He also claimed that both the parties have "hatched a conspiracy" to snatch these departments from the AAP government.
Addressing his first public meeting in North Delhi's Burari area in the run up to the civic polls, Kejriwal said if his party wins MCD polls, it will not allow this to happen.
He appealed the voters to "strengthen the hands" of the AAP government in developing Delhi by voting for the party.
Kejriwal also promised a cleaner city if his party is voted to power in MCD polls.
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"Two days back, a senior BJP leader met me and said, a big conspiracy is being hatched. He told me that if either Congress or BJP comes to power in the MCD, they will increase rates of power and water in Delhi.
"When I asked how it could be possible, he told me that the Centre will snatch power and water departments from the Delhi government and give them to MCDs. These both departments were with MCD before 1998," Kejriwal told the gathering.
"We have done a lot of work over the last two years. We have sent government teachers abroad for training and have been providing free medicines. Our government revamped the education and health sector," he said.
During his 35-minute speech, he dared BJP chief Amit Shah for a debate on achievements of BJP-ruled states as against the AAP government's here.
"In Mumbai, the power bill for the same units is Rs 4,000 while in Karnataka, it comes to Rs 2,700," he said.