Lightning-Induced Overvoltages: Trees & Protection

Section 1

Nearby Objects Change Lightning Performance

This second part looks at what happens when an overhead line is not alone — when trees, forests, buildings, poles, towers or other elevated structures stand near it — and how those objects, line geometry, striking-distance models, surge arresters and real field data all shape the lightning performance.

Continuing from Part One

Part One built the basic mechanism — the Rusck induced voltage \(V_c\), the flashover condition \(V_c \ge \text{CFO}\), the maximum distance \(X_m\), the ground boundary \(D_g\), the critical current \(I_{sc}\), the induced-voltage flashover rate IVFOR, and the neutral / ground-wire effect, alongside the backflashover rate BFR. It considered the basic case where nearby ground strokes induce voltages on the line and where flashover is controlled by the relationship between \(X_m\), \(D_g\), CFO and IVFOR. In this second part the same logic is extended to more realistic corridors where trees, forests, buildings or other elevated objects sit near the line. These objects change where lightning terminates, and therefore they change the balance between direct-stroke flashovers and induced-voltage flashovers.

Objects near a line can act as natural shielding objects: a stroke that would otherwise have hit the conductor may instead terminate on the nearby object. At first this looks purely beneficial, because fewer direct strokes reach the line and the direct-stroke flashover rate falls.

But the picture is not that simple. A stroke that terminates on a nearby object is often still close enough to induce a voltage on the line. So while the direct-stroke flashover rate may fall, the induced-voltage flashover rate may rise. The total flashover rate may therefore increase, decrease, or stay almost unchanged.

\[ \text{Total flashover rate} = \text{BFR} + \text{IVFOR} \]

\(\text{BFR}\): backflashover / direct-stroke-related flashover rate
\(\text{IVFOR}\): induced-voltage flashover rate

Key idea

Nearby objects may protect the line from direct strokes, but they may increase induced-overvoltage problems. The two effects compete, and the net result depends on the geometry.

What Part Two covers

how trees and forests shield the line yet raise induced voltages;
the striking-distance geometry that decides where a stroke terminates;
the piecewise IVFOR cases and why a computer program is normally needed;
the CIGRE correction to the calculated flash incidence;
worked example flashover rates for shielded and unshielded lines;
surge-arrester spacing for induced surges and comparison with field data.

Section 2

Why Trees and Forests Matter

Trees and forests are common beside distribution and sub-transmission lines, and they can strongly affect lightning performance. A tree has height above ground, so the lightning attachment process may prefer the tree over the line conductor. But even a stroke to the tree produces electromagnetic fields that can induce voltage on the nearby line.

So trees and forests introduce two competing effects:

Beneficial effect — they reduce direct strokes to the line, which lowers \(\text{BFR}\).
Negative effect — they increase nearby terminations close to the line, which can raise \(\text{IVFOR}\).

Which effect wins depends on the height of the trees, the distance between the trees and the line, the conductor height, whether the phase conductor or the ground wire is uppermost, the CFO, the ground flash density, the lightning-current distribution and the shielding geometry.

Section 3

Trees versus Forests

A single row of trees

The “trees” case means a single line or row of trees beside the line — for example one row along a road or right-of-way, or isolated but repeated trees near the line.

A forest

A “forest” means an extended area of trees: after the first row, more trees continue behind it, so the shielding object behaves like an extended elevated surface rather than a narrow row.

The distinction matters because lightning may terminate differently for a single row than for an extended forest. As the examples later show, a single row of trees can actually increase the induced flashover rate more than a forest, because the first row receives many strokes while still being close enough to the line to induce a high voltage.

Section 4

The Geometry Used in the Analysis

The analysis considers an overhead line with nearby trees or a forest. The uppermost line conductor — whichever of the phase conductor or ground wire is higher — controls the lightning attachment:

\[ h_{gc} = \max(h_c,\,h_g) \]

\(h_c\): height of the phase conductor (m)
\(h_g\): height of the ground wire or neutral (m)
\(h_{gc}\): height of the uppermost line conductor (m)
\(h_T\): height of the trees or forest (m)
\(S_{12}\): horizontal distance from the line to the trees or forest (m)
\(D_g\): ground-termination boundary distance without nearby objects (m)
\(D_o\): modified boundary distance when shielding by objects is considered (m)

The nearby trees modify the effective lightning collection area of the line, which is why both \(D_g\) and the modified distance \(D_o\) appear in the shielded analysis.

