A solar array on the roof of a house is rated at 6 kW (power). The system consis

Question

A solar array on the roof of a house is rated at 6 kW (power). The system consists of the solar panels and a power inverter that converts the DC power generated by the solar cells into AC power. The power generated will vary during the day due to several factors, including the angle of solar incidence.

On a particular day, the power generated is recorded and a mathematical model is created to predict the power generation as a function of time: P = A sin(Bt), where P is the AC power in watts, t is time in minutes A and B are constants: A = 4500 W, B = 0.005712 radians/min.
Use numerical integration in Excel to determine the amount of energy in kiloWatt Hours (kWh) and in kilo Joules (kJ) for a period of 540 minutes (~9 hours). Note that power is the rate of energy generation, thus if E represents energy, P = dE/dt. Energy generated during a time step is dE/dt * t.

Explanation / Answer

Total kWhs generated in 540 minutes = 9 hours is E = 26.24 kWhs

Total Energy generated E = 26.24 kWhs

A = 4500 W B = 0.005712 rad/min P = A*sin(Bt) E = integral P dt Wh E = A/Bcos(Bt) Time (min) P (W) E (Wh) E (kWh) 1 25.70 0.42839767 0.000428398 2 51.41 0.856781364 0.000856781 3 77.11 1.285137103 0.001285137 4 102.81 1.713450912 0.001713451 5 128.50 2.141708816 0.002141709 6 154.19 2.569896844 0.002569897 7 179.88 2.998001023 0.002998001 8 205.56 3.426007388 0.003426007 9 231.23 3.853901972 0.003853902 10 256.90 4.281670816 0.004281671 11 282.56 4.709299963 0.0047093 12 308.21 5.136775459 0.005136775 13 333.85 5.564083359 0.005564083 14 359.47 5.991209721 0.00599121 15 385.09 6.418140608 0.006418141 16 410.69 6.844862091 0.006844862 17 436.28 7.271360248 0.00727136 18 461.86 7.697621163 0.007697621 19 487.42 8.123630929 0.008123631 20 512.96 8.549375647 0.008549376 21 538.49 8.974841425 0.008974841 22 564.00 9.400014383 0.009400014 23 589.49 9.824880647 0.009824881 24 614.97 10.24942636 0.010249426 25 640.42 10.67363766 0.010673638 26 665.85 11.09750072 0.011097501 27 691.26 11.5210017 0.011521002 28 716.65 11.94412678 0.011944127 29 742.01 12.36686217 0.012366862 30 767.35 12.78919406 0.012789194 31 792.67 13.21110868 0.013211109 32 817.96 13.63259227 0.013632592 33 843.22 14.05363107 0.014053631 34 868.45 14.47421134 0.014474211 35 893.66 14.89431936 0.014894319 36 918.84 15.31394143 0.015313941 37 943.98 15.73306385 0.015733064 38 969.10 16.15167296 0.016151673 39 994.19 16.56975508 0.016569755 40 1019.24 16.98729659 0.016987297 41 1044.26 17.40428385 0.017404284 42 1069.24 17.82070327 0.017820703 43 1094.19 18.23654125 0.018236541 44 1119.11 18.65178423 0.018651784 45 1143.99 19.06641866 0.019066419 46 1168.83 19.48043102 0.019480431 47 1193.63 19.89380779 0.019893808 48 1218.39 20.30653549 0.020306535 49 1243.12 20.71860065 0.020718601 50 1267.80 21.12998982 0.02112999 51 1292.44 21.54068959 0.02154069 52 1317.04 21.95068656 0.021950687 53 1341.60 22.35996735 0.022359967 54 1366.11 22.7685186 0.022768519 55 1390.58 23.17632698 0.023176327 56 1415.00 23.58337919 0.023583379 57 1439.38 23.98966195 0.023989662 58 1463.71 24.39516201 0.024395162 59 1487.99 24.79986613 0.024799866 60 1512.23 25.2037611 0.025203761 61 1536.41 25.60683376 0.025606834 62 1560.54 26.00907094 0.026009071 63 1584.63 26.41045954 0.02641046 64 1608.66 26.81098644 0.026810986 65 1632.64 27.21063858 0.027210639 66 1656.56 27.60940293 