From the Department of Neighborhoods, information on eligibility requirements for help with utility bills:

Financial Assistance Available for Utility Bills
Fall is just around the corner, and for many that means a choice between higher utility bills or colder homes. However, the City is working hard to make sure that families will not have to make that choice. Mayor Murray has challenged city departments to double enrollment in the Utility Discount Program (UDP) by the end of 2018.
The City estimates that up to 72,000 residents qualify for this program, yet less than 16,000 are currently enrolled. If you know of someone who may benefit from this program – please help spread the word!
The Utility Discount Program (UDP) offers a 60 percent discount for electrical service, and a 50 percent discount for combined utility services, one of the most generous plans for either type of service available in the country. To be eligible for the program, a customer must have a household income less than 70 percent of the State Median Income (see table below) be a SCL and/or SPU customer; and not reside in housing operated by the Seattle Housing Authority, King County Housing Authority, or federal government that includes a utility allowance.