The Nutshell Essays: |
Power and Energy (Watts and kWh) in a NutshellLarry Miloshevich ON THIS WEBSITE, as in life, it's often useful to understand how electricity is measured (Watts and kWh). This is useful, for example, to plan the size of a solar PV system using information from electricity bills, or to calculate the payback time for a compact fluorescent light (CFL), or to appreciate how your electricity use compares to the national average or to mine, or to be convinced and confident that it's financially wise to replace an older refrigerator. Following the description of Power and Energy below, there are three examples titled The Interesting Microwave Calculation, CFL Payback Time, and The Enlightening Refrigerator Calculation. Electricity measurements are not difficult to understand and the math is easy, but they're often misunderstood because most people have never heard a good explanation, which is the goal of this essay. 1. For our purposes, there are only two electrical measurements of concern:
A RATE is an amount of something per unit time, like SPEED is distance per unit time (miles/hour or meters/second), FLOW RATE is volume per unit time (gallons/minute), and ELECTRICAL POWER is amount of electrical charge (electrons) per second (Watts)[1]. A RATE is an instantaneous value, like 30 miles/hour or 100 Watts, and the TOTAL amount of something after a period of time, such as distance traveled after 10 hours or energy consumed after 10 hours, is the rate multiplied by the amount of time. If a car's speed of 30 miles/hour is constant, then the total distance traveled after 10 hours is: distance = speed x time, or d = (30 miles/hour)x(10 hours) = 300 miles. For the energy example:
TO CLARIFY, using 1 kWh (1000 Wh) of energy might involve EITHER using 1000 W of power for 1 hour, or using 1 W of power for 1000 hours, or using 200 W of power for 5 hours, or any other combination where the product of power in Watts and time in hours is 1000. If my utility bill shows that I consumed 300 kWh of electricity last month, then my AVERAGE POWER if it was constant all month, given that there are 720 hours in a month, is: P = E/t = (300 kWh)/(720 hours) = 0.417 kW = 417 W. So, a monthly energy consumption of 300 kWh is equivalent to consuming 417 W of power continuously for the month. Of course, my actual power use varies all the time, but if I added up Power times Time for each second of the month, it would sum to 300 kWh. THIS FIGURE SHOWS my whole-house power consumption (red) and the power produced by my solar panels (blue), both in Watts, for a 24-hour period. Four main categories of electricity consumption are seen. 1) The sawtooth pattern is the refrigerator's compressor cycling on and off. 2) The baseline load, when the fridge is off and it's the middle of the night, is the continuous electricity consumption from clocks, wall-cube devices, modem, phone, and especially the DVR, and these total about 70 W for me. 3) Constant loads like lights, computer, tv, and stereo that raise the baseline level, and 4) Everything else (microwave, coffee maker, etc., plus larger loads not shown like the washing machine and stove). The blue curve is the production from my solar panels on a clear day in March. The PV production changes continuously as the angle between the direction the panels face and the direction of the sun changes throughout the day. PV production reaches a maximum of about 1700 W (1.7 kW) around noon. The numbers show the total energy consumed and produced over the 24-hour period in kWh, namely 5.08 kWh used by the house and 11.33 kWh produced by the solar panels. This power data from my monitoring system consists of 1-minute averages in Watts, so the energy totals are calculated by summing up "power times time" for each minute of the day. This also happens to be the "area under the curve", since the vertical axis is power and the horizontal axis is time, so calculating the energy involves summing up narrow rectangles each having a width of 1 minute (1/60 hour) and a height equal to each 1-minute average power measurement. Similarly, the total PV production for the day is the area under the blue curve. In case you didn't know, this is the essence of calculus, the mathematics of rates. 2. The Interesting Microwave Calculation Using a Watts Up? electrical meter (standard model) [3], I measured my microwave and found that it uses 1300 W of power when cooking, and 3.3 W of power continuously to run the display and clock. The amount of energy used to run the clock for 1 day is: (3.3 W)x(24 hr) = 79.2 Wh. The amount of energy used to cook for 1 minute is: (1300 W)x(1/60 hr) = 21.7 Wh. I would need to use the microwave for 3.7 minutes to equal the energy consumed by the clock each day (79.2 / 21.7 = 3.7), so if I use the microwave less than 3.7 mins/day then I'm using more electricity to run the clock than to cook food. I already have a clock in the kitchen, so the microwave clock is pure waste, and it's a substantial portion of the total energy that the microwave uses. There are many things in a house that draw small amounts of power continuously, sometimes called "vampire loads" for the power they suck, and it can add up to a lot of hidden energy consumption. Such things include clocks, chargers and other wall cubes that draw power whenever they're plugged in whether it's needed or not, and tvs and other things with remote controls that must constantly check to see if you pushed a button. These functions can be done using extremely little "standby power" if they're well designed, and these days SOME OF THEM are well designed. It would be thoughtful microwave design to have an option to turn it really off (0.0 W) and forego the display until a button is pushed. 