Step and Touch Voltages in Substation Grounding per IEEE 80

Electrophysiological Fundamentals and Ventricular Fibrillation Criteria

The primary objective of designing grounding systems in high-voltage and extra-high-voltage electrical substations is the preservation of human life and the integrity of electromechanical assets. During a ground fault event, the massive injection of current into the soil generates surface potential gradients that can subject an operator's or bystander's body to critical potential differences.

The physiological response of the organism to the passage of industrial-frequency alternating current (50/60 Hz) is dominated by the induction of ventricular fibrillation, a phenomenon in which ventricular myocytes lose contractile synchronism, thereby abolishing cardiac output. The probability of fibrillation depends on the magnitude of the current, the duration of exposure, the pathway through the body, and the phase of the cardiac cycle in which the electrical shock occurs (especially the vulnerable phase corresponding to the T-wave in the electrocardiogram).

I_B = \frac{k}{\sqrt{t_s}}

Within the framework of IEEE Std 80, Charles Dalziel empirically established that the energy tolerable by the human body before reaching the threshold of a 0.5% probability of ventricular fibrillation is governed by a shock energy constant $S_b$ . For a population with a body weight of approximately 50 kg, the constant $k50 = 0.116$ , whereas for 70 kg, $k70 = 0.157$ . Therefore, the maximum allowable body current $I_B$ for a shock duration $t_s$ (where 0.03 \le t_s \le 3.0 s ) is analytically defined by:

I_{B,50 kg } = \frac{0.116}{\sqrt{t_s}} \quad [ A ], \qquad I_{B,70 kg } = \frac{0.157}{\sqrt{t_s}} \quad [ A ]

On the other hand, the International Electrotechnical Commission in its technical report IEC/TS 60479-1 adopts a non-linear probabilistic approach based on Biegelmeier's studies. Instead of assuming a purely inversely proportional relationship to the square root of time for the entire time window, IEC 60479-1 defines time/current curves (zones AC-1 to AC-4.3). The human body impedance $Z_B$ in IEC is not a fixed resistance, but rather a resistive-capacitive network dependent on the touch voltage ( $U_T$ ), the current pathway, and the skin's moisture state:

Z_B(U_T) = Rint + \frac{Rp1(U_T)}{1 + j\omega Rp1 Cp1} + \frac{Rp2(U_T)}{1 + j\omega Rp2 Cp2}

While IEEE Std 80 conservatively simplifies the human body resistance to a constant, purely resistive value R_B = 1000\,\Omega (representative of the hand-to-foot or hand-to-hand pathway with firm contact), standard IEC 61936-1 and IEC 60479-1 apply dynamic curves where the effective impedance decays drastically as the touch voltage exceeds 200 V, reaching asymptotic values close to the internal body resistance (Rint \approx 500 to 750\,\Omega).

Mathematical Framework and Rigorous Formulation per IEEE Std 80

Thevenin Equivalent Circuits and Tolerability Limits

To determine allowable step ( $Vstep$ ) and touch ( $Vtouch$ ) voltages, a Thevenin equivalent circuit is utilized, viewed from the points of contact between the body and the soil surface and the grounded structures. The equivalent soil resistance seen by an individual's feet is modeled by considering each foot as a flat conductive disc of radius $b = 0.08 m$ resting on the surface of a homogeneous half-space of apparent resistivity \rho_s.

The self-resistance of a foot in contact with the ground is given by Maxwell's formulation:

Rfoot = \frac{\rho_s}{4b} \approx \frac{\rho_s}{4(0.08)} = 3.125 \rho_s \approx 3\rho_s

Under the touch voltage condition, both feet are in parallel at a practically identical distance from the fault point, so the equivalent resistance is R_{eq,touch} = \frac{Rfoot}{2} = 1.5\rho_s. For the step voltage condition, the feet are in series separated by a standardized distance of 1 meter, resulting in R_{eq,step} = 2 Rfoot = 6\rho_s.

