Superiority clinical trials are often designed with a planned interim analysis for the purpose of sample size re-estimation (SSR) when limited information is available at the start of the trial to estimate the required sample size. Typically these trials are designed with a two-arm internal pilot where subjects are enrolled to both treatment arms prior to the interim analysis. Circumstances may sometimes call for a trial with a single-arm internal pilot (enroll only in the control group). For a dichotomous outcome, Herson and Wittes proposed a SSR method (HW-SSR) that can be applied to single-arm internal pilot trials using an unblinded estimate of the control group outcome rate. Previous evaluations of the HW-SSR method reported conflicting results regarding the impact of the method on the two-sided Type I error rate and power of the final hypothesis test. In this research we evaluate the HW-SSR method under the null and alternative hypothesis in various scenarios to investigate the one-sided Type I error rate and power of trials with a two-arm internal pilot. We find that the one-sided Type I error rate is sometimes inflated and that the power is sometimes reduced. We propose a new method, the Critical Value and Power Adjusted Sample Size Re-estimation (CVPA-SSR) algorithm to adjust the critical value cutoff used in the final Z-test and the power critical value used in the interim SSR formula to preserve the nominal Type I error rate and the desired power. We conduct simulations for trials with single-arm and two-arm internal pilots to confirm that the CVPA-SSR algorithm does preserve the nominal Type I error rate and the desired power. We investigate the robustness of the CVPA-SSR algorithm for trials with single-arm and two-arm internal pilots when the assumptions used in designing the trial are incorrect. No Type I error inflation is observed but significant over- or under-powering of the trial occurs when the treatment effect used to design the trial is misspecified.