Cables in air extended

Introduction

The thermal rating of cables in parallel arrangements is a critical consideration in electrical engineering design. For decades, the IEC 60287-2-2 standard has provided engineers with reliable methods to calculate cable ampacity for grouped cable systems. However, this standard has a significant limitation: it only covers configurations with up to three groups of cables arranged horizontally and two groups arranged vertically.

The extended module introduces a solution to this constraint, extending the capability to handle up to 10 groups horizontally and 6 groups vertically. At the same time more types of cable configurations than the threecore and trefoil single-core cables as covered in the IEC standard can be analyzed. The module covers 2-/3-/4-/5-core cables, two-cable groups (widely used in large-scale PV installations), trefoil arrangements, flexible duct configurations and all with the same extended parallel system capability. In addition GIL/PAC are also covered for a single group.

Cable Rating of Cables in Air

The IEC 60287-2-2 Standard and its Limitations

The IEC 60287-2-2 standard provides thermal proximity correction factors (denoted as $h_{lg}$) for multiple cable groups arranged in various configurations. These correction factors account for the thermal interaction between adjacent cable groups, which reduces the heat dissipation capability of each individual group.

The standard defines specific arrangements:

• Horizontal arrangements: Side-by-side configurations with up to 3 multicore cables or 3 trefoils
• Vertical arrangements: Stacked configurations with up to 3 multicore cables or 2 trefoils

For each configuration, the standard provides empirical correction factors based on the spacing ratio ($d_o/D_o$, where $d_o$ is the spacing between groups and $D_o$ is the cable outer diameter). These factors are derived from experimental data and theoretical calculations limited to the configurations listed in the standard.

Beyond the Standard: A Comprehensive Expansion

The extended module expands IEC 60287-2-1 for a single group and IEC 60287-2-2 for multiple groups in several aspects:

The improved calculation extends from the standard's 3 horizontal / 2 vertical limitation to:

• Up to 10 horizontal groups
• Up to 6 vertical groups

This enables analysis of:

• Large-scale solar farms with dozens of parallel cable strings
• Industrial installations with multiple cable bundles
• Data center cooling systems with extensive cable routing
• Power stations and substations where multiple parallel cable systems are the norm

The improved calculation supports an unprecedented range of cable configurations:

• Multicore cables: Traditional 2- and 3-phase cables as covered by IEC but now extended to allow for support many more parallel systems
• Trefoil arrangements: Three cables arranged in a triangular configuration as covered by IEC, the traditional approach for three-phase systems
• Two-cable groups: Particularly important for large-scale photovoltaic (PV) plant installations where two-cable arrangements are standard practice
• Groups of single-core cables: Individual conductors that can be arranged in groups, offering maximum flexibility for large installations
• 4-/5-core cables: Cables with four conductors (three phases and neutral) and five conductors (three phases plus neutral and earth conductor) are possible too
• GIL/PAC: Instead of cables you can also model gas insulated lines (GIL) and pressurized air cables (PAC, e.g. Hivoduct)

The extended module introduces sophisticated duct modeling:

• Separate ducts: Each cable or cable group in its own duct, allowing independent thermal analysis
• Common ducts: Multiple cables (2 or 3) of the same system within a single duct, reducing installation complexity while maintaining accurate thermal calculations
• Multiple systems in duct: Multiple systems with up to 12 separate cables within a single duct can be visualized accurately and calculated correctly

This flexibility is crucial for modern installations where cost and space constraints often dictate duct sharing arrangements. The ducts themselves are treated as thermal entities in the calculation, providing realistic representation of the thermal environment

Methodology

The extended module addresses the limitation of standard data through a two-stage curve fitting and extrapolation approach. The methodology works as follows:

1. Calculate IEC-compliant values for the first three groups
2. Estimate the fourth value with a constrained anchored model
3. Fit a logarithmic curve to all four points and extrapolate

1) Calculate IEC-Compliant Values

The improved calculation first computes the thermal correction factors for the first three parallel groups using the standard IEC 60287-2-2 methods. These three data points serve as the foundation for extrapolation, ensuring that the calculations remain grounded in the proven standard.

The method also detects when thermal proximity effects are negligible and treats this as a linear (constant) case, using the average of the three points rather than forcing a curve fit that would produce artificial extrapolation artifacts.

2) Estimate the Fourth Value with a Constrained Anchored Model

Fitting a three-parameter curve to only three data points is an exactly-determined problem (zero degrees of freedom), which leads to an unstable and unreliable fit. To resolve this, the fourth value is first estimated independently using a dedicated two-parameter anchored model before any multi-point curve fitting is performed.

