Within six years, the sunny Middle East will have as much solar power as the US has today, and more than Germany.  By 2030, it will be approaching China’s current level. After a slow start, this oil and gas-dominated region has woken up to the potential of the sun

Like other solar markets coming of age, the MENA region has seen a significant shift in attitude by the utilities and power companies, which have gone from seeing solar as an expensive gimmick to a resilient renewable energy juggernaut that can economically compete with more traditional sources of energy—and provide a sustainable decarbonization alternative to the oil and gas status quo.

The report predicts unprecedented growth in solar projects and related industries in the next 10 years in neighbouring countries.  More importantly, it is no longer just a hype but a reality that’s already started in many of the MENA countries.  The report gives highlights of MENA’s leading solar markets, covering 11 countries that include Algeria, Bahrain, Egypt, Jordan, Kuwait, Morocco, Oman, Pakistan, Saudi Arabia, Tunisia, and UAE.  Sadly, Libya is missing on the radar for now and the near foreseeable future, unless by some miracle the country pulls itself together and starts building for the future instead of deepening the hole its already in.

In the forefront, the Renewable Energy Authority of Libya (REAOL) bears the responsibility and burden of shaping Libya’s future in energy supply, which could turn it into the next National Energy Corporation of Libya (NECOL).

Recently, I was working on an implementation of ‘single diode model’ to evaluate the performance of solar cells/modules under real operating conditions and as part of the solution to the model I got introduced to the LambertW function (also known as the Omega function).

Everything went fine initially until I started to change the operating conditions and discovered that my single diode model was going out of bound under certain conditions.

After some investigation, I discovered that the term passed to the LambertW function was becoming too large and causing an overflow, even when evaluated as double precision.  In fact, it was raising (e) to powers greater than ~710 which was the real problem.  The result from the LambertW function would still be relatively small if only I could get past the overflow error of the input term.  This lead me to think that perhaps I could scale down the input term in a way that will allow me later to scale up the LambertW function result without introducing errors.  Looking closely at the inners of the LambertW function, it became obvious that the problem was not so simple.  So what is the LambertW function?, you may ask.

You can find a good explanation of the LambertW function in the following link: "Cleve Moler on Mathematics and Computing".

There is a fast solution developed by Toshio Fukushima “Precise and fast computation of Lambert W-functions without transcendental function evaluations, Journal of Computational and Applied Mathematics 244 (2013) 77–89”.  It relies on lookup tables to avoid slow transcendental functions (evaluating expensive exponentials) but has a limitation on the range of values it can cover (typically -65<x<65).

Solving the overflow error problem:

Given the LambertW function form y = w.e^w, it stands to reason that log(y) = log(w) + w.  This should solve the overflow problem since the problem only occurs when we raise (e) to the power of w > ~710.  All that is needed now is to solve the logarithmic form of the equation to determine the value of w for a given input value log(y).

First, since y in our ‘single diode model’ is of the form y=A.e^B, log(y) is simply log(A) + B.  Hence we avoid the overflow problem altogether. Next, we turn our attention to developing a logarithmic LambertW (LambertWLog) function solution.

The solution involves using one of the Monte Carlo methods to perform iterative root-finding using derivatives, such as Newton-Raphson, Halley or Schröder iterations.

Using Halley’s iteration with an initial guess for w is:

w’ = w – f(w) / (f’(w) – f(w).f”(w) / [2f’(w)])

Applying some substitution and simplification, where:

     log(y) = log(w) + w
     0 = log(w) + w – log(y),         where:  log(y) = log(A) + B, a constant
     f(w) = log(w) + w – log(y)
     f’(w) = (1+w)/w                     1st derivative
     f”(w) = -1/(w.w)                    2nd derivative

Results in:

     w’ = w – fn / (1/w + 1 + fn / [2w(w + 1)])        where:  fn = log(w) + w – log(y),  log(y) = log(A) + B

Given a good initial approximation, a solution can be reached within 2 or 3 iterations.  As it turns out, log(y) is a very good initial approximation of w and works equally well.  The LambertWLog function can resolve very large w values > 1,000,000 when the regular LambertW would crash and burn at ~710.