Unbalance Calculation for Fuseless and Unfused Banks

Short answer

A bank without fuses does not isolate the damaged element: it leaves it shorted. The voltage the element used to carry is shared among those in series with it, which end up more stressed. The unbalance protection must trip before that overvoltage exceeds what the elements can withstand.

IEEE C37.99-2000 covers two types of bank without fuses. In the fuseless bank (clause 8.5) the units are connected in series strings from phase to neutral. In the unfused bank (clause 8.6) the units are grouped as in a conventional bank, but without fuses. Both use per-unit equations similar to those of fused banks.

Fuseless bank

How many elements are in series

The calculation depends on E, the total number of elements in series in a phase-to-neutral string. It is the number of series units multiplied by the series groups of elements in each unit. A string of six units with eight groups each has 48 elements in series. If one is shorted, the remaining 47 carry 48/47 of the previous voltage.

If the manufacturer does not supply that figure, the standard allows an estimate. Each element has a rated voltage of 1800 to 2400 V. Divide the unit voltage by 2400 and round up. A 7960 V unit would have four elements in series. The standard points out that, if the real unit has five, the setting made with four still protects.

The equations

Table 10 of the standard calculates, for each number e of shorted elements in a string:

• Capacitance of the affected string: Cst = E / (E − e)
• Capacitance of the affected wye: Cy = (Sl − 1 + Cst) / Sl
• Capacitance of the phase: Cp = (Cy × Sl + Sp − Sl) / Sp
• Neutral-to-ground voltage: Vng = G × (1 − 3 / (2 + Cp))
• Voltage of the affected phase: Vln = 1 − Vng
• Voltage across the remaining elements: Ve = Vln × E / (E − e)

Here Sp is the total number of strings per phase, Sl the strings in the affected wye, and G is 0 with ground or 1 without ground. The standard assumes that all faults occur in the same string. If a string has a different rating, it is counted as a fractional string: a string of 600 kvar units, next to another of 400 kvar, counts as 1.5 strings.

Table 11 gives the protection signals: affected wye current, phase current, change in ground current, neutral current between wyes and difference current between equal wyes.

A worked example

A string of 54 elements in series (E = 54), single-wye grounded bank. Here Vln is 1 and the voltage on the remaining elements is E / (E − e):

With the fifth element, the remaining ones exceed 110 %. The trip must sit between the signal with four and the signal with five. If the bank is ungrounded, the neutral shift means the affected phase receives less voltage (Vln below 1) and the remaining elements rise less.

Trip and alarm criteria

• Trip. The voltage on the remaining elements must not exceed what industry standards or the manufacturer recommend. The standard places it halfway between the signal with four and with five elements in the example, or at the unbalance that 110 % produces.
• One whole unit. To avoid case rupture on a terminal-to-case fault, the bank must trip on the loss of elements equivalent to one shorted unit. With more than 10 units per string, that trip happens below 110 %.
• Alarm. Above the natural errors, so that it operates with the first or the second element. The highest reliable sensitivity gives the best chance to repair in a scheduled outage.

To understand the design, see the fuseless bank; the other, without fuses but with parallel groups, is the unfused bank.

Unfused bank

How it is calculated

Here what is counted is the number e of shorted element groups inside a unit with Su series groups. Table 12 calculates:

• Capacitance of the unit: Cu = Su / (Su − e)
• Capacitance of the parallel group: Cg = (Pa − 1 + Cu) / Pa
• Capacitance of the affected wye: Cs = S × Cg / (Cg × (S − 1) + 1)
• Capacitance of the phase: Cp = (Cs × Pa + Pt − Pa) / 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 on the remaining elements of the unit: Ve = Vcu × Su / (Su − e)
• Highest voltage on other units, with several series groups: Vh = (S − Vcu) / (S − 1)

Capacitance rises with each shorted group, so the voltage on the phase and on the affected unit falls, while the remaining elements of the unit rise.

A worked example

A grounded wye bank, S = 2 series groups of units, 6 parallel units per group and 4 series groups of elements in each unit (Su = 4):

With a single shorted group, the remaining elements are already at 130 %. The other units are still at 103 %. The example shows why, in an unfused bank, the first shorted group can be the one that decides the trip.

Trip and alarm criteria

• Trip. The standard says that, for simplicity, it may be set to trip on the first shorted group. If calculated, the limits are the maximum the manufacturer recommends for the remaining elements and 110 % for the healthy units.
• Element capability. If the manufacturer does not supply it, the standard says that a value near 125 % is reasonable when a breaker without restrikes or arresters of 2 per unit or less are used. An energization voltage of 2 per unit on an element at 125 % gives 2.5 per unit, which a standard unit must withstand.
• Alarm. Normally between 50 % and 75 % of the signal of one shorted element.

The calculations for banks with external and internal fuses are in unbalance calculation with external fuses and unbalance calculation with internal fuses.

Timing and resetting

For both types, the standard gives an intentional trip delay of 0.01 to 0.05 s, which with the lockout relay and the breaker gives total times of about 0.1 s. The alarm carries a delay of about 10 s.

A bank is not reset or rebalanced without first removing and replacing every unit with failed elements. If any unit with one or two shorted elements is left in, the trip must be lowered so that its elements do not exceed their capability.

The calculated signal is compared with the inherent unbalance and measurement error; alarm and trip times are covered in alarm and trip times of unbalance protection, and the function is the capacitor bank unbalance protection (ANSI 60).

Common mistakes

• Miscounting the elements in series. It changes the whole table; confirm it from the nameplate or with the manufacturer.
• Applying the fuseless equations to an unfused bank, or the reverse. Each one counts different things.
• Forgetting strings of different ratings. They are counted as fractions of a string.
• Resetting without changing units. The bank keeps running with overstressed elements.

What we do at Tiza Group

In our engineering design and studies service we calculate the unbalance table for your bank, fuseless or unfused, 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
• Fuseless capacitor bank: series strings and unbalance
• Unfused capacitor bank: series-parallel without fuses
• Unbalance calculation for a bank with external fuses
• Unbalance calculation for a bank with internal fuses

References

• IEEE Std C37.99-2000, IEEE Guide for the Protection of Shunt Capacitor Banks, clauses 8.5.1 to 8.5.4 and 8.6.1 to 8.6.4, Tables 10 to 13.
• IEEE Std 18-1992, IEEE Standard for Shunt Power Capacitors.

Frequently asked questions