Described by Coello and Boyle (2019), the Humboldt State University (HSU) soiling model predicts time series soiling ratio SR based on accumulated particulate mass density $$\omega$$ (g/m2):
$$
\DeclareMathOperator\erf{erf}
SR = 1 – 34.37 * \erf(0.17 * \omega^{0.8473})
$$
where $$\erf$$ refers to the Gauss error function. Accumulated mass density $$\omega$$ is the cumulative sum of interval accumulation $$m$$, resetting to zero on cleaning events. $$m$$ is calculated using assumptions of airborne particulate concentrations $$C$$ and settling velocities $$v$$ for PM2.5 and PM10:
$$m = (v_{10} C_{10} + v_{2.5} C_{2.5}) \delta T \cos \theta_T$$
where $$\delta T$$ is the timestep length and $$\theta_T$$ is the array tilt.
Coello and Boyle define three methods for determining the velocities $$v$$, indicating that assuming fixed settling velocities results in the most reasonable predictions of soiling ratio. Note that, because the input particulate concentrations can be time-varying quantities, the rate of soiling accumulation can vary with time as well.
A peculiarity of the equation predicting $$SR$$ as a function of accumulated mass $$\omega$$ is that the predicted soiling ratio can drop no lower than 0.6563. Boyle et al. (2015) place an upper limit of $$\omega = 10$$ g/m2 on the valid range for this equation, corresponding to a soiling ratio of roughly 0.6875.
The HSU soiling model is implemented in PVLIB_MATLAB with the `pvl_soiling_hsu` function and in pvlib-python with pvlib.soiling.hsu.
References
[1] M. Coello and L. Boyle, “Simple Model for Predicting Time Series Soiling of Photovoltaic Panels,” IEEE Journal of Photovoltaics, vol. 9, no. 5, pp. 1382–1387, Sep. 2019. doi: 10.1109/jphotov.2019.2919628.
[2] L. Boyle, H. Flinchpaugh, and M. P. Hannigan, “Natural soiling of photovoltaic cover plates and the impact on transmission,” Renewable Energy vol. 77, pp. 166–173, May 2015. doi: 10.1016/j.renene.2014.12.006