THE DYNAMIC VALUE OF INTERMITTENT RENEWABLE ENERGY By Miguel Castro A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Agricultural, Food, and Resource Economics Doctor of Philosophy 2018 AB STRACT THE DYNAMIC VALUE OF INTERMITTENT RENEWABLE ENERGY By Miguel Castro I ntermittent renewable energy sources have significant local air pollution reduction and climate change mitigation benefits. However, their irregular generation creates challenges for integrating these resources in to the power grid. Valuing wind and solar power requires addressing both issues, especially in l ight of the policies and incentives aimed at promoting their large scale adoption. This dissertation values the environmental and economic benefits of wind and solar power by modelling their daily intermittency and interactions with hydropower in Californi a and storage in Texas. In Chapter 2 , I use random fluctuations in hourly wind and solar generation in California to estimate how much they reduce emissions of carbon dioxide, sulfur dioxide, and nitrogen oxides. These offsets depend on the direct displacement of high - cost natural gas generato rs, and on the hydropower reallocation that occurs to the hours with the lowest increase in renewable generation. Solar power daily intermittency causes a shift in hydro from the afternoon to the evening, which increases its emissions offsets since the gas generators displaced in the evening are dirtier than those kept running in the afternoon. In contrast, wind offsets are less sensitive to hydropower reallocation, since wind leads to a substitution of generators with similar emissions intensities . This c h apter highlight s the importance of accounting for interactions between wind, solar, and hydro capacity in assessing their environmental benefits. While Chapter 2 uses time series econometrics to model the dynamics of hydropower storage and renewable energ y, Chapter 3 simulates the interactions between projected u tility - scale batteries and emissions regulations for a ssessing the value of wind and storage in Texas . Wind power can reduce grid - level electricity generation costs and emissions but its large - scale adoption will require electricity storage to deal with intermittency. I model the ERCOT daily electricity market to estimate the value of wind generation , the value of storage capacity (based on hourly arbitrage) and the impact of wind and storage on emissions (CO 2 , NOx, and SO 2 ) under different policy scenarios combining storage availability and emissions taxes. Wind and storage capacities are complements since intermittency raises arbitrage benefits, whic h in turn enhances llocating power based on wind cycles. Emissions taxes increase net welfare and the value of storage. Taxing emissions leads to a larger welfare gain than just installing the planned storage levels in ERCOT (324 MWh). Under current technology and cost trend s, implementing a carbon pricing scheme that delivers stable prices larger than 40 USD/tCO 2 can induce wind to supply 30% of the load in Texas. Finally in Chapter 4 , I extend the daily model to a weekly planning horizon and find that interday arbitrage requires storage capacities larger than 11,000 MW h . For these large capacities , the value of storage increases since it arbitrages a larger gap between weekend off - peak and weekday peak demands and prices. H owever, half of the time the battery is filled with less than 50% of its capacity . C Copyright by MIGUEL CASTRO 2018 v A mi madre Bertha por todo el amor, generosidad, trabajo, cuidado y apoyo, a mi padre Luis por su amor, esfuerzo, calma y cariño y a mi hermano Luis por su amor, ejemplo, enseñanzas y todo lo compartido. Gracias! vi ACKNOWLEDGMENTS It is almost impossible to list everyone whose support and advice contributed during my time in Michigan and while writing this Di ssertation. I thank Dr. Jinhua Zhao and Dr. Soren Anderson for their support and advice throughout these years, and for all their hard work and patie nce with feedback and ideas for this work. I thank Dr. Robert Myers and Dr. Joseph Herriges for their teachings in class and all their insights and suggestions for this research. I thank Dr. Richard Horan f or his teachings in class and for the opportun ity to teach the Natural Resource Economics class . I thank Dr. Scott Swinton for the opportunity that he and the application s committee gave me five years ago when accepted into the program and for all the support throughout tho se years . I thank all facul ty and staff in the Department of Agricultural, Food and Resource Economics (AFRE) for their teachings and support. Thanks to Adrián, Nathy , Dylan, Asa, Byron, Sebasti án , Jungmin, Giri, Richard and all peers for everything shared in and out of class. I thank my family for all their love , support and advice while living abroad. vii TABLE OF CONTENTS LIST OF TABLES ................................ ................................ ................................ .......................... x LIST OF FIGURES ................................ ................................ ................................ ....................... xi CHAPTER 1. INTRODUCTION ................................ ................................ ................................ ... 1 CHAPTER 2: IS A WETTER GRID A GREENER GRID? ESTIMATING EMISSIONS OFFSETS FOR WIND AND SOLAR POWER IN THE PRESENCE OF LARGE HYDROELECTRIC CAPACITY . ................................ ................................ ................................ . 6 Introduction ................................ ................................ ................................ ......................... 6 Estimating intermittent renewable energy carbon and pollution offsets. ........................... 9 A dynamic model of electricity generation, emissions, hydropower and renewables. ..... 11 Data ................................ ................................ ................................ ................................ ... 20 Wind, solar and hydropower trends in CAISO ................................ ................................ . 22 Power generation trends ................................ ................................ ................................ 22 Hydropower institutional and minimum flow constraints ................................ ............. 25 Identification ................................ ................................ ................................ ..................... 26 Static average generation and emissions offsets ................................ ............................... 28 Econometric specification ................................ ................................ ............................. 28 Ge neration and emissions offsets ................................ ................................ .................. 31 Dynamic average generation and emissions offsets ................................ ......................... 34 Econometric specification. ................................ ................................ ............................ 34 Ge neration and emissions offsets ................................ ................................ .................. 35 Valuing emissions offsets and policy implications ................................ ........................... 40 Conclusions ................................ ................................ ................................ ....................... 44 APPENDICES ................................ ................................ ................................ .............................. 47 APPENDIX 1. Theory derivations ................................ ................................ ................... 48 APPENDIX 1a. Derivation of the simplified one period system of equations and implicit function theorem. ................................ ................................ ............................. 48 APPENDIX 1b. Derivation of the two period system of equations and implicit function theorem. ................................ ................................ ................................ ......................... 48 APPENDIX 1c. Hydropower reallocation due to the difference between off - peak and peak renewable generation. ................................ ................................ ........................... 50 APPENDIX 2. Regression Results ................................ ................................ ................... 51 APPENDIX 2a. Robustness check specifications for static marginal generation and offsets. ................................ ................................ ................................ ........................... 51 APPENDIX 2b. Static results ................................ ................................ ........................ 52 APPENDIX 2c. Instrumental variable specification results ................................ ........ 54 APPENDIX 2d. Structural break robustness check of the linear no int eractions static model ................................ ................................ ................................ ............................. 55 APPENDIX 2e. Hydropower reallocation estimates ................................ .................... 56 APPENDIX 2f. Robustness check specifications for the dynamic marginal generation and emissions offsets. ................................ ................................ ................................ .... 57 APPE NDIX 2g. Dynamic estimates ................................ ................................ ............. 58 viii APPENDIX 2h. Dynamic yearly and seasonal estimates ................................ ............. 59 APPENDIX 2i. Estimates of imports offsets ................................ ................................ 60 APPENDIX 2j. External benefits estimates ................................ ................................ . 62 CHAPTER 3: ELECTRICITY STORAGE, EMISSIONS TAXES, AND THE VALUE OF RENEWABLE ENERGY ................................ ................................ ................................ ............. 63 Introduction ................................ ................................ ................................ ....................... 63 Literature review ................................ ................................ ................................ ............... 68 Theoretica l Model ................................ ................................ ................................ ............. 71 Model setup and optimal storage ................................ ................................ .................. 71 The marginal value of wind ................................ ................................ .......................... 75 Effect of storage on the marginal value of wind ................................ ........................... 77 The marginal value of storage ................................ ................................ ....................... 79 The marginal value of wind and storage under large renewable energy penetration levels ................................ ................................ ................................ .............................. 80 Data ................................ ................................ ................................ ................................ ... 82 Electricity demand, generation, and wind power in ERCOT ................................ ........... 82 Empirical Model ................................ ................................ ................................ ............... 86 ................................ ................................ .................. 86 Demand and fossil generation cost calibration ................................ ............................. 88 Emissions functions ................................ ................................ ................................ ....... 91 Wind power and storage parameters ................................ ................................ ............. 92 Policy scenarios and Monte Carlo simulations ................................ ............................. 92 Robustness checks ................................ ................................ ................................ ......... 95 Accounting for regulating reserves costs ................................ ................................ ... 95 Results ................................ ................................ ................................ ............................... 96 The marginal value o f wind generation ................................ ................................ ......... 96 The marginal value of storage ................................ ................................ ..................... 101 Welfare and allocations ................................ ................................ ............................... 105 Emissions offsets ................................ ................................ ................................ ......... 109 Robustness checks ................................ ................................ ................................ ....... 111 Accounting for regulating reserves costs ................................ ................................ . 111 Conclusions ................................ ................................ ................................ ..................... 111 APPENDICES ................................ ................................ ................................ ............................ 114 APPENDIX 1. Solv ing the model for the optimal storage equation ........................... 115 APPENDIX 2. Derivation of the value of wind power ................................ ............... 118 APPENDIX 3. Derivation of the value of wind power with and without storage ...... 120 APPENDIX 4. Derivation of the value of storage ................................ ...................... 121 APPENDIX 5. Model results additional tables and Figures ................................ ....... 124 CHAPTER 4: A WEEKLY HORIZON MODEL OF THE VALUE OF RENEWABLE ENERGY AND ELECTRICITY STOR AGE ................................ ................................ ............ 132 Introduction ................................ ................................ ................................ ..................... 132 Theoretical Model ................................ ................................ ................................ ........... 134 Model setu p and optimal storage ................................ ................................ ................ 134 The marginal value of wind ................................ ................................ ........................ 137 ix The marginal value of storage ................................ ................................ ..................... 138 Data ................................ ................................ ................................ ................................ . 140 Empirical Model ................................ ................................ ................................ ............. 142 Overview ................................ ................................ ................................ ..................... 142 Weekly Demand and wind power calibration ................................ ............................. 142 Policy scenarios and Monte Carlo simulations ................................ ........................... 143 Results ................................ ................................ ................................ ............................. 144 The marginal value of wind generation ................................ ................................ ....... 144 The marginal value of storage ................................ ................................ ..................... 147 Storage allocations ................................ ................................ ................................ ...... 148 Conclusions ................................ ................................ ................................ ..................... 152 APPENDICES ................................ ................................ ................................ ............................ 154 APPENDIX 1. Solv ing the model for the optimal storage equation ........................... 155 APPENDIX 2. Derivation of the value of wind power ................................ ............... 160 APPENDIX 3. Derivation of the value of storage ................................ ...................... 161 APPENDIX 4. Unconstrained storage charging /discharging actions ........................ 163 APPENDIX 5. Utilization rate of the battery in the unconstrained storage scenario (12% wind) ................................ ................................ ................................ .................. 164 CHAPTER 5. CONCLUSIONS ................................ ................................ ................................ . 164 REFERENCES ................................ ................................ ................................ ........................... 167 x LIST OF TABLES Table 1. Summary statistics on CAISO hourly generation data (2013 - 2015) .............................. 21 Table 2. Static average generation offsets by type. ................................ ................................ ...... 31 Table 3. Dynamic average hydropower generation and emissions offsets ................................ ... 36 Table 4. Average marginal external benefits, incentives and costs of renewables in California . 41 Table 5. Static marginal generation and emissions offsets results. ................................ ............... 52 Table 6. Hausman test of endogeneity of hourly wind and solar generation ................................ 54 Table 7. Hydropower reallocation (static estimator) for each season ................................ ........... 56 Table 8. Dynamic marginal generation and emissions offsets results ................................ .......... 58 Table 9. Dynamic yearly and seasonal emissions offsets estimates ................................ ............. 59 Table 10. Estimates of electricity imports carbon emissions offsets. ................................ ........... 60 Table 11. Total carbon emissions offsets (California power plants and imports) . ....................... 61 Table 12. Estimates of external benefits of emissions offsets (USD) ................................ ........... 62 Table 13. The marginal value of increasing wind power and storage under different emissions taxes and storage scenarios ................................ ................................ ................................ ........... 94 Table 14. Average changes in daily welfare and emissions from implementing storage and taxes with respect to the baseline v0 ................................ ................................ ................................ .... 105 Table 15 Marginal value of wind estimates for different scenarios ................................ ............ 124 Table 16. Average changes in daily welfare and emissions from implementing storage and taxes with respect to the baseline v0 ................................ ................................ ................................ .... 125 xi L IST OF FIGURES Figure 1. Two period linear demand and marginal costs model of hydro reallocation ................ 18 Figure 2. Hourly average wind and solar generation 2013 vs 2015 ................................ ............. 22 Figure 3. CAISO average aggregate hydropower hourly generation ................................ ........... 24 Figure 4. Relative change of hydropower shares for each hour of the day ................................ .. 25 Figure 5. Dynamic average carbon emissions offsets throughout years ................................ ....... 38 Figure 6. Dynamic average carbon emissions offsets and load throughout seasons .................... 39 Figure 7. Value of wind and solar power external benefits (no imports) throughout years and seasons ................................ ................................ ................................ ................................ .......... 43 Figure 8. The equilibrium price with storage is a linear combination of off - peak and peak prices of the no storage scenario. ................................ ................................ ................................ ............ 78 Figure 9. The value of wind power for increasing capacity levels ................................ ............... 81 Figure 10. Load and wind average daily variations and intermittency in Texas. ......................... 84 Figure 11. Private and Social Marginal Generation Costs in ERCOT ................................ ......... 90 Figure 12. CO2 emissions function for a market dispatch based on the private MC in ERCOT . 91 Figure 13. The marginal value of wind generation under increasing capacity ............................. 98 Figure 14. Average eq. prices and marginal generation from 1 MW of new capacity. ................ 99 Figure 15. Value of wind power (no storage) for different policy scenarios. ............................. 101 Figure 16. Storage economic value and gains in surplus for the 12% wind baseline ................. 102 Figure 17. Storage economic value under increasing wind capacity ................................ .......... 104 Figure 18. Private marginal cost and emissions function at 5 AM and 4 PM ............................ 107 Figure 19. Optimal average (dis)charging actions for a 324 MWh storage. ............................... 108 Figure 20. Average emissions offsets of wind generation (no storage) ................................ ...... 110 Figure 21. Emissions function at 5AM and average net load with 40% wind ............................ 126 xii Figure 22. Unconstrained storage charging and discharging and state of charge ....................... 127 Figure 23. Emissions function for dispatch based on private MC under 12% and 30% wind during peak hours ................................ ................................ ................................ ........................ 128 Figure 24. Social merit order marginal cost curve at 5 AM ................................ ....................... 12 9 Figure 25. Average emissions offsets of wind generation with unconstrained storage .............. 130 Figure 26. Wind power and storage marginal values in scenarios that include regulating reserves costs ................................ ................................ ................................ ................................ ............. 131 Figure 27. Average Weekly Electricity Demand and Wind Power in ERCOT .......................... 141 Figure 28. The value of wind generation under increasing capacity for different scenarios ...... 146 Figure 29. The value of storage for the 12% wind and emissions taxes scenario ...................... 147 Figure 30. Unconstrained State of charge (battery capacity) for the 12% wind scenario .......... 149 Figure 31. State of charge for the planned 324 MWh capacity in the 12% wind scenario ......... 150 Figure 32. State of charge for 12% wind scenario and 85% round - trip efficiency .................... 152 Figure 33. Unconstrained storage charging /discharging actions ................................ ............... 163 Figure 34. Utilization rate of the battery in the unconstrained storage scenario (12% wind) .... 