The design of a flat-field XUV spectrograph is optimized for a high throughput, aberration-corrected spectral image in the wavelength region of 50-300 $AA$ The varied-line spacing (VLS) concave grating theory for an XUV spectrograph with a toroidal mirror in front of an entrance slit is derived. Since the derived theory includes the arbitrary shaped source, it is able to correct the limit of the simple optimization theory which considers only a point source at the center of the entrance slit. The reflection matrix at the toroidal mirror and the diffraction matrix at VLS grating are derived and compared with those of a holographic grating. The absolute energy efficiency of a flat-field spectrograph is calculated by considering the reflectivities of the toroidal mirror and the Au coated concave grating and the grating efficiency. The alignment sensitivity of the toroidal mirror and the concave grating is investigated, and the method to achieve the best imaging of XUV spectrum is discussed. The calculated resolving power of the flat-field XUV spectrograph is more than 4000 in the aberration-corrected wavelength range. The focused spot size at the dispersion plane is less than $20mu extrm m imes mu extrm m$at the wavelength 100$AA$ It is shown that a high throughput characteristic can also be achieved through a careful adjustment of alignment parameters.