Approach 2:

Including existing load profiles – but how?

The potential saving therefore rises and falls with the load profile. The duration of the – usually unsteady – load is to be integrated over the square of the load current, to obtain a measure of whether there is a potential for a massive saving that is better exploited now than later, or whether this idea is absurd. The task now is to find out whether this finding can be confirmed approximately using existing standard load profiles and whether these can be used for a corresponding, rather more accurate estimate. The question therefore is whether the above method of the geometric average of two definable unreal extreme scenarios can be used as an adequate approximation of the undefinable real scenario sought. This will be investigated below.

“Annual RMS value”

Table 6: Data underlying the calculations according to Table 7
Table 6: Data underlying the calculations according to Table 7

The next issue that arises is that the load current is not only distributed unevenly temporally over the year, but also spatially over the individual circuits. If the calculation is made on the basis that distribution is always evenly over the final circuits, an embellished result is obtained – which nonetheless should be determined here as a benchmark.

Conversely, all final circuits are hardly ever fully loaded in any system, neither simultaneously nor consecutively, let alone continuously. However, if we assume that in each final circuit the highest load occurring in the year is also the highest permissible, we obtain an overly pessimistic result for the lost energy. This should therefore serve as a benchmark for the “opposite corner”. Up to here, the method still resembles Approach 1.

Table 7: Annual losses of cables and lines depending on selected standard load profiles – in the upper half the maximum annual load = Iz of the relevant line and installation method to VDE 0298-4; in the lower half the mean annual load (for a dwelling – column H0) was selected such that the peak load matches the geometric mean from Approach 1
Table 7: Annual losses of cables and lines depending on selected standard load profiles – in the upper half the maximum annual load = Iz of the relevant line and installation method to VDE 0298-4; in the lower half the mean annual load (for a dwelling – column H0) was selected such that the peak load matches the geometric mean from Approach 1

However, the standardised load profiles are now assumed. While these refer to the typical annual course of the active power drawn or the active energy (work) of a particular type of load, it is assumed here that this power is drawn with a power factor of λ = 1 (P = S) and at nominal voltage. This allows the power values of the load profile – the 15-minute averages of the active power – to be converted directly into the current values in the line. The inaccuracy arising from the fact that the line loss, of course, depends on the apparent power, or more precisely on the apparent current, while the electricity bill only shows the active energy, must be borne here. Other uncertainties in this assessment are even greater in any case.

It is also assumed that a cable with a cross-section of 3*1.5 mm² or 5*1.5 mm² in installation method B1 can be loaded at 17.5 A or 15.5 A, respectively (Table 6). It is now subjected to the various standardised load profiles such that the highest load current occurring in a given year corresponds to the maximum permissible operating current Iz of this cable in the relevant installation method.

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Annual peak factor

One of the difficulties is that the terms – including the symbols – have already been defined for the “normal” RMS value, which is not, however, meant here. Symbols that are as similar as possible are therefore invented, albeit without reassigning existing ones, hoping the reader does not become too confused.

The maximum current Iz from the standard, divided by the average annual current Iz_mean (converted from the load profile), yields the so-called “annual peak factor” FP = Iz/Iz_mean. In other words: If the maximum current occurring in a given year equals the maximum permissible current Iz for that cable, then the mean annual current in the respective load profile is equal to Iz_mean. So the factor FP is determined from the respective tables of the annual load profiles, which cannot be shown again here – not even in extracts – due to their excessive size (a year in 15-minute averages gives 35,040 lines).

Annual form factor

This is not, however, sufficient to take the discontinuous current amplitude into account appropriately. Since the heating increases not linearly but as the square of the current, the divergence contained in the annual load profile has to be factored in here. Exactly like when determining the TRMS value of an alternating current over a period depending on the shape of the curve, the 35,040 current values must be squared, the squares added together and the root derived from the total. This was also done in the support table that is not shown here, and was taken into account in the calculations in Table 7 and Table 11 in the form of the form factor FF, which is also listed there informatively.

For a “base load profile” characterised by a load that is constant throughout the year, you would get back precisely to where you came from; the calculation process would be superfluous (FF = 1; FP = 1) – as for the TRMS value of a smooth direct current. However, the less evenly the current amplitude spreads across the year, the higher the actual heat dissipation will rise above the value that would be produced with a constant current corresponding to the arithmetic annual average Iz_mean.