164 1 CHAPTER 1. INTRODUCTION What is the value of intermittent renewable energy sources, such as wind and solar? These sources can reduce grid - level electricity generation costs and emissions. But their intermittency both cyclical and random can make it difficult to integrate them int o conventional electric grids (Joskow, 2011; Baker et al., 2013). Electricity storage could facilitate this integration by allowing renewable electricity to be used at the times it is most valuable. But simply pushing storage capacity onto the grid could a ctually decrease welfare if intra - day storage arbitrage leads to a large increase in off - peak coal generation in place of cleaner, on - peak natural gas generation. Therefore, assessing the full value of intermittent renewables requires a dynamic framework t hat takes into account all private and external benefits and costs. This dissertation will assess the dynamic value of intermittent renewable electricity in California and Texas, which are early adopters of renewable generation. In the Chapter 2 , I researc h how the interactions between hydro, wind , and solar power affect the economic benefits of the latter two. I use random fluctuations in hourly wind and solar generation in California to estimate how much they reduce emissions of carbon dioxide , sulfur dioxide , and nitrogen oxide s . These offsets depend on the direct displacement of high - cost natural gas generators and on the hydropower reallocation that occurs to the hours with the lowest increase in renewable generation. I address the shift in hydro wi th a dynamic econometric model that accounts for the interactions betw een the three renewable sources. W ind, solar , and hydro generation trends in California for 2011 - 2015 show that reallocation follows daily patterns . Hence, using a dynamic model that agg regates hourly into daily data, I estimate the appropriate average marginal carbon 2 dioxide emissions offsets of solar ( tCO2/MWh) to be larger than those of wind ( tCO2/MWh). S olar power delivers most emissions offsets since its generation peaks at the midday and decays in the afternoon , which leads to hydro arbitrage that substitutes gas turbines with the cleaner steam turbines. On the other hand, wind power peaks at midnight and decreases during the day which leads to a substitution of generator s with similar emissions intensities: combined cycle for steam turbines. Therefore, solar power delivers larger carbon and sulfur offsets than wind. These findings highlight that accounting for the dynamic interactions between the three renewable sources i s necessary for assessing the proper external benefits and value of intermittent renewables. In California , hydropower enhances the emissions offsets of solar power whose external benefits range between 9.78 and 30.08 per USD/MWh while for wind the range i s 8.31 to 19.2 USD/MWh. 1 One of the key insights from this chapter is that hydropower in California is already helping to smooth the integration of wind and solar generation. Since potential hydroelectric generation (i.e., water behind the dam) can be built up when prices are low and drawn down when pri ces are high, hydroelectric capacity acts much like storage capacity in this regard. As Chapter 2 makes clear, the impact of intermittent renewables on costs and emissions will likely depend on the presence of storage capacity. Therefore in the third chapt er I simulate the impact of utility - scale storage in the Texas electricity mark et (ERCOT). While in the c hapter 2 I used historical data and time series econometrics to value the environmental benefits of wind and solar power, in the c hapter 3 , I develop an electricity market model that accounts for environmental benefits and fuel cost reductions in assessing the value of wind power and storage. 1 With a confidence interval of 95% built using the Newey West standard errors of the dynamic offset estimates. 3 Furthermore, I explore how the value of wind, storage , and ( dis)charging allocations depend on emissions pricing using a theoretical model and an empirical model of welfare maximization in the daily electricity market. One key feature of the model , consistent with the Texas grid, is that wind generation is larger in the off - peak period and it is negatively correlated with demand. I show that wind generation has a declining marginal value since as its capacity increases, incoming generation will replace the most expensive fossil power, which leads to lower equilibrium prices and to a lower value of wind. The value of s torage depends on the gap between peak and off - peak prices and as its capacity increases, arbitrage opportunities fall along with its marginal value. Nevertheless, I find that wind power and storage are complements since the intermittency of wind raises th e benefits of storage. Emissions taxes increase the value of storage since the endogenous arbitrage decisions account for private costs as well as emissions damages. Without storage and a t current wind capacity ( serving 12% of load) economic value breaks even with its levelized cost in all scenarios ( 60 USD/MWh). 2 pollution and CO 2 mitigation benefits leads to private surplus gains ( 26.3 - 27.4 USD/MWh on average ) which are only half the levelized cost. Thu s, pricing the externalities is vital to provide the correct incentives to finance the development of wind power. A large storage capacity can slightly boost the value of wind ( 0.86 USD/MWh on average ). Under current technology and cost trends , the decline in the marginal value of wind as its capacity increases from 12 - 30% of the load is smaller than the projected levelized cost reductions by 2025 (IRENA, 2016). Therefore, wind can supply 30% of the load in Texas 2 cial Cost of Carbon for a discount rate of 3% measured in 2015 USD (40 USD/tCO 2 ) 4 The economic value of storage almost double s its private surplus gains. Even under the large intermittency of the 30% wind scenario, estimated economic value lower than its levelized cost of 420 - 950 USD/kWh (Lazard, 2016) . However, the implemented model onl y accounts for hourly intermittency smoothing. The value is likely to be larger if we include minute level intermittency smoothing (1 min, 5 min) and frequency regulation services. I show that storage moves power from the off - peak to the peak. This arbitrage is based on the differences in electricity demand, wind power generation , and emissions intensities throughout periods. Taxing emissions reduces storage and arbitrage by accounting for the higher emissions intensity of off - peak generators. Simulation results show that simply taxing emissions leads to a larger welfare gain than environmental taxes. Both, low and unconstrained storage capacit ies , have less than a 50% utilizat ion rate since hourly arbitrage mostly charges during several hours in the morning to release power in the afternoon and evening. Finally, the fourth chapter explores the value of wind and storage with a weekly horizon electricity market model that capture s the differences in demand between weekdays and the weekend. I find that the value of wind power is basically the same due to its daily generation patterns. T he value of storage is higher for capacity levels larger than 11,000 MW since it captures a large r gap between interday demand and prices. Nevertheless, this value is still lower than the levelized cost of storage which further highlights the need to account for minute level intermittency. Interday arbitrage requires storage capacities larger than 11,000 MWh with a utilization level lower than 50% : half of the time the battery is filled with less than 50% of its capacity. Only . Given the nascent stage of 5 the industry and high costs, it is very likely that intraday arbitrage will dominate the market. Storage is sensitive to roundtrip efficiency: a decline of efficiency from 90% to 85% reduces interday arbitrage from all 7 days t o just 3 days in the scenario with emissions taxes 6 CHAPTER 2 : IS A WETTER GRID A GREENER GRID? ESTIMATING EMISSIONS OFFSETS FOR WIND AND SOLAR POWER IN THE PRESENCE OF LARGE HYDROELECTRIC CAPACITY . Introduction In order to achieve carbon dioxide and local pollutants abatement goals , an electric grid powered by a significant share of renewable energy is one of the proposed alternatives (Ca rdell and Anderson, 2015). Nevertheless, the intermittent nature of solar an d wind power and the nascent utility scale electricity storage make it challenging to assess the emissions offsets of these power sources. Overcoming these methodological difficulties is necessary to estimate the economic value of renewable energy and to e valuate related policies such as feed - in tariffs, renewable portfolio standards and subsidies . In this paper, using historical data on the randomness of hourly solar and wind generation from 2013 to 2015 , I estimate how much CO 2 , SO 2 and NOx these sourc es abate in the electric grid in California, a worldwide leader in renewable energy adoption. The novel challenge lies in identifying the marginal generation and emissions offsets caused by adding two intermittent sources to a grid that has a significant s hare of hydropower. Previous literature, centered on grids with small fractions of hydro generation, has identified emissions offsets related to the instantaneous displacement of the highest marginal cost fossil generators by zero marginal cost renewables through the merit order effect (Kaffine et al., 2012, Cullen, 2013, and Novan, 2015). 3 3 The merit order effect refers to the ordering of all dispatchable power plants in increasing order according to their margina l cost of generation. 7 Nevertheless, daily solar and wind power cycles introduce dynamics that reallocate hydropower, alter fossil generation and emissions. Responding to arbitrage incentives, incoming wind and solar generation cause hydro to shift to those hours with the lowest increase in renewable output . Hydro reservoirs play the role of facilitating the intertemporal allocation . Capturing this dynamic effect on emissions offsets and on the value of wind and solar requires modelling not only the contemporaneo us or static effect but the overall dynamic effect . Using a time series regression , I model the static displacement of the highest marginal cost fossil generators and the existence of hydropower reallocation. The results show that, on average, each addit ional MWh of solar generation instantaneously displaces 0.565 MWh of fossil generation and reallocates 0.129 MWh of hydropower while each additional MWh of wind displaces 0.731 MWh of fossil power and reallocates 0.044 MWh of water generation. Considering the change in electricity imports causes the full effect to be a one to one displacement. T he reallocated hydropower is switched to a different hour of the day, where it displace s a nother fossil fuel plant and offsets even more emissions . Wind and solar g eneration exhibit mostly a daily intermittency pattern. Hence, using a dynamic model that aggregates hourly into daily data, I estimate the appropriate average marginal carbon dioxide emissions offsets of solar ( tCO 2 /MWh) to be larger than those of wind ( tCO 2 /MWh). Results from a Hausman test on the endogeneity of wind and solar generation, using their weather based hour - ahead forecasts as instruments, show that their hourly and daily variation is exogenous and does not affect the identificatio n of marginal generation and emissions offsets. 8 Furthermore, these offsets vary throughout time: they increase during spring and summer since more polluting fossil generators supply the seasonal larger demand, and t hey slightly decline as years go by due to new renewable capacity substituting these polluting plants. Accounting for displaced electricity imports, I estimate solar and wind carbon offsets to increase substantially to 0.615 and 0.465 tCO 2 /MWh, respectively. It is worth noting that the dynamic model complies with the net zero hydropower displacement condition for a fixed reservoir : the gain in one period has to be compensated with a loss at a different time . Results show that , on average, the larger daily solar intermittency leads to hydro real location within one day while the less pronounced wind power variation leads to a two - day arbitrage. Since solar power delivers most generation at the midday and decays in the afternoon it leads to hydro arbitrage that substitutes gas turbines with the cleaner steam turbines. On the other hand, wind power peaks at midnight and decreases during the day, w hich leads to a substitution of generators with similar emissions intensities: combined cycle for steam turbines. Therefore, solar power delivers larger carbon and sulfur dioxide offsets than wind. These dynamics were contingent on having dry years during 2013 - 2015, which allowed a significant share of hydro to be reallocated due to the less stringent flow requirements during dry periods. These findings highlight that accounting for the dynamic interactions between the three renewable sources is necessary for assessing the proper external benefits and value of intermittent renewables. In California hydropower enhances the emissions offsets of solar power. Using the US social cost of carbon central estimate (IAWG, 2015) and marginal damages estimates for SO 2 and NOx (Muller and Mendelhson, 2009) , solar power external benefits range between 9.78 and 9 30.08 per USD/ MWh while for wind the range is 8.31 to 19.2 USD/ MWh. Summing up, the external benefits of renewable energy vary per technology and throughout time, and their incentives and subsidies should reflect this. From a broader perspective, the proposed dynamic modelling is key for understanding electricity generation and emissions in grids with increasing adoption of storage technologies since the same ins ights about hydropower reallocation would apply to profit maximizing storers. Furthermore, several emerging economies with electric grids powered by a significant share of hydropower are increasingly adopting wind and solar plants. To the extent that these countries dispatch generators based on the lowest marginal cost (via wholesale market) this methodology is a good approach for assessing the heterogeneous value of their renewable energy emissions offsets and guiding the (re)design of related incentives and policies. Estimating intermittent renewable energy carbon and pollution offsets. Using either historical data or projections of fossil fuel generation, load and intermittent renewable energy, several studies have quantified the pollution and carbon offsets that occur when electricity coming from solar or wind power plants substitutes for any fossil based electricity on the grid. Callaway et al. (2017 ) focus on how additional intermittent renewable generation (solar and wind) and energy ef ficiency measures displace carbon emissions in six power system regions of the United States (CAISO, ERCOT, ISONE, MISO, NYISO, PJM). They estimate the marginal emissions for each hour by regressing emissions on dispatchable fossil generation, and then c ompute 10 the previous estimate and projections of renewable energy production. Cullen (2013) recognizes that adding wind power (intermittent supply) has a different effect on the electricity grid and dispatch schedule than reducing load (demand) or fossil generation (dispatchable supply). Using historical data for Texas (ERCOT), the author regresses conventional generation types on the exogenous wind electricity production and other controls to infer what chan ges occurr to the power mix when an intermittent supply of renewable energy is average annual emission rates for fossil fuel plants is used to compute offset emissions. The study highlights that in order to capture t he dynamic factors such as startup, shut down, and ramping effects rate lags of wind generation as controls in the econometric model. The static and dynamic model s yield different results : the latter finds fewer emissions offsets coming from coal and more coming from the expensive and inefficient steam and gas turbine generators. Neither model finds significant hydropower offsets, but it is worth noting that hydro is less than 1% of total c apacity and generation in Texas (Cullen, 2013). Kaffine et al. (2013 and 2012) uses historical patterns of wind power in ERCOT to directly estimate emissions offsets by regressing the amount of pollutants and carbon on renewable energy production, demand, temperature , and other controls. Novan (2015) captures t he hourly variation in electricity demand, wind generation and marginal generators throughout the day by modelling generation and emissions as a function of the interactions between wind power genera tion and load, while controlling for certain fixed effects. 11 Hence, this research finds that wind power causes larger emissions offsets than solar in Texas, given that the former displaces coal base power during low demand night hours. Furthermore, new cap acity increases in wind power would bring larger emissions offsets while solar capacity increases would not. The author also considers a dynamic model , which renders similar resu lts to its static counterpart. This implies that wind power causes mainly inst antaneous emissions offsets. It is worth noting that wind causes practically zero hydropower offsets in a grid where water has a share of less than 1%. In this paper I estimate how much CO 2 , SO 2 and NOx wind and solar power abate in California, a worldwide leader in renewable energy adoption. The novel challenge lies in identifying the marginal generation and emissions offsets caused by adding two intermittent sources to an electric system that has a significant share of hydropower. H ence, I contribute to the literature by estimating emissions offsets in the presence of dynamics that reallocate hydropower, alter fossil generation and emissions . Capturing this dynamic effect on emissions offsets and on the value of wind and solar requir es modelling not only the contemporaneous or static displacement of fossil generation ( merit order effect ) but also the hydropower reallocation . These ideas are further developed and supported in the following sections. A dynamic model of electricity gen eration, emissions, hydropower and renewables . I capture the main insights of hydropower reallocation , caused by adding wind or solar energy, and its effects on fossil generation and emissions by developing a two period ( off - peak o , the initial period followed by the peak p ) stylized short run partial equilibrium model of the electricity market. For simplicity, the model assumes perfect competition, no startup costs or 12 dynamic frictions, no transmission externalities, no intermitten cy costs , no discounting and fixed capacities for all generation types. The objective function of the planner is to maximize Social Welfare at the initial time by choosing the amount of fossil fuel and hydropower generation . Intermittent renewab le energy is produced at different rates during both periods of the day following the cyclic stochastic process , which consists of recurring realizations every day for each period t. 4 This is a natural approach for understanding wind and solar generation patterns . 5 On average, renewable generation is larger during the off - peak and each addition of capacity also leads to a larger increase during this period . 6 The main insights and intuition of how these heterogeneous increases in renewable generation reallocate hydropower, alter fossil generation and emissions can be described using expected values. Thus, to further simplify the analysis, I assume that renewabl e generation is constant with a larger off - peak value . Assuming no losses, total load is given by . There is a representative consumer with a utility function for total electricity consumed for each period of the day with the conventional properties . 4 Intermittent renewable energy can represent either just one power source (wind) or both sources combined (wind and solar). 5 W allow for different iid distributions and means of renewable generation for each period of the day. For example, the mornin g period has one distribution of possible renewable energy generation outcomes and the night period has a different one. The next day, the same distributions determine the possible outcomes again. This recurring or cyclic nature of renewable generation an d demand determines that model ling just the subset of two periods of one day is a valid simplification to understand the dynamic process. 6 This is a key feature of wind and solar power, and it is also found in CAISO as described with more details in the Section Wi nd, solar and hydropo wer trends in CAISO. 13 Wholesale market f ossil fuel generation has the usual convex costs of production and hydropower has no marginal cost. This is in line with previous work which capability and ramping flexibility allow an almost frictionless generation reallocation to those time periods with the largest value ( Borenstein et al. 2002, Thompson et al., 2004). Notice that fossil costs could represent not only the private costs but also the social costs, which includes the damages from carbon and pollution emissions. For simplicity, assume hydropower reservoir levels depend only on the endogenous extraction and that t otal reservoir capacity is fixed : Hence, th e problem becomes: 7 s.t: , The optimality conditi ons at the interior solution are given by : 8 The equilibrium allocation of fossil and hydropower generation is determined by the fossil generation optimality condition (equation 2) that equates the marginal utility of electricity consumption to the marginal cost of fossil generation; and the no arbitrage condition (equation 3) which equates the marginal utility of electricity consumption throughout periods. Notice that by 7 Bold characters denote vectors 8 Corner solutions are not of particular interest in this research since they imply that either all hydro is allocated in just one period or neither hydro nor fossil generation occurs. Any of those cases could not picture all the dynamics and insights of ad ding intermittent renewables to the grid and rarely occur in wholesale power markets. 14 assuming no losses, t he reservoir equation implies that any change in hydr opower allocation at one period has to be compensated with a counterbalancing action in the next period: . Thus, there is no net hydropower displacement. To disentangle the effects of heterogeneous increases in generation from new renewable off - peak period when there is an increase in wind or solar generation and no change in the following peak. This simplification allows us to illustrate the effects of a larger off - peak contribution . Since we assumed perfect to solving the decentralized problem. In the one period decentralized problem we have the fossil generation and the consumer utility maximization equations defining equilibrium fossil generation and prices. Using the implicit function theorem we can show the effects of an exogenous i ncrease in off - peak renewable generation: 9 An exogenous increase in off - peak renewable generation will lead to a decrease in fossil generation and equilibrium prices. This occurs since adding a zero marginal cost source of power shifts the wholesale market supply curve, or the marginal cost , to the right. Thus, less fossi l generation, at a lower marginal cost, is required to meet an unchanged demand. This instantaneous 9 Details of the derivation are on Appendix 1a. 15 displacement of fossil generation when we increase renewable power is the usual merit order effect. Notice that the amount of fossil power displaced depends on the ratio between the rates of changes, or slopes, of fossil power supply and the marginal utility. Hydropower is exogenous in this one period case since its ro le is to arbitrage interperiod benefits. Hence, without any changes in the peak renewable power, since off - peak prices decreased some off - peak hydro will be reallocated to the peak time until marginal benefits and prices are equalized agai n. Basically, the free incoming renewable power decreases the off - peak marginal benefits and price which creates an incentive to store some water and release it during the peak time, at a higher price. Hydropower producers will reallocate until there are n o arbitrage opportunities. This reallocated hydro can displace even more fossil generation and its corresponding emissions. This depend s on the underlying power plant structure in each period. This is further explored by the end of this section. To analyze the full dynamics of adding heterogeneous renewable generation in both periods we can resort back to the original problem, whose optimality is determined by the fossil and no arbitrage conditions (equations 2 and 3), and use the implicit functio n theorem to show : 10 10 Details of the derivation are in Appendix 1b. Assume that the exogenous random shocks are evaluated at their expected v alue, leading to a certainty equivalent steady state solution. 