0.027609403 67 1680.44 28.00726646 0.028007266 68 1704.25 28.40421621 0.028404216 69 1728.01 28.80023922 0.028800239 70 1751.72 29.19532257 0.029195323 71 1775.37 29.58945336 0.029589453 72 1798.96 29.98261875 0.029982619 73 1822.49 30.37480589 0.030374806 74 1845.96 30.76600201 0.030766002 75 1869.37 31.15619432 0.031156194 76 1892.72 31.5453701 0.03154537 77 1916.01 31.93351666 0.031933517 78 1939.24 32.32062133 0.032320621 79 1962.40 32.70667148 0.032706671 80 1985.50 33.09165451 0.033091655 81 2008.53 33.47555787 0.033475558 82 2031.50 33.85836902 0.033858369 83 2054.40 34.24007548 0.034240075 84 2077.24 34.6206648 0.034620665 85 2100.01 35.00012455 0.035000125 86 2122.71 35.37844236 0.035378442 87 2145.34 35.75560588 0.035755606 88 2167.90 36.13160281 0.036131603 89 2190.39 36.50642088 0.036506421 90 2212.80 36.88004786 0.036880048 91 2235.15 37.25247155 0.037252472 92 2257.42 37.62367982 0.03762368 93 2279.62 37.99366054 0.037993661 94 2301.74 38.36240165 0.038362402 95 2323.79 38.72989112 0.038729891 96 2345.77 39.09611695 0.039096117 97 2367.66 39.4610672 0.039461067 98 2389.48 39.82472996 0.03982473 99 2411.23 40.18709336 0.040187093 100 2432.89 40.54814558 0.040548146 101 2454.47 40.90787485 0.040907875 102 2475.98 41.26626942 0.041266269 103 2497.40 41.6233176 0.041623318 104 2518.74 41.97900774 0.041979008 105 2540.00 42.33332824 0.042333328 106 2561.18 42.68626754 0.042686268 107 2582.27 43.03781412 0.043037814 108 2603.28 43.38795651 0.043387957 109 2624.20 43.73668329 0.043736683 110 2645.04 44.08398307 0.044083983 111 2665.79 44.42984454 0.044429845 112 2686.46 44.7742564 0.044774256 113 2707.03 45.11720741 0.045117207 114 2727.52 45.45868639 0.045458686 115 2747.92 45.7986822 0.045798682 116 2768.23 46.13718375 0.046137184 117 2788.45 46.47417998 0.04647418 118 2808.58 46.8096599 0.04680966 119 2828.62 47.14361257 0.047143613 120 2848.56 47.4760271 0.047476027 121 2868.41 47.80689263 0.047806893 122 2888.17 48.13619837 0.048136198 123 2907.84 48.46393358 0.048463934 124 2927.41 48.79008757 0.048790088 125 2946.88 49.11464968 0.04911465 126 2966.26 49.43760934 0.049437609 127 2985.54 49.75895601 0.049758956 128 3004.72 50.0786792 0.050078679 129 3023.81 50.39676848 0.050396768 130 3042.79 50.71321347 0.050713213 131 3061.68 51.02800385 0.051028004 132 3080.47 51.34112934 0.051341129 133 3099.15 51.65257973 0.05165258 134 3117.74 51.96234487 0.051962345 135 3136.22 52.27041463 0.052270415 136 3154.61 52.57677898 0.052576779 137 3172.89 52.88142791 0.052881428 138 3191.06 53.18435149 0.053184351 139 3209.13 53.48553983 0.05348554 140 3227.10 53.7849831 0.053784983 141 3244.96 54.08267154 0.054082672 142 3262.72 54.37859543 0.054378595 143 3280.36 54.67274512 0.054672745 144 3297.91 54.96511101 0.054965111 145 3315.34 55.25568356 0.055255684 146 3332.67 55.54445329 0.055544453 147 3349.88 55.83141078 0.055831411 148 3366.99 56.11654667 0.056116547 149 3383.99 56.39985165 0.056399852 150 3400.88 56.68131648 0.056681316 151 3417.66 56.96093198 0.056960932 152 3434.32 57.23868902 0.057238689 153 3450.87 57.51457854 0.057514579 154 3467.32 57.78859155 0.057788592 155 3483.64 58.06071909 0.058060719 156 3499.86 58.33095229 0.058330952 157 3515.96 58.59928234 0.058599282 158 3531.94 58.86570048 0.0588657 159 3547.81 59.13019802 0.059130198 160 3563.57 59.39276632 0.059392766 161 3579.20 59.65339683 0.059653397 162 3594.72 59.91208103 0.059912081 163 3610.13 60.16881049 