances of the pipeline to the power line. The constraint of Equation (10.72b) is applied in order to maintain sufficient accuracy in calculating the mutual parameters between the pipeline and the power line. This constraint effectively places a limit on the length of a non-parallel pipeline section which necessitates dividing the pipeline into a number of sections each of which are converted to a parallel section to the power line.

For an insulated pipeline having a number of sections of both parallel and non-parallel exposures, the total pipeline voltage to earth can be calculated as the mean of the voltages in each section weighted by its length to the pipeline’s total length as follows:

     (10.73)

This voltage can be used to calculate the current that would flow through the body of a person that touches or comes into contact with the pipeline. The matrix analysis technique presented above can be extended and applied to double-circuit power lines.

10.15.3 Electromagnetic or inductive coupling from power lines to pipelines

General

Since the earth does not act as an electromagnetic shield, both above ground and buried pipelines in proximity to overhead lines are exposed to inductive coupling from the power line. The coupling exists under both normal steady state and faulted power system conditions and induces longitudinal voltages or electromotive forces (EMFs) on the pipeline. These EMFs produce voltage stresses on the pipeline and can also cause currents to circulate in the pipeline. These voltages (to earth) can reach several tens of volts under steady state conditions and a few kilovolts under fault conditions. The latter can damage the pipeline’s insulation coating and cathodic protection systems. The longitudinal induced EMF depends on the distance between the power line and pipeline and the length of parallelism. The inductive zone of influence of a power line increases with the earth’s resistivity and is generally taken as where y is in metres. For example, for The inductive coupling mechanism and zone of influence are illustrated in Figure 10.23 where the sections of the pipeline that fall within the zone of influence are AB, BC and CD. Each pipeline section presents an oblique exposure that can be converted to a parallelism using Equation (10.72).

Figure 10.23 Illustration of electromagnetic or inductive coupling from power line to pipeline and power line’s inductive zone of influence

Analysis of longitudinal induced EMFs on the pipeline

Mutual impedance between power line and pipeline

The EMF induced on a pipeline can be calculated under either normal steady state or faulted power system operating conditions. In both cases, the mutual impedances between the pipeline and the relevant power line conductors are required. Figure 10.24(a) illustrates the mutual inductive coupling between a single-circuit overhead power line with one earth wire and a buried pipeline parallel to the power line within the zone of influence.

Figure 10.24 Calculation of induced EMF on a pipeline from a power line: (a) illustration of mutual coupling between pipeline and power line, (b) induced EMF under normal operation and (c) induced EMF under fault condition

The mutual impedance between the pipeline p and an overhead line phase or earth wire conductor j, with earth return, is calculated using Equation (3.20a) as follows:

     (10.74)

where d_pj is the distance between the centres of the pipeline and line conductor j, f is system frequency and D_erc is the depth of equivalent earth return given in Equation (3.15). If the distances between the pipeline and the power line’s conductors are greater than the distance given in Equation (3.20b), i.e. , Equation (10.74) results in an error that generally increases with distance particularly at low earth resistivity. The following equation gives an alternative formula for the mutual impedance and is valid for any distance between the pipeline and the power line:

     (10.75)

The use of Equation (10.75) may result in a safe upper overestimate in the magnitude of the mutual impedance that is always lower than 8% provided that Equation (10.72) is used.

Induced EMFs during steady state unfaulted system conditions

In Figure 10.24(a), we consider a general case of a single-circuit overhead line with one earth wire. However, the analysis approach is general and applicable for more than one earth wire, double-circuit lines and underground cables. The phase and earth wire currents are illustrated in Figure 10.24(b). Each current induces a voltage on the pipeline through the appropriate mutual impedance between the pipeline and the conductor. Therefore, the total longitudinal EMF induced on the pipeline due to the three-phase currents and the earth wire current is given by

     (10.76)

where the currents are in amps. The series voltage drop across the earth wire conductor is given by giving

Substituting I_E into Equation (10.76), we obtain

     (10.77)