Section 5

Striking Distance and the Consequence of Shielding

Lightning attachment is studied with the striking-distance model: the distance at which a downward leader attaches to an object. Three striking distances matter near a shielded line:

\[ r_{cc}\;(\text{to conductor}) \qquad r_{ct}\;(\text{to tree top}) \qquad r_{g}\;(\text{to ground}) \]

\(r_{cc}\): striking distance to the line conductor or overhead ground wire
\(r_{ct}\): striking distance to the tree top
\(r_{g}\): striking distance to ground

If a tree is close enough and tall enough, its striking-distance zone overlaps that of the line conductor, and the tree shields the line. Some strokes that would have hit the line now hit the tree — but they can still induce voltage on the line. The result is the central trade-off of this whole part:

When trees shield the line, direct strokes to the line decrease, so \(\text{BFR}\downarrow\). But strokes terminating on the nearby trees stay close to the line, so \(\text{IVFOR}\uparrow\). The total depends on which change is larger: if the fall in BFR exceeds the rise in IVFOR, the total drops; if not, the total rises; if they roughly cancel, the total is nearly unchanged.

Not a contradiction

The tree removes some direct strokes from the line, so BFR decreases. But those same strokes may now terminate on a nearby elevated object. If that object is still close to the line, the stroke can induce a voltage large enough to cause flashover. Shielding therefore improves one mechanism (direct strokes) while worsening another (induced voltages) — the two are separate effects, not a paradox.

Section 6

Using the Rusck Equation for Strokes to Elevated Objects

Induced overvoltages from strokes to elevated objects, such as trees, have not been studied as thoroughly as strokes to flat ground. Rusck’s equation was developed for nearby strokes generally treated as ground strokes, so applying it to elevated objects is an approximation. The practical position is: the direct use of Rusck’s equation for strokes to elevated objects is questionable, but no better simple method exists, so it may be used with caution. The result is a practical estimate, not an exact prediction.

The underlying Rusck relationship, carried over from Part One, is:

\[ V_c = \frac{30\,I\,h_c\,K_v}{x} \qquad\Rightarrow\qquad V_c \propto I,\quad V_c \propto h_c,\quad V_c \propto \frac{1}{x} \]

\(V_c\): induced voltage on the conductor
\(I\): lightning stroke current
\(h_c\): conductor height
\(K_v\): return-stroke velocity correction factor
\(x\): distance between the stroke and the line

Why this matters near trees

Because \(V_c \propto 1/x\), a high-current stroke to a tree that is close to the line can still induce a significant voltage on the conductors. Distance to the termination point, not just current, controls the induced stress.

Section 7

Where Strokes Terminate, and the Flashover Condition

The line, the trees and the ground divide the nearby area into regions. A stroke may terminate on the line conductor or ground wire, on the first tree or row of trees, on the forest, or on the ground beyond the trees. The induced-flashover contribution depends on which region the stroke hits, and on the relative position of \(X_m\), \(D_o\), \(S_{12}\) and \(D_g\).

As in Part One, induced flashover is possible only when the induced voltage reaches the CFO:

\[ V_i \ge \text{CFO} \quad\text{(or, in conductor-voltage form, } V_c \ge \text{CFO)} \;\;\Rightarrow\;\; X_m = \frac{30\,I\,h_c\,K_v}{\text{CFO}} \]

\(V_i\): induced voltage across the insulation
\(X_m\): maximum distance at which the induced voltage still equals the CFO
\(D_o\): modified shielding boundary due to the nearby object
\(S_{12}\): distance from the line to the trees or forest
\(D_g\): original ground-stroke boundary distance

If the tree is close to the line, a stroke to it can create a high induced voltage; if the tree is far away, the induced voltage is lower. So the position of the tree relative to \(X_m\) is decisive.

Distance glossary — keep these four apart

Before the tree and forest cases, it is useful to separate the main distances:

\(D_g\) — the original ground-stroke boundary without nearby shielding objects.
\(D_o\) — the modified boundary with trees or nearby objects shielding the line.
\(S_{12}\) — the horizontal distance from the line to the trees or forest.
\(X_m\) — the maximum distance at which a stroke can still induce a voltage equal to or greater than the CFO.