16 The results display the usual merit order effect or instantaneous displacement of costly fossil generation, in the same time period, to accommodate renewable power . Adding renewable generation in the same period displaces hydropower , as described in previous lines, but adding it to the following period increases current hydro generation . In the latter case, if hydropower producers expect an increase in renewable generation at the peak, which will lead to a reduction in prices, they will generate more at the off - peak and p rofit from this differential until marginal benefits are equated. This increase in hydro generation also leads to another decrease in (current) off - peak fossil generation and is captured by the cross partial derivatives Adding renewable generation has opposite effects on hydropower depending on whether the increase occurs at the peak or off - peak . Since new renewable capacity delivers power un evenly throughout the d ay, whether hydropower is reallocated to the peak or off - peak depends basically on the difference in generation between periods. Using the implicit function theorem results we can show that hydro will be reallocated to the peak time as long as off - peak gains in generation 17 from new capacity are larger than pe ak gains 11 As argued previously, even when capacity increases generation during both periods, the larger increase at the off - peak reduces marginal benefits particularly during this period, creating an incentive to shift hydro to the peak unt il there are no arbitrage opportunities. Hydropower is just reallocated and there is no net displacement as argued previously . With linear demand and marginal costs, the two period dynamic solution can be depicted as the intersection of pe ak and off - peak marginal utilities or benefits (Figure 1) . To illustrate this, since utility is a function of electricity consumption, which is supplied by three sources, we can fix the renewable generation and the optimal fossil allocation in order to graph the declining marginal utility as a function of hydropower allocation . Before adding new renewable capacity, the equilibrium hydropower allocation is determined by the no arbitrage condition, graphically the intersection of marginal utilities (represented with black letters and lines). 11 Details of the derivation are shown in Appendix 1c. To formally show this we need to assume the same rate of change in the marginal utility of electricity consumption in both periods , which does not prohibit consumers from getting a larger total utility from peak electricity consumption. 18 Figure 1 . Two period linear demand and marginal costs model of hydro reallocation The black colors denote the baseline and the green colors deno te the post - renewables shift The blue line represents equilibrium prices after the renewables shift New capacity adds more generation at the off - peak and shifts down the marginal utility curves. The larger shift at the off - peak represents the larger decline in marginal benefits. Thus, some off - peak hydro will be reallocated to the peak in the new equilibrium allocation (represented by the green lines and letters). Since the Social Planner Problem is equivalent to the decentralized market, off - peak and peak equilibrium prices will be the same and equal to the marginal utility at the optimal allocations . If hydropower producers are constrained by downstream flow requirements, the above problem is basically the same with added constraints. 19 generation schedule is in line with flow regulations, and to no reallocation when requirements bind in the two - period model. Similar insights hold when we have more than two periods: during those periods in which the flow requirements restrict the profit maximizing allocation, hydro reallocation will be restricted, while for those periods in which the optimal schedule is in line with the requirements, hydro reallocat ion will take place. The previously described dynamics of hydropower reallocation will also introduce dynamics when assessing grid level emissions, renewable energy emissions offsets and also the value of renewable power . Emissions are an increa sing function of fossil fuel use . For the previous two period example we can show that the average marginal renewable energy emissions offsets are: Assessing wind or solar emissions offsets requires accounting for the contemporaneous displacement of fossil fuels , the usual merit order effect. This is the main underlying mechanism in the literature reviewed in the previous section ( Graff Zivin et al., 2014 ; Kaffine et al., 2013; Novan, 2015) . However, we need to account for the hydropower reallocation and i t s related fossil fuel and emissions offsets to compute the full value of these offsets. 20 Notice that emissions offsets also depend on the underlying power plant structure of each period, which is captured in . It could be the case that generators during the off - peak (coal) have larger emissions than those at the peak (natural gas). Finally, if there is no hydropower or storage, the dynamic problem becomes static and accounting just for contemporaneous emissions offsets will deliver an unbiased estimator. In summary, a dding renewable energy capacity leads to heterogeneous increases in generation during the day which reallocate hydropower, alter fossil generation and emissions. In the previous stylized model, hydropower will be reallocated to the period with th e lowest increase in renewable generation until marginal benefits are equated in both periods. Reservoir storage plays the role of facilitating the intertemporal allocation by saving water which will be released at a later time, replacing fossil fuel gener ation and offsetting emissions in the next period . A similar pattern was found by Green and Vasilakos (2012) between Denmark and the Scandinavian countries, where the former exports wind generation on breezy days and imports electricity on calm days. The a uthors argue that Scandinavian hydropower basically acts as a storage for Denmark, smoothing the intermittency of its wind power. Capturing this dynamic effect on emissions offsets and the value of renewables requires modelling more than the conventional s tatic emissions offsets. Data For the econometric models, t his research uses hourly data on load and all generation types ( fossil, hydro, nuclear, wind, solar, imports and other renewables , which comprises geothermal, biomass and biogas ) from 2013 to 20 15 in California. The data comes from publicly available 21 sources: the California Independent System Operator (CAISO) web database OASIS and its renewables watch portal (CAISO, 2016a; 2016b and 2016c). I scraped data from the OASIS server in order to get 26 ,280 observations (Table 1) . 12 Missing observations represent 0.11 % of the total number of hours in the five year span. 13 I use hourly wind and solar power hour - ahead forecasts from OASIS but this data was only publicly available from December 2014 onwards. Table 1 . Summary statistics on CAISO hourly generation data (2013 - 2015) I also use hourly average t emperature data from several California weather stations (IPM, 2016) ; and hourly carbon dioxide , sulfur dioxide and nitrogen oxide emissions data from EPAs Continuous Emission Monitoring system (CEMS) (EPA, 2016). One limitation with the emissions data is that CEMS only compiles information for fossil fuel powered units whose capacity is greater tha n 25 MW. Nevertheless, most power plants in CAISO report their emissions to CEMS. To account for the carbon offsets coming from electricity imports , I use the GHG emissions inventory of all power plants outside California that sell power to this grid (CARB , 2018) . 14 12 Using web mining codes programmed in the R statistical software. 13 This minimum fraction of missing variables is due to either missing information on the OASIS web server or a minor failure in the web mining code. They show no systematic trend in their unavailability. 14 22 Wind, solar and hydropower trends in CAISO Power generation trends From 2013 to 2015 in the California Independent System Operator (CAISO), a significant share of wind and solar capacity and generation was added to the grid (Figure 2 ). In fact, the average share of solar generation in the power mix grew from less than 1% to 7%, while for wind it grew from 3% in 2011 to 6% in 2014 and then it decreased to 5% in 2015. California has a g oal of reaching a 33% share of renewables in the grid by 2020 (CAISO, 2013). As stated in the theoretical model, this growing share of power does not make an even generation contribution throughout the day. Figure 2 . Hourly average wind and solar generation 2013 vs 2015 Source: CAISO, 2016a 23 In fact, solar generation peaks at midday, with an average generation of 0.51 MWh per MW of capacity, declines in the late afternoon, 0.18 and 0.067 MWh/MW at 6 and 7 pm respectively, and is nonexistent after sunset. 15 On the other hand, wind power peaks after sunset, with an average of 0.34 MWh/MW at 11 pm - 12 am and a bit less from 1 - 3 AM, while it delivers the least at the midday (0.18 MWh/MW). Wind and solar power heterogeneous generation through the day have chang ed hydropower generation incentives and reallocation. As stated in the theory, water is stored while solar and wind power peak and released when their contribution declines. This can be observed on the aggregate hydropower generation curve : in 2015, in com parison to previous years, producers generate less during the morning and midday in order to store water for the peak demand time (6 - 9 PM). During these years, the hydropower curve exhibits an evolution towards the : a concave shape (belly) i n the midafternoon and a quick ramp up (arched neck) during evening hours (CAISO, 2013). Notice that the effects of the California drought push the hydro generation curves downwards throughout the years. Nevertheless, the appearance of the convex belly dur ing the midday is entirely a reallocation effect (Figure 3 ). 15 Average computed with aggregate hourly generation data for CAISO for 2011 - 2015 and capacity data from the California Energy Commission. 24 Figure 3 . CAISO average aggregate hydropower hourly generation Source: CAISO, 2016a Hydropower reallocation can be better depicted by using the relative change in the hourly generation shares of each day (Figure 4). In 2011 solar power was only 1% of the total generation while wind was 3%. In the following years, solar capacity outgrew wi nd power capacity which determined three reallocation patterns. First, from 0 - 7 AM, when the sun is gone and wind peaks there has been a drop in hydro generation which follows a wave pattern: reductions are larger from 1 - 3 AM and as wind power dwindles and solar kicks in, they decrease. The second wave goes from 8AM 3PM: reductions increase and decrease with solar generation, peaking at midday. Finally, the displaced hydro is reallocated during the third wave (4PM 11PM) as the sun sets, and these genera tion increases dwindle as wind ramps up. 16 16 As I depict in the theoretical model, hydropower reallocation occurs due to wind and solar generation cycles during the day b ut real time prices in California do not show any convergence. This could be caused by several factors not captured in the styli ze d model such as transmission constraints and agricultural or ecological flows benefits. 25 Figure 4 . Relative change of hydropower shares for each hour of the day Source: CAISO, 2016a I calculated the relative change as follows: first compute the hourly share of hydro generation using annual totals. Then subtract the hourly shares of each year (2013, 2014 and 2015) from the baseline year 2011 to obtain the relative change. The exact hours of the three reallocation patterns vary each year, but the general trend o f moving power from late night, very early in the morning and midday to the late afternoon and evening is clear. Addressing the daily reallocation patterns is key for estimating wind and solar power emissions offsets in CAISO and assessing their economic v alue. This is the goal of the following sections. Hydropower institutional and minimum flow constraints Compliance with minimum flow requirements can alter the generation schedule of hydropower reservoirs in California. These regulations have several goals: flood management, 26 supporting habitat for fish and wildlife, maintaining water supplies and recreation. The schedule of minimum downstream flow rates varies according to the time of year and the expected amount recreation and environmental standards. These vary wit hin days or weeks (Archsmith, 2018). Minimum and pulsed flows constraints increase with wetter conditions, in a particular month or year, limiting the choice set of hydropower producers (Archsmith, 2018). Thus, during the drought years (2013 - 2015) both f lows constraints were relaxed, in comparison to previous years (Figure 3), and allowed hydro generators to reallocate a larger share of production, and arbitrage differences in wind and solar output. In the theory section I argued that flow constraints wil l reduce reallocation when the requirements differ from the profit maximizing generation schedule, but since the constraints were relaxed due to the drought, the optimal generation was more in line with these lax requirements. Hydropower reallocation play s a key role in the California grid since most plants have a reservoir and only a 2% of its total capacity is run - of - river (CEC, 2018). I dentification I use the short term exogeneity and randomness of wind and solar power t o identify the ir fossil generation offsets and related CO 2 , SO 2 and NOx emissions offsets during 2013 - 2015. Since decision making process that settles electricity generation in the whole sale market, their random hourly variation can identify changes in the electricity dispatch and the subsequent emissions. 27 This identification strategy is the same than the one used by Cullen (2013) and Novan (2015) to identify wind power marginal offsets in Texas. Furthermore, the latter uses wind speed as an instrument to control for possible endogeneity, finding that the offsets estimates are similar using wind generation or speed. This suggests that using hourly random generation does not lead to biase d estimates. Given this , I use the OLS estimator with Newey West standard errors to perform inference robust to heteroscedasticity and seria l correlation in the generation and emissions time series (Newey and West, 1987). I also perform a Hausman test fo r the endogeneity of wind and solar generation using their hourly hour - ahead forecasts as instruments. These forecasts are based on exogenous weather data which makes them uncorrelated with any curtailment decisions that could affect the actual renewable g eneration. In the proposed identification strategy I argue that controlling for weekly fixed effects can address the impact of drought and snow pack loss on hydropower generation. Based on the augmented Dickey - Fuller test, neither of these time series show the presence of unit roots. This allows to identify causal effects without using any cointegration procedures. Finally, I might be underestimating emissions and thermal generation offsets since generation data is at the California grid level , which reflects net generation rather than gross generation, and it does not include details about neither thermal generation displaced at foreign nor the specific electricity imports sources . 17 However, since a small percentage of total generation is exported, this error should not amount to a significant share. Since it is almost impossible to disentangle the origin of imports due to the mixing of electrons in the grid, I compute a bound on their CO 2 em issions offsets by using 17 28 the yearly average carbon emissions rates of all power plants outside California that sell power to this grid (CARB, 2018). 18 Static average generation and emissions offsets Econometric specification The estimating equation that identifies the hourly contemporaneous or static impact of wind and solar power on generation and emissions is: where: represents the m different groups of endogenous generation in CAISO: fossil , hydro, nuclear, other renewables and im ports measured in MWh at hour t, are CAISO aggregate solar, wind g eneration and demand (load) in MWh at hour t , is daily total precipitation and is average daily temperature, stands for weekly fixed effects and for weekend FE, are regression coefficients. I estimate offsets of five conventional generation types: fossil , which in California includes gas combined cycle, combustion turbine and boiler plants; hydropower which includes reservoir and run - of - river plants; other renewables which comprises geothermal, biomass and biogas; nuclear and finally imports from other system operators. Standard errors for hypothesis testing are 18 I compute the yearly average emissions rates dividing the total CO2 emissions of all importing power plants by the total electricity imported. 29 computed using the Newey - We st estimator with a 168 - hour lag to account for possible weekly dependencies ( Cullen and Mansur, 2017) . For identify ing the marginal emissions offsets, I use the same equation (7) but the dependent variables are emissions : hourly carbon dioxide emissions (tCO 2 ), sulfur dioxide (lbs of SO 2 ) and nitrogen oxide (lbs of NOx). The estimating equation models the endogenous generation types and emissions as a function of the exogenous wind, solar power and electricity demand or load. The latter is also exogenous since most end use consumers do not face wholesale market real time price s but rather flat tariffs. F ollowing Novan (2015), the model uses a first degree lineal polynomial to allow interactions between wind, solar and load, which aim to capture the heterogeneity of the marginal generator and its emissions. In the theoretical mo del I argued that emissions are an increasing function of fossil fuel use, which is in turn a function of demand. The interactions capture how the marginal plant displaced by renewable energy, and its corresponding emissions, varies for diff erent levels of load. The instantaneous marginal generation offsets caused by adding wind or solar are identified with the average partial effect (APE) estimators : APE for solar generation: APE for wind generation: 30 These average partial effects estimate the instantaneous displacement of the highest marginal cost generator when we increase wind o r solar power. This is just the merit order effect discussed and labeled in the theory section as . Similarly, t he marginal emissions offsets of solar and wind compute the immediate pollution and carbon emi ssion avoided by the incoming wind and solar generation. The most parsimonious specification stated in equation (7) was chosen using the simplest model that complies with the energy identity condition, which states that 1 MWh of wind or solar power shou ld displace a 1 MWh of electricity produced by all other types combined. I estimated two other functional forms as ro bustness checks: a) the main specification with out interactions and b) a second - degree pol ynomial model with interactions. 19 To test the endo geneity of hourly wind and solar generation I do a Hausman test with the main specification using the hour - ahead forecast s as instruments. 20 Moreover, to assess whether one model is good for all the years or if there are structural breaks throughout ye ars, due to the increasing wind and solar capacity, I multiply a year indicator function with each coefficient of the main specification without interactions. 21 19 See details on Appendix 2a and 2b. 20 See details on Appendix 2c. Since the instruments have to model nonlinear interactions of the suspected endogenous variables, the IV - GMM specification als o models interactions between the instruments for the Hausman test. 21 See details on Appendix 2d. Testing the structural break with the simplest specification, linear no interactions, is the most straightforward way to know if only one model can explain wi nd and solar offsets throughout years. 31 Generation and emissions offsets All static econometric models show that, on average and contr olling for load, temperature, and daily fixed effects, solar and wind generation displace mostly natural gas generation and imports. 22 This instantaneous displacement occurs due to the merit order effect , the most expensive generation gets substituted by the zero marginal cost renewable generation. The preferred specification estimates that one MWh of solar power displaces 0.565 MWh of natural gas, while one MWh of wind replaces 0. 731 MWh, on average (Tab le 2) . Table 2 . Static average generation offsets by type. Significance levels denoted by: *** p< 0.01, ** p< 0.05, * p< 0.1 . Newey West std. errors with 168 lags are displayed RE stands for renewables. Furthermore, all models reject the null hypothesis, at all conventional significance levels, of having no hydropower reallocation due to the incoming wind and solar 22 See Appendix 2b for details. 32 . This is evidence of the d ynamics of hydropower reallocation which move generation from the hours with the largest increase in wind and solar generation to those with the lowest by storing water. This finding is confirmed by all the alternative specifications in cont rast to the results of the Texan wholesale electricity market ERCOT, where adding wind did not displace any substantial amount of hydropower during 2005 - 2011 (Cullen, 2013; Novan, 2015). One possible explanation for this divergence is the low hydropower sh are, less than 1%, in total capacity and generation in ERCOT , which would lead to a static allocation problem as shown in the theory section. On average, the preferred model gives an estimate of 0.129 MWh of displaced hydropower for each additional MWh of solar generation and 0.0 44 MWh for each additional MWh of wind. This outcome points out that a considerable share of hydropow er producers are reacting to change s in incentives from the heterogeneous w ind and solar generation throughout the day by modifying their generation schedule . take into account all wind and solar dynamic effects on hydro. 23 There is barely any base power displacement coming from the nuclear, geothermal, biomass, or biogas generation since they are basically base power. The preferred specification 23 In the theoretical framework we derived that the static estimator only captures the derivatives of peak and offpeak hydro generation with respect to the same period renewable generatio n , but not the cross derivatives. Since , it simply follows that . The summation of the static estimator plus the cross derivatives due to wind and solar dynamic effects on hydro equals zero, which is the appropriate dynamic estimator. 33 yields an almost perfect compliance with t he energy identity requirement, which implies that the first degree polynomial with interactions model captures most of the grid complexity. Using the Hausman test with hour - ahead forecast s as instruments , I cannot reject the exogeneity of wind and solar generation for explaining their static impact on hydropower and foss il generation, as well as on carbon emissions. 24 Basically, the hourly short term variation in wind identification of marginal generation and emissions offsets. The st ructural break test shows the appropriateness of having different models each year to explain wind and solar offsets. This allows a more flexible modelling of changing grid interactions under increasing renewable energy capacities. The results for all thre e years and three functional forms show very similar estimates than the preferred specification with data from all years and confirm the hydropower reallocation. 25 As argued in previous sections, h ydropower reallocation varies each season according to the down stream flows requirements. I find that wind and solar displace less hydro in the summer and spring since dams are more constrained by larger flow regulations during these wetter months. Moreover, the effect of wind on hydro generation is statisticall y insignificant during the summer. 26 Hence, the results show evidence that hydro operators respond to the different incentives from renewable energy generation profiles within a constrained profit maximization scheme. This 24 See Appendix 2c. 25 See details on Appendix 2b 26 Details are on Appendix 2e 34 finding is in line with Archsmith (2018) who argues that tighter constraints during wetter periods is a source of inefficiency in California. The static or instantaneous average marginal emissions offsets show low abat ement potential for solar (0.185 tCO 2 /MWh) and a sensible potential for wind power (0.373 tCO 2 /MWh). 27 This difference can be explained by the 24 hour continuous aggregate wind generation in CAISO , while solar generation is present mostly for ten to twelve hours. However, the key rea son for this difference is the hydropower reallocation that occurs from midday to sunset hours. This displaced hydro is re al located at a different time and it will substitute natural gas generation and reduce even more carbon emissions than what is captured in the static APE . The net effect is captured by the dynamic estimator. Dynamic average generation and emissions offsets Econometric specification. I argued in the theoretical model that given a fixed reservoir, the gain in hydropower at peak has to be compensated with a loss at the off - peak . Hence, a valid dynamic specification has to capture a daily net zero hydropower displacement. The analysis of wind, solar and hydro generation trends in CAISO suggests that hydropower reallocation occurs mostly within one day (Figure 4). Therefore, I estimate a dynamic model by aggregating the hourly data into dail y data and accounting for lags to test if all reallocation occurs within one day . 