The anchored model is defined as:

where:

• $y_1$ = the calculated value for a single group (the anchor point)
• $a \geq 0$, $0 < b \leq 1$ = fitted coefficients
• The minus sign applies when the output is a current rating (decreasing with more groups)
• The plus sign applies when the output is a temperature (increasing with more groups)

This model has two important properties. First, it passes exactly through the first data point at $x = 1$, since $x^b - 1 = 0$ at $x = 1$. Second, constraining $b \leq 1$ ensures the curve decelerates — each additional group has less impact than the previous one — and prevents the model from blowing up for large $x$.

A safety check is applied after the fit: if the estimated fourth value does not respect the required direction (i.e. it is not strictly lower than the third value for current, or not strictly higher for temperature), it is corrected by mirroring the last observed step.

3) Fit a Logarithmic Curve to Four Points and Extrapolate

With the fourth value now established, a logarithmic curve is fitted to all four points (the three IEC-calculated values plus the estimated fourth). Having four points for a one-parameter model makes the fit well-posed and stable.

The logarithmic model is defined as:

where:

• $y_1$ = the anchor value at $x = 1$ (since $\ln(1) = 0$, the curve always passes exactly through the first point)
• $a$ = the single fitted coefficient
• $a < 0$: current rating decreases with each additional group, with progressively smaller steps
• $a > 0$: conductor temperature increases with each additional group, with progressively smaller steps

The logarithmic model is the physically correct choice for both output types. Its second derivative is always $-a/x^2$, which is opposite in sign to $a$. This means the rate of change always decelerates — acceleration is mathematically impossible regardless of the data. This reflects the physical reality that thermal interaction between groups saturates: the first additional group causes the largest change, and each subsequent group contributes progressively less.

Once the curve is fitted to the four points, it is used to extrapolate the ratings for groups 5 through 10 (horizontal) or 4 through 6 (vertical), providing engineers with reliable estimates for configurations previously outside the standard's scope.

Visualization

The accompanying plot provides a compelling visualization of this methodology. Let's examine its key features:

Upper Panel: Rating Extrapolation

The upper panel displays two critical curves:

Calculated Current per System (Blue Solid Dots): These three points represent the actual IEC 60287-2-2 calculations for 1, 2, and 3 parallel systems.

Extrapolation (Blue Dashed Line): This curve represents the fitted logarithmic function. It smoothly extends through the first three calculated points plus the fourth point and continues to the right, showing the extrapolated ratings for up to 10 (6) parallel systems. The first additional group causes significant thermal interaction, but the last group adds minimal thermal burden.

Extrapolated Values (Blue Hollow Circles): These points mark the calculated ratings for systems 4 through 10 (6). Notice how the reduction in current rating becomes progressively smaller with each additional group.

Lower Panel: Cable Group Visualization

The lower panel provides a visual representation of the cable arrangements, showing:

• Flexible grouping: Each position displays the actual cable configuration — whether single cables, two-cable pairs, trefoil arrangements, or multicore cables
• Progressive scaling: As the number of groups increases, the visualization demonstrates how the cables would be arranged in practice
• Duct representation: When cables are in common ducts, the visualization reflects this configuration

This visual element helps engineers immediately grasp the physical layout corresponding to each calculation point and understand how their specific cable configuration scales across multiple parallel systems.

Right Axis: Total Current

The green stepped line on the right axis shows the total current across all parallel systems. This is calculated as:

This metric is crucial for system designers, as it shows the total ampacity available from the multi-group installation. The stepped nature of the curve reflects the discrete addition of each new parallel system, while the individual system ratings (left axis) show the thermal penalty for adding more groups.

Practical Applications

Solar farms often require dozens of parallel cable strings. The extended module enables designers to:

• Model two-cable groups (standard in PV) with up to 10 parallel strings
• Accurately predict thermal behavior across the entire installation
• Optimize cable sizing for cost-effectiveness without over-conservatism

Manufacturing facilities with multiple parallel power feeds can now:

• Analyze complex duct arrangements (separate, common, or mixed)
• Evaluate the thermal impact of adding additional circuits
• Make informed decisions about cable cross-sections and spacing

Rather than applying arbitrary safety factors for configurations beyond the standard, engineers can:

• Use physics-based extrapolation grounded in the first three IEC-compliant points
• Achieve more economical designs without sacrificing safety
• Document their calculations with confidence

The improved calculation's flexibility supports:

• Multicore cables (IEC)
• Trefoil arrangements (IEC)
• Two-cable groups (not covered by IEC, PV standard)
• Single-core cables in separate ducts (not covered by IEC, typical use case)
• 2/3 single-core cables in common ducts (maximum flexibility)