The flashover calculation is controlled by the relative position of these distances. If \(X_m\) is smaller than the relevant shielding boundary, no induced flashover occurs. If \(X_m\) extends beyond the tree or forest boundary, the corresponding exposure region contributes to IVFOR.

Section 8

Case 1 — A Direct Stroke to the Phase Conductor

Very close to the line a stroke may still terminate on the line itself. If the ground wire is uppermost, the stroke terminates on it and performance is assessed by backflashover methods. If the phase conductor is uppermost, the stroke may terminate directly on the phase conductor, launching a surge in both directions:

\[ V = \frac{I\,Z_c}{2} \qquad\Rightarrow\qquad \text{flashover if } \frac{I\,Z_c}{2} \ge \text{CFO} \qquad\Rightarrow\qquad I_c = \frac{2\,\text{CFO}}{Z_c} \]

\(V\): voltage launched on the phase conductor
\(I\): stroke current
\(Z_c\): surge impedance of the phase conductor
\(I_c\): minimum direct-stroke current that causes flashover

The factor 2 appears because the surge travels away from the stroke point in two directions. For low-voltage and distribution lines \(I_c\) is very small, so almost every direct stroke to the phase conductor causes flashover.

Section 9

Flashover Rate from Direct Strokes

If practically all direct strokes flash over, the direct-stroke flashover rate is essentially the number of strokes that terminate on the line:

\[ \text{BFR} = 2\,N_g\,L \int D(I)\,f(I)\,dI \qquad\text{and, empirically,}\qquad \text{BFR} \approx N_g\,L\,\bigl(2.8\,h^{0.6}\bigr) \]

\(N_g\): ground flash density (flashes/km²/year)
\(L\): line length
\(D(I)\): collection distance as a function of stroke current
\(f(I)\): probability density of stroke current
\(h\): line / conductor height (m)

The factor 2 accounts for both sides of the line. The empirical form shows that the number of direct strokes rises with line height: the higher the line, the more attractive it is to lightning.

Section 10

Cases 2–5 — Strokes Near and Beyond the Trees

Case 2 — between the line and the trees

Where the stroke lands between the line and the trees, the termination depends on the shielding geometry. Strokes that might have hit the ground or line can be intercepted by the trees. If a stroke hits the first tree but the resulting distance gives an induced voltage below the CFO, its IVFOR contribution is zero. In particular, if \(D_g < X_m < S_{12}\), strokes in the region may not contribute, because they terminate on the first tree or fall outside the effective flashover band.

Case 3 — the forest

For a forest the trees continue beyond the first row, behaving like an extended elevated surface. If the stroke lands beyond \(S_{12}\) and \(X_m > S_{12}\), induced flashover may occur, with incremental rate:

\[ \Delta\text{IVFOR} = 2\,N_g\,L\,(X_m - D_g)\,f(I) \]

\(X_m - D_g\): effective exposure width where strokes can cause induced flashover

Case 4 — a single row of trees

A single row collects strokes, but beyond it the ground is open again. For \(S_{12} \le X_m \le S_{12} + D'_g\), the incremental rate becomes:

\[ \Delta\text{IVFOR} = 2\,N_g\,L\,(S_{12} + D'_g - D_g)\,f(I) \]

\(D'_g\): modified ground-distance boundary for the tree-shielding condition
\(S_{12} + D'_g - D_g\): effective exposure width set by the tree location, not simply \(X_m - D_g\)

Case 5 — beyond the tree row

For a single row, if \(X_m > S_{12} + D'_g\), the induced-flashover region extends past the immediately shielded zone. The exposure must then be split by region — strokes to the tree row, strokes to the ground beyond, and a region with no induced flashover — so the IVFOR becomes piecewise. The key idea is more useful than memorising the formula: the calculation is no longer a single integral.

Section 11

Why the Tree / Forest Calculation Becomes Complex

Without nearby objects, the induced-flashover region is simply \(D_g < x < X_m\). With trees or a forest, the termination process is modified, and the calculation must now consider whether each stroke attaches to the line, the tree, the forest or the ground, whether the resulting distance is less than \(X_m\), and which is the preferred attachment point.

