I   E   O     R                  1   6   0                                             F i n a l P ro j e c t             ...
Table of ContentsExecutive Summary!                                                               1       Objective!      ...
The Optimal Tilt Angle for Fixed Solar Panels!                                     15Problem Analysis!                    ...
Introduction!                                                33       Model 1!                                            ...
Executive SummaryObjectiveFor this project, our goal was to provide our client with an optimal solar system designfor thei...
SolutionTo answer these questions, we created five models 1:1. Off grid but buying and selling power: This is not a feasibl...
RecommendationsFrom our model, we found that the best option for our client would be to go with theon-grid option and sell...
Task List      1. Research general solar power system background information to see what com-         ponents need to be c...
PERT Chart       Task #    Time (in days)      Corresponding Nodes          Completed By:           1           1         ...
Design Objectives - Solar System BatteriesSolar System BatteriesSince we have decided to go off the grid, a battery backup...
Fig. 1 demonstrates the life span of these batteries if they are used in deep cycle service.According to this plot, “Indus...
Therefore, the user will most probably not want to store them in his house and they willend up costing more to dispose of ...
2) Hiring someone and paying him“$20/hr” for maintenance chores       3) Each battery requires ¼ hour maintenance each mon...
The depth of discharge affects the lifespan of batteries. For example, Fig. 2 demonstratesthe effect of DOD on the lifecyc...
In	  order	  to	  determine	  the	  number	  of	  batteries	  to	  buy,	  it	  was	  necessary	  to	  make	  the	  deJinit...
ing Surrette® 400 series and invest the difference in the market. Hopefully, 8 he is able toat least earn twice as this fu...
Design Objectives - Solar System PanelsChoosing the best solar panelsEvery square meter of the Earth’s surface receives ap...
power within a fixed space. This creates a lower de-rating related to temperature. Inother words, as the temperature increa...
The Optimal Tilt Angle for Fixed Solar Panels             The optimal orientation for solar panels would be to align the f...
Hence, the user should tilt the fixed panel at the latitude angle, which is 37.87 fromhorizontal, because it is easiest, ch...
Problem AnalysisDetermining the Monthly DemandIn order to determine how much demand the client would need monthly, our gro...
class. The Renewable Resource Data Center website provided us with informationabout the available solar insolation in Berk...
Models - IntroductionThese are the variables and parameters that show up in our models. The ones with anasterisk next to t...
*E = maximum useable energy within battery*dod = depth of discharge of batterysun[i] = sunlight availability per day per m...
Model 1Introduction: Off-Grid, Solar Contractor (Buy/Sell Power)In this model, the consumer is off grid but can buy/sell p...
you have to buy them every “ltb” years. We start at time t=0 because you must buyparts for the installation now.The term i...
AMPL Model: Minimizing the Objective FunctionI E O R 1 6 0!                                  BerkeleySOLAR                ...
ConstraintsConstraint #1 is the budget constraint, which says that the initial investment that theconsumer made on the sol...
Model 2Introduction: On-Grid, Solar ContractorThis model is essentially the same as Model #1, except now the person is con...
Model 3Introduction: Off-Grid, Solar Contractor (No Buy/Sell Power)In this scenario, the person is completely off grid and...
AMPL ModelI E O R 1 6 0!        BerkeleySOLAR                 27
Works Cited"Batteries Catalog." Kyocera Solar. N.p.,         "Surrette Rolls are the crown jewels of DCn.d. Web. 6 Dec 201...
Appendix A - Battery SelectionThe Premium Surrette® 500 (bold numbers in the table below) is our final selection for the ba...
Appendix B - Demand Calculations    Month         KWH KWH Billed         Average        Variance   Standard	  De-­‐ Averag...
Appendix C - Weather Calculations                             Average	  Solar	  Insola1on	  (In	  KWH/m^2/day)	           ...
Triangular	  Distribu1on	   Standard	  Devia1on                                       Variance                            ...
Appendix D - AMPL Model OutputsIntroductionAMPL Assumptions      •      In these files, sigmas are not included in the calc...
var LI>=0;var p;minimize cost: (sum{i in 1..12}-net[i]*ce)*(1-1/(1+.04)^ProjectLife)/.04+cp*np+LI+bc*(365*ProjectLife)/360...
Output:MINOS 5.51: optimal solution found.1 iterations, objective 14605.39498Nonlin evals: constrs = 6, Jac = 5.: _varname...
Model 2param ProjectLife;param sun {i in 1..12};param d {i in 1..12};param days {i in 1..12};param ce;param cp;param A;par...
data; ############ DATA STARTS HERE ############param sun:= 1 2.94 2 2.93 3 3.55 4 4.01 5 4.10 6 3.93 7 4.42 8 4.57 9 4.74...
10 net[10] 2578.0611 net[11] 1570.0912 net[12] 1000.6513 np             102.70214 LI            9354.8415 p              1...
var net{i in 1..12}>=0;var np>=0;var nb>=0;var LI>=0;var p;minimize cost:cp*np+LI+bc*(365*ProjectLife)/3600*nb+3000*p-.3*(...
output:MINOS 5.51: optimal solution found.3 iterations, objective 1260743.679Nonlin evals: constrs = 4, Jac = 3.: _varname...
Appendix E - Battery Cost OptimizationMinimizing the Cost per LifetimeThe eventual model that we decided to use in order t...
Appendix F - Solar Panel Cost OptimizationMinimizing the Cost per Watt                                       min u = xy + ...
Appendix G - Solar Installation Costs                                      A1 Sun Inc.                                   A...
Appendix H - Night Hours v Months                  12.500                  12.275    Night Hours                  12.050  ...
Appendix I - kWh Bill for 25 YearsMonth kWH       $    $/kWh           kWH         $    $/kWh Ave.      kwh/        $    $...
Appendix J - Solar Power CalculatorSystem Specifications                                  Berkeley, CASolar Radiance (kWh/s...