When incorporating a surface layer of high-resistivity material (typically crushed rock or gravel of thickness $h_s$ and resistivity \rho_s) over natural soil of resistivity \rho, a reduction or derating factor C_s(h_s, K) must be introduced, which compensates for the reflection effect caused by the dielectric discontinuity:

C_s = 1 + \frac{16 b}{\pi} \sum_{n=1}^{\infty} \frac{K^n}{\sqrt{1 + \left(2 n \frac{h_s}{b}\right)^2}} \approx 1 - \frac{0.09 \left(1 - \frac{\rho}{\rho_s}\right)}{2 h_s + 0.09}

Where the reflection coefficient $K$ between the natural soil and the surface gravel layer is expressed as:

K = \frac{\rho - \rho_s}{\rho + \rho_s}

Integrating these electrodynamic deductions, the maximum tolerable potentials calculated under IEEE Std 80 for a 50 kg and 70 kg operator result in the following fundamental equations:

V_{touch,50} = \left(1000 + 1.5 C_s \rho_s\right) \frac{0.116}{\sqrt{t_s}}, \qquad V_{touch,70} = \left(1000 + 1.5 C_s \rho_s\right) \frac{0.157}{\sqrt{t_s}}

V_{step,50} = \left(1000 + 6.0 C_s \rho_s\right) \frac{0.116}{\sqrt{t_s}}, \qquad V_{step,70} = \left(1000 + 6.0 C_s \rho_s\right) \frac{0.157}{\sqrt{t_s}}

Ground Potential Rise (GPR) and Grid Design Parameters

The design current effectively dissipated through the grid into the infinite earth ( $I_G$ ) is not equivalent to the total symmetrical three-phase or single-phase fault current at the terminals ( $I_f$ ), but is instead affected by the fault current division factor ( $S_f$ ) and the decrement factor ( $D_f$ ) due to the asymmetry of the direct-current component:

I_G = S_f \cdot D_f \cdot I_f

The decrement factor $D_f$ quantifies the aperiodic content of the short-circuit current as a function of the reactance-to-resistance ratio ( $X/R$ ) of the Thevenin impedance at the fault point and the clearing duration $t_f$ :

D_f = \sqrt{1 + \frac{T_a}{t_f} \left(1 - e^{-2 t_f / T_a}\right)}, \qquad T_a = \frac{X}{\omega R} = \frac{X}{2\pi f R}

The maximum potential rise of the system relative to a remote reference point is termed Ground Potential Rise ( $GPR$ ) and is calculated via:

GPR = I_G \cdot R_g

The grounding system resistance ( $R_g$ ) for complex interconnected grids with rods or deep vertical electrodes is determined with high analytical precision using Sverak's generalized formula:

R_g = \rho \left[ \frac{1}{L_T} + \frac{1}{\sqrt{20 A}} \left( 1 + \frac{1}{1 + h \sqrt{20 / A}} \right) \right]

Where $L_T$ is the total length of buried conductors (horizontal grid plus vertical electrodes), $A$ is the area occupied by the grid in square meters, and $h$ is the burial depth of the horizontal grid conductors.

Real Substation Mesh and Step Voltages

The real mesh voltage ( $E_m$ ), representing the worst touch voltage condition at the center of the most critical outer quadrant of the grid, is rigorously formulated by the product of the apparent soil resistivity (\rho), the geometrical spacing factor $K_m$ , the corrective irregularity factor $K_i$ , and the injected linear current density:

E_m = \frac{\rho \cdot I_G \cdot K_m \cdot K_i}{L_M}

The dimensionless factors $K_m$ and $K_i$ synthesize the grid topology and are analytically defined as:

K_m = \frac{1}{2\pi} \left[ \ln\left( \frac{D^2}{16 h d} + \frac{(D + 2h)^2}{8 D d} - \frac{h}{4d} \right) + \frac{Kii}{K_h} \ln\left( \frac{8}{\pi (2n - 1)} \right) \right]

K_i = 0.644 + 0.148 \cdot n

Where $D$ is the spacing between parallel conductors, $d$ is the mesh conductor diameter, $h$ is the burial depth, $n$ is the equivalent geometric factor derived from the dimensions and perimeter shape (n = n_a \cdot n_b \cdot n_c \cdot n_d), K_h = \sqrt{1 + h/h_0} (with $h_0 = 1 m$ ), and $Kii = 1$ for grids with perimeter electrodes or Kii = 1/(2n)^{2/n} for grids without ground rods.