IVFOR is therefore no longer one simple integral — it becomes a set of conditional, piecewise equations. For this reason the original treatment recommends a dedicated computer program (the IVFOR program).

Engineering lesson

The presence of trees or forests makes lightning-performance calculations highly geometry-dependent. Detailed IVFOR studies near vegetation are not normally done by hand.

Section 12

Complete Shielding by Trees or Forests

Trees or a forest may be tall enough and close enough to prevent direct strokes reaching the line. The direct-stroke flashover rate then becomes very small:

\[ \text{BFR} \rightarrow 0 \qquad\Rightarrow\qquad \text{Total flashover rate} \approx \text{IVFOR} \]

This does not mean the line has no flashovers. Instead, the line becomes dominated by induced-overvoltage flashovers. A well-shielded line can still suffer significant flashovers driven entirely by induced voltages.

Section 13

Correcting IVFOR Against the CIGRE Flash Incidence

The calculated IVFOR can be biased because the calculated number of flashes to the line may differ from the widely accepted CIGRE estimate. The two expressions are:

\[ N_L(\text{calc}) = 2\,N_g\,L \int D_g\,f(I)\,dI \qquad\qquad N_L(\text{CIGRE}) = N_g\,\frac{28\,h^{0.6} + S_g}{10} \]

\(N_L(\text{calc})\): calculated number of flashes to the line
\(N_L(\text{CIGRE})\): CIGRE empirical number of flashes to the line
\(h\): height of the line or ground wire (m)
\(S_g\): spacing between two overhead ground wires (m)

Because the CIGRE equation is the accepted estimate of line lightning incidence, the calculated IVFOR is scaled to be consistent with it:

\[ \text{IVFOR}_{\text{corrected}} = \text{IVFOR}_{\text{calc}} \cdot \frac{N_L(\text{CIGRE})}{N_L(\text{calc})} \]

If the calculation underestimates the flashes to the line compared with CIGRE, IVFOR is increased; if it overestimates, IVFOR is reduced. When trees shield the line, the strokes collected by the line also fall, so the effective \(N_g\) used in backflashover calculations may need adjustment too — another step that is far easier inside a computer program.

Section 14

Example Flashover Rates — Lines Without Trees or Forests

The following example values illustrate the method for lines that are not shielded by nearby objects. They are example results, not universal design values; all are in flashovers per 100 km-years.

Table 1 — Example flashover rates for unshielded lines.

Configuration	\(h_c\) (m)	\(h_g\) (m)	IVFOR	BFR	Total
A — No neutral or ground wire	10	0	4.93	11.15	16.08
B — Neutral below the phase	10	8	1.67	11.15	12.82
C — Ground wire above the phase	8	10	0.159	2.73	2.89

Section 15

Lessons from the No-Tree Cases

No neutral or ground wire gives the highest total. Both induced-voltage and direct-stroke problems are significant (Total \(=16.08\)).
A neutral below the phase reduces IVFOR from \(4.93\) to \(1.67\) — a large reduction — but the BFR stays almost the same, because a neutral below the phase gives little direct-stroke shielding.
A ground wire above the phase is best. IVFOR falls from \(4.93\) to \(0.159\) and BFR from \(11.15\) to \(2.73\), so both induced and direct-stroke performance improve. This confirms the practical value of overhead ground wires.

Section 16

Example Flashover Rates — Lines Shielded by Trees or Forests

These example values use trees / forest of height \(h_T = 10\ \text{m}\) at distance \(S_{12} = 20\ \text{m}\) from the line. All values are in flashovers per 100 km-years.

Table 2 — Example flashover rates for shielded lines (\(h_T=10\) m, \(S_{12}=20\) m).

Configuration	\(h_c\) (m)	\(h_g\) (m)	IVFOR	BFR	Total
Forest — phase above neutral	10	8	8.60	2.69	11.29
Forest — ground wire above phase	8	10	4.49	0.24	4.73
Trees — phase above neutral	10	8	13.92	2.69	16.61
Trees — ground wire above phase	8	10	12.6	0.24	12.88

Section 17

Lessons from the Tree and Forest Cases

Trees and forests strongly change the balance between IVFOR and BFR. Without shielding, BFR may dominate. With trees or forests, BFR is reduced because the vegetation intercepts direct strokes — but IVFOR rises, because strokes to the trees are close enough to induce overvoltages. In the shielded examples, IVFOR becomes the dominant part of the total, representing roughly:

\[ \text{IVFOR} \approx 76\%\;\text{to}\;98\%\;\text{of the total flashover rate (shielded cases)} \]

Practical conclusion

For lines near trees or forests, flashovers may be dominated by induced overvoltages rather than direct strokes.