0.06016881 164 3625.41 60.42357683 0.060423577 165 3640.58 60.67637174 0.060676372 166 3655.63 60.92718696 0.060927187 167 3670.56 61.17601433 0.061176014 168 3685.37 61.42284572 0.061422846 169 3700.06 61.66767307 0.061667673 170 3714.63 61.9104884 0.061910488 171 3729.08 62.15128378 0.062151284 172 3743.40 62.39005137 0.062390051 173 3757.61 62.62678336 0.062626783 174 3771.69 62.86147204 0.062861472 175 3785.65 63.09410974 0.06309411 176 3799.48 63.32468889 0.063324689 177 3813.19 63.55320194 0.063553202 178 3826.78 63.77964146 0.063779641 179 3840.24 64.00400005 0.064004 180 3853.58 64.22627039 0.06422627 181 3866.79 64.44644522 0.064446445 182 3879.87 64.66451738 0.064664517 183 3892.83 64.88047973 0.06488048 184 3905.66 65.09432523 0.065094325 185 3918.36 65.30604692 0.065306047 186 3930.94 65.51563787 0.065515638 187 3943.39 65.72309125 0.065723091 188 3955.70 65.9284003 0.0659284 189 3967.89 66.13155831 0.066131558 190 3979.95 66.33255865 0.066332559 191 3991.88 66.53139477 0.066531395 192 4003.68 66.72806018 0.06672806 193 4015.35 66.92254847 0.066922548 194 4026.89 67.11485328 0.067114853 195 4038.30 67.30496834 0.067304968 196 4049.57 67.49288746 0.067492887 197 4060.72 67.67860449 0.067678604 198 4071.73 67.86211339 0.067862113 199 4082.60 68.04340816 0.068043408 200 4093.35 68.22248288 0.068222483 201 4103.96 68.39933172 0.068399332 202 4114.44 68.57394891 0.068573949 203 4124.78 68.74632874 0.068746329 204 4134.99 68.9164656 0.068916466 205 4145.06 69.08435393 0.069084354 206 4155.00 69.24998825 0.069249988 207 4164.80 69.41336316 0.069413363 208 4174.47 69.57447334 0.069574473 209 4184.00 69.73331352 0.069733314 210 4193.39 69.88987852 0.069889879 211 4202.65 70.04416323 0.070044163 212 4211.77 70.19616262 0.070196163 213 4220.75 70.34587173 0.070345872 214 4229.60 70.49328568 0.070493286 215 4238.30 70.63839965 0.0706384 216 4246.87 70.78120892 0.070781209 217 4255.30 70.92170881 0.070921709 218 4263.59 71.05989476 0.071059895 219 4271.75 71.19576224 0.071195762 220 4279.76 71.32930683 0.071329307 221 4287.63 71.46052416 0.071460524 222 4295.36 71.58940997 0.07158941 223 4302.96 71.71596004 0.07171596 224 4310.41 71.84017024 0.07184017 225 4317.72 71.96203652 0.071962037 226 4324.89 72.08155491 0.072081555 227 4331.92 72.19872151 0.072198722 228 4338.81 72.31353248 0.072313532 229 4345.56 72.4259841 0.072425984 230 4352.16 72.53607268 0.072536073 231 4358.63 72.64379463 0.072643795 232 4364.95 72.74914645 0.072749146 233 4371.13 72.8521247 0.072852125 234 4377.16 72.952726 0.072952726 235 4383.06 73.05094709 0.073050947 236 4388.81 73.14678476 0.073146785 237 4394.41 73.24023588 0.073240236 238 4399.88 73.33129739 0.073331297 239 4405.20 73.41996634 0.073419966 240 4410.37 73.50623983 0.07350624 241 4415.41 73.59011504 0.073590115 242 4420.30 73.67158923 0.073671589 243 4425.04 73.75065975 0.07375066 244 4429.64 73.82732402 0.073827324 245 4434.09 73.90157954 0.07390158 246 4438.41 73.97342388 0.073973424 247 4442.57 74.0428547 0.074042855 248 4446.59 74.10986974 0.07410987 249 4450.47 74.1744668 0.074174467 250 4454.20 74.23664378 0.074236644 251 4457.78 74.29639866 0.074296399 252 4461.22 74.35372948 0.074353729 253 4464.52 74.40863436 0.074408634 254 4467.67 74.46111153 0.074461112 255 4470.67 74.51115927 0.074511159 256 4473.53 74.55877594 0.074558776 257 4476.24 74.60396 0.07460396 258 4478.80 74.64670996 0.07464671 259 4481.22 74.68702443 0.074687024 260 4483.49 