Relation between LLF and LF

Load factor (LF) is the ratio of average load to peak load. Loss load factor (LLF) is the ratio of average power loss to the loss at peak load. Because loss is proportional to $I^2$, LLF always lies between $LF^2$ and $LF$. The usual relation is

$$LLF = k\,LF + (1-k)\,LF^2, \qquad k \approx 0.2\text{–}0.3$$

Definitions

For a period $T$ with peak load $P_m$, and assuming constant voltage and power factor:

$$LF = \frac{\frac{1}{T}\int_0^T P\,dt}{P_m}, \qquad LLF = \frac{\frac{1}{T}\int_0^T P^2\,dt}{P_m^2}$$

since loss $\propto I^2 \propto P^2$.

Suitable load curve

Take a two-step daily load curve. The load is at peak $P_m$ for time $t$, and at a lower load $P_1$ for the rest of the time $(T - t)$.

 P
 Pm |-----+
    |     |
 P1 |     +--------------------+
    |                          |
    +-----+--------------------+--> time
    0     t                    T

Then

$$LF = \frac{P_mt + P_1(T-t)}{P_mT}, \qquad LLF = \frac{P_m^2t + P_1^2(T-t)}{P_m^2T}$$

Limiting cases

Case 1: the off-peak load is zero ($P_1 \to 0$, a short sharp peak).

$$LF = \frac{t}{T}, \qquad LLF = \frac{t}{T} \;\Rightarrow\; LLF = LF$$

Case 2: the peak lasts a very short time ($t \to 0$, a nearly flat load at $P_1$).

$$LF = \frac{P_1}{P_m}, \qquad LLF = \frac{P_1^2}{P_m^2} \;\Rightarrow\; LLF = LF^2$$

Every real load curve lies between these two extremes, so

$$LF^2 \le LLF \le LF$$

General relation

LLF is expressed as a weighted combination of the two limits:

$$LLF = k\,LF + (1-k)\,LF^2$$

Here $k$ is found from measured load curves of the system. For typical distribution feeders $k = 0.3$ (Buller and Woodrow), giving $LLF = 0.3LF + 0.7LF^2$. Some utilities use $k = 0.2$ for urban feeders, giving $LLF = 0.2LF + 0.8LF^2$.

Example: for $LF = 0.5$, $LLF = 0.3(0.5) + 0.7(0.25) = 0.325$. The energy loss per year is then $P_{loss,peak}\times0.325\times8760$ kWh.