Section 18

Why a Single Row of Trees Can Be Worse Than a Forest

It seems surprising, but a single row of trees can produce a higher flashover rate than a forest. The reason is where the strokes terminate. A single row close to the line concentrates lightning terminations near the line, and those strokes are close enough to induce high voltages. A forest, by contrast, spreads terminations over a larger elevated area, so some strokes land farther from the line and induce less. In the examples:

\[ \text{IVFOR}_{\text{trees}} > \text{IVFOR}_{\text{forest}} \]

The practical lesson: a single row of nearby trees can be more severe for induced overvoltages than an extended forest.

Section 19

Effect of Distance to the Trees and Tree Height

Distance \(S_{12}\)

If the trees are very close, strokes to them induce large voltages because \(V_c \propto 1/x\). As \(S_{12}\) increases, those strokes are farther away and the induced voltage falls — but the trees also shield the line less, so direct strokes rise:

\[ S_{12}\uparrow \;\Rightarrow\; \text{IVFOR}\downarrow \qquad\text{but}\qquad S_{12}\uparrow \;\Rightarrow\; \text{BFR}\uparrow \]

Tree height \(h_T\)

Taller trees intercept more strokes, so \(h_T\uparrow \Rightarrow \text{BFR}\downarrow\). But taller trees also receive more strokes, and if they are close they induce more, so \(h_T\uparrow \Rightarrow \text{IVFOR}\uparrow\). For a forest the two changes may roughly cancel, leaving the total near the no-forest value; for a single row, the IVFOR rise can dominate and raise the total.

Section 20

Trees up to 60 m Away Can Still Matter

Trees or forests as far as 60 m from the line can still affect performance. Considering only objects directly beside the line is not enough: vegetation outside the immediate right-of-way can still influence lightning behaviour. For distribution lines, vegetation management therefore affects not only mechanical clearance and fault risk, but also lightning flashover performance.

Section 21

Effect of CFO with Trees or Forests

CFO remains one of the most important factors. The maximum flashover distance varies inversely with it:

\[ X_m \propto \frac{1}{\text{CFO}} \]

Raising the CFO reduces the distance over which induced flashover can occur. A low-CFO line is dominated by induced overvoltages; increasing the CFO to roughly \(200\)–\(250\ \text{kV}\) can greatly reduce induced flashovers for typical distribution configurations. However, with trees or forests nearby, induced overvoltages may remain important because many strokes still terminate close to the line.

Section 22

Striking-Distance Equations and Why They Matter

The striking distance is generally written as a function of lightning current, with a height correction applied for elevated objects:

\[ r = a\,I^{\,b} \]

\(r\): striking distance
\(I\): lightning stroke current (kA)
\(a,\,b\): constants depending on the chosen model

Table 3 — Striking-distance models referenced in the analysis.

Model	Striking Distance to Ground
Young	\(r_g = 27\,I^{0.32}\) (with height correction)
Brown–Whitehead	\(r_g = 6.4\,I^{0.75}\) (with height correction)
Love	Power-law form \(r = a\,I^{b}\), commonly used in shielding studies
Substations / Mousa	Power-law form used for substation shielding
IEEE-92	IEEE 1992 lightning-performance method

The choice of model matters because IVFOR depends on the exposure width \(X_m - D_g\), and \(D_g\) depends on the striking-distance model. A larger \(D_g\) makes \(X_m - D_g\) smaller and lowers IVFOR; a smaller \(D_g\) widens the band and raises IVFOR. Two studies with identical line data but different striking-distance equations can therefore give different flashover rates — so the model used should always be checked.

Section 23

Protecting the Line with Surge Arresters

Surge arresters can protect lines against induced overvoltages. Crucially, induced surges are usually less steep than surges from direct strokes to the line, and arrester protection distance depends strongly on surge steepness. A steep surge rises quickly as it travels, so arresters must be close together; a less steep surge rises slowly, so one arrester can protect a longer section. Because induced surges are less steep, arresters do not necessarily need to sit at every pole.

