3D Modeling of Armor Losses in High-Voltage Cables

Wires and cables make up a global industry worth hundreds of billions of dollars. In fact, Infinium Global Research reports that the cable market is poised to reach $220 billion by 2025 (Ref. 1). A major portion of the rapidly growing cable industry’s revenue is from installation, maintenance, and development. For instance, the NorNed cable, a joint cable project connecting the power grids of Norway and the Netherlands, cost roughly €600 million (approx. $700 million USD) to install, and that was back in 2008 (Ref. 2).

When cables of this magnitude need to be repaired or replaced, it can also be expensive. A 2010 report from the SubOptic submarine cable conference estimated that submarine cable repairs can cost more than $12,000 a day, and over $1 million per project. (Ref. 3) Since cable costs are so large, getting a return on investment also takes many years.

Apart from being major project investments, cables are demanding to test experimentally as well (Figure 1). In fact, cables researched by NKT, a global cable supplier, have been tested experimentally for many years, and it has been both time and resource consuming. “Cable losses are a complex thing to measure,” says Ola Thyrvin, senior analysis engineer at NKT.

Figure 1. High-voltage cables connect the world. They can also be expensive to maintain and difficult to analyze. Image courtesy NKT.

One tool that can help in this regard? Electromagnetics modeling, which enables the NKT team in Karlskrona, Sweden to test cable designs virtually, visualize how different cable parameters affect armor losses, and predict cable performance in different installation conditions (Figure 2).

Figure 2. A 3D cable modeled in the COMSOL Multiphysics® simulation software.

With cable costs as steep as they are, designers can, with simulations, analyze the cable losses and reduce the amount of required conductor size and, thereby, cable cost. However, they need to be absolutely confident that their modeling tools can perform the analyses they need — and give them the correct results, since the cable cannot be measured until it is manufactured after the design is sold.

Bypassing Limitations in Cable Modeling

One issue when it comes to testing cable designs is that the standards are a bit outdated. In fact, some IEEE and IEC standards for cables are still based on analytical expressions that were derived about 80 to 100 years ago and simplified to enable hand calculations. Over the last decade, several publications have provided measurements that show that the formulas in the standard overestimates the armor losses.

For some cases, the losses are around 50% of what the IEC standard gives. As the possible current a cable can carry is limited by a maximum allowed conductor temperature, a reduction of the losses enables a possibility to reduce the conductor size. A reduced conductor size means less copper or aluminum, which are expensive metals, and therefore cost savings for the cable project.

It is possible to measure the armor losses accurately with the methods developed ten years ago, but it requires that you have the cable. Almost all high-voltage offshore cables are custom made and therefore not available to test before a project is sold and manufacturing starts, and cables need to be designed already in the tender phase. With the adoption of numerical analysis, the study of cables and armor wires became easier, but still left a lot to be desired. In fact, the first 3D models of a cable were created less than a decade ago.

Even more inhibiting: Models of this kind, up until recently, could take several days to a few weeks to run on a supercomputer. Advancements in both computer hardware and modeling techniques have made cable design and analysis quicker, easier, and more robust. A cable model that used to require a supercomputer, for example, can now be run on a standard laptop and take minutes instead of days. These enhancements have opened up new possibilities for NKT's research.

Modeling an Armored Cable in 3D

Part of the work at NKT in Karlskrona, Sweden involves the electrical simulation of cables, as well as calculating their temperature distribution and corresponding losses. In an armored cable, it is difficult to calculate the losses in the magnetic steel armor. This is because of a complex interaction between active and passive conductors, combined with nonlinear material properties (hysteresis) and temperature dependence. Further, the geometry of an armored cable model (Figure 3) includes small, detailed features, like the narrow gaps between the armor wires, leading to a large number of mesh elements, long computation times, and increased memory requirements.

Figure 3. The 3D cable model geometry, which includes the basic features of an armored submarine cable; the main conductors, the screens, and the armor.

To address these challenges, NKT set to find out if they could use a coarse mesh for their cable model (Figure 4) while still accurately describing the nonlinear magnetic behavior of the steel material, a strongly magnetic soft steel with high permeability and large hysteresis losses.

Figure 4. Different degrees of meshing for the cable model, from one to four mesh elements per wire diameter from left to right.

The group turned to the COMSOL Multiphysics® simulation software, as well as the add-on AC/DC Module, which is especially suited for cable analysis. This software enables the 3D modeling of an armored cable in order to analyze the magnetic fields and compute the armor losses (Figure 5).

Figure 5. The geometry of the 3D magnetic flux in the air gap between the conductors in the armored cable model.