Equation (10.77) can be used whether the phase currents I_R, I_Y and I_B are balanced or not. If the power line has no earth wire, the Z_pEI_E term in Equation (10.76) disappears and the induced EMF is given by

     (10.78)

Induced EMFs during a short-circuit fault in the power system

In many practical power systems, balanced three-phase short-circuit faults on high voltage power systems result in fault currents having a small degree of unbalance caused by the unbalanced network such as due to untransposed overhead lines. Therefore, the vector sum of the induced EMFs on an adjacent pipeline results in a small net induced EMF on the pipeline because the phase angle displacements between the three currents are not significantly different from 120°. The highest induced EMF on the pipeline will, therefore, occur under an unbalanced one-phase to earth short-circuit fault on the power system. The high ZPS current that flows on the power line induces a high EMF on a nearby pipeline depending on its proximity to the power line. Figure 10.24(c) illustrates a one-phase to earth short-circuit fault on phase conductor B, i.e. the nearest phase to the pipeline. The fault location is assumed at one extremity of the parallelism. For a fault within the length of the parallelism, the net EMF is calculated as the vector sum of the EMFs due to the opposing fault currents within the parallelism. From Figure 10.24(c), the series voltage drop across the earth wire is given by or The EMF induced on the pipeline is given by and using we obtain

     (10.79a)

where

     (10.79b)

and k is the screening factor. Equation (10.79b) is similar to Equation (10.40) except that the screening factor now includes the effect of both the earth wire and pipeline. Equation (10.79a) shows that since the fault current is constant within the parallelism, the longitudinal EMF per km induced on the pipeline will also be constant.

Analysis of pipeline voltages caused by inductive coupling

The calculation of pipeline voltages caused by the EMFs produced by inductive coupling from the power line, where the pipeline is within an overhead line’s zone of influence, requires the modelling of the pipeline and determination of its electrical characteristics. Like the conductors of an overhead transmission line, a metallic pipeline can be considered as a long lossy transmission conductor of known geometrical dimensions and physical characteristics. The magnetic coupling from the power line to the pipeline may be represented as distributed induced EMF sources on the pipeline. The equivalent circuit of a distributed pipeline section of length L in parallel with the conductors of an overhead power line is shown in Figure 10.25. E(x) is the induced EMF increment from the power line per km, z and y are the pipeline’s series impedance and shunt admittance per km and Z_A and Z_B are the impedances of the pipeline seen from ends A and B, respectively.

Figure 10.25 Equivalent circuit of a distributed metal pipeline inductively coupled to a power line: (a) pipeline section parallel to a power line and (b) pipeline distributed parameter equivalent circuit

From Figure 10.25(b), we can write

     (10.80a)

or

     (10.80b)

and

     (10.81a)

or

     (10.81b)

Differentiating Equations (10.80b) and (10.81b) with respect to x and using the original equations, we obtain

     (10.82a)

     (10.82b)

where

     (10.82c)

γ is the propagation constant of the lossy pipeline. Under normal unfaulted power system operation, the load currents influencing the parallel pipeline section are constant. The current will also be constant under a one-phase to earth fault condition on the power line beyond the section parallel to the pipeline. Under both conditions, the induced EMF is constant, i.e. E(x) = E_o. It can be shown that the solutions of Equations (10.82a) and (10.82b) are given by

     (10.83a)

     (10.83b)

where

     (10.83c)

Z_o is the characteristic impedance of the lossy pipeline. Equation (10.83a) can be used to calculate the pipeline voltage to earth along the pipeline section, i.e. at any value of x between zero and L and Equation (10.83b) can be used to calculate the pipeline current. The terminal impedances Z_A and Z_B are chosen to represent the pipeline’s electrical characteristics at ends A and B of the pipeline section. For a pipeline section that continues for a few kilometres beyond end A, the electrical continuity of the pipeline persists and the pipeline’s impedance seen from point A may be taken as equal to the pipeline’s characteristic impedance or Z_A = Z_o. If the pipeline is solidly earthed at end A then Z_A = 0 but if the pipeline has insulating joints and unearthed at end A, then Z_A = ∞. These characteristics apply equally to end B. The magnitude of the pipeline voltage will obviously vary from zero, where the pipeline is earthed to a maximum where the pipeline is insulated and unearthed. For the case where the pipeline parallel section is insulated and unearthed at end B but extends beyond end A, i.e. Z_A = Z_o and Z_B = ∞, Equations (10.83a) and (10.83b) simplify to the following:

     (10.84a)

     (10.84b)

The maximum pipeline voltage to earth occurs at end B, i.e. x = L and is given by

     (10.85a)

and the current at end B is I(x = L) = 0. At end A, i.e. x = 0, Equation (10.84a) gives

     (10.85b)

Beyond end A where the pipeline extends away perpendicular to the power line, the attenuation of the pipeline voltage can be evaluated to ensure that appropriate safety measures are taken to protect personnel working on the pipeline even many kilometres away. Using Equation (10.85b) that gives the pipeline voltage at end A, the pipeline voltage along a new axis a at a distance u beyond end A is given by

     (10.86)

Where the pipeline extends beyond end A and remains inductively coupled to the power line, this can be considered a new pipeline section having an oblique exposure and converted to a parallelism using Equation (10.72).

10.15.4 Resistive or conductive coupling from power systems to pipelines

In Sections (10.11)-(10.13), we discussed how ZPS earth return currents due to short-circuit earth faults in substations and on overhead line towers cause a rise of earth potential with respect to remote earth over a given area. If a buried pipeline is located in the zone of influence, i.e. earth potential rise, irrespective of whether the pipeline is parallel to the power line or not, the coating insulation will be exposed to a voltage stress since the pipeline’s metal remains at virtually earth potential. If this voltage stress is greater than the dielectric strength of the insulation coating, it may puncture the coating and damage cathodic protection systems. If the pipeline passes through the earth electrode systems of substations or overhead power line towers, the voltage stress may be so high that it may puncture the coating. The intense leakage current may damage the pipeline’s metal. If the pipeline is earthed and connected to the earth electrode of a substation or tower, the rise of earth potential of the substation or tower will be transferred to the pipeline and may endanger safety.

10.15.5 Examples

Example 10.3

Consider a single-circuit 400 kV overhead transmission line with one earth wire and an above ground insulated metal pipeline in the vicinity as shown in Figure 10.26.

Figure 10.26 Power line and above ground pipeline for calculation of capacitive coupling for Example 10.3

The geometrical and physical data of the overhead line circuit is identical to that used in Example 3.4 and system frequency is 50 Hz. The pipeline is parallel to the axis of the power line at a distance of 30 m, has an outer radius of 0.35 m and its height above ground is 1 m. The length of parallel exposure of the pipeline and power line is 4 km. The three-phase voltages on the power line are in a balanced steady state condition. Calculate the touch voltage to earth on the pipeline induced by capacitive coupling with the power line and the discharge current that can flow through a person’s body when touching the pipeline.

The pipeline’s self-potential coefficient and mutual potential coefficients with the power line are

The potential coefficient matrix of the power line and pipeline coupled system is

Thus

Therefore,

As discussed in Section 10.2.3, a steady state current higher than about 5 mA would generally be considered unacceptable from personnel safety viewpoint. Therefore, the pipeline in this example would generally be earthed through an appropriate resistance typically of the order of a few hundred ohms.

Example 10.4

Consider a single-circuit overhead power line and a parallel metal pipeline in the vicinity. The power line carries a balanced set of three-phase steady state currents. Derive an expression for the induced EMF on the pipeline in terms of the distances between the pipeline and the power line’s phase conductors. Consider a power line with and without one earth wire.

We start with case where the power line has no earth wire. The balanced three-phase currents are PPS only thus I_R = I^P, I_Y = h²I^P and I_B = hI^P. Therefore, from Equation (10.78), we have −EMF_p = (Z_pR + h²Z_pY + hZ_pB)I^P. Using Equation (10.74),

we have

or after a little algebra

where the mutual impedances are in Ω/km and I_R = I^P is in amperes.