The protection problem: a surge is induced at some point, travels to a pole with an arrester (which limits the voltage there), while a farther pole may have no arrester. The goal is the maximum arrester spacing such that the voltage at the unprotected pole stays below the CFO. That maximum spacing follows from a travel time:

\[ d = c\,T_i \]

\(d\): maximum protected distance
\(c\): surge propagation velocity (≈ speed of light, \(\approx 300\ \text{m/}\mu\text{s}\))
\(T_i\): allowable travel time from the surge point to the arrester

Section 24

Arrester Spacing — No Ground at the Unprotected Pole

With no ground at the unprotected pole, the allowable travel time is:

\[ T_i = \frac{\text{CFO} - E_A}{2\,(S_c - S_g)} = \frac{\text{CFO} - E_A}{2\,S_c\,(1 - K)} \qquad\text{with}\qquad K = \frac{S_g}{S_c} \]

\(T_i\): maximum travel time from the induced-surge point to the arrester
\(\text{CFO}\): critical flashover voltage of the insulation
\(E_A\): arrester discharge (protective) voltage
\(S_c\): steepness of the induced surge on the phase conductor
\(S_g\): steepness of the induced surge on the neutral
\(K\): steepness ratio \(S_g/S_c\)

The numerator \(\text{CFO} - E_A\) is the voltage margin between the insulation level and the arrester protective level — a larger margin lets the arrester protect farther. The denominator \(2(S_c - S_g)\) is how fast the voltage difference grows with travel; a steeper surge forces arresters closer. So wider arrester spacing is allowed by a higher CFO, a lower arrester discharge voltage, a lower surge steepness, and stronger coupling between conductor and neutral (which reduces the differential steepness \(S_c - S_g\)).

Section 25

Arrester Spacing — With Ground at the Unprotected Pole

If a ground exists at the unprotected pole, part of the neutral surge discharges through the grounding resistance, and the analysis changes. The modified expression for \(T_i\) involves the conductor / neutral surge impedance \(Z\), the mutual surge impedance \(Z_m\), the grounding resistance \(R_2\), the time to crest \(t_f\), the time \(t_A\) to reach arrester discharge, and the modified steepness terms \(S_A\) and \(S'_g\). The full equation is more detailed, but the engineering interpretation is simple:

Interpretation

The grounding condition at an unprotected pole changes the voltage difference between conductor and neutral, and therefore changes the distance over which an arrester can provide protection.

Section 26

Worked Example — Arrester Spacing

Table 4 — Example values for the arrester-spacing calculation.

Quantity	Symbol	Value
Critical flashover voltage	\(\text{CFO}\)	250 kV
Conductor surge crest	\(e_c\)	400 kV
Neutral surge crest	\(e_g\)	320 kV
Footing resistance	\(R_1\)	20 Ω
Surge impedance	\(Z\)	450 Ω
Mutual surge impedance	\(Z_m\)	130 Ω
Arrester discharge voltage	\(E_A\)	46 kV
Time to crest	\(t_f\)	1.6 µs

\[ S_c = \frac{400}{1.6} = 250\ \tfrac{\text{kV}}{\mu\text{s}} \qquad S_g = \frac{320}{1.6} = 200\ \tfrac{\text{kV}}{\mu\text{s}} \]

\[ T_i = 2.04\ \mu\text{s} \;\;\Rightarrow\;\; d = c\,T_i = 300 \times 2.04 = 612\ \text{m} \;\;\Rightarrow\;\; \frac{612}{50} \approx 12\ \text{spans} \]

The calculation suggests protection over about 12 spans (with a 50 m span), yet the practical recommendation is an arrester at every 5th pole. The gap is deliberate margin, because the lightning current, induced waveshape, grounding resistance, arrester discharge voltage, pole geometry, neutral connections and CFO all vary, and insulation can age or become contaminated. Calculated spacing should not be used blindly.

Section 27

Comparison with Field Data

A field study on an 11 kV distribution line in South Africa provides a real-world check. The line was 9.9 km long, three-phase, flat-configured, wood-pole, with a ground wire / neutral 1 m below the phases, arranged for a CFO of 500 kV. The ground flash density was \(N_g = 7.5\ \text{flashes/km}^2/\text{year}\), over undulating grassland with few trees. Over two years, the maximum measured induced voltage was 300 kV, with on average 13.5 voltages per year exceeding 100 kV.