74.7249021 0.074724902 261 4485.62 74.76034174 0.074760342 262 4487.60 74.79334217 0.074793342 263 4489.43 74.82390234 0.074823902 264 4491.12 74.85202124 0.074852021 265 4492.66 74.87769795 0.074877698 266 4494.06 74.90093164 0.074900932 267 4495.30 74.92172154 0.074921722 268 4496.40 74.94006699 0.074940067 269 4497.36 74.95596738 0.074955967 270 4498.17 74.96942219 0.074969422 271 4498.83 74.98043098 0.074980431 272 4499.34 74.9889934 0.074988993 273 4499.71 74.99510916 0.074995109 274 4499.93 74.99877807 0.074998778 275 4500.00 75 0.075 276 4499.93 74.99877492 0.074998775 277 4499.71 74.99510286 0.074995103 278 4499.34 74.98898395 0.074988984 279 4498.83 74.98041839 0.074980418 280 4498.16 74.96940645 0.074969406 281 4497.36 74.9559485 0.074955948 282 4496.40 74.94004496 0.074940045 283 4495.30 74.92169637 0.074921696 284 4494.05 74.90090332 0.074900903 285 4492.66 74.87766649 0.074877666 286 4491.12 74.85198664 0.074851987 287 4489.43 74.8238646 0.074823865 288 4487.60 74.7933013 0.074793301 289 4485.62 74.76029772 0.074760298 290 4483.49 74.72485495 0.074724855 291 4481.22 74.68697415 0.074686974 292 4478.80 74.64665654 0.074646657 293 4476.23 74.60390345 0.074603903 294 4473.52 74.55871626 0.074558716 295 4470.67 74.51109646 0.074511096 296 4467.66 74.4610456 0.074461046 297 4464.51 74.40856531 0.074408565 298 4461.22 74.3536573 0.074353657 299 4457.78 74.29632336 0.074296323 300 4454.19 74.23656537 0.074236565 301 4450.46 74.17438527 0.074174385 302 4446.59 74.1097851 0.074109785 303 4442.57 74.04276696 0.074042767 304 4438.40 73.97333303 0.073973333 305 4434.09 73.90148559 0.073901486 306 4429.63 73.82722697 0.073827227 307 4425.03 73.7505596 0.07375056 308 4420.29 73.67148599 0.073671486 309 4415.40 73.5900087 0.073590009 310 4410.37 73.50613041 0.07350613 311 4405.19 73.41985384 0.073419854 312 4399.87 73.33118181 0.073331182 313 4394.41 73.24011722 0.073240117 314 4388.80 73.14666303 0.073146663 315 4383.05 73.0508223 0.073050822 316 4377.16 72.95259814 0.072952598 317 4371.12 72.85199378 0.072851994 318 4364.94 72.74901248 0.072749012 319 4358.62 72.64365761 0.072643658 320 4352.16 72.53593261 0.072535933 321 4345.55 72.42584098 0.072425841 322 4338.80 72.31338633 0.072313386 323 4331.91 72.19857233 0.072198572 324 4324.88 72.0814027 0.072081403 325 4317.71 71.96188129 0.071961881 326 4310.40 71.84001199 0.071840012 327 4302.95 71.71579878 0.071715799 328 4295.35 71.5892457 0.071589246 329 4287.62 71.4603569 0.071460357 330 4279.75 71.32913656 0.071329137 331 4271.74 71.19558898 0.071195589 332 4263.58 71.05971852 0.071059719 333 4255.29 70.9215296 0.07092153 334 4246.86 70.78102673 0.070781027 335 4238.29 70.63821449 0.070638214 336 4229.59 70.49309756 0.070493098 337 4220.74 70.34568066 0.070345681 338 4211.76 70.1959686 0.070195969 339 4202.64 70.04396626 0.070043966 340 4193.38 69.88967862 0.069889679 341 4183.99 69.73311069 0.069733111 342 4174.46 69.57426759 0.069574268 343 4164.79 69.41315449 0.069413154 344 4154.99 69.24977667 0.069249777 345 4145.05 69.08413945 0.069084139 346 4134.97 68.91624822 0.068916248 347 4124.77 68.74610848 0.068746108 348 4114.42 68.57372576 0.068573726 349 4103.95 68.3991057 0.068399106 350 4093.34 68.222254 0.068222254 351 4082.59 68.04317641 0.068043176 352 4071.71 67.86187879 0.067861879 353 4060.70 67.67836705 0.067678367 354 4049.56 67.49264718 0.067492647 355 4038.28 67.30472524 0.067304725 356 4026.88 67.11460735 0.067114607 