The weighted effective conductor length for mesh voltage calculation ( $L_M$ ) incorporates the contribution of the vertical grounding electrodes ( $L_r$ ):

L_M = L_c + \left[ 1.55 + 1.22 \left( \frac{L_r}{\sqrt{L_x^2 + L_y^2}} \right) \right] L_R

Similarly, the calculated step voltage ( $E_s$ ) at the outer perimeter of the substation is evaluated using the geometric step coefficient $K_s$ :

E_s = \frac{\rho \cdot I_G \cdot K_s \cdot K_i}{L_S}

K_s = \frac{1}{\pi} \left[ \frac{1}{2h} + \frac{1}{D + h} + \frac{1}{D} \left( 1 - 0.5^{n-2} \right) \right]

L_S = 0.75 \cdot L_c + 0.85 \cdot L_R

Transient Behavior and Lightning Impact per IEC 62305

Impulse Impedance vs. Industrial-Frequency Resistance

The response of a grounding grid to the injection of lightning currents regulated by the IEC 62305 series of standards differs radically from behavior at 50/60 Hz. Direct or induced atmospheric discharges present ultra-fast normalized wave fronts (10/350 \mu s waveform for the first lightning stroke per IEC 62305-1, and 8/20 \mu s for subsequent strokes), with frequency contents extending into the hundreds of kilohertz to megahertz range.

At these frequencies, the distributed longitudinal inductive reactance ( $L'$ ) of the copper or galvanized steel conductors becomes the dominant factor over the transverse conductance of the soil ( $G'$ ). Consequently, the current lacks the physical time to distribute throughout the entire grid geometry, concentrating instead in the immediate vicinity of the impact point.

Zimp(t) = \frac{u(t)}{i(t)} \neq R_g

The effective length of the buried conductor $l_e$ , beyond which any further extension of the electrode does not reduce the apparent impulse impedance for a front time $T_1$ , is expressed analytically according to IEC 62305-3 as:

l_e = \sqrt{\frac{\rho \cdot T_1}{\mu_0}} \approx 1.6 \cdot \sqrt{\rho \cdot T_1}

Where \mu_0 = 4\pi \times 10^{-7} H/m is the magnetic permeability of free space and $T_1$ is the front time of the impulse in microseconds (\mu s ).

Lightning-Induced Step and Touch Voltages

Under the action of an impulse lightning current i(t), the radial potential gradient on the soil surface generates extreme transient voltages that can cause sparkovers in air along the human body's lower extremities (surface arc or flashover). The transient impulse touch voltage ( $U_{touch,imp}$ ) and transient step voltage ( $U_{step,imp}$ ) per IEC 62305-3 Annex E are coupled through pure inductive and resistive components:

U_{touch,imp}(t) = Rimp \cdot i(t) + Lgrid \cdot \frac{di(t)}{dt}

U_{step,imp}(r) = \frac{\rhoeff \cdot i(t)}{2\pi} \left( \frac{1}{r} - \frac{1}{r + \Delta r} \right)

In the case of lightning strikes on the shielding structure (Franklin rods, overhead ground wires, or gantries), high temporal derivatives (di/dt > 100 kA/ \mu s ) generate parasitic inductive potentials in the metallic down-conductors that exceed the dielectric strength of air, requiring safe separation distances ( $s$ ) calculated in accordance with the formulation:

s \ge k_i \cdot \frac{k_c}{k_m} \cdot l

Where $k_i$ is the coefficient depending on the lightning protection level (LPL I to IV), $k_c$ is the geometric current partitioning factor in the down-conductors, $k_m$ is the insulating material coefficient (air or solid), and $l$ is the linear distance along the down-conductor to the nearest equipotential bonding point.

Normative Comparative Matrix: IEEE Std 80 vs. IEC 60479-1 / IEC 61936-1 / IEC 62305-3

Parameter / Design Criteria	IEEE Std 80 (2013)	IEC 60479-1 / IEC 61936-1 (2021)	IEC 62305-3 (Lightning Protection)
Body Weight Model	Discrete: 50 kg ( $k=0.116$ ) or 70 kg ( $k=0.157$ )	Statistical population (5th, 50th, 95th percentile)	Not applicable (Risk evaluated by impulse shock)
Body Impedance ( $R_B$ / $Z_B$ )	Constant: R_B = 1000\,\Omega (purely resistive)	Non-linear: Z_B = f(U_T), 500 to 1000\,\Omega	R_B \approx 0\,\Omega (High-frequency electric arc)
Critical Time Regime	Industrial frequency (50/60 Hz), 0.03 \le t_s \le 3.0 s	Industrial frequency (50/60 Hz), 0.01 \le t_s \le 10.0 s	Impulse transient (10/350\,\mu s , 8/20\,\mu s )
Surface Layer Resistance	Formal analytical derating C_s(h_s, K)	Similar or empirical surface layer factor	Control insulation: asphalt $5 cm$ (\ge 100 k \Omega\cdot m )
Touch Voltage Limit	Vtouch = (1000 + 1.5 C_s \rho_s) \frac{k}{\sqrt{t_s}}	Defined by limit curve U_{v,tol} = f(t_f, Z_B, Rfoot)	Utouch \le 100 kV (with equipotential bonding)
Fault Division Factor ( $S_f$ )	Explicit calculation with shield wires and neutrals	Reduction factor $r$ (function of shields and guards)	Division factor $k_c$ based on down-conductor topology
Soil Behavior	Multi-layer resistivity stratification at 50 Hz	Bi-layer and homogeneous half-space models	Soil ionization effect (E_0 \approx 300 kV/m )