One correction is essential before comparing with calculation. The measured voltage is on the phase conductor, but the calculation refers to the voltage between the phase conductor and the neutral. Because the neutral is coupled to the phase and also experiences induced voltage, the insulation voltage is lower than the phase-to-remote-ground voltage:

\[ V_i = e_c - e_g \]

\(V_i\): voltage across the insulation (phase-to-neutral)
\(e_c\): voltage on the phase conductor
\(e_g\): voltage on the neutral

In the study, a 100 kV phase-conductor surge corresponds to about 89.7 kV between phase and neutral — which is why field measurements must be interpreted carefully.

Section 28

Calculated versus Measured Results

Using assumed parameters — \(R = 20\ \Omega\) and a coupling factor \(C = 0.37\) — the calculated number of voltages exceeding 100 kV ranged from about 6.5 to 10.7 per year depending on the striking-distance equation, against a measured 13.5 per year. The difference was roughly 22%–35% for most models (IEEE-92 gave a lower value).

This level of agreement is considered acceptable given the uncertainty in lightning current, return-stroke velocity, exact stroke location, line parameters, grounding resistance, coupling factor, terrain and measurement interpretation. The method predicts the right order of magnitude, which is what an engineering performance estimate needs.

Section 29

Main Conclusions — Lines Without Trees or Forests

No neutral or ground wire

The CFO to ground may be very high, so induced flashover to ground may not be the main issue — the danger is a direct stroke to the phase conductor, which can produce a very high voltage \(V = I\,Z_c/2\). This creates severe incoming surges to substations and high arrester energy duty, which is why the IEC application guide recommends protective gaps near substations to limit the incoming surge.

Neutral below the phase conductor

A neutral below the phase significantly reduces IVFOR. A CFO of about \(200\)–\(250\ \text{kV}\) can essentially eliminate induced-voltage flashover for typical cases; below about \(200\ \text{kV}\), flashovers may be mainly induced-voltage driven — particularly important for distribution lines.

Ground wire above the phase conductor

A ground wire above the phase reduces induced voltages further and lowers the total flashover rate through better direct-stroke shielding. From a lightning-performance viewpoint, the best arrangement is generally \(h_g > h_c\) — the ground wire above the phase conductor.

Section 30

Main Conclusions — Lines With Trees or Forests

The total may rise, fall or stay the same. Nearby trees cut direct strokes but raise induced flashovers, so there is no universal answer — the geometry must be studied.
Flashovers become mainly induced-voltage flashovers. With direct strokes reduced, IVFOR dominates the total in the shielded examples.
A single row of trees may be worse than a forest, because strokes are concentrated on nearby trees still close enough to induce high voltages.
Objects up to 60 m away may matter, so the study area should not be limited to the immediate clearance zone.

Section 31

Protection Conclusion

Protection against induced-voltage flashover can be provided by line surge arresters. Because induced surges are less steep than direct-stroke surges, arresters can protect longer distances and need not sit at every pole.

The worked example supports wide arrester spacing for induced surges, but this should be treated as a practical indication, not a universal design rule. The required spacing depends on CFO, arrester protective level, surge steepness, grounding, span length, insulation condition and the required reliability target. With that caution understood, a typical arrangement is:

\[ \text{practical example: arrester every 5th pole} \]

Final spacing should always be confirmed by study and utility practice rather than copied from the example.

Section 32

Practical Engineering Checklist

When assessing induced overvoltages on an overhead line, check:

Line geometry

Phase-conductor, neutral and ground-wire heights; phase configuration; span length.

Insulation

Phase-to-neutral and phase-to-ground CFO; contamination or ageing; wood-pole contribution where applicable.

Grounding

Neutral grounding arrangement; pole footing resistance; counterpoise; grounding at arrester locations.

Lightning environment

Ground flash density; lightning-current distribution; return-stroke velocity assumption.

Nearby objects

Trees, forests, buildings, nearby poles or structures; object height and distance from the line.

Protection

Arrester locations, discharge voltage and energy capability; protective gaps; substation-entrance protection.

Calculation assumptions

Striking-distance equation; Rusck applicability; CIGRE flash-incidence correction; whether tree/forest effects are included.