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The perfect solar design for a berkeley home

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The perfect solar design for a berkeley home

  1. 1. I E O R 1 6 0 F i n a l P ro j e c t BERKELEYSOLAR PREPARED FOR: BERKELEY CLIENT, PROFESSOR GLASSEY PREPARED BY: NASSIM FARROKHZAD, REGINE LABOG, KENNETH LEE, RHONDA NASSAR, MIRANDA ORTIZ, CHRISTINA YOUUniversity of California, Berkeley IEOR 160 FINAL PROJECT
  2. 2. Table of ContentsExecutive Summary! 1 Objective! 1 Goals! 1 Solution! 2 Recommendations! 3Task List! 4PERT Chart! 5Design Objectives - Solar System Batteries! 6 Solar System Batteries! 6 Maintenance Cost! 8 Acid Leakage and Durability! 9 Determining the Optimum Battery Capacity! 9 Minimizing the Lifetime Cost of the Battery System! 10 Battery Cost Optimization Results! 11 Conclusion! 12Design Objectives - Solar System Panels! 13 Choosing the best solar panels! 13U C B e r k e l e y! IEOR 160 Final Project i
  3. 3. The Optimal Tilt Angle for Fixed Solar Panels! 15Problem Analysis! 17 Determining the Monthly Demand! 17 Determining the Average Amount of Sunlight in Berkeley! 17Models - Introduction! 19 Variables! 19 Parameters! 19Model 1! 21 Introduction: Off-Grid, Solar Contractor (Buy/Sell Power)! 21 AMPL Model: Minimizing the Objective Function! 23 Constraints! 24Model 2! 25 Introduction: On-Grid, Solar Contractor! 25 AMPL Model! 25Model 3! 26 Introduction: Off-Grid, Solar Contractor (No Buy/Sell Power)! 26 AMPL Model! 27Works Cited! 28Appendix A - Battery Selection! 29Appendix B - Demand Calculations! 30Appendix C - Weather Calculations! 31Appendix D - AMPL Model Outputs! 33U C B e r k e l e y! IEOR 160 Final Project ii
  4. 4. Introduction! 33 Model 1! 33 Model 2! 36 Model 3! 38Appendix E - Battery Cost Optimization! 41 Minimizing the Cost per Lifetime! 41Appendix F - Solar Panel Cost Optimization! 42 Minimizing the Cost per Watt! 42Appendix G - Solar Installation Costs! 43Appendix H - Night Hours v Months! 44Appendix I - kWh Bill for 25 Years! 45Appendix J - Solar Power Calculator! 46U C B e r k e l e y! IEOR 160 Final Project iii
  5. 5. Executive SummaryObjectiveFor this project, our goal was to provide our client with an optimal solar system designfor their Berkeley residence. Because every city has a different policy on solar panel in-stallation and gets varying amounts of sunlight, we had to tackle many variables toprovide our client with his best options.GoalsThe owner outlined the following: ‣ Off the grid where he would have no connection to PG&E and would require a self-sustainable solar panel system, even during consecutive cloudy days where there would be little sunshine. ‣If possible, he would like to sell excess power and buy it if necessary. ‣The solar system must fit the needs of his home with a working space of 1500 sq ft of roof space.Before even attempting to address the owner’s concerns, we needed to take multiplevariables out of the equation:! How many hours of sunshine does the house get every day?! How much energy does the owner need every month? What kind of batteries and solar panels would serve the client’s needs while minimizing the cost?I E O R 1 6 0! BerkeleySOLAR 1
  6. 6. SolutionTo answer these questions, we created five models 1:1. Off grid but buying and selling power: This is not a feasible option for the user be-cause the cost of the batteries needed to support the house during cloudy days exceedsthe payout from switching to solar.2. On grid and selling excess power: In this situation, the user should install 103 solarpanels while selling to PG&E excess power during months when he doesn’t use asmuch energy. After 25 years, the cost of the solar system panels would be $913. Also, be-cause the cost of maintenance will be marginal compared to the cost of installation andthe lifetime of the solar panels are longer than what we outlined, the user would bemaking revenue after 25 years. Also, we based our interest rate on 4% which couldchange over time and did not factor the refinancing value of the home after switching tosolar.3. Completely off grid getting nothing for excess power: This model was very difficultto justify. Although it is possible to go off grid with the size of the roof and the amountof sunshine Berkeley gets, we had to completely omit the budget constraint due to thelarge costs.4. Optimizing the battery’s cost over lifetime: To factor in key characteristics for ourideal battery, we researched multiple batteries and created a model that used cost overlifetime and added multiple constraints. After putting the batteries through this model,we found that the Premium Surrette performed the best.5. Minimizing the cost of solar panels: Before subjecting our choices through the model,we limited our options to panels that fit four key characteristics that will be outlinedlater. We finally settled on the Evergreen 210W panels due to it’s lower cost per watt,but still decent efficiency and a low impact on the environment.1 See appendix D-F for all models.I E O R 1 6 0! BerkeleySOLAR 2
  7. 7. RecommendationsFrom our model, we found that the best option for our client would be to go with theon-grid option and sell excess power to PG&E. Although the client can feasibly go off-grid, it would be in his best interest to stick with the second model because it is the mostlikely to fit in his budget. The key issue with solar panels is that their cost does not jus-tify their low efficiency and the main barrier in this is a lack of technological research insolar panels. If the user goes on-grid, he will be able to get governmental aid in the formof the California Solar Initiative as well as tax rebates. That way, he will reach his break-even point sooner and can later invest in cheaper solar technology.I E O R 1 6 0! BerkeleySOLAR 3
  8. 8. Task List 1. Research general solar power system background information to see what com- ponents need to be considered. 2. Determine how much power the system needs to provide to be fairly certain that he will not ever use more than this amount of power. 3. Research all things about batteries including information about their capacities, lifetimes, sizes, brands, costs, etc. 4. Find optimal battery. 5. Research different panels including their wattage, size, efficiency, costs, etc. 6. Research costs projected by different contractors for the average home in Ber- keley. 7. Research federal tax deductions and California Solar Initiatives. 8. Research variances in weather and sunlight availability in Berkeley. 9. Create models for different options: 1. Have contractor build your system design, on grid. 2. Build your own system, off grid 3. Have contractor build your system, off grid, but can buy and sell power 10. Solve models. 11. Write executive summary and recommendations. 12. Create PERT chart.I E O R 1 6 0! BerkeleySOLAR 4
  9. 9. PERT Chart Task # Time (in days) Corresponding Nodes Completed By: 1 1 Start, A Rhonda, Regine, Kenneth, Nas- sim, Miranda 2 0.2 A,B Miranda 3 5 B,D Rhonda, Kenneth, Nassim 4 0.5 D,E Kenneth 5 2 B,C Regine 6 0.5 B,F Regine and Rhonda 7 0.2 B,G Regine, Miranda, and Rhonda 8 1 B,H Miranda 9 2 I,J Miranda and Regine 10 1 J,K Rhonda, Regine, Kenneth, Nas- sim, Miranda 11 1 K,L All 12 0.2 L,M MirandaI E O R 1 6 0! BerkeleySOLAR 5