3. CFL Payback Time CFLs (the corkscrew bulbs) use only one-fourth the electricity of old-fashioned incandescent (glowing wire) bulbs to produce the same amount of light. A CFL uses only 15 W of power to produce as much light as a 60 W incandescent bulb, a savings of 45 W. CFLs cost more up front, but they cost less to operate and they also last longer than incandescents. Assuming that a CFL costs about $2 more than an incandescent (they can be found for less), and assuming that the price of electricity is 10¢/kWh, we want to know how long it will take to receive $2 in electricity savings from switching to a CFL. If the light is typically on for 5 hours/day, then the energy savings each day is (45 W)x(5 hours) = 225 Wh per day saved. Putting the energy savings in terms of kWh, it is (225 Wh/day)*(1 kWh/1000 Wh) = 0.225 kWh/day. The savings in one month is (0.225 kWh/day)x(30 days/month) = 6.75 kWh/month, which is a monthly dollar savings of (6.75 kWh)x($0.10/kWh) = $0.675 (67.5¢ per month). The number of months needed to save $2 is $2/($0.675/month) = 3.0 months. So, after only 3 months the CFL has paid for its higher initial cost with the electricity savings, and for the rest of its long life (10 times longer than an incandescent) it continues to save $2 in electricity every 3 months. This calculation was for a light that's on 5 hours/day, so the payback time is longer for bulbs that are on less than 5 hours/day, and shorter for bulbs that are on more than 5 hours/day ("inversely proportional"). A CFL that's on for 2 hours/day will pay for itself in (3 months)x(5 hours/2 hours) = 7.5 months. A CFL that's on continuously will pay for itself in (3 months)x(5 hours/24 hours) = 0.6 months (18 days). The total electricity savings for converting many lightbulbs in a house to CFLs would involve summing up the savings for each bulb according to how long it is used each day. The lesson is clear: it makes financial sense to immediately replace incandescents with CFLs, even before the incandescent burns out. I often see houses with incandescent porch lights that are left on 24/7, and I wonder if they know that each year this costs $52 for just one 60 W bulb (assuming a low rate of 10¢/kWh, although it may be several times this cost in states or regions with higher electricity rates). This amount would be reduced from $52 to $13 if it was a CFL, or to $4.40 if the CFL was left on just 8 hours at night, or to less than $1 if the CFL was left on just when it's needed. 4. The Enlightening Refrigerator Calculation In 2004 I replaced my 1980s model refrigerator (classic gold) with a new, more efficient model, after measuring with my Watts Up? meter (standard model) [3] that the refrigerator alone consumed almost half of my total electricity use, plus I learned that new refrigerators are much more efficient thanks to regulations in 1993 that were further improved in 2001 (reference). The caption on this plot of my electricity use since 1997 explains that replacing the old fridge decreased my average electricity consumption by 87 kWh/month, or a little over 1000 kWh/year. At 10¢/kWh, that's $100 in electricity savings per year, so my new $600 refrigerator paid for itself in electricity savings in 6 years. Now I have a newer, nicer refrigerator FOR FREE, and for the rest of its life I will continue to save $100/year on electricity. The payback would be quicker and the savings greater in states or regions with higher electricity rates than 10¢/kWh. Clearly there is financial wisdom in replacing any refrigerator produced before 1993, and it's probably also true for 1993-2001 models but with a longer payback period. Efficient appliances are even more important for a house that produces its own power. In 2007 I installed a 2 kW solar PV system (10 solar panels) on my roof, and annually it produces slightly more than my total annual electricity consumption of about 2700 kWh (or 225 kWh/month). If I still had the old inefficient refrigerator, I would have needed to produce an extra 1000 kWh/year (37% more) that doesn't even go toward cooling food but is simply turned into waste heat that is dumped into my kitchen. This would have required an extra 3.7 solar panels, which costs far more than a new refrigerator. I'm glad I opted for a new, free refrigerator that is now saving me $100/year in electricity. However, there are different levels of efficiency, and next time I'd start by searching among the most efficient models at the Energy Star refrigerator database, and I'd consider the tips from the Union of Concerned Scientists that include: freezer-on-top is more efficient than side-by-side, and automatic icemakers and through-the-front water and ice dispensers substantially decrease the efficiency. Footnotes 1. Like other RATE quantities, electrical power in Watts is an amount of something per unit time. The something is "electrical charge", or electrons, which is the negatively-charged part of atoms that can move through a metal wire as electric current, and give up their energy to do something like run a motor or make light or heat. The movement of electrical charge is what electricity is, and the rate of movement is the electrical power. One Watt of power is the movement of one Coulomb of electrical charge per second, where one Coulomb of electrical charge is 6.24151 x 10^18 electrons. 1 Watt = 1 Coulomb/sec. Almost no one actually needs to know more than that 1 Watt is a very small amount of electrical power for household purposes. 2. In the METRIC system, any unit of measure can be preceded by a prefix that is some multiple of 10. The most common prefixes are: kilo means 1000, mega means 1,000,000 (million), giga means 1,000,000,000 (billion), and on the small end, centi means 1/100 (0.01), milli means 1/1000 (0.001), micro means 1 millionth (0.000001), and nano means 1 billionth (10^-9). For example, a centimeter is 1/100 meter, a kilogram (kg) is 1000 grams (g), and a megawatt-hour (MWh) is a million Wh or a thousand kWh. 3. Anyone in an Xcel Energy service area can borrow a Watts Up? power and energy meter from a participating library. |