EX 2.36 solar cells

The article “Reliability Evaluation of Solar Photovoltaic Arrays” (Solar Energy, 2002: 129-141) presents various configurations of solar photovoltaic arrays consisting of crystalline silicon solar cells.

Figure 2.14

System configurations for Example 2.36: a. series-parallel; b. total-cross-tied

Consider first the system illustrated in Figure 2.14 (a).

• There are two subsystems connected in parallel,
  • each one containing three cells.
• In order for the system to function,
  • at least one of the two parallel subsystems must work.
• Within each subsystem,
  • the three cells are connected in series,
  • so a subsystem will work only if all cells in the subsystem work.

Question

• Consider a particular lifetime value $t_0$
• Suppose we want to determine the probability that the system lifetime exceeds $t_0$.

• Let $A_i$ denote the event that the lifetime of cell $i$ exceeds $t_0\left(i=1,2,\dots ,6\right)$ .
• We assume that the $A_i^{\prime }s$ are independent events
  • whether any particular cell lasts more than $t_0$ hours has no bearing on whether or not any other cell does
  • and that $P\left(A_i\right)=.9$ for every $i$ since the cells are identical.
• Then

$$\begin{array}{rl}& P\left(\text{system lifetime exceeds }{t}_0\right)\\ =& P\left[\left(A_1\cap A_2\cap A_3\right)\cup \left(A_4\cap A_5\cap A_6\right)\right]\\ =& P\left(A_1\cap A_2\cap A_3\right)+P\left(A_4\cap A_5\cap A_6\right)\\ & -P\left[\left(A_1\cap A_2\cap A_3\right)\cap \left(A_4\cap A_5\cap A_6\right)\right]\\ =& \left(.9\right)\left(.9\right)\left(.9\right)+\left(.9\right)\left(.9\right)\left(.9\right)-\left(.9\right)\left(.9\right)\left(.9\right)\left(.9\right)\left(.9\right)\left(.9\right)\\ =& .927\end{array}$$

• Alternatively,

$$\begin{array}{rl}& P\left(\text{system lifetime exceeds }{t}_0\right)\\ =& 1-P\left(\text{both subsystem lives are}\le t_0\right)\\ =& 1-{\left[P\left(\text{ subsystem life is }\le t_0\right)\right]}^2\\ =& 1-{\left[1-P\left(\text{ subsystem life is }>t_0\right)\right]}^2\\ =& 1-{\left[1-{\left(.9\right)}^3\right]}^2\\ =& .927\end{array}$$

Next consider the total-cross-tied system shown in Figure 2.14 (b),

• obtained from the series-parallel array by connecting ties across each column of junctions.
• Now the system fails as soon as an entire column fails,
• and system lifetime exceeds $t_0$
  • only if the life of every column does so.
• For this configuration,

$$\begin{array}{rl}& P\left(\text{system lifetime is at least }{t}_0\right)\\ =& {\left[P\left(\text{column lifetime exceeds }{t}_0\right)\right]}^3\\ =& {\left[1-P\left(\text{ column lifetime is }\le t_0\right)\right]}^3\\ =& {\left[1-P\left(\text{ both cells in a column have lifetime }\le t_0\right)\right]}^3\\ =& {\left[1-{\left(1-.9\right)}^2\right]}^3\\ =& .970\end{array}$$