  This paper describes a new determination of the primordial helium abundance
(Y_P), based on the abundance analysis of five metal-poor extragalactic HII
regions. For three regions of the sample (SBS 0335-052, I Zw 18, and H29) we
present tailored photoionization models based on improved calculations with
respect to previous models. In particular, we use the photoionization models to
study quantitatively the effect of collisional excitation of Balmer lines on
the determination of the helium abundance (Y) in the individual regions. This
effect is twofold: first, the intensities of the Balmer lines are enhanced with
respect to the pure recombination value, mimicking a higher hydrogen abundance;
second, the observed reddening is larger than the true extinction, due to the
differential effect of collisions on different Balmer lines. In addition to
these effects, our analysis takes into account the following features of HII
regions: (i) the temperature structure, (ii) the density structure, (iii) the
presence of neutral helium, (iv) the collisional excitation of the HeI lines,
(v) the underlying absorption of the HeI lines, and (vi) the optical thickness
of the HeI lines. The object that shows the highest increase in Y after the
inclusion of collisional effects in the analysis is SBS 0335-052, whose helium
abundance has been revised by Delta Y = +0.0107. The revised Y values for the
five objects in our sample yield an increase of +0.0035 in Y_P, giving Y_P =
0.2391 +/- 0.0020.