27 Using the preferred specification. For robustness checks see Appendix 2b. 35 Basically the same estimating equation specified previously but using daily data ( t ) and lags with weekly ( ), weekend ( ) and temperature effects ( ) can capture the hydro reallocation. To account for weekly dependence or serial correlation I use Newey West standard errors with seven lags. Using the average partial effect I compute the dynamic estimator of solar and wind power emissions offsets. These estimators account for the instantaneous displacement of fossil fuels , through the usual merit order effect, as well as for the hydropower reallocation and i t s related fossil fuel and emissions offsets , whose effect was labeled in the stylized model using the cross partial derivatives . Thus it computes the net effect of adding one MWh of wind and solar power on fossil generation and emissions offsets . Generation and emissions offsets Once I control for hydropower reallocation by using daily aggregate data, the results show that all hydropower displaced by solar power and wind power is reallocated within one and two days respectively (Table 3 ). This is formally tested with the null hypothesis of having net zero 36 hydro offsets due to wind and solar , which cannot be rejected at any conventional significance levels. 28 The natural variation in renewable generation follows daily repetitive patterns, which provide incentives to use hydropower for arbitrage. Results show that , on average, the larger daily solar intermittency leads to hydro reallocation within one day while the less pronounced wind power variation leads to a two - day arbitrage. Further evidence of the daily net zero hydropower displacement comes from the robustness checks. Results from the Hausman test show that the daily variation is also exogenous and does not affect the identification of the appropriate marginal emissions offsets. 29 The structural break test shows that one model with all years is appropriate for estimating the dynamic emissions offsets. 30 Table 3 . Dynamic average hydropower generation and emissions offsets Significance levels denoted by: *** p< 0.01, ** p< 0.05, * p< 0.1 . Newey West std. errors with 7 lags are displayed 28 Results on the other generation types are on Appendix 2g. Basically, as shown in the theory section, hydro is the only techno logy whose dynamic offsets cannot be rejected to be differen t from zero in most robustness checks. 29 See Appendix 2c. 30 Appendix 2d 37 As argued in the theory section , the larger solar generation at the midday will lead to a hydr o reallocation from the morning and noon to the late afternoon and early evening hours (Figure 2 and 4). Thus, hydropower generation is basically arbitraging midday generation (where steam turbine is the marginal unit) for late afternoon generation (gas tu rbines being the marginal generator). On the other hand, wind power peaks at late night and early morning (combined cycle as the marginal generator) causing hydro to arbitrage those hours for daylight hours (steam turbines as marginal). Hence, solar power is basically substituting late afternoon gas turbine for midday steam gas turbine generation while wind replaces daylight steam turbine for late night combined cycle generation. Since gas turbines have high er carbon and sulfur emissions intens ities than steam gas turbine s while combined cycle and steam turbine generation have similar CO 2 , SO 2 emissions intensities , solar power delivers larger CO 2 , SO 2 emissions offsets than wind. 31 Notice that the dynamic estimates of solar power emissions offse ts correct the low carbon abatement assessment from the static model and they slightly reduce the wind abatement coefficients in all specifications (Table 3 ). 32 On the other hand, since steam turbines emit more NOx per MWh than combined cycle units and ga s turbines, wind delivers larger nitrogen oxide emissions offsets than solar. Actually, solar power NOx emissions offsets are not statistically different from zero. These findings 31 Emissions intensity is measured as units of pollution per units of electricity supplied (tCO 2 /MWh, lbs SO 2 /MWh, lbs NOx/MWh) 32 Compare appendix 2b and 2g for details. 38 highlight that accounting for the dynamic interactions between the three ren ewable sources, wind, solar and hydro, is necessary for assessing their proper external benefits and value. Emissions offsets also vary each year and season since they are a function of total installed capacity and load. As a larger wind and solar capac ity join the grid, the most expensive, inefficient and dirty fossil generators get replaced (merit order effect) leading to lower marginal emissions (Hirth, 2013, Novan, 2015). Hence, new renewable energy capacity will offset a lower amount of emissions. T his trend is clear for solar power carbon emissions offsets during all three years, and for wind in the last two years. 33 Figure 5 . Dynamic average carbon emissions offsets throughout years Errors bands are based on 95% CI with 7 lags Newey West std. errors 33 Not having a declining trend all years for wind power could have been caused by the restructuring of the grid and inclusion of fossil generation units in the years after the closure of the San Onofre Nuclear Generation Station in February 2012 (Davis a nd Hausman 2016) . 39 Furthermore, carbon emissions offsets peak during summer due to the large electricity demand in those months, which lead to a larger use of fossil generation and higher emissions than th e rest of the year. On the other hand, sulfur and nitrogen oxides emission offsets show less seasonal variability and show a slight decrease during the summer. The seasonality of emissions offsets depends mostly on load than on hydro reservoir levels and g eneration (Figure 6). Nevertheless, controlling for the dynamics of hydropower reallocation is still necessary to estimate offsets adequately. Summing up, the external benefits of renewable energy vary per technology and throughout time. Figure 6 . Dynamic average carbon emissions offsets and load throughout seasons Offsets were estimated using data of each season for all three years Load is on the left axis and emissions offsets on the right axis 34 34 Details on estimates are in Appendix 2h. 40 Finally, I account f or the emissions offsets of imports and compute the bound on total carbon emissions of solar and wind power to be substantially larger: 0.615 and 0.465 tCO 2 /MWh respectively. While including these external benefits is not straightforward, it is important s ince a significant amount of imported electricity is displaced by renewables. 35 Valuing emissions offsets and policy implications Estimating the abatement potential of wind and solar power is necessary to assess the economic value of their external benefits and evaluate current adoption policies. Using the US EPA social cost of carbon (IAWG, 2015) 36 , sul fur dioxide a nd nitrogen oxide marginal damages from Muller and Mendelhson (2009), and the dynamic offset estimates we can infer that s olar power external benefits are almost, on average, one and a half times larger than those of wind for different scenarios (Table 4 ) . The bulk of external benefits (99%) come from carbon reductions since SO 2 and NOx offsets are minimal due to the clean ga s fleet of California. 37 35 See table 2 for import electricity offsets and Appendix 2i for their emissions offsets 36 The central estimate for a discount rate of 3% expressed in 2015 USD. 37 Since I use time series on generation and emiss ions for the entire California grid, I cannot trace the location of each displaced fossil power plant and account for the spatial heterogeneous damages from sulfur and nitrogen oxides. However, I use the aver age marginal damages from these emissions for al l counties in California based on Muller and Mendelhson (2009). 41 Table 4 . Average marginal external benefits, incentives and costs of renewables in California 1 3 Based on Callaway et al . 2017 . 2 Heeter et al. 2014 . I report larger solar and wind power external benefits than those of Kaffine et al. (2012), who used daily aggregate data for 2009, to control for hydro dynamics and reallocation, and Callaway et al. (2017), who did not explicitly model hydropower reallocation. The estimates of benefits are much large r if we account for offsets from electricity imports. The environmental and economic benefits of solar and wind power in grids with a significant share of hydro depend on how their daily generation cycles alter economic incentives 42 and lead to reallocatio n which substitutes different types of marginal fossil generators. Thus, t he se external benefits ultimately depend on the grid fossil fuel configuration . My approach and estimates allow to account for the interactions between the three renewables, which en hance the economic value of solar power in California. These dynamics were contingent on having dry years during 2013 - 2015, which allowed a significant share of hydro to be reallocated facilitating power arbitrage. With more stringent flow requirements d uring wetter years, it is likely to expect less hydro arbitrage and a lower value of solar power external benefits. On the other hand , Texas has no large scale hydropower and a significant share of its electricity comes from coal. Novan (2015) finds that the external benefits of solar power are lower than those of wind since the latter replaces more coal during its generation peak at night. Moreover, the interaction between the three renewables in California At the central estimate of the social cost of c arbon (SCC), even accounting for imports offsets, the average external benefits of neither solar nor wind in California are large enough to compensate for the incentives they receive from the renewable portfolio standard or federal tax credits. Considering imports, the external benefits of solar represent a bit more than a third (35%) of its levelized cost and almost a quarter (23%) in the case of wind. If we consider the high impact estimate of the SCC, then the external benefits of solar surpass all of it s costs and incentives, while those of wind cover only its tax credit incentive and around a 70% of its LCOE and RPS credits. 43 It is worth noting that since emissions offsets vary throughout seasons and years, so does the value of the external benefits, w hich increase during spring and summer and slightly decline as years go by, due to the less polluting marginal generator. In spite of this seasonality, the band . Figure 7 . Value of wind and solar power external benefits (no imports) throughout years and seasons Errors bands are based on 95% CI with 7 lags Newey West std. errors While the RPS credits give the same incentives to both technologies, the federal tax credit does a better targeting by favoring solar over wind since the latter has a higher levelized cost and delivers lower carbon emissions offsets. Ideally, the RPS credi ts could use trading ratios to 44 recognize the larger benefits of one technology and provide the optimal signals for investments in new renewable capacity. In principle, a price on emissions, through a tax or cap and trade, should automatically send the co rrect signals on the value of renewables. Nevertheless, if this price is lower than the estimated marginal damage of the externality, it could lead to undervaluation and underinvestment. Average carbon prices in California for 2015 were 12 USD/tCO 2 , which reduces significantly the value of the external benefits of renewables (Table 4 ) and undermines the rationale for justifying their incentives (Climate Policy Initiative, 2017). Borenstein et al. (2016) argue that these low carbon prices are caused partially by overlapping environmental policies and incentives. Harmonizing these different incentives to provide clear signals on the value of wind and solar is a pending challenge. Finally, the current policies and incentives should reflect the varying level of benefits of renewables throughout seasons and years. This is difficult to implement with the RPS and tax credits but can be more easily implemented in the case of feed - in - tar iff contracts. Therefore, the above lessons can shed some light to new contracts and schemes that want to take into account the heterogeneity of benefits among technologies and throughout time. Conclusions This paper tackles the novel challenge of identifying the marginal generation and emissions offsets caused by adding two intermittent renewable energy sources to an electric grid that has a significant share of hydropower. Using a static model based on recent (201 3 - 2015) hi storical data 45 of hourly random solar and wind generation, I estimate the instantaneous displacement of the highest marginal cost generator, through the merit order effect, but also the dynamic hydropower reallocation that occurs due to the uneven wind and solar generation throughout the day. Hence, I control for hydropower reallocation using a dynamic model which aggregates hourly into daily data. The results allow us to infer the correct marginal external benefits of wind and solar due to CO 2 , SO 2 and NO x emissions reductions. In California, hydropower dynamics particularly enhance the external benefits of solar power. From a broader perspective, the proposed dynamic modelling is key for understanding electricity generation and grid level emissions in systems with increasing a doption of storage technologies since hydro is already an imperfect form of storage whose charging functions by keeping water in the reservoir. Such devices can help managing the short term (day vs night) intermittency of wind and solar power . To the extent that the main cost of operating storage comes from the capital investment with negligible marginal costs , the derived insights about hydropower arbitrage and reallocation would also apply to profit maximizing storers. Thus, in f uture backed storage grids, economic dynamics will also play a key role in determining renewable energy emissions offsets, external benefits and ultimately its economic value. Furthermore, several emerging economies with electric grids powered by a signif icant (or even mostly) share of hydropower are increasingly installing wind and solar plants (some examples include China, Brazil and India To the extent that these countries dispatch generators based on the 46 lowest marginal cost (via wholesale market) emissions offsets offers a feasible replication methodology for assessing the heterogeneous value of wind and solar power. Due to the differences between the external benefits of renewable technologies and their variation throughout seasons and years, assessing emissions offsets is necessary to target new policies and incentives accordingly, or to redesign long term inefficient schemes. 47 A PPENDICES 48 APPENDIX 1. Theory derivations APPENDIX 1 a. Derivation of the simplified one period system of equations and implicit function theorem. Equilibrium allocations are defined by the fossil generation and the utility maximization conditions: Using the implicit function theorem, where the endogenous variables are fossil generation and price and the exogenous are renewable generation and hydro we get: And using the conventional properties of utility . and cost functions we get: APPENDIX 1 b. Derivation of the two period system of equations and implicit function theorem. The solution to the two period problem proposed in equations 2 and 3 can be represented by the system . The endogenous variables are fossil generation and hydropower ( ) and the exogenous variables are renewable generation . 49 The implicit function theorem states: where And using the conventional properties of utility . and cost functions we get: 50 APPENDIX 1c. Hydropower reallocation due to the difference between off - peak and peak renewable generation. The reservoir equation implies that any change in hydropower allocation at one period has to be compensated with a counterbalancing action in the next period: Using the total derivative we can show When hydro is taken from one period, the differential is negative . Then, w e can show the key condition for hydro reallocation from off - peak to peak by stating that the peak delta has the positive sign or is larger than the reduction at the off - peak : And using the implicit function results leads to: Since if we assume the same rate of change of the marginal utility for changes in electricity consumption in both periods, hydro will be reallocated to the peak time as long as off - peak gains in generation from new capacity are larger than peak gains 51 APPENDIX 2. Regression Results APPENDIX 2a. Robustness check specifications for static marginal generation and offsets. I estimated three functional forms : a) the main specifications, first degree polynomial model with interactions , b ) a first degree polynomial model with out interactions , and c) a second - degree polynomial model with interactions, a) First degree polynomial without interactions: (t stands for one hour) b) First degree polynomial with interactions c) (t stands for one hour) c) Second degr ee polynomial with interactions: t s tands for one hour. i,j,k index the exponents of solar wind and load 52 APPENDIX 2b. Static results Table 5 . Static marginal generation and emissions offsets results. Newey West std. errors with 168 lags are displayed RE stands for renewables. 53 Table 5 . Static marginal generation and emissions offsets results. ) Newey West std. errors with 168 lags are displayed RE stands for renewables. 54 APPENDIX 2c. Instrumental variable specification results Table 6 . Hausman test of endogeneity of hourly wind and solar generation 55 APPENDIX 2d. Structural break robustness check of the linear no interactions static model (33) For the static model (with hourly data) The results show that we reject the null of having no difference between the coefficients of each year. Thus, estimating models with the data for each year separately is appropriate. For the dynamic model (with daily data) The results show that we cannot reject the null of having no difference between the coefficients of each year 56 APPENDIX 2 e. Hydropower reallocation estimates Table 7 . Hydropower reallocation (static estimator) for each season Newey West std. errors with 168 lags are displayed Estimates are computed with the preferred specification (linear with interactions) using hourly data from all years and then evaluating the marginal generation offsets with the average partial effect for each season: for solar and for wind 57 APPENDIX 2f. Robustness check specifications for the dynamic marginal generation and emissions offsets. a) First degree polynomial without interactions: t stands for one day b) Second degr ee polynomial with interactions: t s tands for one day. i,j,k index the exponents of solar wind and load 58 APPENDIX 2g. Dynamic estimates Table 8 . Dynamic marginal generation and emissions offsets results Newey West std. errors with 7 lags, are displayed. 59 APPENDIX 2h. Dynamic yearly and seasonal estimates Table 9 . Dynamic yearly and seasonal emissions offsets estimates Newey West std. errors with 7 lags are displayed Newey West std. errors with 7 lags are displayed 60 APPENDIX 2i. Estimates of imports offsets Table 10 . Estimates of electricity imports carbon emissions offsets. The product of the marginal import displacements and the yearly average carbon emissions rates yield the average marginal emi ssions offsets from imports. Then adding those estimates to the dynamic average emissions offsets estimates of previous sections yi elds: 61 T able 11 . Total carbon emissions offsets (California power plants and imports) . 62 APPENDIX 2j. External benefits estimates Table 12 . Estimates of external benefits of emissions offsets (USD) 63 CHAPTER 3 : ELECTRICITY STORAGE, EMISSIONS TAXES, AND THE VALUE OF RENEWABLE ENERGY Introduction What is the value of intermittent renewable energy sources, such as wind and solar? These sources can reduce grid - level electricity generation costs and emissions. But their intermittency both cyclical and random can make it difficult to integrate them int o conventional electric grids (Joskow, 2011; Baker et al., 2013). Electricity storage could facilitate this integration by allowing renewable electricity to be used at the times it is most valuable. But simply pushing storage capacity onto the grid can inc rease emissions if intra - day storage arbitrage leads to a large increase in off - peak coal generation in place of cleaner, on - peak natural gas generation. Thus , assessing the full value of intermittent renewables requires a dynamic framework that takes into account all private and external benefits and costs. In this research, I assess the value of wind generation and storage capacity by developing theoretical and empirical model s of the Texas (ERCOT) electricity grid. The models account for the interactio ns between wind , storage , a nd emissions taxes. The main theoretical insights show that the value of wind is exogenously determined by electricity demand and its cycles while the value of storage depends on the timing of charging and discharging events. Th e empirical results show that wi nd and storage are complements since intermittency raises arbitrage benefits, which leads storers to reallocate power based on wind cycles increasing its value. Emissions taxes increase net welfare and the value of st orage since they provide the correct incentives to dispatch generation that maximizes social benefits . At 2015 wind power levels in ERCOT, its marginal 64 value breaks even with its levelized cost of electricity (LCOE) if carbon offset benefits are valued at least at 40 USD/tCO 2 . 38 Previous studies have estimated the costs of solar power intermit tency (Gowrisankaran et al, 2016 ), the impact of storage on generation and investment in generating capacity (De Sisternes et al., 2016), the value of wind generation in the presence of storage under market power (Sioshansi, 2011), and the long - term dynamic effects of carbon taxes on electricity markets (Cullen and Reynolds, 2016). These stud ies have not, however, accounted for the interactions between storage capacity and emissio ns regulations in assessing the value of intermittent renewables. I develop a two - period ( off - peak and peak) analytical model of an electricity market with a given l evel of renewable energy capacity and unconstrained storage. The model considers private generation costs as well as emissions damages. The social planner maximizes welfare by choosing fossil generation and storage actions (charging or discharging) for eac h period. One key feature consistent with the Texas grid is that wind generation is larger in the off - peak period and it is negatively correlated with demand. I show that the optimal storage will move power from the off - peak to the peak, arbitraging the di fferences in electricity demand, wind power generation , and emissions intensities between the off - peak and peak periods . Therefore, if emissions externalities are not considered, there is a distortion in the charging and discharging actions, which carries excessive electricity from the off - peak to the peak period . 