We systematically study the strong localization of the light in a two-dimensional, randomly disordered, dielectric medium, that is periodic in average. We consider a simple scattering geometry consisting of an infinite array of infinitely long, parallel, dielectric rods of square and circular cross section embedded in a vacuum. In the case of the parallel rods of square cross sections the dielectric constant of each rod is independent random variable whose values are uniformly distributed about an average value $\epsilon_a$. In systems formed by the rods of circular cross section the disorder is introduced by statistically uncorrelated dielectric constant of each rod, positions of the scatterers and radius of each rod all of which are small deviations satysfying a gaussian probablity distribution. We calculate photonic band structure of average periodic system. and by using finite-difference time-domain methods based on the numerical excitation of the mode by virtual oscillating dipole embedded in the center of the computational domain we evaluate the electromagnetic energy stored in consecutive pairs of cylinders centered at the position of the dipole. By inspecting the time-averaged energy associated with the modes with the frequencies close to the band gap edge we observe the strong localization characterized by an exponential decay. We simultaneously calculate the optical transmission of a slab of disordered 2D systems to show the corresponedence between the Anderson localized wave functions and the resonances in energy transfer to the transmitted Bragg waves.