Forensic Failure Analysis and Potential Transfer Phenomena

Mechanisms of Transferred Potential Propagation

One of the most devastating failure modes in substation forensic engineering corresponds to the inadvertent export or import of Ground Potential Rise ( $GPR$ ) outside the facility's perimeter boundaries. When a ground fault occurs inside the substation, the entire grounding grid rises to a potential $V = GPR$ relative to zero reference potential in remote earth.

Vtrans = GPR - \Delta Vdrop \approx I_G \cdot R_g

The main physical vectors of galvanic coupling for transferred potentials include:

Medium- and Low-Voltage Network Neutral Conductors: If the neutral of the auxiliary transformer secondary winding or a rural distribution line is solidly grounded to the internal grid and extends outside the substation with multiple grounding, the full $GPR$ is applied directly across end-user network insulators, resulting in residential meter explosions, appliance punch-through, and massive electrocution risk.
Power and Telecommunication Cable Metallic Shields: Underground medium-voltage cable shields grounded at both ends transfer soil return current to remote terminals, causing circulating currents of hundreds of amperes that melt the outer polyethylene jacket (thermal failure via $I^2 t$ ) and induce overvoltages that puncture digital instrumentation interfaces.
Water or Gas Pipeline Conduits and Railway Rails: Continuous metallic elements buried without dielectric insulating joints act as infinite-extension electrodes, exposing operators at pumping plants or distant level crossings to the entirety of the $GPR$ , where the transferred touch voltage V_{touch,trans} \approx GPR lacks the mitigating effect of the substation gravel.

Behavior in Gas-Insulated Substations (GIS)

In sulfur hexafluoride-insulated ( $SF _6$ ) substations, disconnector switching operations or the occurrence of sparkover discharges generate Very Fast Transients (VFT) with rise times in the nanosecond range ( $t_r < 10 ns$ ). These wave fronts fail to circulate to ground through conventional grounding connections due to the parasitic inductive reactance of connection busbars (L \approx 1\,\mu H/m ).

Consequently, the Transient Enclosure Voltage (TEV) phenomenon arises, wherein the external aluminum enclosure of the GIS experiences potential oscillations of up to several tens of kilovolt relative to supporting metallic structures and surrounding floors, generating secondary micro-arcs that destroy numeric control and protection system (SAS) control cables and cause contact discharges to plant inspectors.

Soil Stratification Methodology and Grid Optimization

Vertical Electrical Sounding (VES) Curve Inversion

The accuracy of $V_m$ , $V_s$ , and $R_g$ calculations is strictly linked to the formulation of the subsurface geoelectric model. Prosection via the Wenner four-electrode method consists of injecting direct or very low-frequency current through outer electrodes ( $C_1, C_2$ ) and measuring the potential drop at inner electrodes ( $P_1, P_2$ ), symmetrically arranged with a spacing $a$ :

\rho_a(a) = 2\pi a \frac{\Delta V}{I}

Since soil is rarely homogeneous, the apparent resistivity \rho_a varies as a function of current line penetration depth, which is proportional to spacing $a$ . To model a bi-layer structure (top layer of resistivity \rho_1 and thickness $h$ , over a deep half-space of resistivity \rho_2), Stefanescu's integral equation is solved analytically via the zero-order Hankel transform:

\rho_a(a) = \rho_1 \left[ 1 + 4 \sum_{n=1}^{\infty} \left( \frac{K^n}{\sqrt{1 + \left(\frac{2 n h}{a}\right)^2}} - \frac{K^n}{\sqrt{4 + \left(\frac{2 n h}{a}\right)^2}} \right) \right]

Where the geoelectric reflection factor is:

K = \frac{\rho_2 - \rho_1}{\rho_2 + \rho_1}

The numerical optimization algorithm (Levenberg-Marquardt nonlinear damped least squares) seeks to minimize the root-mean-square (RMS) error objective function:

\varepsilonRMS = \sqrt{\frac{1}{N} \sum_{i=1}^{N} \left( \frac{\rho_{a,meas}(a_i) - \rho_{a,calc}(a_i)}{\rho_{a,meas}(a_i)} \right)^2} \times 100\%

Advanced Surface Gradient Mitigation Topologies

When surface resistivity is unfavorable (\rho_1 \gg \rho_2) or fault current $I_G$ is massive, simply increasing uniform grid conductors yields diminishing returns. Advanced electromagnetic optimization techniques include:

Gradual Densification Grid (Perimeter Compression): Exponential reduction of spacing between conductors as the outer boundary of the substation is approached. This homogenizes the current density expelled per linear meter, flattening the peripheral potential peaks that trigger exterior touch and step voltages.
Deep Vertical Perimeter and Corner Electrodes: The insertion of deep driven rods (15 to 30 meters) drilled at grid corners allows current to drain preferentially toward lower, more conductive strata (\rho_2), collapsing the overall $GPR$ and substantially reducing the surface gradient at critical edges.
High-Performance Dielectric Layer: Application of washed granite gravel layers (\rho_s \ge 3000\,\Omega\cdot m ) or asphalt carpeting (\rho_s \ge 10000\,\Omega\cdot m ) with a thickness $h_s = 0.10 to 0.15 m$ . It must be strictly verified that the reduction factor $C_s$ maintains its efficacy throughout the plant's operational lifespan, mitigating fine sediment and weed contamination via impermeable geotextiles.

Implementation and Computational Modeling in Vexten Suite

Short-Circuit Integration and Parametric Evaluation

Within the engineering design workflow inside the comprehensive Vexten Suite platform, grounding sizing is natively synchronized with three-phase and asymmetrical short-circuit calculation modules under IEC 60909 and IEEE 141 (Red Book) standards. Exact determination of the zero-sequence current component ( $3I_0$ ) and Thevenin equivalent impedance at the substation node feeds directly into the earth optimization engine.

The following matrix formulation is implemented within Vexten Suite to resolve inductive and galvanic coupling of multiple overhead lines with shield wires (OPGW/steel) connected to the substation, allowing rigorous determination of the actual division factor $S_f$ :

\begin{bmatrix} \mathbf{I}_{g1} \\ \mathbf{I}_{g2} \\ \vdots \\ \mathbf{I}_{gn} \end{bmatrix} = \begin{bmatrix} Z11 & Z12 & \cdots & Z1n \\ Z21 & Z22 & \cdots & Z2n \\ \vdots & \vdots & \ddots & \vdots \\ Zn1 & Zn2 & \cdots & Znn \end{bmatrix}^{-1} \begin{bmatrix} \mathbf{V}_{GPR} - \mathbf{E}_{ind,1} \\ \mathbf{V}_{GPR} - \mathbf{E}_{ind,2} \\ \vdots \\ \mathbf{V}_{GPR} - \mathbf{E}_{ind,n} \end{bmatrix}

D Surface Gradient Mapping and Normative Verification

The Vexten Suite electromagnetic engine subdivides the earth grid and vertical electrodes into discrete cylindrical segments, applying the Method of Moments (MoM) to calculate longitudinal charge density $q_j$ and radial leakage current across each segment. The potential distribution at any surface coordinate (x, y, 0) is evaluated analytically via numerical integration:

\Phi(x, y, 0) = \frac{1}{4\pi \sigma} \sum_{j=1}^{M} \int_{L_j} \frac{I_j(\ell')}{\sqrt{(x - x')^2 + (y - y')^2 + (z')^2}} \, d\ell'

Once the three-dimensional potential map is computed, Vexten Suite automatically processes plant topology by superimposing switchgear equipment, power transformers, perimeter fences, and operating pathways. The system generates 1-meter spatial vectors in all radial directions from accessible conductive masses to cross-check point by point:

\max\left( \Phistructure - \Phi(xfoot, yfoot, 0) \right) \le V_{touch,adm}(t_s)

\max\left( \Phi(x+1, y, 0) - \Phi(x, y, 0) \right) \le V_{step,adm}(t_s)

Any transgression of IEEE Std 80 or IEC 61936-1 regulatory margins is highlighted in the graphical environment via high-resolution heatmaps, allowing the engineer to interactively apply auxiliary rods, perimeter gradient equalizers, or gravel layer thickness modifications until full facility compliance is certified against maximum human safety and operational robustness requirements.