CHAPTER 9: THEORY OF OPERATION OVERVIEWC70 CAPACITOR BANK PROTECTION AND CONTROL SYSTEM – INSTRUCTION MANUAL 9-59Substituting these results into the first equation, the initial operating signal is:Eq. 9-12The match factor setting kA is chosen as:Eq. 9-13Therefore, as can be seen from the previous two equations, the initial operating signal is zero.9.1.3.3 SensitivityNow consider the consequences of an element failure in the upper sub-string of leg A, making a small capacitance changein C1A . The effect on the operating signal can be calculated by taking the derivative of equation 9-12 with respect to CA ,holding C 2A constant. In the general case, the derivative of the absolute value function is messy, but in our case where theinitial value is zero, the derivative of the absolute function is simply the absolute value of the derivative of its argument. Thederivative is thus:Eq. 9-14Substituting equation 9-13, we have:Eq. 9-15This can be written as:Eq. 9-16In this equation, ΔC A (pu) is the capacitance change as a per-unit of the leg capacitance, VSpg is the system phase-to-ground voltage, and VOP(1A) (pu) is the operating signal resulting from the failure in the upper sub-string. Both voltages arein per-unit of nominal bus phase-to-ground voltage, so V Spg can be taken as 1 when the system is normal (not faulted).Note however that under external fault conditions, sensitivity can be much different from the non-fault sensitivity.Instead of a failure in the upper sub-string, now suppose a failure in the lower sub-string. The same approach is used,except to make the algebra easier, the initial operating signal is converted to be in terms of C1A using:Eq. 9-17
