# Pacific Gas & Electric (PCG)

## PCG»Topics» UCLs, LCLs, UWLs and LWLs for Index 3 when Number of Lines is less than or equal to 125 and greater than or equal to six.

This excerpt taken from the PCG 10-K filed Feb 18, 2005.

UCLs, LCLs, UWLs and LWLs for Index 3 when Number of Lines is less than or equal to 125 and greater than or equal to six.

The UCLs, LCLs, UWLs, and LWLs for the control charts for each voltage class shall be based on exact binomial probabilities for those voltage classes having equal to or more than six but less than or equal to 125 median transmission lines per year.

A customized macro and a statistical software package approved by the ISO creates the proportion control charts. The macro determines the control limits and use of the exact binomial or the normal approximation to the binomial for computing the control chart limits. This macro ensures the UCL and LCL contains about 99.5% and the UWL and LWL contains about 95% of the binomial distribution. The percentile values of the UCL, UWL, LWL, and LCL are respectively 99.75%, 97.5%, 2.5%, and 0.25%.

The UCL, UWL, LWL, and LCL are calculated using the following formulas:

UCL = (X1+ (P2 - P1)/(P3 - P1))/ n

UWL = (X1+ (P2 - P1)/(P3 - P1))/ n

LWL = (X1+ (P2 - P1)/(P3 - P1))/ n

LCL = (X1+ (P2 - P1)/(P3 - P1))/ n

Where

62

P2
=  A cumulative binomial probability equal to the 0.9975, 0.9750, 0.025, and 0.0025 values used respectively in the UCL, UWL, LWL, and LCL above formulas (i.e. P2 = 0.9975 in the UCL formula and = 0.025 in the LWL formula)

P1
=  A cumulative binomial probability that if not representing the percentile value is representing the percentile value that is less than and closest to the 99.75, 97.50, 2.5, and 0.25 percentile values used respectively in the UCL, UWL, LWL, and LCL formulas(e.g. if P1 = 0.74 and is closest to the 99.75 percentile value and represents the 99 percentile then P1 = 0.74 should be used in the UCL formula).

P3
=  A cumulative binomial probability that if not representing the percentile value is representing the percentile value that is greater than and closest to the 99.75, 97.50, 2.5, and 0.25 percentile values used respectively in the UCL, UWL, LWL, and LCL formulas(e.g. if P3 = 0.82 and is closest to the 99.75 percentile value and represents the 99.85 percentile then P3 = 0.82 should be used in the UCL formula).

X1
=  The number of lines with no outages associated with the P1 cumulative binomial probability values used respectively in the UCL, UWL, LWL, and LCL formulas.(e.g. If P1 = 0.74 and represents the 99th percentile for the case where 78 lines didn't have any outages then X1 = 78 should be used in the UCL formula).

n
= The median number of lines that are in service in a given year. This number remains the same in each of the UCL, UWL, LWL, and LCL formulas

More information on the calculations of the proportion control chart limits is in the current ISO Transmission Facility Availability Performance Monitoring System Handbook.

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