38 Social cost of carbon at a 3% discount rate and in 2015 USD, equivalent to 35 USD/tCO 2 measured in 2007 USD. 65 The model shows that t he marginal value of wind is the increase in generation, from additional capacity, valued at equilibrium prices. This value entails the benefits of a voided fossil generation and the value of emissions offsets . Since wind capacity generates electricity based on exogenous and stochastic patterns, once a turbine is built, its value depends on the correlation between wind cycles and the electricity demand patterns. T he marginal value of storage depends on the difference between peak and off - peak prices. Using a constrained version of the model, I show that the value of storage depends directly on its allocation decisions since it profits from arbitrage opportunities. As wind capacity increases, wind generation replaces the most expensive fossil power plants, which leads to lower equilibrium prices and to a lower value of wind regardless of stor age. On the other hand, a s wind capacity increases so does its hourly intermittency and the gap between equilibrium peak and off - peak prices widens, which leads to a larger value of storage. I calculate the value of wind power and storage in the Texas e lectric grid (ERCOT) by developing an empirical short term dynamic model, in which a social planner maximizes welfare by choosing fossil generation and storage. Since wind power has daily cycles , the model assumes a one day horizon with hourly time steps. It also considers carbon dioxide, sulfur dioxide , and nitrogen oxide emissions damages, and it represents wind power as a stochastic process whose uncertainty comes from iid forecast errors. I model storage with an equation of motion that relate s the state of charge each hour to charging, discharging , and roundtrip efficiency losses . I calibrate parameters using hourly data on load, fossil generation, wind power, and emissions from ERCOT and EPA for 2015. I then implement a thousand Monte Carlo simulations based on 66 different possible realizations of daily wind and demand profiles of 2015. I explore four scenarios combining storage a vailability and emissions taxes: no storage and no taxes (baseline), no storage and taxes, storage and no taxes, and storage and taxes. I find that t he value of wind power at current levels (12% of load) depends largely on the social cost of carbon (SCC) : with the 40 USD/tCO 2 39 used in the range between 60.34 and 61.58 USD/MWh for the averages ) breaks even with its levelized cost (60 USD/MWh) in all scenarios USD/tCO 2 in 2015, The marginal value of wind generation is downward - slopin g since as more zero marginal cost generation enters the grid, there will be a lower marginal cost fossil generator setting the equilibrium price. I find that (from 12% to 30%) leads to a 15 % loss in its average marg inal value. The marginal value of storage capacity is also downward - sloping since increased arbitrage reduces the gap between peak and off - peak prices. Wind and storage capacities are complements. There is a positive feedback since increasing wind capacity exacerbates intermittency, which leads to a larger gap between peak and off - peak prices, and raises the value of storage (12% gain when the share of wi nd increases by 18%). Meanwhile, since storage allocates intermittent wind generation to periods with high prices, storage also increases the value of wind. This increases is almost 1 USD/MWh with wind as 12% of load for the unconstrained storage scenario. 39 Average value for a discount rate of 3% measured in 2015 USD (IAWG, 2015). 67 Emissions taxes increase net welfare and the value of storage since the endogenous arbitrage decisions account for private costs as well as emissions damages. Without emissions taxes, the economic value of wind breaks even with its levelized cost (for a S CC of 40 USD/ tCO 2 . However, the increase in private surplus (26.3 - 27.4 USD/MWh) generation costs is less than half the levelized cost . Thus, pricing externalities is vital to send the correct signals and incentives to spur investment in wind capacity. I find that s torage capacity and emissions taxes are complements in driving welfare since accounting for the external and intertemporal costs leaves no room for welfare losses that could be exacerbated either by competitive storers or by the lack of flexibility in allocating energy when it is most of 324 MWh with no environmental taxes leads to an increase in CO 2 , SO 2 , and NOx emissions. The estimated value of stor age for the planned capacity falls short of its levelized cost ( 39.05 USD/kWh for the 324 MWh in the scenario with taxes ) but it is likely to be larger if we account for short term intermittency smoothing (1 min, 5 min) and frequency regulation services. Both, low and unconstrained storage capacit ies , have less than a 50% utilization rate since hourly arbitrage mostly charges during several hours in the morning to release power in the afternoon and evening. Under current technology and cost trends, implementing a carbon pricing scheme with stable prices of around 40 USD/tCO 2 can induce wind to supply 30% of the load in Texas. However, the estimated value of storage for hourly intermittency smoothing is lower than its cost, 68 even under the 30% wind scenario . Thus additional benefits (eg. short term intermittency) would be needed to justify investments or incentives for storage capacity. Literature review Gowrisankaran et al. (2016) assessed the additional intermittency costs of incorporating solar power at different integration levels (0 % , 10 % , 15 % and 20%) with detailed operational data for southeastern Arizona. solar generators produce only when the sun is shining, adding to social costs and requiring electricity system operators to ( Gowrisankaran et al. , 2016 , p.1187 ) . The authors document the additional grid level costs, such as increasing o perating reserves to support the intermittent solar power, and demand curtailment costs or paying consumers not to use electricity at peak hours in order to reduce the risk of failing to meet load. Gowrisankaran et al . (2016) build a model of an electrici ty system operator maximizing surplus by choosing fossil generation and demand curtailment levels subject to the intermittency of solar power, the stochasticity of demand, generation investment costs, and fixed goals of solar power capacity. The model spec ifies a short - run component (usual daily operation) and a long - term component (choosing capacities). This formulation captures approximat ions of operating reserve costs but does not account for dynamic linkages from period to period determined by ramping a nd start - up costs ( Gowrisankaran et al. , 2016) . The authors find that reaching a 20% integration of solar power is socially costly ($138.4 per MWh), with cyclic intermittency representing around $46/MWh and unforecastable 69 intermittency accounting for $6. 10/MWh. The key parameter for enabling a large adoption of solar power and reducing grid level and demand curtailment costs is solar installation costs. If installation costs could be reduced from $4.41/W to $1.52/W, and with carbon dioxide social costs o f $39, the authors project that solar power would be welfare neutral. De Sisternes et al. (2016) explores the long - run impact and value of storage in the ERCOT grid by using a detailed capacity expansion model with unit commitment constraints of the powe r grid. The authors determine the optimal fossil and renewable energy generation mix for meeting the projected 2035 electricity demand by exogenously varying the storage capacity (0 - 30 GW). This research accounts for a chronological or cyclic intermittency of demand and renewables as well as system requirements for operating reserves. Rather than accounting for the external damages from emissions for choosing the optimal allocation, the authors constrain the optimal solution to comply with emissions intensi ty targets in the range of 0.02 - 0.05 tCO 2 /MWh, which are much lower than those for the average combined cycle unit in 2015 (0.47 tCO 2 /MWh) (EIA, 2017; EPA, 2017). The model results show in great detail hourly dispatch for weekly timeframes of all resource s, including the charging and discharging of storage. It incorporates technical details about storage such as the 80% round - trip efficiency and power ratios for maximum hourly discharge. The authors conclude that storage increases the cost - effective penetr ation of renewables, reducing capacity investments in nuclear and gas peaking units. Its adoption ultimately will depend on reducing technology costs and improving power ratios. 70 Sioshansi (2011) was one of the first papers to model the value of wind powe r in the presence of storage under market power. This paper uses a Stackelberg model in which wind producers are the leaders since they operate a monopoly of storage capacity. Thus, wind generators choose storage allocations and then fossil fuel producers react and set their generation profiles. Nevertheless, the model does not consider the external costs of fossil generation and, by modelling a market power setting, it cannot render the Pareto o ptimal allocation and the first - best value of wind power. Lam adrid et al (2015) simulate the impact of wind power and energy storage in ERCOT with a detailed model of the grid that accounts for carbon dioxide emissions. The model takes an engineering approach with a very detailed account of ramping and reserve costs as well as hourly wind stochasticity. Using hourly data for 2011 , the authors find that taxing carbon increases welfare , and that the net benefits of wind and storage are much larger than their impact on reserve and ramping costs. While t he authors model fossil generation, storage allocation , and emissions, they do not explicitly model the marginal value of increasing wind capacity or storage. The previous works provide insightful and detailed modelling of the challenges in adopting wind and solar powe r . With the exception of Lamadrid et al (2015) they do not explicitly model the external benefits of carbon and other emissions reductions in the operator objective, nor do they account for storage dynamics in the allocation problem. To the best of my knowledge, the dynamic model of the Texas (ERCOT) electricity grid proposed here is the first in the literature to account for the interactions between storage capacity and emissions regulations in assessing the perform ance and value of i ntermittent renewables. I do this by developing a discrete time welfare - 71 maximizing dynamic model of ERCOT using hourly data for 2015 on demand, fossil generation , wind generation, and emissions of carbon dioxide, sulfur diox ide, and nitrogen oxides . Theor etical Model Model setup and optimal storage I develop a partial equilibrium, short term, dynami c model of electricity dispatch. First, I define a social benefit function of electricity consumption which captures total welfare to consumers from u sing power at each period t . It is assumed to be concave and twice continuously differentiable. Demand for electricity is quasilinear and can be obtained by differentiating this function . 40 Electricity is supplied using fossil fuels generation , wind power and storage , where the latter is negative when charging and positive when discharging. I assume two time periods, off - peak followed by the peak Second, w ind power is a function of its installed capacity with recurring realizations every day for the off - peak and peak . F or all capacity levels, its generation is larger in the off - peak period . D emand for electricity is larger during the peak period but wind generation is larger during the off - peak period s . This feature captures a key characteristic of several power systems in which the daily intermittency of wind power results in a larger availab ility usually at midnight and early in the morning. Furthermore, this recurring or cyclic nature of renewable generation and demand determines that modelling just the subset of two periods of one day is a valid simplification to understand the dynamic pro cess. 40 Using this assumption, the integral of the Marshallian demand will recover the benefit function and can be used for welfare comparisons. 72 To further operationalize the model, I assume linear demand , and private marginal wholesale costs , where the latter is the same function during both periods . D emand is assumed to have the same slope in both p eriods but different intercepts, where because electricity demand is higher during the peak period . Carbon emissions and pollution are a by - product of fossil generation with their intensity varying during the day according to the fuel type used. 41 For simplicity, I assume that emissions are linear in fossil fuels and that their intensity is larger in the off - peak period, . Marginal social costs of emissions are represented by . The charging state follow s the equation s of motion and , where is the battery roundtrip efficiency. Notice that when recharging power is withdrawn from the grid and the stock or state of charge increases, while the opposite occurs when discharging . I assume no startup costs or dynamic frictions , no discounting , no constraints on storage actions and a zero initial charging state , . There are fixed capacities for all generation types since the model captures the insights of allocating power within a day . The social p lanner consumption at the initial time by choosing the amount of fossil fuel generation and storage for each period. 42 41 For example, in ERCOT, emissions per MWh are usually larger early in the morning, since the marginal generators are coal power plants whereas in the afternoon the marginal generators are gas plants. While these features and the increasing private marginal costs for different fuel types at the wholesale level can be captured with a nonlinear function, a stylized linear model can render the main insights by assuming different emissions intensities . The empirical model in the next section uses nonlin ear functions for both costs and emissions. 42 considered and assuming no other market imperfections (e.g. no transmission externalities). 73 The equilibrium allocation is determined by the fossil generation optimality condition (2) , which equates the marginal benefits (or prices) of electricity consumption to the marginal social cost of fossil generation, and the no arbitrage condition (3), which equates the benefits of electricity consumption throughout peri ods: Solving the system, the optimal storage expression becomes: 43 Since there are only two time periods, and no scrap value or final incentive to store any power, peak and off - peak storage actions will have opposing signs. The optimal storage arbitrages across periods the differences in incentives from electricity demand, wind power ge neration and emissions intensities . Given higher marginal benefit s of consuming electricity during the peak 43 Solving for the interior solution, details are on Appendix 1. 74 period , there is an incentive to recharge the battery during the off - peak and discharge this power during the peak period . Wind capacity delivers a larger generation at the off - peak and since the marginal benefits of electricity consumption are decreasin g, this implies that there is a further incentive to move power from the off - peak to the peak. If there were no differences in the emissions intensities, the optimal storage would only require recha rging during the off - peak and discharging during the peak . Since generating fossil electricity at the peak is cleaner than at the off - peak , this creates a n incentive that reduces the amount of arbitrage . When the planner reallocates power from the off - peak to the peak period , via storage, he re quires more fossil generation in the off - peak to meet demand. However, using more fossil generation implies more c arbon and pollution emissions in the off - peak. The optimal storage weighs this tradeoff by considering the d ifference in emissions intensities . If emissions externalities are not considered, there is a distortion in the charging and discharging actions, which carries excessive power from off - peak to peak p eriod, by only arbitraging differences in demand and wind power but not in carbon and pollution emissions intensities . Furthermore, the amount of the optimal storage decreases with larger private marginal costs of fossil generation c and with higher roundt rip storage efficiency . 44 44 In the current setup, with neither startup costs nor ramp up constraints, there is no difference in the marginal private cost s between hours of the day. In the empirical model, this assumpt ion is partially relaxed by modelling different hourly marginal costs. In this case, the difference between hourly wholesale marginal costs will also influence the optimal storage allocatio ns. 75 The addition of marginal wind power capacity will increase the optimal storage since adding capacity magnifies the gap between peak and off - peak generation given the recurrent wind patterns . 45 As more wind capacity is built, more zero marginal cost generation is added to the grid. This reduces the price of electricity during both periods. However, the larger addition of wind power in the off - peak reduces its prices more than during the peak , an d creates an incentive to move power between periods until both marginal benefits are equal and the no arbitrage equilibrium condition is satisfied. This feature of daily wind patterns i s explained in the section on trends in Texas electricity demand, gene ration, and wind power . The marginal value of wind In the previous model , with unconstrained storage and emissions taxes, we can infer the value of renewable energy under storage and emissions taxes by using the optimized objective function and its deriva tive with respect to capacity. 46 The marginal value of wind generation is det ermined by the marginal benefit s of electricity consumption and the marginal impact of capacity on wind power generation during both periods of the day. Intuitively, equation 6 describes the marginal change in benefits due to a change 45 . As capacity increases , this gap also increases and the optimal allocation changes to comply with the arbitrage condition equating marginal benefits of power consumption. 46 Details are on Appendix 2. 76 in wind capacity, which is assessed considering the impact of capacit y on wind generation valued at equilibrium prices. Notice that peak generation is divided by storage losses while off - peak generation is not . This allows expressing both quantities in equivalent units since transferring power from off - peak to peak incurs in losses. Hence, increasing wind generation in 1 MWh during the peak has a larger impact on its value than increasing 1 MWh in the off - peak period. Notice that the value of wind depends on equilibrium marginal benefits, which in the previ ous model are social marginal benefits since they include the cost of fossil fuel generation (private marginal costs) and the damage from emissions (externalities). I n the no taxes scenarios, equilibrium prices do not reflect the full marginal social cost of power generation but only the marginal fossil generation cost. Hence, the marginal value of wind expression for the unconstrained storage and no tax scenario is : 47 The marginal value of wind is comprised of the benefits of a voided fossil generation and the value emissions offsets . These benefit s of avoided fossil generation are the private surplus 47 See Appendix 2. 77 gains when there is no tax on emissions to internalize the emissions offsets benefits. Notice that these private gains are lower t han the full economic marginal value of wind. Effect of storage on t he marginal value of wind So far the model has mainly considered the scenario with storage and emissions taxes. We the no storage and tax scenario as , then the value o f wind is defined by an expression similar to equation 6 but peak and off - peak prices do not equate. The difference in the value of wind power between the storage and tax and the no storage and tax scenario is given by: 48 In the scenario with storage, the equilibrium price is a weighted average of the prices of the no storage scenario due to the no arbitrage e quilibrium equation that transfers power to equate marginal benefits between periods ( Figure 8 ). Then , the equilibrium price under storage is larger than the off - peak price without storage but smaller than the peak one . The value of wind with storage can be larger than its counterpart without storage since in the former scenario, the larger off - peak marginal increase in generation is valued at a higher price. As equation 5 and 7 state , the value of wind depends on the contributions of capacity to additional peak and off - peak generation. In the no storage scenario, the largest increase during the off - peak 48 Derivation details are given on Appendix 3 under the no u ncertainty assumption. 78 is valued at the low off - peak prices, while the small peak increase is valued at the high peak prices. With st orage, on the other hand, both generation increases a re valued at equilibrium prices: off - peak prices increase (Figure 8 ) and the total value of wind as well. Figure 8 . The equilibrium price with storage is a linear combination of off - peak and peak prices of the no storage scenario. With storage the equilibrium price is a weighted average of the prices of the no storage scenario Notice that the previous comparison between storage and no stora ge scenarios is developed considering a tax on emissions. Nevertheless, the same results and insights will apply to the value of wind when both scenarios do not tax emissions. Th is insights from equation 7 are based o n the assumption of negative correlation between demand and windpower: high peak demand with low wind and viceversa for the off - peak. If demand and wind are positively correlated, then storage could decrease the value of wind. 79 The marginal value of stor age If we constrain the amount of power that can be transferred between periods, the shadow value of the constraint will capture the value of a marginal increase in storage capacity: 49 The value of storage lies in the price differential between off - peak and peak prices discounting for power losses when reallocating energy. The larger this difference, the higher the ma rginal value of storage is. The marginal value of storage has a downward slope: as c apacity increases, arbitrage grows, which reduces the gap between off - peak and peak prices, and the value of storage. If we extend the model to compare the value of storage with and without emissions taxes we find: The value of storage in the scenario with an emissions tax is larger than its counterpart if adding the tax does not increase off - peak prices in a larger amount than the difference in marginal damages of off - peak and peak emissions. Using the linear functional forms we can show that , as long as demand is not too elastic (b>1), the value of storage under an emissions tax is always larger (equation 10) . 50 An emissions tax increases the value of storage since the amount of power 49 Derivation details are given on Appendix 4 50 An inelastic demand (b>1) is a plausible assumption for the case of Texas and most power grids in the US since most customers do not yet face real time prices. 80 arbitraged is a decision variable in the model and by incorporating all social costs, storers can allocate energy accordingly. Unlike the marginal value of wind power, the marginal value of storage capacity does depend on its allocation decisions. Storers can actively decide when to charge and sell power back to the grid, given differences in peak and off - peak p rices determined by demand and wind intermittency. The marginal value of wind and storage under large renewable energy penetration levels Understanding the marginal value of wind under large scale adoption is key to assessing the challenges of a grid powered mostly by intermittent renewables. U sing equation 5, which describes the value of wind capacity , we can state that equilibrium prices determine the value of additional renewable capacity. As more zero marginal cost wind enters the grid, there will be a lower marginal cost fossil generator setting the price. Hence, the value of wind power decreases with a larger share of capacity (equation 11). 81 Using the linear functional forms we can show that t he value of wind power is decreasing with respect to capacity 51 as long as demand is not too elastic (b>1), a condition that is met in most electricity markets without real time pri cing. H aving a large wind capacity built brings significant renewable generation to the grid, which substitutes the most expensive fossil power plants and sets prices at a lower value than a counterfactual without wind ( Figure 9 ). The derived marginal value of wind power determines the economic marginal benefits delivered by its capacity. Then, the equilibrium wind capacity is set at the level where its value equates its full cost (technically called the levelized cost of electri city LCOE ). Figure 9 . The value of wind power for increasing capacity levels This reduction in the value of wind, for increasing capacity levels, can be partially compensated by storage. However, it cannot be rever s ed since s torage only increases t he value of wind by raising off - peak prices up to a certain level. As more renewable cap acity serves the grid, both off - peak and peak prices have a downward trend and so does the value of wind power. On 51 See Appendix 2 for detai ls. 82 the other hand, an increase in wind capacity leads to an increase in intermittency with a larger gap between its off - peak and peak generation which implies a wider gap between both prices and a higher value of storage. Hence, while the value of wind decreases with its capacity and intermittency , the value of storage capacity increases. Data Electricity dema nd and generation costs are para meterized with publicly available information for 2015 on hourly load, generation, fuel use , and prices from the US EIA and ERCOT (EIA, 2017, ERCOT, 2016a). H ourly CO 2 , NOx, and SO 2 emissions come from the Continuous Emissions Monitoring System of US EPA ( EPA, 2017) and hourly wind power output from ERCOT (ERCOT, 2016b) . Storage efficiency parameters are based on existing storage projects and the literature ( Byrne et al., 2018, De Sisternes et al., 2016; Lamadrid et al., 2015) and its initial capacity is based on the 324 MWh ERCOT projection for 2020. Electricity demand, generation , and wind power in ERCOT From 2011 to 2015 there has been a significant increase in wind capacity and generation in ERCO T. In fact, wind is the major source of renewable ener gy in Texas and grew from supplying 8% to 12% of total load as its capacity increased from to 9,380 MW to 16,170 MW. It is worth noting that wind capacity generates po wer unevenly throughout the day, peaking late at night and with its lowest generation at midday ( Figure 9 ). 