SANA said the blasts were caused by two improvised explosive devices which went off on Khalid Bin Walid street and the nearby Bab Mesalla Square.
It said the bombs were planted by "terrorists," a term the government uses to describe rebels fighting to topple the Syrian leader.
The Britain-based Syrian Observatory for Human Rights however said the Bab Mesalla explosions were due to rockets that fell in the area. It said initial information indicated that there were casualties, but the number could not be obtained immediately.
The Observatory said police sealed off Bab Mesalla, which has restaurants, shops and a main public transportation station linking Damascus with the southern provinces of Daraa and Sweida.
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In the other incident, the Observatory said a bomb exploded near police headquarters on Khalid Bin Walid Street. It said several people, including children, were wounded in the blast.
No other details were immediately available.
Assad's visit to the power station came just a day after a powerful bomb hit the capital.
A broadcast on Syria's state television showed Assad speaking to staff on the occasion of International Workers Day, or May Day, at the Umayyad Electrical Station in the Tishrin Park district.
"They want to scare us, we will not be scared ... They want us to live underground, we will not live underground," Assad was shown on TV, telling a group of workers who gathered around him.
"We hope that by this time next year we will have overcome the crisis in our country," he added.
At least 14 people were killed in yesterday's blast, the second in the heart of the capital in two days.
Rebels seeking to topple Assad have been trying to create a supply line from Jordan, so that arms bought by Saudi Arabia and Qatar can be shipped in for assaults on the city they hope to capture.