357 4015.34 66.92229973 0.0669223 358 4003.67 66.72780864 0.066727809 359 3991.87 66.53114043 0.06653114 360 3979.94 66.33230152 0.066332302 361 3967.88 66.1312984 0.066131298 362 3955.69 65.92813762 0.065928138 363 3943.37 65.72282581 0.065722826 364 3930.92 65.51536968 0.06551537 365 3918.35 65.30577598 0.065305776 366 3905.64 65.09405156 0.065094052 367 3892.81 64.88020333 0.064880203 368 3879.85 64.66423826 0.064664238 369 3866.77 64.4461634 0.064446163 370 3853.56 64.22598586 0.064225986 371 3840.22 64.00371283 0.064003713 372 3826.76 63.77935156 0.063779352 373 3813.17 63.55290938 0.063552909 374 3799.46 63.32439366 0.063324394 375 3785.63 63.09381186 0.063093812 376 3771.67 62.86117151 0.062861172 377 3757.59 62.6264802 0.06262648 378 3743.38 62.38974559 0.062389746 379 3729.06 62.15097539 0.062150975 380 3714.61 61.9101774 0.061910177 381 3700.04 61.66735948 0.061667359 382 3685.35 61.42252954 0.06142253 383 3670.54 61.17569559 0.061175696 384 3655.61 60.92686566 0.060926866 385 3640.56 60.67604788 0.060676048 386 3625.40 60.42325043 0.06042325 387 3610.11 60.16848156 0.060168482 388 3594.70 59.91174958 0.05991175 389 3579.18 59.65306287 0.059653063 390 3563.55 59.39242987 0.05939243 391 3547.79 59.12985908 0.059129859 392 3531.92 58.86535906 0.058865359 393 3515.94 58.59893846 0.058598938 394 3499.84 58.33060596 0.058330606 395 3483.62 58.06037031 0.05806037 396 3467.29 57.78824034 0.05778824 397 3450.85 57.51422491 0.057514225 398 3434.30 57.23833298 0.057238333 483 1680.41 28.00675534 0.028006755 484 1656.53 27.60889064 0.027608891 485 1632.61 27.21012514 0.027210125 486 1608.63 26.81047186 0.026810472 487 1584.60 26.40994385 0.026409944 488 1560.51 26.00855415 0.026008554 489 1536.38 25.60631589 0.025606316 495 1390.55 23.17580297 0.023175803 496 1366.08 22.76799362 0.022767994 497 1341.57 22.35944142 0.022359441 498 1317.01 21.95015971 0.02195016 499 1292.41 21.54016183 0.021540162 500 1267.77 21.12946116 0.021129461 501 1243.08 20.71807111 0.020718071 502 1218.36 20.30600509 0.020306005 503 1193.60 19.89327654 0.019893277 504 1168.79 19.47989895 0.019479899 505 1143.95 19.06588578 0.019065886 506 1119.08 18.65125056 0.018651251 507 1094.16 18.2360068 0.018236007 508 1069.21 17.82016806 0.017820168 509 1044.22 17.40374791 0.017403748 510 1019.21 16.98675992 0.01698676 511 994.15 16.56921772 0.016569218 512 969.07 16.1511349 0.016151135 513 943.95 15.73252513 0.015732525 514 918.80 15.31340206 0.015313402 515 893.63 14.89377936 0.014893779 516 868.42 14.47367072 0.014473671 517 843.19 14.05308985 0.01405309 518 817.92 13.63205047 0.01363205 519 792.63 13.21056632 0.013210566 520 767.32 12.78865115 0.012788651 521 741.98 12.36631873 0.012366319 522 716.61 11.94358283 0.011943583 523 691.23 11.52045725 0.011520457 524 665.82 11.0969558 0.011096956 525 640.39 10.67309229 0.010673092 526 614.93 10.24888055 0.010248881 527 589.46 9.824334414 0.009824334 528 563.97 9.399467746 0.009399468 529 538.46 8.974294403 0.008974294 530 512.93 8.548828257 0.008548828 531 487.38 8.12308319 0.008123083 532 461.82 7.697073092 0.007697073 533 436.25 7.270811862 0.007270812 534 410.66 6.844313409 0.006844313 535 385.06 6.417591648 0.006417592 536 359.44 5.9906605 0.005990661 537 333.81 5.563533897 0.005563534 538 308.17 5.136225772 0.005136226 539 282.53 4.708750069 0.00470875 540 256.87 4.281120734 0.004281121 Total 26.241165 kWh

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.