83 This is captured by capacity factors, which represent the ratio of power generated to maximum capacity. 52 Since each new wind turbine generates more power at night, the large increase in capacity has led to a widening gap between peak and off - peak generation as discussed in the theoretical model. The difference between the average generation at midnight and midday increased from 1,301 to 2,133 MWh between 2011 and 2015. Hence, the hourly intermittency of wind power grows with capacity ( Figure 10 ). On the other hand, load or electricity demand increased but it shows basically the same intermittency, the gap between peak (at 5 pm) and the lowest demand ( at 4 am ) is almost the same in 2011 and 2015 (14,653 vs 14,219 MWh respectively). Whi le this gap has not widened, it is almost ten times larger than the gap between the highest and lowest wind generation. In fact, if we dep ict the hourly wind and demand o n the same scale, the intermittency of wind is dwarfed by that of demand . It is worth noting that in Texas wind generation and dem and have a negative correlation, l oad peaks in the afternoon when wind power is low and wind generation peaks at midnight when load is low (Graph 6 b). 52 For example, a 5 MW wind turbine during the midday when wind is low might only generate 1.25 MWh, which implies a capacity factor of 0.27. 84 Figure 10 . Load and wind average daily variations and intermittency in Texas. a) Average hourly wind generation from 2011 to 2015 b) Average load and wind generation in 2015 *For 2012 - 2015, generation is given by the cumulative area Source: ERCOT, 2016 a and 2016b . MWh 85 T o handle the increasing intermittency of wind power and the large difference between peak and off - peak demand, ERCOT plans to install 324 MWh of electricity storage by 2020 (ERCOT, 2016c). This s torage will help smoothing wind power cycles but mostly differences in demand, which explain most of the price gap between peak and off - peak hours . As wind capacity grows in the coming years, its intermittency and its influence on arbitrage decisions will also grow. On the other hand, demand w ill also grow but its patterns will not chang e much unless an effective real - time pricing scheme gets implemented and even then load need not be flat. Therefore, new wind capacity will be constantly affecting prices and leading to a re - optimization of arbi trage and storage decisions. Natural gas is the main power source in ERCOT. Its share grew from 40% in 2011 to 48% in 2015 due to lower prices remains the second power source. Nuclear maintained at 12%. Except for natural gas replacing some coal generation, the fuel mix basically kept the same structure. The very low levels of electricity imports and hydropower generation make ERCOT a good case study of the effects of wind power on fossil generation (Cullen, 20 13; Novan, 2015). Furthermore, it is also a good case study for simulating the economics of storage, arbitrage and its interactions with emissions taxes. 86 Empirical Model I implement a short term dynamic model of the wholesale electricity market in ERCOT in which a Social Planner maximizes welfare by choosing fossil generation and storage (equation 11). The model implements the key characteristics and assumptions of its theoretical counterpart: given the daily intermittency and recurring wind power cycles it assumes a one day horizon with hourly time steps t , and no discount . Electricity demand is represented with a linear functional form and private generation and social costs are captured wit h an exponential function. Social costs include damages from carbon dioxide, sulfur dioxide , and nitrogen oxide emissions. Wind power is modelled as a daily cyclic process that implement s the key feature of having larger generation during off - peak hours. Since the model has a daily horizon I assume that the P lanner knows the generation profile for the entire day. N is a constant capturing the average hourly base nuclear generation of 4,500 MWh in 2015. The goal of this empirical model is to assess the marginal value of wind generation and storage, the (dis)c harging actions and welfare for four scenarios combining storage a vailability and emissions taxes: (1) no storage no taxes (baseline), (2) no storage and taxes, (3) storage and 87 no taxes, and (4) storage and taxes. T he model solves for the optimal daily fos sil generation and storage allocations for given win d power and storage capacities. Also, I compute the marginal value of wind generation and storage by marginally increasing their capacities and rerunning the simulations . These economic values can be comp ared to the levelized costs to draw conclusions on optimal adoption levels. Notice that while the theoretical model derived all insights using wind capacity, this installed capacity leads to realized generation, which is what gives utility to end consumers . Therefore, I report results in terms of the marginal value of wind generation (USD/MWh) but the same economics insights and intuition derived in the theory using capacity apply. 53 I calibrate demand and costs using hourly data of 2015. I implement a Mo nte Carlo simulation with 1000 random draws of 24 - hour - wind capacity factor profiles and demand parameters taken from all 365 days during 2015. 54 Thus, the model computes the average value of wind generation and storage, for four scenarios combining storage a vailability and emissions taxes, for the grid configuration and costs of ERCOT in 2015. I solve the optimization using the Kuhn - Tucker method via discrete non - linear programming. It is worth noting that the model estimates u pper bounds on welfare and the marginal values of wind and storage since it assumes no ramp - up, startup and transmissions costs. Furthermore, it also does not consider long term investments in fossil and nuclear capacity, and it assumes a limited role for demand response, very inelastic demand, since there is barely any electricity real time pricing in ERCOT. 53 The literature also reports results in terms of wind generation (USD/MWh). 54 Random sampling with replacement . 88 Demand and fossil generation cost calibration Average electricity demand and generation costs are parameterized with publicly available information on hourly load, generation, fuel use , and prices for 2015 from the US EIA 923 Form, the and ERCOT. Demand is calibrated with a linear functional form and an elasticity of - 0.0 3 which is a low estimate taken from the literature in order to represent the low response of electricity demand to wholesale prices (Cullen and Reynolds, 2016) . I compute the demand parameters for each hour t of each day d of the month m using average load and prices to solve the two - linear - equation system comprised of the demand function and the elasticity equation . Hence, there are 1994 different intercepts and slopes . I use this variation along actual wind capacity factors for the Monte Carlo Simulation random draws of daily parameters . Using hourly fossil generation, heat input data , and fuel costs for 136 plants, I build the hourly marginal generation cost (MC) of the wholesale market electricity dispatch (EIA, 2017; EPA, 2017; ERCOT, 2017a and 2017b). I also build the hourly wholesale marginal social cost (MSC) using EPA - CEMs plant - level information on emissions, the social cost of carbon (IAWG, 2015) and the marginal damages of smo kes tack SO 2 and NOx emissions (Muller and Men delsohn, 2009). For the social cost of carbon (SCC) , I use values of 40 USD/tCO 2 , 55 which is the conventional estimate used in the literature (Cullen, 2013; Novan, 2015), and 12 USD/tCO 2 , which carbon price in 2015 (Climate Policy Initiative, 2017). 55 Average value for a discount rate of 3% measured in 2015 USD. 89 The approach consists in finding the average heat intensity (mmBTU/MWh) for each hour in 2015 using the EPA CEMs data. Then, I compute the hourly average generation cost for each plant by multiplyi ng this heat intensity by the average fuels costs in 2015 from EIA (2017). I order the plant s and their dispatches using this average generation cost to estimate the wholesale marginal cost curve (MC) for each hour . The functional form for the wholesale ma rket marginal cost that yields the best adjustment is exponential (R 2 >0.98), and it captures the inelasticity of supply at peak demand ( Figure 11 ). I calculate the wholesale marginal social cost (MSC) using a similar procedure but adding total damages due to CO 2 , SO 2 average generation cost and then reor dering the dispatch based on this social merit ( Figure 11 ). This approach incorporates Pigouvian taxes directly on that the use of tax revenue does not cause any allocation or efficiency distortions. Therefore, the tax revenue do es not enter welfare computations. 90 Figure 11 . Private and Social Marginal Generation Costs in ERCOT I calibrate marginal private and social costs of electricity for each hour of the day since the same power plant infrastructure and machinery serve demand throughout the week and the year . The empirical model and the marginal cost calibration assume no dynamic frictions due to start - up costs, no transmission losses and constraints , and ramp - up constraints. F urthermore, when simulating non - marginal increases in wind capacity, the model as sumes that the same 2015 fossil and nuclear capacity levels will be available. Nevertheless, supporting large wind capacities might require investing in more flexible natural gas units whose generation cost is not captured. Thus , the model is likely to underestimate some of the integrati on costs of supporting wind power. 91 Emissions functions Using the ordering from the marginal private and social costs I compute the emissions functions for both cases by regressing the total market emissions on their corresponding generation quantities. I model aggregate emissions using a fifth order polynomial for the case of CO 2 emissions and sixth order polynomials for NOx and SO 2 (Figure 12 ). 56 Thus, I have two emissions functions for each pollutant in each hour, one corresponding to the ordering of the plants based on the private marginal cost and another corresponding to ordering of the plants based on the social marginal cost. Figure 12 . CO2 emissions function for a market dispatch based on the private MC in ERCOT 56 I chose the polynomial functions that maximize the R 2 of each regression. 92 Wind power and storage parameters I use efficiency and cost parameters from the literature (Byrne et al., 2018, De Sisterne s et al., 2016 ) . The s torage round - trip efficiency 57 parameter is 90%, the battery lifetime is 10 years , and levelized cost of storage ranges from 420 to 950 USD/kWh ( Lazard, 2016; Tesla, 2017). I implement Monte Carlo simulations by randomly drawing 24 - hour - wind capacity factor profiles from all 365 days during 2015. To simulate the value of wind under different integration levels (non marginal increases in wind capacity ) I assume the optimistic case that the hour ly capacity factors will be kept at larger wind capacities. Hence, I compute wind generation profiles by multiplying projected capacity by these hourly factors. I compute the value of wind for capacity levels that generate wind power that represents 12% to 30% of load in ERCOT, on average. Policy scenarios and Monte Carlo simulations I implement the model for four scenarios combining storage a vailability and emissions taxes: no storage no taxes (baseline), no storage and taxes , storage and no taxes , and storage and taxes . I compute average allocations, welfare and emissions (CO2, NOx, and SO2) using Monte Carlo simulations (1000 draws) for each scenario. T he algorithm basically consists o f the following steps : 1) Draw 24 - hour - wind capacity factor profiles and demand parameters ( ) from all 365 days of 2015 . Multiply by the total fixed capacity to obtain wind generation for each hour t 57 Total efficiency that includes power losses due to charge and discharge actions. 93 2) Solve the maximization problem with the Discre te Non - Linear Programming Solver (DNLP) from GAMs and find the optimal fossil generation and storage for each hour ( t ) of the i th draw . F or the scenarios simulating an unconstrained storage we solve (13) without further constraints. 58 In the case of constrained storage we add constrain ts for each hour. 3) Compute the average and stan dard deviations of hourly optimal fossil gen eration , storage actions , emissions , and daily welfare . I n the case of the model with MSC I use the value of the maximized objective directly. For the model with marginal private costs, the optimization renders the surplus S * and I compute welfare by subtracting the damage of the externality , which is the total social cost minus the total private cost : 58 The unconstrained storage co nsists on the vector of allocations hi that maximize equation 13 without any constraints on the amount of power withdrawn from the grid. These values represent an unconstrained storage that does not consider any investment costs. Rather, it represents the maximum amount of power that should be arbitraged to increase the short term daily welfare of equation 13. 94 I n order to assess the marginal value of increasing wind generation and its associated displacement of emissions and fossil generation reductions, I run a new Monte Carlo s imulation using the projected increase in wind power generation with an additional 1% of capacity (100 MW) 59 and I compare the results against the scenario that used the baseline wind generation . This procedure is the same for all scenarios. The marginal value of wind power , in each scenario, is calculated using the dif ference in the average daily welfare of the increased wind and baseline simulations (Table 7 ). Notice that the reported marginal value is that of generation (USD/MWh) Table 13 . The marginal value of increasing wind power and storage under different emissions taxes and storage scenarios No tax With tax Value of wind Value of wind with storage Value of storage I compute the marginal value of wind for increasing capacity levels from the current 12% (percentage of wind serving load) up to 30% using two percent intervals . The results show the willingness to pay for wind power at different integration levels. Finally, I compute the value of wind power for t he planned 324 MWh and the unconstrained storage levels. 59 This delta increase of 100 MW renders stable numerical solutions for the value of wind in the scenario with taxes and its difference with respect to the value of wind using an increment of 1 MW in the no tax scenarios is minimal. 95 To compute the value of storage I use intervals of 500 MW h of capacity starting from 100 MW h until 11,000 MW h . 60 At each interval a delta increment of 10 MW h to the battery constraint allows simulating the marginal increase in value. I run Monte Carlo simulations to find the expected marginal value of storage for fixed levels of wind integration (12, 20 and 30%). The results allow me to trace a storage demand curve which can be compared against current and projected costs to determine the optimal investment in capacity. Robustness checks Accounting for regulating reserves costs D ealing with wind power intermittent generation requires paying a larger amount of fossil generators to be ready to supply power in case of forecast errors. These are called regulating reserves and they smooth out fluctuations in load and renewable output. Thus, a larger wind capacity will increase regulating reserves costs (Gowrisankaran et al., 201 6 ; Ela et al., 2011). I account for these increasing costs using a formula from a National Renewable Energy Laboratory (NREL) study on Operating Reserves and Variable generation , which states that regulating reserves each hour are set using a 1% of load plus the stochasticity of wind captured by its forecast error standard deviation times its squared generation (Ela et al., 2011). The marginal cost of reserves is a portion of the marginal generation cost since a fraction of the generator is set to be under reserve. Based on ERCOT data, Gowrisankaran et al. (2016 ) calculated this ratio of 60 Following the engineering literature on storage and wind generation, I refer to wind capacity in power units (MW) whereas I refer to storage capacity in energy units (MW h). 96 margi nal reserve to marginal generation costs to be . The hourly regulating reserves costs formula is: I do not account for contingency reserves costs since it is reasonable to assume that new wi nd capacity will not affect the sudden failure of the other generators. While regulati ng reserves costs of equation ( 15 ) make the programming of equation ( 13 ) more complex to be solved the same insights and intuition of the theoretical model and of the simpler empirical model are still valid to interpret the results on the value of wind power and storage. Results The marginal value of wind generation For the 2015 wind capacity levels, serving 12% of load, t he average estimate of the marginal value of its generation is around 60 USD/MWh. As derived in the theoretical model, th is marginal value is downward sloping since as more zero marginal cost generation enters the grid, there will be a lower marginal cost fossil generator setting the equilibrium price. Hence, as wind capacity increases, in absolute value and as a share of load, the marginal value of its generat ion decreases. This reduction has almost a unit elasticity since an 18% increase in the wi % loss in its value (Figure 13 ). This finding is in line with previous research (Hirth, 2013; Baker et al., 2013). 97 The results corroborate that wind power becomes more valuable with unconstrained storage since its generation patterns are negatively c orrelated with demand (Figure 10b and 14 ). However, this increase is not large : appro ximately 0.86 USD /MWh between the average estimates of the unconstrained and no storage scenarios with taxes for a wind capacity of 12% of the load . 61 The moderate increase in the value of wind from storage occurs since i n the unconstrained case, off - peak prices rise, making the very early morning (1 - 5) increase in generation more valuable but onpeak prices decrease and the late evening generation increase losses value (21 - 23). The overall effect is a low gain in the value of wind power since equilibrium prices are driven by demand rather than by increases in wind generation from new capacity ( Figure 13 ). With low levels of storage, such as the planned 324 MWh, the difference in the value of wind with and without it is negligible. 62 61 See Appendix 5a 62 See Appendix 5. 98 Figure 13 . The marginal value of wind generation under increasing capacity a) Value of wind and surplus gains without taxes b) Value of wind power with taxes * The band depicts a 95% confidence interval, while the solid lines represent averages. Carbon offsets are valued at 40 USD/tCO2 99 As wind adoption increases, its intermittency has a larger effect on arbitrage decisions, compared to the stagnant demand patterns. Thus, equilibrium prices will have a stronger correlation with the marginal generation gains from new wind capacity . However , this leads to only a small increase in the contribution of storage to the value of wind power, which cannot overcome the declining value trend . While for a wind capacity of 12% of the load , the increase from unconstrained storage was 0.86 USD/MWh, for a 30% capacity , the increase is 1.13 USD/MWh. Figure 14 . Average eq. prices and marginal generation from 1 MW of new capacity. Unlike the theoretical model, prices with unconstrained storage are not exactly equal due to the battery round - trip inefficiency imputed in a discrete form in the programming of equation 13. For the conventional 3% discount rate carbon price (40 USD/tCO 2 ), the economic value of wind breaks even with its levelized cost (60 USD/MWh) in all scenarios at the 2015 capacity levels (IRENA, 2016; UT, 2016) . N evertheless, without emissions taxes, wind producers would not capture the environmental benefits since there is no pricing mechanism for the externalities 100 . Rather they would only receive payments associated with the wholesale price of electricity based on fo ssil generation costs (Figure 13 ). As derived in the theory, t hese private surplus gains are lower than the full economic value of wind. In fact , they are also less than half the levelized cost (27.4 and 26.3 USD/MWh, with and without storage respectively) and they would reflect the price paid to wind producers at the wholesale market if no emissions are taxed. Thus, pricing externalities is vital to send the correct signals or incentives to spur the development of wind power. Current average investment costs for wind power in North America and Texas are 60 USD/MWh and they are projected to decline to 44.4 USD/MWh by 2025 (IRENA, 2016; UT, 2016). On the other hand, the value of wind power does not decline as much for increasing capacity levels up to a 30% (average estimates being larger than 50 USD/MWh in all scenarios). Thus, with stable fossil fuel prices and a carbon price of 40 USD/tCO 2 , wind capacit y could serve a 30% of the load in ERCOT. T he marginal value of wind power depends primarily on the benefits of avoided fossil generation and on the value of emissions offsets. Therefore, estimates are sensitive to assumptions on the social cost of carbon . With a social cost in line with the 2015 California carbon prices (12 USD/tCO 2 ), estimates are lower than the average levelized cost (LCOE) of wind of 60 USD/MWh . For the 40 USD/tCO 2 carbon price used in the literature, and for higher values, the value of wind covers its levelized cost . 63 My estimates of the marginal value of wind are larger than those from 63 The SCC of 40 USD/tCO2 is measured in 2015 dollars. 101 previous works since they add the benefits of avoided fossil generation, emissions offsets, a nd arbitrage, while other studies only assessed the environmental benefits ( Figure 15 ). Figure 15 . Value of wind power (no storage) for different policy scenarios . Values computed for 2015 wind integration levels (12% of Load). The estimates from Novan (2015) are based on 10,000 MW and those from Cullen (2013) on the High benefit scenario. All prices are deflacted to 2015 levels. The marginal value of storage The previous section assessed the value of wind under different policy and storage scenarios, but the value of storage itself also depends on those levers. The value of storage depends on the arbitrage opportunities created by the difference between peak a nd off - peak prices. Hence, as storage capacity increases, the volume of power traded between hours increases and the gap between prices decreases. This leads to the usual downward sloping demand for storage curve. As the theoretical model and Figure 16 ill ustrate , the full economic value of storage is larger under the emissions pricing scenario since the amount of power arbitraged is based on all social costs, 102 and energy is allocated accordingly . Similar to the case of wind, t he gains in private surplus from storage are much less than its economic value (Figure 16 ). 64 This highlights the significance of assessing and accounting for the environmental benefits of arbitrage, which leads to the substitution of dirtier peak gas turbi nes for the cleaner off - peak steam turbines. Figure 16 . Storage economic value and gains in surplus for the 12% wind baseline *Solid lines are media n values and the bands are for the 25 th and 75 th quartiles. As wind capacity increases so does its hourly intermittency and the gap between equilibrium peak and off - peak prices widens. 65 Since storers can actively decide when to 64 As defined in the empirical framework this considers consumer surplus and producer surplus without discounting for emissions damages 65 This gap disappears with the unconstrained storage level due to the no - arbitrage optimality condition. 