Reducing carbon emissions and electricity costs in industry is a major challenge to ensure competitiveness and compliance with new climate policies. Photovoltaic power offers a promising solution but also brings considerable uncertainties and risks that may endanger the continuity and quality of supply. From an operational point of view, large-scale integration of solar power could result in unmet demand, electrical instabilities and equipment damage. The performance and lifetime of conventional fossil equipment are likely to be altered by repeated transient operations, making it necessary to adopt specific modeling tools. Control strategies and sizing methodologies must be adapted to account for the strong reliability constraint while dealing with significant production uncertainties. In addition, conventional mitigation technologies, such as storage and load flexibility, have limited potential in these applications and may result in high investments or penalties if they are not properly assessed. This study provides an overview of these challenges by providing a transversal analysis of the scientific literature from fossil engine thermodynamics to control system theory applied to industrial systems. The main characteristics of reliability-constrained microgrids are identified and a conceptual definition is proposed by analyzing state-of-the art studies of various industrial applications and taking oil-and gas microgrids as an enlightening example. Then follows a review of the challenges of accounting for dynamical behavior of fossil equipment, PV and storage systems, ending with the identification of several research gaps. Finally, applicable control strategies and sizing techniques are presented.