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Anomalies, boundary conditions, assessment

Several minor influencing factors were not taken into account, since even the major ones are coined by simplifications and assumptions.

  • Generally, for instance, the cold resistances of the lines in question from VDE 0295 were taken into account, even though the line naturally heats up during operation. The average annual temperature is, however, significantly below the maximum permissible temperature, which only occurs during peak load. The inaccuracy taken into consideration is correspondingly small.
  • As previously with the washing machine, the fact that the heating of the upgraded line is slightly lower than that of the line replaced was therefore also ignored. The error that thus occurs is extremely small.

It may seem rather surprising at first glance that the annual form factor FF is particularly large in the case of night storage heating. However, this load only exhibits a high power consumption for very short periods of time:

  • in winter only at night,
  • in summer not at all,
  • in the transitional period not for the whole night at reduced power, but at full power for just part of the night.

Night storage heating therefore represents an extremely discontinuous load. The mean power is less than ⅛ of the peak value. However, this is also due to the existence of this standardised profile, which only encompasses a single type of load. All other profiles, whether household, industry or agriculture, contain a mix of appliances that serve different purposes. If, for instance, there were a “standard load profile electric cooker”, its annual form factor would be even higher.

Results

Let us then assume that a final circuit is loaded in accordance with the load profile of the system which it is part of. The highest occurring current is equal to the maximum permissible current Iz. For a cable of 3*1.5 mm² cross-section in installation method B1 with normal load profiles for homes, industry and agriculture, this produces lost energies of around 12 kWh a year for every metre of cable length. The results are similar for a five-conductor cable with three loaded wires (Table 7, upper section “Permissible load”). While it is true that there is power loss in three instead of just two wires here, this is less per wire in accordance with the lower current carrying capability.

If the cross-section is increased by just one standard step to 2.5 mm², the lost energy drops to values of around 7 kWh/(m*a). The additional costs may pay back in around four months!

For the reasons given, night storage heating represents an exception. Here, the payback periods are around 2 years. Nonetheless, for this reason upgrading the supply line would be of limited use, since here electrical energy is used to generate heat (the sense of doing this could also be “placed under general suspicion”; unless you live in Norway). Moving the release of heat from night to day justifies an economy measure only to a – very – limited extent: Almost as an aside it should also be noted here that the differences between electrical heating tariffs and the normal household tariffs are no longer as great as they once were. If you want to save not (just) money but (also) fuel and CO2, the equation no longer applies that power stations are fully utilised at peak load times at midday or in the early evening while being underutilised at night. That a “green” supply is not in principle provided at night is only true of solar energy; with wind, this is merely dependent on the weather, in other words on chance. This reduces the lump-sum diurnal spreading of the value of electrical energy as well as the prices to be paid for it (which does not necessarily correlate) in the direction of stochastic parameters.

A further “outlier” is load profile G1 for a facility that only operates on working days – and then only during business hours. Because of the relatively short loading times, to some extent the same applies here as for night storage heating. The payback periods for G1 are just 1/3 as long as with HZ0, but around 3 times as long as in the other profiles.

The corresponding observation can be made for the other industrial and agricultural profiles, but there, too, the results tend to correspond to the selection made here by way of example.

When “playing” with the Excel table it can also be seen that even an upgrade from 1.5 mm² to 16 mm² is worthwhile! Even then, the payback periods are only around 1.5 years, for profile G1 barely 4 years and for profile HZ0 around 13 years. On the other hand, in profile G3, which approximates base load, the upgrade pays back in just 0.6 years.

However, as mentioned, this only applies assuming that the valid standardised load profile for a system equally applies to each final circuit in the system and the maximum permissible current is reached at least once a year. However, this is hardly ever the case; simply because cables and lines are only available with specific standard cross-sections and the next larger one must be selected. Furthermore, reserve and safety factors are always built in. If the cable is only utilised up to half its current carrying capability in practice, the heating drops to a quarter, and the payback period rises e.g. from six months to two years. And yet: What is two years in the life of an installation cable? This generally lasts as long as the entire building. So one number larger is always worthwhile!