103 charge and sell power back to the grid, they can take advantage of this wider gap ( Figure 17 b ). Thus, the value of storage increases with wind capacity (Figure 17 a ). Notice that unlike the value of wind power, the value of storage does depend on its allocation decisions and it can adjust to the exogenous random intermittency of wind. Storage creat es a positive feedback to the extent that increasing wind capacity raises its value and by allocating power responding to the increasing generation intermittency of wind , it also increases the value of wind but to a lower extent. For the planned 324 MWh of storage in ERCOT, the median value is 39.05 USD/kWh in the emissions taxes scenario under 2 015 wind power levels (Figure 17 ) . This value is based on hourly intermittency smoothing and it falls short of covering its corresponding levelized cost, whose range is 420 - 950 USD/kWh (Lazard, 2016 ) . The estimated value of storage is likely to be large r if we include its short - term intermittency smoothing (1 min, 5 min) and frequency regulation services. 104 Figure 17 . Storage economic value under increasing wind capacity a) The increasing value of stor age b) Avg electricity prices no storage and tax scenario *Solid lines are median values and the bands are for a 25 th and 75 th quartiles. 105 Welfare and allocations Implementing storage and emissions taxes jointly leads to the largest welfare increase compared to the current baseline of neit her since e nergy is allocated when it is most socially valuable. Electricity storage and emissions taxes are complements since accounting for the external and intertempora l costs leaves no room for welfare losses that could be exacerbated by competitive storers (Table 13) . Given the large marginal and total benefits of reducing emissions, both scenarios that include a tax ( and ) yield significant offsets compared to the baseline irrespective of the social cost of carbon. Table 14 . Average changes in daily welfare and emissions from implementing storage and taxes with respect to the baseline v0 * For a social cost of carbon of 40 USD/tCO 2 . O n the other hand, i mplementing limited storage (324 MWh) without environmental taxes, leads , on average, to an in crease in all emissions since power is moved from early morning to the afternoon, arbitraging the cleaner steam turbine gas plants for the dirtier peak gas units . This finding is in line with Carson and Novan (2013) who estima te marginal emissions based on 106 econometric modelling and use the results along a two - period model of the economics and dynamics of power generation, storage, and emissio ns to compute substitution between power sources and the associated impact on emissions. However, my research develops more detail than the previously cited work by simulating all 24 hours of the day and calibrating hourly emissions functions. Figure s 18a and 18 b illustrate how arbitrage leads to combined cycle gas plants substituting for the dirtier peak gas units. Without emissions taxes, 265.6 MWh of power are charged at 5 AM , on average . This electricity would be mainly supplied by combined cycle gas , the marginal generator, and it would replace peak gas turbines at 4 PM . Electricity prices are higher in the afternoon due to a larger demand and increasing fuel cost s of supplying electricity. Hence, there are gains from reallocating power between hours b ut an increase in emissions as a byproduct. T axing emissions leads to reduced fossil generation and a socially optimal allocation of storage. Thus, as described in the theoretical model, a smaller amount of power will be taken from the grid at 5 AM, once marginal damages are taken into account (Figure 19 ) . Notice that the storage and tax scenario has a different dispatch order than the no tax scenarios since the marginal cost curve is ordered using generation costs plus marginal damages . 66 The early morning charging serve s the peak load during the afternoon (4 PM) and take s advantage of the relative lower prices and marginal CO 2 emissions. For the 324 MW h of storage, the battery has an average utilization rate of 48% with and without emissions taxes. 67 Also, half of the time it is filled with less than 50% of its total capacity. 66 See Figure 24 in Appendix 5. 67 I obtain the utilization rate by dividing the current state of charge in hour t by the total maximum capacity of the ba ttery. 107 Figure 18 . Private marginal cost and emissions function at 5 AM and 4 PM a) Marginal cost function (Wholesale merit order) b) Emissions function * Average Net Load corresponds to demand minus electricity supplied from nuclear and wind power Gas turbines Combined cycle gas Steam turbine gas Coal steam turbine Gas turbines Combined cycle gas Steam turbine gas Coal steam turbine Average Net Load at 5 AM Average Net Load at 5 AM Average Net Load at 4 PM Average Net Load at 4 PM 108 Figure 19 . Optimal average (dis)charging actions for a 324 MWh storage. * The band depicts a 95% confidence interval, while the solid lines represent averages. Finally, implementing the unconstrained storage and equating prices throughout the day requires arbitraging , on average, almost 28 % of daily load , with emissions taxes, and 25 %, without taxes. Ignoring the externality leads to excessive power arbitrage and emissions. As wind capacity grows, the total unconstrained arbitrage increase s : under 30% wind, the unconstrained storage arbitrages 48 % of the daily load. 68 In the unconstrained storage scenarios , the battery has a n average utilization rate of 51% and 52% for the scenarios with and without emissions taxes respectively . The low utilization rates of the 32 4 MWh and the uncon strained storage capacities 68 For details see Appendix 5. 109 occur since hourly arbitrage mostly charges during several hours in the morning to release pow er in the afternoon and evening . Emissions offsets Increasing wind capacity lead s to decreasing CO 2 , NOx and SO 2 emissions offsets in the scenarios with no tax on the externalities and to increasing SO 2 offsets when pollution is taxed (Figure 20 ) . Without a tax, the merit order based on the private marginal cost sets gas turbines as the marginal generators during peak hours. Incoming wind generation replaces these plants and offsets their emissions. As wind capacity increases, the cleaner combined cycle generators become marginal and the additional wind power that replaces them offsets less emissions than with lower capacities . 69 On the other hand, with environmental taxes, dispatch is based on the social marginal cost, and the merit order of pow er plants changes. Thus, gas turbines are still the marginal generators for current wind levels but coal plants are on the margin for large adoption levels (30%). 70 Hence, as wind generation increases, it replaces coal, delivering increasing sulfur emission s offsets. In spite of these increasing external benefits, the total economic value of wind is decreasing in the the storage and tax scenario since fossil generation costs, which drive equilibrium prices, decline at a faster rate. These previous results ho ld with low levels of storage and without storage since limited arbitrage does not move enough generation to alter average marginal emissions. 69 See Figure 15 above and Figure 21 and 23 in Appendix 5. 70 Combined cycle plants would act as base power in this grid. As argued in the empirical model section, this assumption has its limitations given the start - up co sts and ramp - up constraints of coal, see graph on Appendix 9. 110 With the unconstrained storage level and no taxes, CO 2 , NOx and SO 2 wind emissions offsets also decrease but the environmental benefits are initially larger than those in the no storage scenarios. 71 This occurs since by charging during several off - peak hours, instead of one or two as in the constrained case, a larger portion of the peak gas turbines generation is replaced by cleaner steam turbines. 72 Figure 20 . Average emissions offsets of wind generation (no storage) *Solid lines are median values and the bands are for a 25 th and 75 th quartiles. 71 The ideal storage and tax scenario also delivers increasing sulfur emissions offsets but at a larger rate than the scenario w ith a tax but no storage. See Appendix 10. 72 See Figure 25 in Appen dix 5. 111 Robustness checks Accounting for regulating reserves costs Once I account for the additional regulating reserves costs due to wind power intermittency, I find that the average value of wind power with the unconstrained storage and taxes is 60.44 USD/MWh for the curre nt 12% share. This value is similar to the the main specification and the levelized cost of wind. The marginal value of wind still has a downward but less steep slope since equilibrium prices decline at lower rate when accountin g for additional reserves co sts. 73 The value of storage decreases in all scenarios compared to those of the main specification. significantly the gains in value from a larger gap between off - peak and peak prices. 74 It is worth noting that extrapolating the regulating reserves formula from the NREL study (Ela et al., 2011) to compute aggregate reserves costs at the wholesale level could lead to overestimation of these expenses. To sum up , th e main insights on the marginal value of wind and storage (and their complementarity) from the theoretical model and the main empirical specification still hold valid in the reserves scenarios . Conclusions Large scale adoption of wind and solar power is one of the key policies for a cleaner grid that improves local air quality and mitigates climate change. As renewable energy supplies a larger share of power and its intermittency brings new integration challenges, a significant storage capacity will be re quired to smooth its cycles and supply energy when it is most valuable. 73 See Figure 26 in Appendix 5. 74 See Figure 26 in Appendix 5. 112 Nevertheless, while the transition to a 100% renewable grid takes place, pricing the emissions externalities is necessary to avoid inefficiencies in the power supply mix, excessive car bon and pollution. Ideally , battery storage should be jointly implemented with emissions taxes in order to align all incentives tow ards a sustainable transition and signal entrant storers the socially optimal (dis)charging times (Figure 16 ). Furthermore, if storage cannot reach a large scale adoption, wind continues to supply a larger share of power and no emissions taxes are implemented, power arbitrage can actually lead to an increase in carbon and sulfur dioxide emissions by substituting of f - peak coal for peak natural gas generation. This is likely to occur in a significant scale when wind capacity supplies 40% of the total load. However, if the power mix changes due to further natural gas price reductions or the development of more flexible generators then coal generation might become minimal and an emissions increase less of a concern. In order to create renewable energy finance schemes that recognize the full economic value of wind power and facilitate covering its levelized cost of energy ( LCOE), we need a carbon pricing scheme which can deliver at least stable prices around 40 USD/tCO 2 . Current schemes in North America, such as in California, Alberta and Quebec, and China have carbon prices lower than 18 USD/tCO 2 while some initiatives in Scandinavia and Sweden were well above 40 USD/tCO 2 (WB, 2016). Nevertheless, China and the USA have to do most of the heavy lifting for transitioning to a clean power grid, which makes the case for higher carbon prices in the curr ent schemes. 113 This research results show that when wind serves a 30% of the load, its average value declines to a bit less than 50 USD/MWh . 75 In an ideal case, carbon prices should rise at the rate of the discount rate as the Hoteling Model requires for a limited atmospheric sink. This rise could compensate the decline in the value of wind capacity. In reality, having a stable and medium carbon price seems to be enough of a challenge but it can secure investment for considerable wind capacity since its LCOE is also declining. Current average investment costs for wind power in North America and Texas are 60 USD/MWh and they are projected to decline to 44.4 USD/MWh by 2025 (IRENA, 2016; UT, 2016), which is lower than its simulated value. Therefore, with a ca rbon price of at least USD 40 USD/tCO 2 and under current technology trends, wind power can reach a 30% generation share in ERCOT. It is worth noting that the computed wind values are an upper bound since they assume no ramp - up, start - up , or transmissions c osts. Furthermore, these technical constraints plus transmission constraints make it The estimated storage value is lower than its investment cost even for the 30% wind scenario . While storage value increases with a larger wind capacity and intermittency, this boost is not enough to cover current costs. However, as explained in the empirical model section, one ittency smoothing. The estimated value of storage is likely to be larger if we include its short term intermittency smoothing (1 min, 5 min) and frequency regulation services. Furthermore, the carbon tax revenue could be used to set up not only renewable e nergy support schemes but storage schemes as well. 75 Also under a carbon price of 40 USD/ tCO 2 114 A PPENDICES 115 APPENDIX 1. Solving the model for the optimal storage equation The Social Planner (or decentralized outcome) are the same when emissions externalities are considered and assuming no other market imperfections. Assuming no initial storage, the off - peak equation of motion is , which implies . Assuming no scrap value or final value to any power left in the battery after the peak implies . Adding the fossil generation non negative constraint the Lagrangian becomes: Deriving the above with respect to fossil generation we get: Optimal fossil fuel condition: Which holds with equality for an interior solution. I explore corner solutions regarding in later paragraphs. D eriving the lagrangian with respect to we get: No arbitrage condition: S olving both equilibrium conditions leads to : 116 The fossil fuel condition leads to Plugging this back into the no arbitrage equation gives: Now for the case of corner solutions. In the above setup and derivation there are no constraints for charging and discharging , we assume an unlimited storage. Charging could also be zero, this (considering the discount for battery efficiency at the peak ) and no charging/discharging i s necessary or optimal. On the other hand, fossil generation is zero only in the case of a corner solution. Then the optimal fossil generation condition s become: And the no arbitrage condition remains the same : 117 W ith a corner solution for fossil generation charging/discharging arbitrages only differences in electricity demand and wind generation. It does not arbitrage differences in emissions intensity since there is no fossil generation and emissions at all. 118 APPENDIX 2. Derivation of the value of wind power Optimal storage and fossil allocation render t he maximized objective function : The value of wind power is the increase in the optimized value function for a marginal increase in its capacity And using the fossil fuel optimality condition s we get: Using the no arbitrage condition in the case of unconstrained storage (prices equalized ) we get: In the scenario without storage but with emissions taxes we get Now for t he scenario wit h neither storage nor emissions taxes we have : 119 And for the scenario with storage but no taxes Notice that the since the addition of renewable capacity displaces fossil generation the value of emissions offsets terms are positive and increase the marginal value of wind. The second derivative of the value of wind power shows whether the marginal benefits of adding capacity are homogenous or vary: And using the linear model optimal solutions and parameters (all second derivatives are zero from linearity in the functional forms) we get: when b>1 as in the ca se of an inelastic Demand for power . 120 APPENDIX 3. Derivation of the value of wind power with and without storage From the previous section we know t he value of wind with storage : And its value without storage is : 121 APPENDIX 4. Derivation of the value of storage We can use the same objective function set up with a constraint on the charging action, which is a constraint on storage itself. Hence, the v alue of increasing storage capacity is : To show that the value of storage is larger with a tax on emissions consider the following notation: and the superscript 4 represents the tax scenario and number 3 the no tax scenario : Solving the optimal storage and fossil allocation problem with constrained storage we get : 122 And then solve for the difference between peak prices: w here 123 And since b>1 for an inelastic demand we get: 124 APPENDIX 5. Model results additional tables and Figure s Table 15 Marginal value of wind estimates for different scenarios 125 Table 16 . Average changes in daily welfare and emissions from implementing storage and taxes with respect to the baseline v0 * For a social cost of carbon of 40 USD/tCO 2 . Notice that the average welfare increase of the scenario with unconstrained storage and no tax is lower than the welfare increase from the scenario with no storage and tax. 126 Figure 21 . Emissions function at 5AM and average net load with 40% wind * Average Net Load corresponds to demand minus electricity supplied from nuclear and wind power 127 Figure 22 . Unconstrained storage charging and discharging and state of charge a) Unconstrained storage charging and discharging (12% wind) b) State of charge (battery capacity) * The band depicts a 95% confidence interval, while the solid lines represent averages. 128 Figure 23 . Emissions function for dispatch based on private MC under 12% and 30% wind during peak hours a) Emissions function 6 PM and avg. net load wind 12% b) Emissions function 9 PM and avg. net load wind 30 % * Average Net Load corresponds to demand minus electricity supplied from nuclear and wind power 129 Figure 24 . Social merit order marginal cost curve at 5 AM 130 Figure 25 . Average emissions offsets of wind generation with unconstrained storag e *Solid lines are median values and the bands are for a 25 th and 75 th quartiles. 131 Figure 26 . Wind power and storage marginal values in scenarios that include regulating reserves costs a) Value of wind power with environmental taxes b) Value of storage * The band depicts a 50% confidence interval, while the solid lines represent averages. 132 CHAPTER 4 : A WEEKLY HORIZON MODEL OF THE VALUE OF RENEWABLE ENERGY AND ELECTRICITY STORAGE Introduction In this chapter , I value wind power and storage by extend ing the theoretical and empirical daily model of the previous chapter to a weekly horizon model that captures weekend and weekday differences in load. First, I study the main insights of the weekly power allocation pr oblem using a two - day four - period analytical model . The key features of this model are a lar ger weekday th an weekend demand ( for both peak and off - peak periods) and recurring daily wind cycles. As in Chapter 3 , this model represents private generation costs (fuel costs) and social costs (fuel costs and emissions damages). I show that the optimal storage involv es an arbitrage of intraday differences in peak vs off - peak demand, wind generation , and emissions intensities. Furthermore, for a large storage capacity the optimal allocation involves an arbitrage of interday differences in demand. Given the larger gap b etween weekday peak and weekend off - peak demand, there is a larger price gap and a higher value of storage in the weekly model than in its daily counterpart . Second, I simulate the weekly problem by extending the programming that computes hourly fossil generation and storage allocations that maximize welfare in the Social Planner Problem in the daily model . I use the same calibration for the demand and wind parameters as in Chapter 3 to implement Monte Carlo simulations by randomly drawing weekly (168 ho urs) parameters from actual 2015 data. I assess the value of wind power and storage for policy scenarios 133 that combine emissions taxes and battery availability: no storage and no taxes (baseline), no storage and taxes, storage and no taxes, and storage and taxes. Wind and storage are also complements in the weekly model. The simulated value of wind power is very similar to that in Chapter 3 since wind follows repetitive daily generation cycles. On the other hand, the value of storage increases due to the la rger demand and price gaps between the weekend and weekdays. As in the daily model, accounting for emissions taxes in the power dispatch reduces the amount of electricity arbitraged and increases the value of storage. For the planned 324 MWh battery capaci ty, inter day arbitrage is not optimal and storage just reallocates power within the same day. Simulations results suggest that full interday arbitrage requires capacity levels larger than 11,000 MWh, which are unlikely to be installed in the short run due to the nascent stage of the industry. In the unconstrained storage scenarios interday arbitrage moves power from the weekend to the weekdays but utilization levels are low since half of the time the battery stores less than 50% of its capacity. Interday a rbitrage is also sensitive to the storage roundtrip efficiency. A decl ine of efficiency from 90% to 85 % reduces inter day arbitrage from all 7 days to just 3 days in the scenario with emissions taxes. Finally, even though the marginal value of storage can b e larger in the weekly model, for large capacity levels, it is still below its levelized cost. 76 Accounting for frequency regulation and minute level intermittency should increase the economic value of storage. 76 Which has a 90% roundtrip efficiency. 134 Theoretical Model Model setup and optimal storage I extend the setup of Chapter 3 to a two - day - model (four time periods) t hat illustrates the tradeoffs a planner faces when allocating power between hours (peak vs off - peak ) and days (weekdays vs the weekend). Electricity is supplied u sing fossil fuel generation , wind power and storage . There is a social benefit function of electricity consumption , its associated linear demand , , private marginal wholesale costs and the state of charge or stock of the battery . Wind power is a function of its installed capacity with daily cycles . There are two days, a weekend day followed by the weekday, each has two periods, off - peak followed by the peak. Hence we have four time periods in the following order: weekend off - peak , weekend peak , weekday off - peak , weekday peak . Formally, . In both days, d emand for electricity is larger during the peak period and peak weekday load is the largest one: . On the other hand, wind power has the same cycles every day with a larger generation during the off - peak . Emissions are linear in fossil fuels and their intensity is larger at the off - peak . Marginal social costs of emissions are represented by . In addition to capturing the daily intermittency of wind power, this model represents the weekly intermittency of de mand with larger load requirements during the weekdays compared to the weekend. Therefore, this longer planning horizon allows intra and interday storage. The social p lanner at the initial time by choosing the amount of fossil fuel generation and storage in each period: 135 The optimal allocation is determined by the fossil generation optimality condition (2) that equates the marginal benefits (or prices) of electricity consumption to the marginal social cost of fossil generation, and the arbitrage condition which equates the benefits of electricity consumption throughout periods (3): 136 Solving the system, the expression for the optimal storage is: 77 The optimal storage arbitrages differences in the weekend and weekday demands, wind power and emissions intensities. First, storage moves excess power from the off - peak to the peak within the same day (intraday arbitrage). The larger the difference between the periods, the more power it will withdraw from the initial off - peak weekend period. Second, it stores some power to arbitrage the difference between peak and off - peak in the next weekday (inter day arbitrage). Third, it smoothes wind power generation by arbitraging the difference between intraday periods. Fourth, it arbitrages the gap between peak and off - peak emissions intensities. This final term has a different sign than the previous ones since a larger off - peak emissions intensity reduces the incentives to transfer power between periods. Therefore, emissions taxes signal socially optimal charging actions and discourage moving an inefficient amount of emissions from the dirtier off - peak generators into the peak. Wind pow er and emissions arbitraging have no inter day 77 Solving for the interior solution and assuming no roundtrip efficiency losses for simplicity of deri vation. Details on this and on corner solutions are on Appendix 1. > 0 < 0 < 0 < 0 137 arbitrage given the assumptions of daily recurring wind patterns and using the same fossil g eneration infrastructure for producing power in each period . 