Going back to the computational expense of cable modeling, Ola Thyrvin mentions a feature from the COMSOL® software that he found particularly helpful: the Periodic boundary condition, which enabled the team to model a small piece of the cable, keeping it as short as possible. The reduced size of the model saves on computational time and memory requirements that are specific to this application area, while also ensuring that all of the relevant physics are captured in the model. “The model needs to capture one conductor meeting one armor wire up until they meet again,” says Thyrvin. Another memory-saving modeling approach is the use of infinite elements, which lets the designers include a sufficient amount of air around the cable in the modeling domain, while still limiting the required mesh and memory.

Increased Performance, Accurate Computations

The NKT team's modeling approach involved three main stages. First, they set up a current-driven model with predefined temperatures. The current is not affected by the cable impedance or variations in temperature and is instead controlled by the system load. Next, the team calculated the eddy current losses as losses that are induced by local currents flowing in the armor wires at the predefined temperature.

They found that the losses are dominated by the screening currents around the armor wire perimeters, in the wire sections near the phase conductors. Third, they calculated the magnetic hysteresis losses by integrating a function of the magnetic B-fields over the armor wire volume (Figure 6).

Figure 6. Magnetic properties of the cable calculated from the hysteresis curves.

Figure 6. Magnetic properties of the cable calculated from the hysteresis curves.

In their 2019 paper “Fast Modelling of Armour Losses in 3D Validated by Measurements” at the 10^th International Conference on Insulated Power Cables in 2019 (Ref. 4), NKT demonstrate additional ways to increase performance without significantly harming accuracy. First, even without resolving the skin depth in the armor, they have discovered that with the proper geometric correction factors and fitted material parameters, it is still possible to compute realistic loss values — typically more realistic than what the IEC standard provides, and in several cases, within the measurement accuracy.

Furthermore, while running the model with a coarse mesh, they used a uniform, real μ-value that has been fitted to experimentally obtained material data by considering only the average H-field in the armor wire, not the local one. Therefore, the permeability is not nonlinear or imaginary. Instead, it is set to the correct value for the average armor wire H-field, given the particular operating point of the cable.

Once the solution has been obtained, the losses can be computed afterward as a postprocessing step. This is because from measurements, they know precisely what losses they get for a certain field intensity. So in their models, the hysteresis losses are not electrically linked to either the voltage or the current response of the cable.

To get the correct effective permeability, the team ran the 3D model for different μ-values for each modeled current. They calculated and averaged the H-values from each solution and took into account the reduction in armor wire cross section, when using coarse meshes. Then, the μ-values and average H-values were plotted on the measured μ_r(H) wire curve. The team found that higher μ-values meant lower average H-values in armor, and vice versa. Finally, the intersection of the curves with the measured one gave the correct effective value at the cable's operating point (Figure 7).

Figure 7. Modeled μ(H_ave) for three different currents in a cable design, as well as the measured μ-H curve for the armor wire.

Validating the Cable Model Results

All of the modeling in the world will not matter, however, if the results of the model do not accurately represent the physics of the device in reality. To make sure that the simulation results for the cable analyses are accurate, Thyrvin and his team validated them with the existing cable data. When computing the armor losses of the cable, they found that the modeled results were within 3% of the losses measured from cables experimentally (Figure 8). While that sounds impressive in itself, these results are actually more accurate than the IEC standard for the type of cable being modeled, in which the total loss differs between 10 and 30% compared to measurements.

Figure 8. Validated results of IEC, measured, and modeled losses in five cable designs.

Putting Trust in Cable Analyses

The validated results of the 3D cable model proved to NKT that simulation is a reliable and trustworthy way to study cable designs. This knowledge has had far-reaching effects for the organization. For one, they feel confident studying cables without comparing to measurements each time, because they have already confirmed that the simulations are accurate based on the previously validated results. “We can now simulate instead of measure,” says Thyrvin. “You can simulate before making, but you can't measure before making.” Now, with simulation software, NKT knows how large the losses are in a cable before manufacturing, based on the simulation analyses.

References

1. Wire and Cable Market (Type - Wire, and Cable; Voltage Type - Low Voltage, Medium Voltage, and High and Extra High Voltage; Applications - Power Transmission and Distribution, Transport, Data Transmission, Infrastructure): Global Industry Analysis, Trends, Size, Share and Forecasts to 2024”, Infinium Global Research, 2020. https://www.infiniumglobalresearch.com/ict-semiconductor/global-wire-and-cable-market
2. M. Ardelean and P. Minnebo, “HVDC submarine power cables in the world”, Institute for Energy and Transport, pp. 50–51, 2017. https://op.europa.eu/en/publication-detail/-/publication/78682e63-9fd2-11e5-8781-01aa75ed71a1/language-en
3. G. White, “Insurance and Risks in the Underground Cable World”, SubOptic, 2013. http://www.suboptic.org/wp-content/uploads/2014/10/PD05Poster131.pdf
4. D. Willen, C. Thidemann, et al., “Fast Modelling of Armour Losses in 3D Validated by Measurements”, 10th International Conference on Insulated Power Cables, C7-4, 2019. https://www.jicable.org.

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