If the line has one earth wire, using Equation (10.77), and by analogy with the above equation, we obtain

Example 10.5

Consider the single-circuit 400 kV overhead transmission line of Example 10.3 and a non-parallel buried coated steel pipeline in the vicinity as shown in Figure 10.27. The pipeline is insulated and unearthed at end B and extends perpendicular to the overhead power line at end A.

Figure 10.27 Power line and buried coated steel pipeline for calculation of inductive coupling for Example 10.5

The geometrical and physical data of the pipeline is as follows:

Outer radius = 0.25 m, depth = 1 m, resistivity and relative permeability of steel are 0.138 μΩm and 300, respectively. Insulation coating is polyethylene with thickness = 5 mm, resistivity = 25 × 10⁶ Ωm and relative permittivity = 5.

Earth resistivity is 100 Ωm. System frequency is 50 Hz. Balanced steady state currents on the power line of 2000 A rms. One-phase to earth short-circuit fault current on phase R at the edge of zone of influence is 13 000 A rms.

Calculation of mutual impedances

The non-parallel pipeline is converted to a parallel pipeline at a geometric mean distance from the power line equal to . The zone of influence of the power line is . Hence, the pipeline is well within the zone. The depth of earth return is . The distances between the pipeline and the power line’s phase and earth wire conductors are , similarly d_pY = 164.63 m, d_pB = 169.56 m and d_pE = 178.72 m. The mutual impedances with earth return between the pipeline and the power line’s conductors are

Similarly

and

From Example 3.4, we have

Electrical characteristics of the pipeline

The pipeline’s shunt admittance is

The series impedance of the pipeline with earth return is

The pipeline’s propagation constant is

The pipeline’s characteristic impedance is

Steady state unfaulted power system operation

Induced EMF

The balanced three-phase currents are PPS only thus I_R = 2000 A, I_Y = 2000 Ae^−j2π/3 and I_B = 2000 Ae^j2π/3. The induced EMF on the pipeline is

Therefore, EMF_p = −16.7 − j8.32 = 18.66∠ − 153.5° V/km. If the overhead power line had no earth wire, the induced EMF is EMF_p = -3.57 − j0.182 = 3.57∠−177.1° V/km. It is interesting to note that the effect of the power line’s earth wire under steady state conditions is to increase the induced voltage on the pipeline. This is opposite to the effect under fault conditions as shown in Equation (10.79).

Pipeline voltage to earth

The pipeline voltage to earth along the pipeline between ends A and B as well as beyond end A can be calculated using Equation (10.84a) between ends A and B and Equation (10.86) beyond end A where E_o = (−16.7 − j8.32)V/km and γ = (0.05525 + j0.06295)km⁻¹. The maximum pipeline voltage at end B at x = 4 km is equal to −62.95 − j22.4 = 66.8∠ −160.4° V and the voltage at end A at x = 0 is equal to 4.82 + j8.83 = 10.06∠61.37° V.

One-phase to earth short-circuit fault condition

Induced EMF

The overhead line’s screening factor is

The induced EMF on the pipeline is −EMF_p = Z_pRI_Fk or

Pipeline voltage to earth and pipeline current

We now calculate the profile of the pipeline voltage to earth along the pipeline between ends A and B as well as the attenuation beyond end A using Equations (10.84a) and (10.86) respectively. We also calculate the pipeline current along the pipeline between ends A and B using Equation (10.84b). In these Equations E_o = (−446.5 − j1117.5)V/km, γ = (0.05525 + j0.06295)km⁻¹ and Z_o = (5.478 + j3.11)Ω. The results of |V(x)| and |I(x)| are shown in Figure 10.28.

Figure 10.28 Calculated voltages to earth and current profiles on a steel pipeline inductively coupled to a 400 kV power line under a one-phase to earth fault condition. Corresponding to Figure 10.27