78 The optimal storage in the two - day - model is larger than in its daily counterpart in Chapter 3 since it arbitrages the same intraday wind power and emissions intensities differences but it als o considers interday differences. Hence, it will store more power during the initial low demand off - peak weekend period to serve the larger and more valuable demand during the weekday. On the other hand, the addition of marginal wind power capacity has the same effect in both models. It will increase the optimal storage since adding capacity intensifies the gap between peak and off - peak generation given the intermittent wind patterns . The marginal value of wind The derivative of the optimized function (welfare) with respect to new wind capacity represents the marginal value of increasing wind power. 79 The expression in equation 5 shows that the value of wind comes from the expected increase in generation in each period valued at equilibriu m prices . The concept and intuition are the same as the daily model from the previous chapter. 78 The recurring daily wind generation assumption is relaxed in the empirical model simulation. 79 Details are on Appendix 2. 138 T he value of wind depends on the total marginal increases in generation during the two off - peak and peak periods and on roundtrip efficiency losses. As discussed in Chapter 3 , notice that the marginal wind generation from later periods is divided by the efficiency parameter to account for the losses from moving moving power to later hours an d days. The value of capacity in the two - day - model is larger than in the daily one, since it accounts for increased generation during more periods. The correct metric for comparison is the marginal value of wind generation which is basically obtained by accounting for the total amount of power added by the new capacity . I use this metric in the empirical model. The marginal value of storage I derive the value of storage by solving a constrained version of the two - day - model. The Lagrange multiplier on the storage constraint gives the increase in welfare from augmenting its capacity. 80 Assuming no roundtrip efficiency losses and that the difference between peak weekday and off - peak weekend demands is larger than the difference between any other time period s we get: In this case, t he optimal storage moves power from the first period to the last and the value of its capacity is the related price gap. Therefore, the larger the demand and price difference between the off - peak weekend and the peak weekeday, the more valuable electricity storage 80 Details are on Appendix 3. 139 becomes. In Chapter 3 , I argued that this gap incre ases with a larger wind capacity, due to the uneven increases in off - peak and peak generation. Then, the value of storage in this case also increases with a larger wind capacity. Storage is more valuable when it arbitrages the larger gap between weekend and weekday demands than when it only arbitrages power within the same day. Since having the largest difference between peak weekday and off - peak weekend demands implies that the largest price gap is also among those two periods, then arbitraging interday differences delivers a larger value of storage in the two - day - model than in the daily model. In Appendix 3, I show and discuss the value of storage for the other cases in which the gap between the first and the last period is not as large as the summation of the other differences. I find that the marginal value of storage is the sum of the discoun ted prices in all periods, but the initial, minus the initial period price and the discounted equation of motion constraints. The main intuition of this value is the same: storage is valuable since it moves power from the low price initial period to the ot her higher price three periods. 140 Data Electricity dema nd and generation costs are para meterized with publicly available information for 2015 on hourly load, generation, fuel use and prices, from the US EIA and ERCOT (EIA, 2017, ERCOT, 2016a). H ourl y CO 2 , NOx, and SO 2 emissions come from the Continuous Emissions Monitoring System of US EPA (EPA, 2017) and hourly wind power output from ERCOT (ERCOT, 2016b) . Storage efficiency parameters are based on existing storage projects and the literature (Byrne et al., 2018, De Sisternes et al., 2016; Lamadrid et al., 2015) and its initial capacity is based on the 324 MWh ERCOT projection for 2020. Figure 27 depicts weekly electricity demand and w ind generation. On average, demand for power is lower during the weekends, notice the smaller peaks at the beginning of the figure (Saturday and Sunday). On the other hand, there are no obvious weekly patterns for wind, since it depends mostly on daily cycles while load depends on preferences and consumption patterns which clearly drive a larger demand during weekdays. Daily load and w ind power follow countercyclical patterns: wind peaks late at night when demand declines, and during the morning and afternoon wind declines and demand peaks. 141 Figure 27 . Average Weekly Electricity Demand and Wind Power in ERCOT Source: ERCOT, 2016 a and 2016b . *Hour 1 is Saturday 12H00 AM . Net load refers to load minus wind generation. As explored in the previous section , these weekly load and recurring daily wind patterns can drive storage to move excess generation from the weekend to the weekday s in order to capitalize on the larger gap between off - peak weekend and the peak weekeday prices. The next section describes the empirical model to simulate the electricity market with a weekly planning horizon. 142 Empirical Model Overview The empirical model implements the main characteristics and insights of its theoretical counterpart maximizing welfare by choosing fossil generation and storage fo r the week. Similar to Chapter 3 , there is a linear demand ( , expo nential total costs and the state of charge of the battery during each hour. There is an exogenous nuclear generation N and the roundtrip efficiency of the battery is again . Finally, the planning horizon st arts Saturday at 12H00 AM to take advantage of having the usual wind generation but a lower weekend demand for power, which should lead to storing the excess power to serve a larger weekday demand. Weekly Demand and wind power calibration I use the same calibration methodology for the demand parameters as in Chapter 3 but instead of computing parameters for each hour and each day in 2015, I calculate intercepts and slopes of the linear demand function for each hour of the average week of 2015. The marginal private and social costs of electricity (MC) are the same expon ential functional forms calibrated in Chapter 3 since the same power plant infrastructure and machinery serve demand throughout the week and the year . 143 The model uses average hourly wind power ge neration for each hour of the week based on the 2015 data. I use hourly wind power forecast errors to generate the random draws of the Monte Carlo Simulations. Therefore each wind generation realization i for hour t becomes . The hourly forecast errors are drawn from uniform distributions whose upper and lower bounds are based on the actual maximum and minimum errors registered for each hour in the ERCOT data. The strong assumption in the Monte Carlo Simulations is that at the beginning of the week the Planner knows the wind projection for all seven days and there is no learning in between. The realization of previous days does not affect the forecast of subsequent days. This assumption requires wind forecasts to be accurate enough and capture already all dynamics and dependencies so that t heir errors are just random. Similar to Chapter 3 , I assume the optimistic case that the hourly capacity factors will be kept constant. Hence, I compute wind generation profiles by multiplying projected capacity by these hourly factors. I compute the value of wind for capacity levels that generate wind power that represents 12% to 30% of load in ERCOT, on average. Policy scenarios and Monte Carlo simulations I assess the value of wind power and electricity storage in fou r scenarios using the same algo rithm of Chapter 3 for a 1000 Monte Carlo Simulations. The only difference is that instead of drawing 24 - hour wind capacity factors I draw 168 - hour wind forecast errors to compute the 168 - hour wind generation profiles . The four scenarios comb ine storage a vailability and emissions taxes: no storage no taxes (baseline), no storage and taxes , storage and no taxes , and storage and taxes . I compute the marginal value of wind power using the projected 144 increase in wind generation for the entire average week (168 hours) with an additional one percent of capacity (100 MW). 81 Similar to Chapter 3 , t he marginal value of wind is the difference in the expected welfare from the increased wind relative to the baseline simulations ( see Table 7 ). I compute the marginal value of wind for increasing capacity levels from the current 12% (percentage of wind serving load) up to 30% using two percent intervals . These results show the marginal benefit of wind power at different integration levels. Finally, I compute the value of wind power for the planned 324 MWh and the unconstrained storage levels. I calculate the value of storage at intervals of 500 MW h of capacity starting from 100 MW h until 11,000 MW h . At each interval , I add an incr ement of 10 MW h to the battery capacity to calculate the marginal increase in value. I run Monte Carlo simulations to find the expected marginal value of storage for fixed levels of wind integration (12 % , 20 % and 30%). This process delivers a storage demand curve which can be compared against current and projected costs to determine the optimal investment in capacity. Results The marginal value of wind generation As in the case of the daily model, t he marginal value of w ind generation is downward sloping since as more zero marginal cost generation enters the grid, there will be a lower marginal cost fossil generator setting the equilibrium price. This weekly model also shows the positive 81 This delta increase of 100 MW ren ders stable numerical solutions for the value of wind in the scenario with taxes and its difference with respect to the value of wind using an increment of 1 MW in the no tax scenarios is minimal. 145 feedback between wind and storage: for the uncon strained storage levels, night prices increase as a result of arbitrage, which raises the value of the marginal peak wind generation at night and the overall value of wind. Both d aily and weekly models have very similar values of wind powe r in each of the four scenarios (Figure 28 ). This occurs since the weekly model captures the changing patterns of demand throughout the day but wind follows the same natural daily cycles. The weekly model mechanically produces a larger total welfare contribution of wind (7 days) . However, when it is expressed in terms of USD/MWh, the marginal value of wind is almost the same as in Chapter 3 . The weekly model also shows that the private surplus gains to wind producers are half the levelized cost. Therefore, without emissions taxes or other instrument that can internalize the environmental externa lities and signal correct price and incentives, wind developers would receive low incentives leading to underdeveloped capacity . 146 Figure 28 . The value of wind generation under increasing capacity for different scenarios a) Value of wind power with emissions taxes b) Value of wind power without emissions taxes * The band depicts a 95% confidence interval, while the solid lines represent averages. 147 The marginal value of storage The marginal value of storage is larger in the weekly model than in its daily counterpart for capacities larger than 11,000 MW that can take exploit the larger gap between weekend off - peak and weekday peak demand and prices. The simulations confirm the insights of the theoretical model. Thus, the demand for storage when it arbitrages both interday and intraday price differences is a right shift of the intraday o nly arbitrage curve ( Figure 29 ). Similarly to the daily model, the value of storage is larger with emissions taxes since the amount of power arbitraged is based on all social costs. Figure 29 . The value of storage for the 12% wind and emissions taxes scenario *Solid lines are median value s and the bands are for the 25 th and 75 th quartiles. 148 For the planned 324 MWh of storage, the median value in the weekly model ( 37.51 USD/kWh) is similar to its daily counterpart ( 39.05 USD/kWh) since only intraday arbitrage is optimal fo r low capacity levels (Figure 32 ). The median value in the weekly model is also lower than the levelized cost in all scenario s . However, this value does not capture the frequency regulation and minute - le vel intermittency smoothing capabilities of storage. Storage allocations The unconstrained storage basically uses excess wind power on the weekend , relative to a lower demand, to attenuate peak demand throughout weekdays. It also arbitrages intraday differences in peak vs off - peak wind generation and demand. Each of the waves in Figure 3 1 represents the intraday cycle: recharging during the first hours of the day to serve the peaking demand in the evening/night. This patte rn is clearly marked for the weekdays but less pronounced during the weekends. The smallest power discharges occur Saturday and Sunday evenings/nights while Monday and Thursday evening/nights show the largest withdrawals since the two latter days have the largest demand of all weekdays. 149 Figure 30 . Unconstrained State of charge (battery capacity) for the 12% wind scenario *Solid lines are me di a n values and the bands represent a 95% confidence interval. As derived in the theoretical model, accounting for the social damages via a tax on emissions leads to lower levels of arbitrage at both intra and interday levels (Figure 31 ). 82 The largest state of charge with unconstrained storage has a median value of 96.1 GWh in the no emissions tax scenario. However, since this stock is built - up during approximately two and a half weekend days and then discharged throughout the rest of the week , it has an average utilization rate of 46% and 49 % for the scenarios with and without taxes respectively . I n fact, half of the 82 For detailed Graphs with the charging and discharging ac tions see Appendix 4. 150 hours show a capacity utilization level lower than 50%. Sunday, Monday and Tuesday have the best rates (>80%) and Saturday and Friday the lowest (<35%). 83 While the optimal capacity level is determined by the value of storage derived in the previous section and its levelized cost, it is key to recognize that if costs come down and batteries reach a significant integration into the grid, there will be a subst antial spare capacity during half of the hours and days of the week. Capacity will be used for arbitrage but overall capacity will be driven by a desire to arbitrage electricity from Sunday to Wednesday. Figure 31 . State of charge for the planned 324 MWh capacity in the 12% wind scenario *Solid lines are median values and the bands represent a 95% confidence interval. 83 See Appendix 5. A 50% capacity utilization level means that out of the 96.1 GWh battery capacity only 48.05 GWh are filled at a particular hour. 151 Since the planned 324 MWh storage capacity is a low level relative to the demand differences throughout the week, the optimal charging/discharging patterns only move power within the same day. Thus, the weekly profile is basically a repetition of daily patterns simil ar to those derived in Chapter 3 : charging during the early hours and discharging in the evening. With low storage capacities, interday arbitrage is not likely to occur. Interday arbitrage is optimal only for storage capacities larger than 11,000 MWh. R esults are sensitive to battery round - trip efficiency. The main specification considers the 90% e fficiency of the literature. R esults for 85% show significant decreases in total battery capacity and interday arbitrage, in the unconstrained storage scenario . W hen dispat ch incorporates emissions taxes, interday arbitrage only occurs during 3 days of the week (Sunday - Tuesday) (Figure 32 ). 152 Figure 32 . State of charge for 12% wind scenario and 85% round - trip efficiency *Solid lines are median values and the bands represent a 95% confidence interval. Conclusions In this final chapter I extended the daily model and explored the implications of a weekly counterpart whose key feature is the difference between weekday and weekend loads while wind power has the same daily cycles. The theoretical and empirical model show that the value of wind power is basically the same in both cases since a larger planning horizon does not alter the recurring daily wind patterns. On th e other hand, the value of storage is larger in the weekly model since it captures the larger gap between weekend off - peak and weekday peak (especially Monday and Thursdays) demand s and prices. 153 Howeve r, for interday arbitrage to have any benefit in the short run, storage capacity must be larger than 11,000 MWh. The very large capacities implemented by the unconstrained storage scenarios lead to low utilizations levels: half of the time the battery is filled with less than 50% of its capacity and it only fills up when moving power from Sunday to Monday. The 324 MWh storage capacity planned by ERCOT would only move power within the same day. Even in the weekly model, the value of storage based only on hourly intermittency is less than its levelized cost. Th erefore, at the nascent stage of the industry , intraday arbitrage will very likely dominate the landscape. Finally, storage capacity and (dis)charging are very sensitive to roundtrip efficiency. The optimal arbitrage with unconstrained storage is reduce d substantially when efficiency is lower than 90%. There is no arbitrage at all with an 80% roundtrip efficiency in the scenario with emissions taxes since power losses nullify all incentives from peak and off - peak differences in marginal benefits . 154 A PPENDICES 155 APPENDIX 1. Solving the model for the optimal storage equation The Social Planner solves: And , assuming zero initial reservoir and no final or scrap value , the stock equations become : Which leads to the Lagrangian: Optimizing with respect to fossil generation and charging/discharging leads to the following equilibrium conditions : 156 Optimal fossil fuel condition: where No arbitrage condition s : For an interior solution f rom the optimal fossil fuel equation we get : where Substituting the no arbitrage condition s until equating the first and the last periods : Assume no round - trip losses to simplify the expression: Using the other no arbitrage equations and assuming no round - trip losses, we get: 157 Which leads to: (1) - (3)= (3)*2bc - (2)= Using the result from 4: 158 159 Similar to Chapter 3 , in the previous model there are no corner solutions to charging/discharging since storage capacity has no constraints. For a fossil generation corner solution without any output in any period, we will get a similar expression to the one derived above but since there is no fossil power, it will not contain the emissions intensity arbitrage term. The corner solutions with zero output during only certain periods will also yield similar expressions but with a modified emissions intensity arbitrage term. 160 APPENDIX 2. Derivation of the value of wind power Deriving with respect to wind capacity and u sing the optimality condition for fossil fuel allocation we get: Using the no arbitrage equations and the equations of motion from the previous section we get: 161 APPENDIX 3. Derivation of the value of storage and and the equations of motion of the state charge Assuming no initial stock and no final value, the equation of motion of the state of charge or battery is: Assume no round - trip losses and that the difference between demand in the first and last periods is much larger than in any other periods we get : 162 Value of increasing storage capacity: Where since For the other cases we get: Plugging in the equations of motion and d eriving with respect to we get: The above expression computes the value of storage as the discounted value summation of all the constraints on its capacity which equals the difference between the sum of the discounted prices in all period s, but the initial, minus the initial period price and the discounted equation of motion constraints . Notice that the above formula is applicable to interior solutions as well as to corner solutions combining any periods, just three or any two. Thus, the v alue of storage will be given by a modified expression based on the above. 163 APPENDIX 4. Unconstrained storage charging /discharging actions Figure 33 . Unconstrained storage charging /discharging actions 164 APPENDIX 5. Utilization rate of the battery in the unconstrained storage scenario (12% wind) Figure 34 . Utilization rate of the battery in the unconstrained storage scenario (12% wind) CHAPTER 5. CONCLUSIONS 1 65 Throughout this dissertation I have argued and shown evidence that the value of wind and solar power depend on having storage that smooth s their intermittency. Chapter 2 showed how reservoir hydropower is already acting as storage in California by shifting water generation from the hours with the largest solar and wind output to those with the lowest. This hydro reallocation helps to integrate solar and wind into the grid by handling their intermittency. As a result of the reallocation, the carbon offsets and the economic value of solar power increases relative to those of wind . Chapter 3 simulates the operation of utility scale batteries in Texas to show that when demand and wind are negatively correlated, storage increases the value of wind power by ra ising prices during hours of high wind generation and low demand. Furthermore, storage helps to manage and arbitrage differences in demand, wind generation and emissions intensities of fossil fuel plants. Internalizing the environmental benefits of wind and solar t hrough emissions pricing is key for guiding the power generation and investment decisions of generators , and the (re) design of incentives by policy makers . In Chapter 2 , I argued that CO 2 , SO 2 , and NOx emissions offsets benefits of solar and wi nd power are larger than their Renewable Portfolio Standard and Investment Tax Credit incentives. In Chapter 3 , I showed evidence that without pricing emissions, wind generators would only capture a half of the actual economic value of wind power and a hal f of its levelized cost. Furthermore, private storers would cause an increase in all emissions by arbitraging price differences that do not reflect the marginal social cost. 166 This research has also shed light on the v alue of storage. While Chapter 2 shows with historical data how hydr o accommodates wind and solar, Chapter 3 explicitly quantifies the value of power arbitrage. The value of storage increases for larger wind capacities since the intermittency of wind raises the benefits of arbitrage . Thi s value also increases with a tax on emissions since storers can respond to prices reflecting all social costs and allocate energy efficiently. Accounting only for hourly intermittency smoothing, I find that the value of storage is lower than its current levelized cost. The marginal value of wind generation is downward - sloping since as more zero marginal cost generation enters the grid, there will be a lower margin al cost fossil generator setting the equilibrium price. However, the larger projected declines in its levelized cost by 2025 suggest that with stable fossil fuel prices and a carbon price of 40 USD/tCO 2 , wind capacit y could serve a 30% of the load in ERCOT . cannot revert the downward slope of the marginal value of wind generation. Hence, wind power and storage need sustained cost reductions in the coming year s, for wind to serve more than 4 0% of load. Finally, in Chapter 4 , I showed that accounting for a weekly rather than daily planning horizon slightly increases the value of storage for capacity levels larger than 11,000 MWh. In both, weekly an daily horizon models, the average battery u tilization level s are lower than 50% . 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