Capacity Outage Probability Table (COPT) The COPT is a table contains all the capacity states in an ascending order of outages magnitude. Each outage (capacity state) is multiplied by its probability. If the system contains identical units, Binomial distribution can be used. If the units are not identical, the procedure in the following example can be used.
Example (1): A generating system has the following units: (a) 10 MW (FOR = 0.02). (b) 15 MW (FOR = 0.03). (c) 20 MW (FOR = 0.05). It is required to build the COPT for the system. solution solution: COPT For units (a) will be as follows: cap out 0 10 15 20 25 30 35 45
probability 0.98 x 0.97 x 0.95 = 0.90307 0.02 x 0.97 x 0.95 = 0.01843 0.98 x 0.03 x 0.95 = 0.02793 0.98 x 0.97 x 0.05 = 0.04753 0.02 x 0.03 x 0.95 = 0.00057 0.02 x 0.97 x 0.05 = 0.00097 0.98 x 0.03 x 0.05 = 0.00147 0.02 x 0.03 x 0.05 = 0.00003 --------1.00000
Example (2): A generating system has the following data: (a) 2X20 MW and (b) 1X30 MW. The FOR for each unit is 0.1. It is required to establish the COPT for the system. Solution: COPT For units (a) will be as follows: cap out probability 2 0 0.9 = 0.81 20 2(0.9 x 0.1) = 0.18 2 40 0.1 = 0.01 -------1.00
COPT For units (b) will be as follows: cap out probability 0 0.9 30 0.1 --1.0
The two COPT's can be combined in one single COPT, as follows: Cap out 0 20 30 40 50 60 70
probability 0.81 x 0.9 = 0.729 0.18 x 0.9 = 0.162 0.81 x 0.1 = 0.081 0.01 x 0.9 = 0.009 0.18 x 0.1 = 0.018 --0.01 x 0.1 = 0.001 ----1.000
Loss of Load Probability (LOLP)
The LOLP is not widely used, at the present time, as a criterion for power system planning because it indicates only the probability of system failure. The LOLP can be defined as: n
LOLP =
∑ pi
( L max > C)
i=1
Loss of Load Expectation (LOLE)
The LOLE risk index is the most widely accepted an used probabilistic method in system reliability evaluation for generating systems. Two models are required and employed. One is the previously studied Load Duration Curve (LDC), and the other is the COPT. These two models are convolved (combined) in the process. The units of the LOLE is in days per year (d/y). The LOLE evaluation method is expressed in the following mathematical formula: n
LOLE = ∑ t i pi (d/y) ( L max > C) i=1
It is clear from the above load characteristics that capacity outages less than the reserve will not cause a loss of load. Consider now: Oi = the ith outage state in the COPT. pi = the probability of this ith outage. 2
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ti = the number of time units for which this outage cause loss of load. EXAMPLE (1): Consider a system having 5 x 60 MW units, each with an FOR of 0.03. The load model can be assumed as linear with maximum load (L max) of 240 MW, and a minimum load (Lmin) of 100 MW. Calculate the system LOLE.
Solution:
The COPT and expected load loss (or loss of load expected) can be done as follows: (1) (2) (3) (2x3) cap out probability time t i loss of load expected (MW) (%) -----------------------------------------------------------------------------------0 0.858734 0 -60 0.132794 0 -120 0.008214 42.86 0.352052 180 0.000254 85.71 0.021770 240 0.000004 100.00 0.000400 ----------------------1.000000 0.374222% -----------------------------------------------------------------------------------If 100% of time represents 365 days, The LOLE will be: LOLE = 0.374222 x 365/100 = 1.365910 d/y In system planning, the above figure is considered to be excessive (above normal), Therefore, a value often considered to be normal LOLE risk index is considered as 0.1 d/y.
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EXAMPLE (2): Consider now the same system, but with each unit is having an FOR of 0.01. Calculate the LOLE of the system. Solution:
(1) (2) (3) (2x3) cap out probability time t i loss of load expected (MW) (%) -----------------------------------------------------------------------------------0 0.950990 0 -60 0.048030 0 -120 0.000970 42.86 0.041574 180 0.000010 85.71 0.000857 240 0.000000 100.00 0.000000 ---------------------1.000000 0.042431% -----------------------------------------------------------------------------------If 100% of time represents 365 days, The LOLE will be: LOLE = 0.042431 x 365/100 = 0.154874 d/y The next flowchart shows the systematic steps implemented (followed) in the evaluation of the reliability level (LOLE) in power system planning process in order to determine the appropriate capacity reserve margin in each year of the planning horizon. [the LOLE level is based on the level prescribed by the utility management decision (i.e. the LOLE p)].
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Expected Demand Not Served (εDNS) In power system planning, we need sometimes another reliability index beside the 5
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LOLE, to know the magnitude of load that has been lost due to a severe outages (i.e. when Lmax > C). So, the εDNS can be found as follows: n
DNS = ∑ Oi
ε
pi
MW/y
( Lmax > C)
i=1
Expected Energy Not Served (εENS) Since the power systems are in fact energy system, where energy sale is the real revenue for the electric company, so, another essential and most needed reliability index known called the εENS can be deduced as follows: n
ENS = ∑ ENS i
ε
pi
MWh/y
( L > C &) max
i=1
Energy Index of Reliability (EIR) The ratio of expected energy not served ( εENS) to the system Total Energy Demanded (TED) can be found as: ENS pu =
ε
ENS
ε
TED
This ratio, in fact is so small because of the small nature of the εENS and large nature of the TED, so, we can deduce another important reliability index called the EIR, and can be expressed as follows: EIR = 1 - ε ENS pu
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