  10. 10. Design Objectives - Solar System BatteriesSolar System BatteriesSince we have decided to go off the grid, a battery backup system is required to save theexcess energy gained during the day for nights and cloudy days. This means the batter-ies would be deeply discharged on regular basis. For a solar system, the following bat-teries are offered by different vendors: • Lead-acid Batteries are made of lead electrode plates submerged in dilute sulfuric acid as an electrolyte. They are readily available in the market and have low ini- tial cost. These batteries can be designed for either shallow or deep cycle usage. oShallow cycle batteries are designed to supply a large amount of current for a short time but they cannot tolerate being deeply discharged frequently. oDeep cycle batteries are designed to be repeatedly discharged by as much as 80% of their capacity (Depth of Discharge, DOD). • Marine batteries are made of lead sponge electrodes and considered to be a “hy- brid” of starting and deep cycle battery. • Gelled Deep Cycle contains an acid gel which means if it’s broken, the acid does not leak. But, this type of batteries must be charged at a slower rate and lower voltage to prevent excess gas from damaging the cells. • Absorbed Glass Mat (AGM) Batteries are made of fine fiber Boron-Silicate glass mats which contains the acid. These batteries also don’t leak acid if broken.I E O R 1 6 0! BerkeleySOLAR 6
  11. 11. Fig. 1 demonstrates the life span of these batteries if they are used in deep cycle service.According to this plot, “Industrial Deep Cycle” battery demonstrates the longest life 30 Years  (max) Years  (min) 23Years 15 8 0 Marine Gelled  Deep  Cycle Industrial  Deep  Cyclespan, followed by “Rolled Surrette® Deep Cycle”.Figure 1: Life span of batteries used in "Deep Cycle Services" (Source:www.windsun.com, deep cycle battery FAQ)Choosing the Best Battery In general, Lead-acid batteries are cheaper and last longer than Marine, AGMand Gelled batteries, but they are not as safe as the latter ones. So, in order to make theright selection, first we need to make some assumptions for our problem:Assumption 1) The batteries will be used in an off-grid, full-time home for an indefi-nitely long time, therefore, capacity and long term cost will be the most important fac-tors.Assumption 2) The batteries will be placed in the resident’s home and not in a remotesite; therefore, maintenance is not much of a concern in our choice of batteries. Priceand lifetime is valued over maintenance in our research.Assumption 3) The batteries are stored in a place where the temperature does not fellunder 50°F below which the batteries capacity starts to decline.Assumption 4) Nickel-Cadmium batteries were not considered in our analyses becausethey are “extremely toxic to the environment and require very expensive disposal” (3).I E O R 1 6 0! BerkeleySOLAR 7
  12. 12. Therefore, the user will most probably not want to store them in his house and they willend up costing more to dispose of than buying new lead-acid batteries.Assumption 5) We do not consider Lithium ion batteries because they are extremely ex-pensive for this specific type of application.Assumption 6) The user have no constrains on his/her initial capital investment andhas enough and appropriate space to store the batteries and the solar panels.Within different brands of lead-acid batteries, Surrette® repeatedly is reported as themost efficient and economical choice by vendors and contractors.2 Our initial calcula-tion for lifetime cost has confirmed3. In particular, Premium Surrette 500 (12CS11PS)excels over all the other batteries in an economic sense. The initial up front expense maybe out of reach for some customers, but given its lifetime, it is the cheapest.4In contrary to “sealed” batteries, Surrette® batteries require frequent maintenance,which after researching and discussing in detail below does not alter our choice of bat-tery.Maintenance Cost As mentioned earlier, AGM and Gel batteries are almost maintenance free. There arealso so-called Lead-Acid “Sealed” batteries which needs to be replaced every 5-7 yearsin the exchange of no maintenance throughout these years. These batteries are not eco-nomically suited for our purpose.The maintenance of Flooded Lead Acid-Surrette 500 batteries require “watering, equal-izing charges and keeping the top and terminals clean” (7). One of the websites ourgroup researched that supported AGM batteries, conducted numerical analyses of theprice difference between Surrette® and AGM. Even after adding the electrolyte mainte-nance costs, the Surrette Premiums 500® still remained the cheapest. (8)Their calculation costs were based on some assumptions: 1) Cycle once a day2  www.solarinfo.com,  www.rollsbattery.com,  www.dcbatteries.com  3  For  further  detail  on  battery  calculation  refer  to  page  274  Based  on  Assumption  1I E O R 1 6 0! BerkeleySOLAR 8
  13. 13. 2) Hiring someone and paying him“$20/hr” for maintenance chores 3) Each battery requires ¼ hour maintenance each month 4) Each cell requires ¼ qt of $ 1/qt distilled water that equates to $0.000583/(Ah*cycle) for Surrette 400 and $0.00049 (Ah*cycle) for Surrette 500Table 1: Maintenance cost per cycle for various batteries (Source: www.vonwentzel.net) Lifeline  AGM Surrette  400 Surrette  500 Adjusted  Cost($) 0.0015 0.00147 0.00108Acid Leakage and DurabilityThe Premium Surrette 500 (12CS11PS) utilizes the new generation “dual containermodular construction” (6). This feature eliminates breakage and subsequently acidleakage due to rough handling or abuse. Even if the outer container were to break, thebattery would still operate without any acid spills (7). Therefore, Premium Surrettes aresafe for our user to store in his garage or a battery room in his house. In addition, thesebatteries can be installed without any special skills or tools. Therefore, our user is goingto highly value this option, since he wants to save as much money as possible. Our analyses and assumptions show that Premium Surrette 500 (12CS11PS) cellsare unsurpassed in the qualities they offer. Their higher cycle lives compared to theirbudget competition, their durability, their thick lead plates and not having to replacethem every few years makes them an attractive economic choice, even if their up-frontprice is not the most economical (4).Determining the Optimum Battery CapacityIn order to find the optimum battery capacity, first we looked at customer’s averagedaily usage based on Kwh-hr. In order to be in safe side, we decided to design a storagesystem that would provide up to 5 times of this capacity in case of an emergency. Thisnumber is an industry standard. Next, given their ampere-hour5, depth of discharge6,and the cost of the batteries, we calculated the lifetime cost of different batteries.5  Ampere-­hour  is  a  measure  of  a  battery’s  capacity  (e.g.  6  Amp-­hr  battery  can  maintain  a  current  of  1  Ampere  for  6  hours)  6  Depth  of  Discharge  (DOD)  is  the  extend  at  which  a  battery  is  being  dischargedI E O R 1 6 0! BerkeleySOLAR 9
  14. 14. The depth of discharge affects the lifespan of batteries. For example, Fig. 2 demonstratesthe effect of DOD on the lifecycle of Surrette 400 and 500 series. 5000 3750 #  of  cycles 2500 1250 0 0 25 50 75 100 DOD  (%) 500  Series 400  SeriesFigure 2: Surrette(R) batteries’ lifecycle vs. %DOD (Source: www.surrette.com)Since we will have batteries for 5 times of the user’s average usage, we assume the bat-teries rarely go beyond DOD of 50%. Now, our goal is to find which battery would offerthe minimum lifetime cost.Minimizing the Lifetime Cost of the Battery System7   Given  the  ampere-­‐hour  and  the  hour  rating,  we  were  able  to  determine  the  maxi-­‐mum  current  that  would  be  pulled  from  the  battery  to  last  for  20  hours.    Then,  with  the  given  voltage  and  the  calculated  current,  we  were  able  to  calculate  the  maximum  energy  in  Kilowatt-­‐hours  by  multiplying  the  voltage  and  current  and  dividing  by  1000  to  convert  it  to  the  correct  units.7 Refer to the appendix for the numerical results of the modelI E O R 1 6 0! BerkeleySOLAR 10
