# Loss of Load Probability (LOLP)

Over the years, LOLP has been used as the single most important metric to estimate overall power system reliability and it may simply be described as a projected value of how much time, in the long run, the load on a given power system is expected to be greater than the capacity of the generating resources [8]. Various probabilistic methods/techniques are used for calculating LOLP.

In the setting up of an LOLP criterion, it is always assumed that an electric power system strong enough to have a low LOLP, can probably withstand most of the foreseeable peak loads, contingencies, and outages. Thus, a utility is always expected to arrange for resources (i.e., generation, load management, purchases, etc.) in such a way so that the resulting system LOLP will be at or lower than an acceptable level.

Generally, the common practice is to plan to power system for achieving an LOLP of 0.1 days per year or lower. All in all, past experiences over the years indicate that there are many difficulties with this use of LOLP. Some of these are as follows [8]:

- • Major loss-of-load incidents usually occur because of contingencies not modeled effectively by the traditional LOLP calculation.
- • Different LOLP estimation techniques can result in different indices for exactly the same electric power system.
- • LOLP does not take into consideration the factor of additional emergency support that one region or control area may receive from another, or other emergency measures/actions that control area operators can exercise for maintaining system reliability.
- • LOLP itself does not state the magnitude or duration of the electricity’s shortage.

# Availability Analysis of a Single Power Generator Unit

There are a number of mathematical models that can be used for performing availability analysis of a single generator unit. Three of these mathematical models are presented below.

**8.5.1 Model I**

This mathematical model represents a single power generator unit that can either be in operating state or failed state. The failed power generator unit is repaired. The power generator unit state-space diagram is shown in Figure 8.1. The numerals in the boxes denote the power generator unit state.

The model is subjected to the following assumptions:

- • The power generator unit failure and repair rates are constant.
- • The repaired power generator unit is as good as new.
- • The power generator unit failures are statistically independent.

The following symbols are associated with Figure 8.1 diagram and its associated equations:

*Pj* (/) is the probability that the power generator unit is in state *j* at time; for *j* = 0 (operating normally),/= 1 (failed).

*X _{pg}* is the power generator unit failure rate.

*H _{pg}* is the power generator unit repair rate.

Using the Markov method presented in Chapter 4, we write down the following equations for Figure 8.1 state-space diagram [11]:

Solving Equations (8.7)—(8.8) by using Laplace transforms we obtain

The power generator unit availability and unavailability are given by

and

where

*A V _{pg} (/)* is the power generator unit availability at time

*t.*

*UA _{pg}* (/) is the power generator unit unavailability at time

*t.*

For large *t,* Equations (8.11)—(8.12) reduce to

and

where

*A V _{pg}* is the power generator unit steady state availability.

*UA _{pg}* is the power generator unit steady state unavailability.

SinceA,,,, =---andu„„ =---. Equations (8.13)—(8.14) become

^{ps} MrTF_{pg} ^{ps} MTTR_{pg}

and

where

*MTTF* is the power generator unit mean time to failure.

*MTTRp _{g}* is the power generator unit mean time to repair.

Example 8.2

Assume that constant failure and repair rates of a power generator unit are *X _{p}g =* 0.0005 failures/hour and

*fi*0.0008 repairs/hour, respectively. Calculate the steady state unavailability of the power generator unit.

_{pg}=By substituting the given data values into Equation (8.14), we obtain

Thus, the steady state unavailability of the power generator unit is 0.3846.

**8.5.2 Model II**

This mathematical model represents a power generator unit that can be either operating normally (i.e., generating electricity at its full capacity), derated (i.e., generating electricity at a derated capacity, e.g., say 300 megawatts instead of 500 megawatts at full capacity), or failed. This is depicted by the state-space diagram shown in Figure 8.2. The numerals in the boxes denote system state.

The model is subjected to the following assumptions:

- • All power generator unit failure and repair rates are constant.
- • The power generator unit failures are statistically independent.
- • The repaired power generator unit is as good as new.

The following symbols are associated with Figure 8.2 diagram and its associated equations:

*Pj* (/) is the probability that the power generator unit is in state /' at time *t;* for / = 0 (operating normally),/ = 1 (derated),/' = 2 (failed).

*X _{p}* is the power generator unit failure rate from state 0 to state 2.

*K _{pd}* is the power generator unit failure rate from state 0 to state 1.

*X _{pl}* is the power generator unit failure rate from state 1 to state 2.

*U _{fi}* is the power generator unit repair rate from state 2 to state 0.

*H _{ixl}* is the power generator unit repair rate from state 1 to state 0.

*U _{fil}* is the power generator unit repair rate from state 2 to state 4.

Using the Markov method presented in Chapter 4, we write down the following equations for Figure 8.2 state-space diagram [11]:

Solving Equations (8.17)—(8.19) by using Laplace transforms, we obtain where

where

where

The power generator unit operational availability is expressed by
For large *t,* Equation (8.32) reduces to

where

*A V _{pg}* is the power generator unit operational steady state availability.

**8.5.3 Model III**

This mathematical model represents a power generator unit that can either be in operating state or failed state or down for preventive maintenance. This is depicted by the state-space diagram shown in Figure 8.3. The numerals in the boxes and circle denote the system state.

The model is subjected to the following assumptions:

- • The power generator unit failure, repair, preventive maintenance down, and preventive maintenance performance rates are constant.
- • After preventive maintenance and repair, the power generator unit is as good as new.
- • The power generator unit failures are statistically independent.

The following symbols are associated with the state-space diagram shown in Figure 8.3 and its associated equations:

*P:* (r) is the probability that the power generator unit is in state ) at time /; for / = 0 (operating normally),) = 1 (down for preventive maintenance),) = 2 (failed).

Ay is the power generator unit failure rate.

*X _{pm}* is the power generator unit (down for) preventive maintenance rate.

*fi*is the power generator unit repair rate.

_{f}*fi _{pm}* is the power generator unit preventive maintenance performance (repair) rate.

Using the Markov method presented in Chapter 4, we write down the following equations for Figure 8.3 state-space diagram [11]:

Solving Equations (8.34)—(8.36) by using Laplace transforms, we obtain

where

The power generator unit availability, *A V _{pg}* (/), is given by

It is to be noted that the above availability expression is valid if and only if *k _{x}* and

*k*are negative. Thus, for large

_{2 }*t,*Equation (8.42) reduces to

where

*A V _{pg}* is the power generator unit steady state availability.

**Example 8.3**

Assume that for a power generator unit we have the following data values:

Ay = 0.0001 failures/hour *f.ij* = 0.0004 repairs/hour *X _{pm}* = 0.0006/hour

*ц*0.0008/hour

_{рт}=Calculate the power generator unit steady state availability.

By inserting the stated data values into Equation (8.43), we obtain

Thus, the steady state availability of the power generator unit is 0.5.