I would not count on them actually paying you for every kilowatt hour generated. Instead, I expect it is a net metering agreement. That is, they only pay you for the amount you generate beyond what you consume, the months when you are a net producer of power. If you generate 900kWH in a month, but use 600kWH that month, you will only be paid for 900-600=300kWH that month at the $0.30 rate.
Newport, Rhode Island averages 4.23 solar hours per day, averaged over a year. So on an averaged month, you will produce 4.23*3.7*365/12=476 kWH with a 3.7kW solar installation. When you account for the fact that summer production will be higher and winter lower, you can only really expect them to pay you in summer time, unless you are a very effective power miser. If you use air conditioning, heat pump, or electric heat, expect no payment.
http://www.solardirect.com/pv/systems/gts/gts-sizing-sun-hours.htmlThat affects the payback substantially. You should probably not use the $0.38 rate in your calculations, but instead the $0.18 billed rate. This is because you are likely to still be using more power than you produce, each month. Even using the summer solar hours, you can only expect to generate about 528kWH per month in the summer (403 in winter). Most people have electric consumption higher than that.
Lets assume you perfectly target your power consumption for maximum payback (because use too little and they pay you less, oddly. Use too much and you pay them). So for your average 476kWH solar production month you hit your target of using 381kWH, for max payback. You will see a payment from the utility of 95kWH at $0.38, or $36.10
Expect that they will still charge you their "Customer fee" and whatever else charges they have that are not consumption based. In my area, south jersey, those fixed fees are about $12. Probably $15 in RI. So, if you game it perfectly you will have a net payment of $21 a month from the utility company, instead of whatever they charge you now. You will probably not game the system perfectly.
So, does the amount of your average power bill, plus $21, give you your ROI period you need? Probably not. Assume out of pocket of $10k, needed payback of 15 years. Loan/opportunity cost of 4.5%/yr.
Interest first year of $450. Your max utility cost for the same ideal case usage would be 381kWH/month * $0.18 = $68.58 a month (is your average bill above or below this?). $823 dollars in electric a year if no solar system but same ideal usage. Net cash flow change from utility company to you of $844, savings of $394 that first year.
You miss the 15 year ROI. A really rough approximation puts the ROI at somewhere around year 18, if I did the math right. The actual ROI calculation is not my strong point. Overall though, that is the absolute best case scenario, assuming you had a perfect robot running your power consumption.
Only do this if the non financial aspects appeal to you.