  In its long-duration observation phase, the PLATO satellite will observe two
non-overlapping fields for a total of 4 yr. The exact duration of each pointing
will be determined 2 yr before launch. Previous estimates of PLATO's yield of
Earth-sized planets in the habitable zones (HZs) around solar-type stars ranged
between 6 and 280. We use the PLATO Solar-like Light curve Simulator (PSLS) to
simulate light curves with transiting planets around bright (m_V > 11) Sun-like
stars at a cadence of 25 s, roughly representative of the >15,000 targets in
PLATO's high-priority P1 sample (mostly F5-K7 dwarfs and sub-dwarfs). Our study
includes light curves generated from synchronous observations of 6, 12, 18, and
24 of PLATO's 12 cm aperture cameras over both 2 yr and 3 yr of continuous
observations. Automated detrending is done with the Wotan software and
post-detrending transit detection is performed with the Transit Least Squares
(TLS) algorithm. We scale the true positive rates (TPRs) with the expected
number of stars in the P1 sample and with modern estimates of the exoplanet
occurrence rates and predict the detection of planets with 0.5 R_E <= R_p <=
1.5 R_E in the HZs around F5-K7 dwarf stars. For the (2 yr + 2 yr)
long-duration observation phase strategy we predict 11-34 detections, for the
(3 yr + 1 yr) strategy we predict 8-25 discoveries. Our study of the effects of
stellar variability on shallow transits of Earth-like planets illustrates that
our estimates of PLATO's planet yield, which we derive using a photometrically
quiet star like the Sun, must be seen as upper limits. In conclusion, PLATO's
detection of about a dozen Earth-sized planets in the HZs around solar-type
stars will mean a major contribution to this yet poorly sampled part of the
exoplanet parameter space with Earth-like planets.