Section 33

Summary of Key Equations

Equation Summary

Total flashover rate

\( \text{Total} = \text{BFR} + \text{IVFOR} \)

Direct-stroke voltage

\( V = \dfrac{I\,Z_c}{2} \)

Critical direct current

\( I_c = \dfrac{2\,\text{CFO}}{Z_c} \)

Direct-stroke flash rate

\( \text{BFR} \approx N_g L\,(2.8\,h^{0.6}) \)

Incremental IVFOR (forest)

\( \Delta\text{IVFOR} = 2 N_g L (X_m - D_g) f(I) \)

CIGRE line incidence

\( N_L = N_g\,\dfrac{28\,h^{0.6} + S_g}{10} \)

IVFOR correction

\( \text{IVFOR}_{\text{corr}} = \text{IVFOR}\,\dfrac{N_L(\text{CIGRE})}{N_L(\text{calc})} \)

Arrester travel time

\( T_i = \dfrac{\text{CFO} - E_A}{2(S_c - S_g)} \)

Protected distance

\( d = c\,T_i \)

Section 34

Overall Final Understanding

Nearby objects change lightning performance in two opposite ways. They may shield the line from direct strokes, reducing \(\text{BFR}\); but they may receive strokes close to the line, increasing \(\text{IVFOR}\). The total flashover rate \(\text{BFR} + \text{IVFOR}\) may therefore rise, fall, or stay almost unchanged.

For unshielded lines, direct strokes and BFR may dominate. For lines near trees or forests, induced-voltage flashovers may dominate — sometimes 76% to 98% of the total. A ground wire above the phase conductor gives the best overall performance, because it reduces both direct-stroke exposure and induced-voltage stress; a neutral below the phase reduces induced voltage but gives little direct-stroke shielding. Surge arresters protect effectively against induced overvoltages and, because induced surges are less steep, may be spaced more widely — with suitable margin.

Table 6 — How Part One and Part Two fit together.

Item	Part One	Part Two
Main focus	Basic induced-overvoltage mechanism	Effect of nearby objects and protection
Main flashover driver	Nearby ground strokes	Strokes to trees, forests or nearby objects
Main equation logic	\(V_c\), \(X_m\), \(I_{sc}\), IVFOR	Modified exposure regions and total flashover rate
Main practical issue	Distribution-line insulation stress	Natural shielding may reduce BFR but increase IVFOR
Main protection topic	Neutral / ground wire and insulation stress	Arrester spacing and field comparison

The central conclusion

Lightning performance cannot be judged by direct strokes alone or induced voltages alone. The total depends on the interaction between line geometry, insulation level, grounding, nearby objects, shielding and arrester protection.

Reader should remember

Nearby objects can reduce direct strokes to the line but increase induced-voltage flashovers. The final result depends on geometry, so trees and forests cannot automatically be treated as either good or bad.

This is Part Two — the final part of the two-part self-study series on lightning-induced overvoltages. Part One covers the Rusck equation, the flashover condition, the maximum distance \(X_m\), the critical current \(I_{sc}\) and the IVFOR integral.

Section 35

Glossary

Table 5 — Key terms used on this page.

Term	Meaning
Natural shielding object	A tree, forest, building or structure that intercepts strokes that would otherwise hit the line.
IVFOR	Induced-Voltage Flashover Rate, normally per 100 km-years.
BFR	Backflashover / direct-stroke flashover rate.
\(S_{12}\)	Horizontal distance from the line to the trees or forest.
\(h_T\)	Height of the trees or forest.
\(h_{gc}\)	Height of the uppermost line conductor, \(\max(h_c,h_g)\).
\(D_g\)	Ground-stroke boundary distance without nearby objects.
\(D'_g\)	Modified ground-distance boundary under tree shielding.
\(X_m\)	Maximum distance at which the induced voltage still equals the CFO.
Striking distance	Distance at which a downward leader attaches to a conductor, tree or ground (\(r = aI^{b}\)).
Surge steepness	Rate of rise of the surge voltage (kV/µs); controls arrester spacing.
\(E_A\)	Arrester discharge (protective) voltage.
CIGRE \(N_L\)	Accepted empirical number of lightning flashes to the line.

Lightning-Induced Overvoltages: Nearby Objects, Trees & Protection