Unbalance Calculation for Internally Fused Capacitor Banks

Short answer

A capacitor with internal fuses has groups of elements in parallel inside the same unit. Each element carries its own fuse link. When an element fails, its link blows and the group is left with one element fewer. The elements remaining in that group carry more voltage than before.

IEEE C37.99-2000 (clause 8.4) says that this overvoltage on the elements adds to the one carried by units with no blown links, and must be taken into account when setting the protection. The calculation gives two figures for each number of blown links: the voltage on the remaining elements and the unbalance signal the relay will see.

Notation in the standard

The standard uses these letters for the capacitor and the bank:

The design being calculated is the internally fused bank. For external fuses see unbalance calculation with external fuses, and for banks without fuses, fuseless and unfused unbalance calculation.

The equations for a wye bank

Table 6 of IEEE C37.99-2000 calculates first the group, then the unit, then the phase. In per unit:

• Capacitance of the affected group: Ci = (N − f) / N
• Voltage taken by the group, with the unit at 1: Vg = Su × N / ((Su − 1) × (N − f) + N)
• Capacitance of the unit: Cu = Su × Ci / (Ci × (Su − 1) + 1)
• Capacitance of the parallel group: Cg = (P − 1 + Cu) / P
• Capacitance of the affected wye: Cs = S × Cg / (Cg × (S − 1) + 1)
• Capacitance of the phase: Cp = (Cs × P + Pt − P) / Pt
• Neutral-to-ground voltage: Vng = G × (3 / (2 + Cp) − 1)
• Voltage of the affected phase: Vln = 1 + Vng
• Voltage across the affected unit: Vcu = Vln × Cs / Cg
• Voltage across the remaining elements: Ve = Vcu × Vg

Table 7 adds the current in the affected string (Cs × Vln), the phase current (Cp × Vln), the change in ground current [(1 − G) × (1 − Iph)], the neutral current between wyes and the difference current between equal wyes. These are the signals of the neutral unbalance, neutral-to-ground current and double-wye schemes.

For a delta the same applies: one leg of the delta is treated as one phase of a grounded wye.

A worked example

A grounded single-wye bank. Each unit has 3 series groups of 14 parallel elements. The bank has 4 series groups of units and 10 parallel units per group. With a ground connection, Vng is zero and Vln is 1. The result for each blown link in the same unit:

The neighboring units barely notice: the increase reaches 2.4 % with eight blown links. The overloaded part is the affected group. In this example the limit is not set by 110 % on the healthy units, but by the number of blown links the manufacturer allows in one unit.

How to use the result

Per clause 8.4.2, the trip must be set so that:

• The number of operated fuses in the affected unit does not exceed the maximum the manufacturer recommends.
• The voltage on the healthy units does not exceed their capability, normally 110 % of rated voltage.

If the manufacturer asks for a trip after the seventh fuse, the setting goes halfway between the signal with six fuses and the signal with seven. The alarm goes above the natural unbalance, so that it operates with the first or the second fuse. The standard places it halfway between the signal for that number of fuses and the previous one.

On timing, the standard gives an intentional delay of 0.01 to 0.05 s for the trip, which with the lockout relay and the breaker gives total times near 0.1 s. For the alarm it gives a delay of about 10 s. In a bank with internal fuses, a delay that hides system faults, which can last several cycles, is not appropriate.

Compare the signal with the inherent unbalance and measurement error before setting the alarm; the function is the capacitor bank unbalance protection (ANSI 60).

H-bridge and voltage tap

The standard includes the calculation for two other schemes using the same method:

• H-bridge (clause 8.4.5, Table 8). A current transformer in the crossbar of the H measures the difference between two halves of the bank. Any capacitance change in one of the legs changes the current in the H. The table calculates the unit capacitance, the bridge-to-neutral capacitance, the voltage of the H leg and the current in the H.
• Voltage tap (clause 8.4.6, Table 9). The voltage at an intermediate point of the phase is measured and compared with the phase voltage. The table calculates the tap voltage and its per-unit change depending on whether the affected unit is inside or outside the measured part.

In the H-bridge, the standard warns that another blown fuse elsewhere in the bank can cancel the signal. For schemes with an ambiguous indication, it asks for an alarm sensitive to the first element, sealed in and reset by hand.

The measurement schemes for these configurations are described in H-bridge unbalance and in voltage-differential unbalance in a grounded wye.

Common mistakes

• Resetting the balance without removing the damaged units. The bank is left with elements already overstressed and no margin.
• Checking only the 110 % of the healthy units. The standard also requires meeting the maximum number of fuses set by the manufacturer.
• Mixing up the number of links with the number of units. The signal gives only the total of failed elements in the unit.
• Not asking the manufacturer for the table. The standard notes that some manufacturers supply the tabulation for their bank; it is useful for checking the calculation.

What we do at Tiza Group

In our engineering design and studies service we calculate the unbalance table for your bank from the unit data and your manufacturer’s, and leave it ready for the protection settings. We work remotely, worldwide. Contact us and we will quote the calculation.

Related guides

• IEEE C37.99: shunt capacitor bank protection guide
• Capacitor bank with internal fuses: how it is protected
• Unbalance calculation for a bank with external fuses
• Unbalance calculation for fuseless and unfused banks
• H-bridge protection in a capacitor bank

References

• IEEE Std C37.99-2000, IEEE Guide for the Protection of Shunt Capacitor Banks, clauses 8.4.1 to 8.4.6, Tables 6 to 9.
• IEEE Std 18-1992, IEEE Standard for Shunt Power Capacitors.
• IEEE Std 1036-1992, IEEE Guide for Application of Shunt Power Capacitors.

Frequently asked questions