B

$\because$题中无直径或其他含有度数的角，
$\therefore$排除选项$A$、$C$，
$\because \angle AEC=\angle ADC+\angle C$，
$\therefore \angle AEC+\angle P=\left(\angle ADC+\angle C\right)+\angle P$
$\because \widehat {BD}=\widehat {BD}$，
$\therefore \angle A=\angle C$，
$\therefore \angle AEC+\angle P=\angle ADC+\angle A+\angle P$，
$\because \angle ADC=\angle A+\angle P$，
$\therefore \angle AEC+\angle P=2\angle ADC$，
$\therefore$选项$B$符合题意；
$\because \angle AEC-\angle P=\left(\angle ADC+\angle C\right)-\angle P=\angle ADC+\angle A-\angle P$，
$\therefore$选项$D$不符合题意.
故选：$B$.