  15. 15. In  order  to  determine  the  number  of  batteries  to  buy,  it  was  necessary  to  make  the  deJinition  of  the  number  of  batteries  to  buy  be  a  function  in  terms  of  the  depth  of  discharge.    Because  each  battery  had  a  different  maximum  energy,  a  different  depth  of  discharge  had  to  be  used  for  each  battery.    Also,  the  chosen  depth  of  discharge  for  each  battery  would  affect  the  number  of  batteries  bought,  with  lower    requiring  more  batteries  due  to  the  low  level  of  drain  on  the  battery.    The  numerator  of  that  function  was  obtained  by  Jinding  the  average  daily  energy  usage  of  the  household  which  ended  up  being  approximately  20KWH.    The  av-­‐erage  daily  energy  usage  was  used  because  that  would  reJlect  the  average  amount  of  energy  drained  from  the  battery  each  day,  which  would  give  a  more  realistic  analysis  of  the  cost  per  cycle  through  a  more  accurate  cost  and  lifetime  determination.    Because  there  are  some  days  where  the  sun  will  not  shine,  and  the  battery  will  not  charge,  the  battery  energy  capac-­‐ity  should  be  greater  than  the  average  daily  discharge.    Five  times  the  average  daily  usage  was  used  because  the  probability  of  having  Jive  days  of  no  sunshine  is  very  small.    There-­‐fore  the  number  of  batteries  required  was  determined  using  the  total  energy  required,  mul-­‐tiplied  by  Jive  and  dividing  it  by  the  total  energy  of  the  battery  that  will  be  taken  from  each  battery  at  that  depth  of  discharge  and  rounding  up. Total  cost  was  then  determined  by  multiplying  the  number  of  batteries  by  the  price  given.    The  lifetime  in  cycles  is  determined  by  using  a  function  which  is  different  for  each  battery,  and  the  depth  of  discharge,  which  determined  the  lifetime  of  the  battery.    The  cost  per  cycle  of  each  battery  was  then  found  and  the  battery  with  the  lowest  cost  per  cycle  is  the  one  chosen  to  be  the  most  optimal,  with  an  optimal  battery  capacity  equal  to  the  com-­‐bined  capacity  of  the  battery  chosen  and  the  number  of  batteries  bought.Battery Cost Optimization Results As it was discussed earlier, we can safely assume that the batteries rarely wouldbe discharge above 50% of their capacity since the user stores electricity five times ofhis/her average daily usage. Based on manufacturers’ data on corresponding number oflifecycles to DOD, we found the minimum cost per cycle that is required for the resi-dence to completely supply his own energy, for approximately four days without re-charging. The cheapest battery cost per cycle according to our calculation is $8.96. Itmeans that we need to buy 42 of the Premium Surrette® 500 (12CS11PS) batteries. We need to mention, there are also some aspects of the battery selection that canaffect our final decision but not easy to incorporate into the model. The saving of using500 series battery is $120 per year which for twenty-five years translates to $1080 (as-suming 10% discount rate). The resident can save on the front cost of batteries by buy-I E O R 1 6 0! BerkeleySOLAR 11
  16. 16. ing Surrette® 400 series and invest the difference in the market. Hopefully, 8 he is able toat least earn twice as this future saving. That decision is based on the customer’s per-sonality and lifestyle.The model also doesn’t take into the account the energy loss by the wires. By havingmore wiring, there is more energy loss during transportation. So a battery of a largervoltage, say 6V, would lose less energy than several batteries of smaller voltage, saythree 2V’s. So, the user may want to consider using the same series of the batteries ourmodel suggest but pick the one with higher voltage. Also there is energy lost during theconversion of DC to AC and that is not taken into account our model either.ConclusionThe total storage capacity would then be 172 KWH with a Depth of Discharge of 50%. Itis feasible to go completely off the grid but it is an ill advice based on the battery costsalone. If one chose to go off grid, one would have to pay at least $8.96 dollars per cycle.Each cycle is one day, so the cost per month would be 268.8 dollars, much more than theprice of electricity from PG&E. Also the weight and volume demand for storage ofbatteries would exceed the typical free space in a typical household. The volumerequired for all of the batteries is 110 Cubic feet and would weigh 11,424 lbs., a space ofwhich one would be hard pressed to find in Berkeley.8  Assuming  no  recession  for  foreseeing  futureI E O R 1 6 0! BerkeleySOLAR 12
  17. 17. Design Objectives - Solar System PanelsChoosing the best solar panelsEvery square meter of the Earth’s surface receives approximately 164W of solar energyfrom the sun. If we could cover 1% of the Sahara desert with solar panels, we couldgenerate enough electricity to power the entire world. Although we could potentiallyharness the sun’s energy to satisfy all of our needs, the technology currently availablecan only harness, at most, 20% of that power. As is frequently said in the solar industry,“not all solar panels are created equal.” Therefore, we based our choice in solar panelson the following four criterions:1. Minimum warranted power rating - This is the amount of power guaranteed by the manufacturer that the solar panel can generate. In some solar panel specification sheets, this was also known as the negative tolerance rating. Generally, a good solar panel would have a negative tolerance rating at 5% or less.2. PVUSA Test Conditions (PTC): PVUSA is an independent lab that releases a PTC rat- ing for all solar panels listed under the California Solar Initiative. Compared to the STC (Standard Test Conditions) rating that manufacturing companies use, the PTC tests the panels under more extreme, real-world conditions.3. Efficiency Rating: This is the most well-known rating since researchers are focused on creating a low-cost high-efficiency solar panel. The higher this efficiency, the more power attainable per square inch of the panel surface.4. UL Listing: Underwriters Laboratories is a product rating company that tests the safety of products. They test solar panels for their mounting method, weather resis- tance, performance, as well as other safety considerations and have a large photovol- taic testing site in Silicon Valley. Products that pass UL’s harsh tests are often adver- tised as UL Listed.After passing the four constraints, we narrowed our options to two solar panels whichexcelled in either efficiency, or environmental impact and affordability.The Sanyo 195W PV module, compared to the average 12% efficiency of most panels,surpasses them with a 19.7% cell efficiency. They do this with a patented HIT (hetero-junction w/ intrinsic thin layer) technology that allows the PV module to obtain maxI E O R 1 6 0! BerkeleySOLAR 13
