Preheat an oven to 325 degrees F (165 degrees C). Grease 16 muffin cups.
Stir the flour, 1 cup sugar, baking powder, and salt together in a large bowl. Make a well in the center of the flour mixture and add the oil, egg yolks, grated orange peel, vanilla extract, orange juice, and water. Mix well by hand until smooth with no lumps.
Beat the egg whites with the cream of tartar until foamy in a large glass or metal mixing bowl. Gradually add the 1/2 cup sugar, continuing to beat until soft peaks form. Lift your beater or whisk straight up: the egg whites will form soft mounds rather than a sharp peak. Fold the flour mixture into the egg whites. Pour the resulting batter into the prepared muffin cups to about 2/3 full.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, about 40 minutes. Cool in the pans for 10 minutes before removing to cool completely on a wire rack.
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