  18. 18. power within a fixed space. This creates a lower de-rating related to temperature. Inother words, as the temperature increases, these solar panels produce 10% or more elec-tricity than conventional crystalline silicon modules. The PV design reduces recombina-tion loss of the charged carrier by surrounding the energy generation layer of single thincrystalline silicon with high-quality ultra-thin amorphous silicon layers. The solar pan-els operate silently with no moving parts and are among the lightest per watt in the in-dustry. They have a PTC rating of 180.9W and its packing density reduces the transpor-tation, fuel, and storage cost per installed watt.Evergreen’s 210W PV modules are ideal for grid-tied solar systems and feature anti-reflective glass, an anodized aluminum frame, 108 cells per panel, and watertight junc-tion boxes that require zero maintenance. All panels have a minimum warranted powerof -0/+5W, have a PTC rating of 180.7W, and are independently tested by four labs thatregularly check panel power so the power given is the power promised. The anti-reflective glass delivers 2-3% more electricity than panels containing standard glass andmaintains 4% higher output than most other crystalline silicon panels under hot condi-tions. The amount of time it takes for the environmental footprint of the manufacturingprocess to be offset by the clean energy created by the PV module is called the “low en-ergy payback.” Evergreen’s products can recoup the environmental impact in a yearwith a combination of efficiency and environmentally responsible manufacturing proc-esses. The Evergreen Spruce PV module produces 30g of CO2 per equivalent kWh aswell as uses less lead than other panels thanks to lead-free solder. BRAND PTC PRICE/ AREA MAX # OF PRICE/ PA N E L (FT^2) PA N E L S WAT T Evergreen 210W 180.7 $643 16.93 93 $3.49 Sanyo 195W 180.9 $915 12.47 125 $5.06I E O R 1 6 0! BerkeleySOLAR 14
  19. 19. The Optimal Tilt Angle for Fixed Solar Panels The optimal orientation for solar panels would be to align the face of the solarpanel with the sun. However, that would require continuous adjustments of the solarpanel. It is too expensive to purchase the equipment to adjust it continuously and, there-fore, changing the tilt angle to its daily and monthly optimal values is not practical, ifthe panels are mounted on the roof, or economical for our user. As professor Glassey atthe Industrial Engineering and Operations Research department at UC Berkeley sug-gested and many solar panel websites our group consulted, tilting the fixed plate by anangle equal to the latitude seems to be the most practical solution. At this tilt, if the col-lector is facing south, our case, since the user lives in the Northern Hemisphere, the sunwill be “normal to the collector at noon twice a year” at the “equinoxes”, when day andnight are equal length. The noontime sun will only vary “above and below this positionby a maximum angle of 23.5 degrees”.8 Our group research presents the results of a study that was conducted on twosouth facing sites in Albuquerque, New Mexico and Madison Wisconsin. Figure 2.4shows that by titling at the latitude, the user will only be slightly below the maximumyearly irradiation optimal position. The figure shows that variations in the tilt angle donot affect the irradiation received by much and therefore, given the amount of moneyand work the user has to invest in order to reach an optimal tilt angle each day, it is notworth his/her effort or money, because the amount of irradiation difference isminimal.8I E O R 1 6 0! BerkeleySOLAR 15
  20. 20. Hence, the user should tilt the fixed panel at the latitude angle, which is 37.87 fromhorizontal, because it is easiest, cheapest and will maximize annual performance. Figure 2.4 Total irradiation south-facing tilted surfaces_________________8. http://www.powerfromthesun.net/Chapter6/Chapter6.htm#6.3.1%20OrientationI E O R 1 6 0! BerkeleySOLAR 16
  21. 21. Problem AnalysisDetermining the Monthly DemandIn order to determine how much demand the client would need monthly, our groupfirst assumed that the given KWH billed for this year and last year have a normal dis-tribution. Using this assumption, the average and standard deviation of the two datasets were calculated. More data would have made the data sets more accurate, but ourgroup was only given two, so we worked with what we had. According to the normaldistribution, approximately 95% of data is located within two standard deviations of themean. Thus, we made our target demand for each month equal to the average plus twotimes the standard deviation, so that we could be 97.5% sure that his demand wouldnever exceed this value.Determining the Average Amount of Sunlight in BerkeleyIn order to determine the average amount of sunlight that was available (kWh/m^2/day) to the solar panels in Berkeley, we used the triangular distribution presented inI E O R 1 6 0! BerkeleySOLAR 17
  22. 22. class. The Renewable Resource Data Center website provided us with informationabout the available solar insolation in Berkeley, taking into account cloudy days andmonthly temperature variations. Since we were only given one set of averages, maxi-mums, and minimums for each month, we used the triangular distribution to find thestandard deviation of the data.Firstly, the website only provided us with the solar insolation values for a 15 degree tiltand a 90 degree tilt. Since our optimal design required an approximately 38 degree tilt,we had to extrapolate the data. Upon making the assumption that the data was ap-proximately linear, we used the degree of tilt as our x value and the solar insolation asour y value and calculated a line for each month passing through the two points (15, in-solation[i]) and (90,insolation[i]). First, the slopes were calculated. Next, using theequation , plugging in the point (15, insolation[i]) for (x1,y1), and then plugging in x=38,we obtained the insolation (y) value for a tilt of 38 degrees. We performed this iterationfor each month’s average, maximum, and minimum insolation values. Next, in order tofind the standard deviation, we assumed that the average given was equivalent to themode, and we found the standard deviation formula on Wikipedia. Using this formula,the averages, the maximums, and the minimums, we calculated the standard deviationfor each month. Adding and subtracting 2*standard deviation from the average, we ob-tained a 95% confidence interval. To be safe, we assumed that the available amount ofsunlight would be equal to the lower bound of this confidence interval. In taking thelower bound of the confidence interval to be our assumed solar availability for eachmonth, we are 97.5% sure that the amount of available solar insolation will never be lessthan this value. Thus, we are 97.5% sure that there will always have enough sunlight toprovide an adequate amount of power to our system.I E O R 1 6 0! BerkeleySOLAR 18
  23. 23. Models - IntroductionThese are the variables and parameters that show up in our models. The ones with anasterisk next to them (*) are the variables/parameters that don’t show up in everymodelVariables*net[i]= If negative, the system did not produce enough energy in month I and the con-sumermust purchase this much. If net[i] is positive in month i, then the system produced morethanneeded and the consumer will sell it.np = number of panels>=0*nb = number of batteries>=0p = If 0, then no panels were produced and therefore no installation costs were incurredand notax can be deducted. If p=0 then they can.Parametersce= cost to purchase electricity/price to sell back electricitycp = cost of each solar panelLI = labor and installation cost (equal to $7-$9 dollars per watt)nmc = number of miscellaneous costs (inverter, controller, maintenance)LT[j] = lifetime of each of the miscellaneous componentsmc[j] = cost of each miscellaneous componentBudget = maximum initial budgetd[i] = demand for each month*bc = cost of each battery*ltb = lifetime of each batteryI E O R 1 6 0! BerkeleySOLAR 19
  24. 24. *E = maximum useable energy within battery*dod = depth of discharge of batterysun[i] = sunlight availability per day per m^2 in month isigmas[i] = standard deviation of available sunlight in month isigmad[i] = standard deviation of demand in month iA = area of one panel in m^2Eff = efficiency of the panelsI E O R 1 6 0! BerkeleySOLAR 20
  25. 25. Model 1Introduction: Off-Grid, Solar Contractor (Buy/Sell Power)In this model, the consumer is off grid but can buy/sell power that he needs/has over-produced. The objective function is to minimize the net present value of the costs in-curred over a 25 year project lifetime.The term is a summation of the consumer’s costs from buying extra energy that heneeds and the revenues from selling power in the months he has excess. If in month inet[i]>0 then this means that he has produced excess power and will sell it at price “ce”(we are making the assumption that the price to sell energy is equal to the price to buyit). Thus, if net[i]>0 then the cost is subtracted, whereas if net[i]<0 then the cost is addedto the total cost.The term is the annuity formula, where “ ” is the money that we dis-count back each year for the duration of 25 years at a rate of 4%. In order to simplify ourcalculations, we assumed that the interest was compounded at the end of each year, sothat the fact we discounted the sum of the payments at the end of each year rather thandiscounting them each month does not make a difference.The term represents the total discounted cost of both the initial batteries and their re-placements over the 25 year period. We made the assumption that you have to buy newbatteries every “ltb” years. So, if the lifetime of the battery is 10 years then we have tobuy a battery every 10 years (i.e. in year 0, 10, and 20). Ceil(25/ltb) is equivalent to 25divided by ltb rounded to the next highest integer (i.e. ceil(25/10)=ceil(2.5)=3). This de-termines, based upon the lifetime of each battery (ltb) in years, how many times youwill have to buy new batteries throughout the project lifetime of 25 years, assuming thatI E O R 1 6 0! BerkeleySOLAR 21
  26. 26. you have to buy them every “ltb” years. We start at time t=0 because you must buyparts for the installation now.The term is a summation of the j miscellaneous parts, such as controllers, inverters,mounting systems, and switches. The parameter LT[j] is the respective lifetime of mis-cellaneous cost j; here we assume again that we must buy a new miscellaneous partevery LT[j] years. Of course, there are more costs, but we are assuming that the rest arenegligible in comparison.The term takes into account the Federal Tax Deduction of 30% of total costs (not includ-ing batteries) for people who go “off the grid”. Unfortunately, when people go off thegrid, they do not qualify for the California Initiative, which compensates you for an ad-ditional 13% of the total after tax rebate costs.-1318*Lastly, the term is the “revenue” that you save by not having to pay your monthly PGE bills. Theterm -1318 is the average amount that Berkeley residents pay for their PG&E bill anddiscounts this annual payment back at a 4% discount rate for the duration of the project lifetime.I E O R 1 6 0! BerkeleySOLAR 22
  27. 27. AMPL Model: Minimizing the Objective FunctionI E O R 1 6 0! BerkeleySOLAR 23
  28. 28. ConstraintsConstraint #1 is the budget constraint, which says that the initial investment that theconsumer made on the solar power system does not exceed the amount (“Budget”)available to him. This brings me to another assumption: in order to simplify our calcula-tions we are assuming that this person has savings from which he can invest thismoney, rather than having to deal with complications of a loan and loan payments.Constraint #2 is binary and is determinant of whether or not certain costs associatedwith actually installing the system will be incurred. In some of the models it was opti-mal to not build the solar powered system, and to instead just stick with PGE bills, sothe installation and labor costs would not be incurred. Constraints and costs multipliedby variable p are the constraints and costs that are only applicable if the system is actu-ally built, and equal to zero if it is not.Constraint #3 makes sure that your power demands are met. As explained earlier, net isthe variable which measures the amount that you must purchase in order to have anadequate amount of power (if negative), and the amount by which you have exceededyour power needs and can sell back (if positive).Constraint #4 ensures that the panels do not exceed the available roof space.Constraint #5 ensures that we have adequate battery capacity to store the energy weneed, and is explained further in the “Battery” section of our paper.Constraint #6 is a measure of installation costs, which we have found is approximately$8 per watt. Thus, constraint #6 finds the wattage of our system and multiplies it by 8dollars to get total installation costs.I E O R 1 6 0! BerkeleySOLAR 24
  29. 29. Model 2Introduction: On-Grid, Solar ContractorThis model is essentially the same as Model #1, except now the person is connected toPGE. We modeled the net again like revenue for two reasons. One, PGE gives you theoption of a plan where they do buy back your excess energy, and sell you energy in themonths that you do not have enough. Two, even if you choose to go with the plan inwhich you buy extra energy and PGE credits you for future electricity (in this case youwould qualify for the California Solar Initiative), the electricity that you don’t have topay in the future is like revenue. For simplicity, we will assume that the person is sellingto PGE excess power and buying power that he did not make enough of himself.Thus, everything is the same as in the previous model except batteries are not includedin the cost or the constraints because the consumer does not need them.AMPL ModelI E O R 1 6 0! BerkeleySOLAR 25
  30. 30. Model 3Introduction: Off-Grid, Solar Contractor (No Buy/Sell Power)In this scenario, the person is completely off grid and does not have the means to buy orsell to anyone; for this reason the variable “net” is not included in the objective function,because whatever he makes extra is lost. Due to the fact that he must sustain himselfcompletely, we have added the constraint that all values of “net” must be greater thanor equal to zero. If during any month net<0, then he did not have enough energy andhis power went out. Lastly, since the panels must be built if he wants any electricity atall, p will equal one no matter what.I E O R 1 6 0! BerkeleySOLAR 26
  31. 31. AMPL ModelI E O R 1 6 0! BerkeleySOLAR 27
  32. 32. Works Cited"Batteries Catalog." Kyocera Solar. N.p., "Surrette Rolls are the crown jewels of DCn.d. Web. 6 Dec 2010. batteries." N.p., n.d. Web. 6 Dec 2010. <www.dcbattery.com>.<www.kyocerasolar.com>."How about Nickel-Cadmium Cells?" N.p., "NASA Surface Meteorology and Solar En-n.d. Web. 6 Dec 2010. ergy." NASA Langley Atmospheric Science<www.vonwentzel.net>. Data Center (Distributed Active Archive Cen- ter). Web. 06 Dec. 2010. <http://eosweb.larc.nasa.gov/cgi-bin/sse/grid.c"Life span of batteries used in "Deep Cycle gi?>.Services" ." Web. 6 Dec 2010.<www.windsun.com>. "Solar Calculator." Solar Power Facts and Helpful Info. Web. 06 Dec. 2010. <http://www.solartradingpost.com/calcu"My third letter." Von Wentzel Family Site. late.php?name=5>.N.p., n.d. Web. 6 Dec 2010. "SOLAR RADIATION FOR FLAT-PLATE<http://www.vonwentzel.net/>. COLLECTORS FACING SOUTH AT A FIXED-TILT." Renewable Resource Data Cen- ter (RReDC) Home Page. Web. 06 Dec. 2010. <http://rredc.nrel.gov/solar/old_data/ns"Renewable Energy 2010 Design Catalog." rdb/redbook/sum2/23234.txt>.N.p., n.d. Web. 6 Dec 2010.<www.aeesolar.com>."Solar Series 5000." Kyocera Solar. N.p.,n.d. Web. 6 Dec 2010.<www.kyocerasolar.com>."Solar Series 5000." Pure Energy Systems.N.p., n.d. Web. 6 Dec 2010.<www.pureenergysystems.com>."Surrette(R) batteries’ lifecycle vs. %DOD." Web. 6 Dec 2010. <www.surrette.com>.I E O R 1 6 0! BerkeleySOLAR 28
  33. 33. Appendix A - Battery SelectionThe Premium Surrette® 500 (bold numbers in the table below) is our final selection for the battery systemTable  2:  Summary  of  calculations  for  battery  selection  based  on  the  model Ampere-­   Hour   #  to   Cost  per   Battery Voltage Current Power Max  KWH DOD KWH Lifetime Price Total  Cost hours rating buy LifetimeLifeline  AGM   12 225 20 11.25 135 2.7 0.5 1.35 64 1000 $387.00 $24,768.00 $24.77 (8D)West  Marine   12 225 20 11.25 135 2.7 0.5 1.35 64 500 $449.00 $28,736.00 $57.47 Gel  (8D)Inexpensive   Trojan   12 225 20 11.25 135 2.7 0.5 1.35 64 500 $152.00 $9,728.00 $19.46 (2xT105) Premium  Surrette  400   12 221 20 11.05 132.6 2.652 0.5 1.32 65 1250 $246.00 $15,990.00 $12.79 (HT8DM) Premium  Surrette  500   12 342 20 17.1 205.2 4.104 0.5 2.05 42 3200 $683.00 $28,686.00 $8.96(12CS11PS) 2-­KS-­33PS   (Surrette   2 1750 20 87.5 175 3.5 0.5 1.75 50 3300 $1,184.00 $59,200.00 $17.94 500  series) 4-­KS-­21PS   (Surrette   4 1104 20 55.2 220.8 4.416 0.5 2.20 39 3300 $1,703.00 $66,417.00 $20.13 500  series) 4-­KS-­25PS   (Surrette   4 1350 20 67.5 270 5.4 0.5 2.7 32 3300 $2,130.00 $68,160.00 $20.65 500  series) 6-­CS-­17PS   (Surrette   6 546 20 27.3 163.8 3.276 0.5 1.63 53 3300 $1,316.00 $69,748.00 $21.14 500  series) 6-­CS-­21PS   (Surrette   6 683 20 34.15 204.9 4.098 0.5 2.04 42 3300 $1,643.00 $69,006.00 $20.91 500  series) 6-­CS-­25PS   (Surrette   6 820 20 41 246 4.92 0.5 2.46 35 3300 $1,905.00 $66,675.00 $20.20 500  series) Surrette  S-­ 460  (Sur-­ 6 350 20 17.5 105 2.1 0.5 1.05 82 1300 $484.00 $39,688.00 $30.53 rette  400   series) Surrette  S-­ 530  (Sur-­ 6 400 20 20 120 2.4 0.5 1.2 72 1300 $550.00 $39,600.00 $30.46 rette  400   series)I E O R 1 6 0! BerkeleySOLAR 29
  34. 34. Appendix B - Demand Calculations Month KWH KWH Billed Average Variance Standard  De-­‐ Average  KWH+2σ Billed Previous year via1on This Year 12 784 776 780 16 4 788 11 665 701 683 324 18 719 10 566 561 563.5 6.25 2.5 568.5 9 557 485 521 1296 36 593 8 396 459 427.5 992.25 31.5 490.5 7 465 526 495.5 930.25 30.5 556.5 6 507 472 489.5 306.25 17.5 524.5 5 421 509 465 1936 44 553 4 374 567 470.5 9312.25 96.5 663.5 3 646 413 529.5 13572.25 116.5 762.5 2 686 654 670 256 16 702 1 795 645 720 5625 75 870I E O R 1 6 0! BerkeleySOLAR 30
  35. 35. Appendix C - Weather Calculations Average  Solar  Insola1on  (In  KWH/m^2/day)   InsolaFon  with   InsolaFon  at   Slope  of  Line  Found    Projected  Insola-­‐ Average  -­‐  2σ 15˚  Tilt 90˚  Tilt From  Two  Points Fon  at  37.87  ˚January 3.7 3.3 -­‐0.005333333 3.578026667 2.939376544February 4.4 3.6 -­‐0.010666667 4.156053333 2.926498113March 5.1 3.7 -­‐0.018666667 4.673093333 3.54510545April 5.6 3.4 -­‐0.029333333 4.929146667 4.012497307May 5.7 2.8 -­‐0.038666667 4.815693333 4.096906619June 5.6 2.5 -­‐0.041333333 4.654706667 3.930582018July 5.9 2.7 -­‐0.042666667 4.924213333 4.418053413August 6.1 3.3 -­‐0.037333333 5.246186667 4.570613507September 6.1 4.1 -­‐0.026666667 5.490133333 4.743232755October 5.5 4.3 -­‐0.016 5.13408 4.312633499November 4.1 3.6 -­‐0.006666667 3.947533333 3.241059534December 3.6 3.3 -­‐0.004 3.50852 2.447386298 Minimum  Solar  Insola1on    (In  KWH/m^2/day)   InsolaFon  with  15˚   InsolaFon  at  90˚   Slope  of  Line  Found    Projected  InsolaFon  at   Tilt Tilt From  Two  Points 37.87  ˚January 2.8 2.5 -­‐0.004 2.70852February 3.1 2.4 -­‐0.009333333 2.886546667March 3.8 2.7 -­‐0.014666667 3.464573333April 4.2 2.6 -­‐0.021333333 3.712106667May 4.8 2.5 -­‐0.030666667 4.098653333June 4.7 2.3 -­‐0.032 3.96816July 5.6 2.6 -­‐0.04 4.6852August 5.3 3 -­‐0.030666667 4.598653333September 5.2 3.5 -­‐0.022666667 4.681613333October 4.3 3.4 -­‐0.012 4.02556November 3.2 2.8 -­‐0.005333333 3.078026667December 2.1 1.9 -­‐0.002666667 2.039013333I E O R 1 6 0! BerkeleySOLAR 31
  36. 36. Triangular  Distribu1on   Standard  Devia1on Variance 0.101968495 0.319325061 0.37795151 0.61477761 0.318089166 0.563993942 0.210061512 0.45832468 0.129163585 0.359393357 0.131089127 0.362062324 0.064049466 0.25307996 0.114099774 0.33778658 0.139465118 0.373450289 0.168693588 0.41072325 0.124776307 0.3532369 0.281501183 0.530566851 Maximum  Solar  Insola1on    (In  KWH/m^2/day)   InsolaFon  with  15˚   InsolaFon  at  90˚   Slope  of  Line  Found    Projected  InsolaFon  at   Tilt Tilt From  Two  Points 37.87  ˚January 4.3 4.2 -­‐0.001333333 4.269506667February 6.1 5.4 -­‐0.009333333 5.886546667March 6.8 4.9 -­‐0.025333333 6.220626667April 6.9 3.8 -­‐0.041333333 5.954706667May 7.1 3 -­‐0.054666667 5.849773333June 7.1 2.6 -­‐0.06 5.7278July 7.2 2.8 -­‐0.058666667 5.858293333August 7.4 3.6 -­‐0.050666667 6.241253333September 7.3 4.7 -­‐0.034666667 6.507173333October 6.4 5.2 -­‐0.016 6.03408November 4.9 4.6 -­‐0.004 4.80852December 4.6 4.7 0.001333333 4.630493333I E O R 1 6 0! BerkeleySOLAR 32
  37. 37. Appendix D - AMPL Model OutputsIntroductionAMPL Assumptions • In these files, sigmas are not included in the calculations because we used the value of sun (calculated in our table) that already accounted for that • We estimated/assumed that the total cost over the lifetime of inverter, controller, mounting system would be approximately 3000 • We estimated/assumed that 1500 would be the initial cost of the inverter, control- ler, mounting system • Also assumed r=0.04 (i.e. 4%) • Assumed sell back cost for electricity = cost to buy electricity which is approxi- mately 12 centsModel 1param ProjectLife;param sun {i in 1..12};param d {i in 1..12};param days {i in 1..12};param ce;param cp;param A;param budget;param sigmad {i in 1..12};param E;param dod;param eff;param bc;var net{i in 1..12};var np>=0;var nb>=0;I E O R 1 6 0! BerkeleySOLAR 33
  38. 38. var LI>=0;var p;minimize cost: (sum{i in 1..12}-net[i]*ce)*(1-1/(1+.04)^ProjectLife)/.04+cp*np+LI+bc*(365*ProjectLife)/3600*nb*+3000*p- .3*(cp*np+LI+1500*p)-1318*p*(1-1/(1+.04)^ProjectLife)/.04;subject to Budget: cp*np+bc*(365*ProjectLife)/3600*nb+LI+1500<=budget;subject to MeetDemand {i in 1..12}: days[i]*(sun[i])*A*np*eff=d[i]+2*sigmad[i]+net[i];subject to Roof: A*np<=1500/0.7894;subject to Battery {i in 1..12}: nb>=if np=0 then 0 else ceil((d[i]+2*sigmad[i]+net[i])/(E*dod));subject to LabIns: LI = max {i in 1..12} ((d[i]+2*sigmad[i])/days[i])*(1000*8/24)*p;subject to cool: p= if np=0 then 0 else 1;data; ############ DATA STARTS HERE ############param sun:= 1 2.94 2 2.93 3 3.55 4 4.01 5 4.10 6 3.93 7 4.42 8 4.57 9 4.74 10 4.31 11 3.24 122.45;param d:= 1 720 2 670 3 529.5 4 470.5 5 465 6 489.5 7 495.5 8 427.5 9 521 10 563.5 11 683 12780;param days:= 1 31 2 28 3 31 4 30 5 31 6 30 7 31 8 31 9 30 10 31 11 30 12 31;param ce:= 0.12;param ProjectLife:= 25;param cp:=868;param A:=1.164;param budget:= 100000;param sigmad:= 1 75 2 16 3 116.5 4 96.5 5 44 6 17.5 7 30.5 8 31.5 9 36 10 2.5 11 18 12 4;param E:=4.104;param dod:=0.5;param eff:=.197;param bc:=683;I E O R 1 6 0! BerkeleySOLAR 34
  39. 39. Output:MINOS 5.51: optimal solution found.1 iterations, objective 14605.39498Nonlin evals: constrs = 6, Jac = 5.: _varname _var :=1 net[1] -8702 net[2] -7023 net[3] -762.54 net[4] -663.55 net[5] -5536 net[6] -524.57 net[7] -556.58 net[8] -490.59 net[9] -59310 net[10] -568.511 net[11] -71912 net[12] -78813 np 014 nb 015 LI 016 p 0;Therefore, if the user is off the grid, he/she has to pay a lot for batteries, so it wouldbe optimal for him to not invest in solar panels and buy all his electricity from PGE.I E O R 1 6 0! BerkeleySOLAR 35
  40. 40. Model 2param ProjectLife;param sun {i in 1..12};param d {i in 1..12};param days {i in 1..12};param ce;param cp;param A;param budget;param sigmad {i in 1..12};param E;param dod;param eff;param bc;var net{i in 1..12};var np>=0;var LI>=0;var p;minimize cost: (sum{i in 1..12}-net[i]*ce)*(1-1/(1+.04)^ProjectLife)/.04+cp*np+LI+3000*p-.3*(cp*np+LI+1500*p)-1318*p*(1-1/(1+.04)^ProjectLife)/.04;subject to Budget: cp*np+LI+1500<=budget;subject to MeetDemand {i in 1..12}: days[i]*(sun[i])*A*np*eff=d[i]+2*sigmad[i]+net[i];subject to Roof: A*np<=1500/0.7894;subject to LabIns: LI = max {i in 1..12} ((d[i]+2*sigmad[i])/days[i])*(1000*8/24)*p;subject to cool: p= if np=0 then 0 else 1;I E O R 1 6 0! BerkeleySOLAR 36
  41. 41. data; ############ DATA STARTS HERE ############param sun:= 1 2.94 2 2.93 3 3.55 4 4.01 5 4.10 6 3.93 7 4.42 8 4.57 9 4.74 10 4.31 11 3.24 122.45;param d:= 1 720 2 670 3 529.5 4 470.5 5 465 6 489.5 7 495.5 8 427.5 9 521 10 563.5 11 683 12780;param days:= 1 31 2 28 3 31 4 30 5 31 6 30 7 31 8 31 9 30 10 31 11 30 12 31;param ce:= 0.13;param ProjectLife:= 25;param cp:=868;param A:=1.164;param budget:= 100000;param sigmad:= 1 75 2 16 3 116.5 4 96.5 5 44 6 17.5 7 30.5 8 31.5 9 36 10 2.5 11 18 12 4;param E:=4.104;param dod:=0.5;param eff:=.197;param bc:=683;OUTPUT:MINOS 5.51: optimal solution found.2 iterations, objective 913.0905062Nonlin evals: constrs = 15, Jac = 14.: _varname _var :=1 net[1] 1276.382 net[2] 1230.073 net[3] 1829.224 net[4] 2169.615 net[5] 2440.256 net[6] 2252.097 net[7] 2670.378 net[8] 2845.889 net[9] 2755.86I E O R 1 6 0! BerkeleySOLAR 37
  42. 42. 10 net[10] 2578.0611 net[11] 1570.0912 net[12] 1000.6513 np 102.70214 LI 9354.8415 p 1;Therefore, the user can maximize his revenue by using 103 panels and producing ex-tra and selling back what he doesn’t need. This way, the cost is only 913 dollars over25 years. If he continued past 25 years his revenue would probably be positive. Also,changes in interest rates over the years could also help.!Model 3**We had to take out constraint for budgetparam ProjectLife;param sun {i in 1..12};param d {i in 1..12};param days {i in 1..12};param ce;param cp;param A;param budget;param sigmad {i in 1..12};param E;param dod;param eff;param bc;I E O R 1 6 0! BerkeleySOLAR 38
  43. 43. var net{i in 1..12}>=0;var np>=0;var nb>=0;var LI>=0;var p;minimize cost:cp*np+LI+bc*(365*ProjectLife)/3600*nb+3000*p-.3*(cp*np+LI+1500*p)-1318*p*(1-1/(1+.04)^ProjectLife)/.04;subject to MeetDemand {i in 1..12}: days[i]*(sun[i])*A*np*eff=d[i]+2*sigmad[i]+net[i];subject to Roof: A*np<=1500/0.7894;subject to Battery {i in 1..12}: nb>=ceil((d[i]+2*sigmad[i]+net[i])/(E*dod));subject to LabIns: LI = max {i in 1..12} ((d[i]+2*sigmad[i])/days[i])*(1000*8/24)*p;subject to blah: p=if np=0 then 0 else 1;data; ############ DATA STARTS HERE ############param sun:= 1 2.94 2 2.93 3 3.55 4 4.01 5 4.10 6 3.93 7 4.42 8 4.57 9 4.74 10 4.31 11 3.24 122.45;param d:= 1 720 2 670 3 529.5 4 470.5 5 465 6 489.5 7 495.5 8 427.5 9 521 10 563.5 11 683 12780;param days:= 1 31 2 28 3 31 4 30 5 31 6 30 7 31 8 31 9 30 10 31 11 30 12 31;param ce:= 0.12;param ProjectLife:= 25;param cp:=868;param A:=1.164;param budget:= 100000;param sigmad:= 1 75 2 16 3 116.5 4 96.5 5 44 6 17.5 7 30.5 8 31.5 9 36 10 2.5 11 18 12 4;param E:=4.104;param dod:=0.5;param eff:=.197;param bc:=683;I E O R 1 6 0! BerkeleySOLAR 39
  44. 44. output:MINOS 5.51: optimal solution found.3 iterations, objective 1260743.679Nonlin evals: constrs = 4, Jac = 3.: _varname _var :=1 net[1] 75.62 net[2] 149.1853 net[3] 379.2964 net[4] 584.6425 net[5] 765.6946 net[6] 698.7427 net[7] 865.1168 net[8] 979.3619 net[9] 882.3610 net[10] 817.73711 net[11] 289.47412 net[12] 013 np 45.245914 nb 71915 LI 9354.8416 p 1;In Model 3, we had to omit the budget constraint because it is so expensive. There-fore, it is ill advisable to go completely off the grid.I E O R 1 6 0! BerkeleySOLAR 40
  45. 45. Appendix E - Battery Cost OptimizationMinimizing the Cost per LifetimeThe eventual model that we decided to use in order to minimize the cost per lifetime ofthe battery was:The depth of discharge is the decision variable.Constraint# 1: The life time of the battery is equal to a function of the Depth of Dis-chargeConstraint # 2: The total cost of batteries from one type = the number of batteriesneeded to meet demand multiplied by the price of one battery of a specific type.Constraint# 3: The number of batteries to buy is calculated by using the Ceiling of thetotal energy required divided by the energy multiplied by the depth of dischargeI E O R 1 6 0! BerkeleySOLAR 41
  46. 46. Appendix F - Solar Panel Cost OptimizationMinimizing the Cost per Watt min u = xy + m subject to x <= 1500 ft2/area of 1 solar panel max wattage > demand x is an integer x = number of panels y = price per panel m = maintenance costs for 25 yearsI E O R 1 6 0! BerkeleySOLAR 42
  47. 47. Appendix G - Solar Installation Costs A1 Sun Inc. ACME Electric Acro Energy Tech, Inc. Advanced Alternative Energy Solutions Advanced Conservation Systems, Inc Akeena Solar, Inc. Albion Power Company, Inc.Company Alliance Solar Services Alter Systems, LLC American Solar Corp. Applied Star Energy Systems Borrego Solar Systems, Inc. CA Solar Systems, Inc. Century Roof and Solar Clean Solar, Inc. Gary Plotner Global Resource Options $0 $12,500.00$25,000.00$37,500.00$50,000.00 CostsI E O R 1 6 0! BerkeleySOLAR 43
  48. 48. Appendix H - Night Hours v Months 12.500 12.275 Night Hours 12.050 11.825 11.600 1 2 3 4 5 6 7 8 9 10 11 12 Months (JAN-DEC)I E O R 1 6 0! BerkeleySOLAR 44
  49. 49. Appendix I - kWh Bill for 25 YearsMonth kWH $ $/kWh kWH $ $/kWh Ave. kwh/ $ $/kWh billed Billed Cur- billed Billed Previ- kWH day Billed for Cur- Cur- rent Previ- Previ- ous for for next 25 rent rent Year ous ous Year next 25 next 25 Years Year Year Year Year Years YearsJan 795.00 $188.00 $0.24 645.00 $128.00 $0.20 720.00 30 $158.00 $0.22Feb 686.00 $145.00 $0.21 654.00 $132.00 $0.20 670.00 27.917 $138.00 $0.21Mar 646.00 $129.00 $0.20 413.00 $55.00 $0.13 529.50 22.063 $89.00 $0.17Apr 374.00 $46.00 $0.12 567.00 $100.00 $0.18 470.50 19.604 $72.00 $0.15May 421.00 $67.00 $0.16 509.00 $93.00 $0.18 465.00 19.375 $79.00 $0.17Jun 507.00 $92.00 $0.18 472.00 $81.00 $0.17 489.50 20.396 $86.00 $0.18Jul 465.00 $79.00 $0.17 526.00 $100.00 $0.19 495.50 20.646 $88.00 $0.18Aug 396.00 $59.00 $0.15 459.00 $78.00 $0.17 427.50 17.813 $69.00 $0.16Sep 557.00 $112.00 $0.20 485.00 $85.00 $0.18 521.00 21.708 $98.00 $0.19Oct 566.00 $116.00 $0.20 561.00 $114.00 $0.20 563.50 23.479 $115.00 $0.20Nov 665.00 $136.00 $0.20 701.00 $151.00 $0.22 683.00 28.458 $144.00 $0.21Dec 784.00 $184.00 $0.23 776.00 $181.00 $0.23 780.00 32.5 $182.00 $0.23Total 6,862.00 6,768.00 6,815 $1,318.00Aver- 571.83 $0.19 564.00 567.92 $109.83ageI E O R 1 6 0! BerkeleySOLAR 45
  50. 50. Appendix J - Solar Power CalculatorSystem Specifications Berkeley, CASolar Radiance (kWh/sqm/day) 5.43Ave. Monthly Usage (kWh/month) 3901System Size (kWh) 29.82Roof Size (sq. ft) 2981Estimated Cost 208,708.60Post Incentive Cost 140,252.18IncentivesFederal IncentivesTax Credit 30%State IncentivesProperty Tax ExemptLocal InventivesRebate (for PG&E) .35/W ACSavingsEstimated Cost 208708.60Post Incentive Cost 140,252.18Ave. Monthly Savings 57025 Year Savings 284,858.0125 Year ROI 203.10%Break Even 15.27 YearsCarbon EmissionsAnnual Carbon Dioxide Usage (pounds) 70,209Driving Equivalent 77,800 milesOffset by planting: 176 trees/yearI E O R 1 6 0! BerkeleySOLAR 46

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