Forecast value considering energy pricing in California

FORECAST VALUE CONSIDERING ENERGY PRICING IN CALIFORNIA Jennifer Luoma Patrick Mathiesen Jan Kleissl University of California, San Diego 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

1.

INTRODUCTION

Forecasting of weather conditions such as global horizontal irradiance (GHI), wind speed, direction, and expected power production is essential for the efficient integration of renewable power into the energy portfolio. For solar, there are two prominent groups that require forecasts: solar power generators and system operators. Solar power generators are entities that own production facilities and profit from the sale of energy. Independent System Operators (ISOs), however, are primarily concerned with grid reliability and require energy forecasts to plan the dispatch of energy, determine spinning reserve requirements, and calculate energy procurement needs. As these two entities have varied interests, there is a mismatch between their individual definitions of the best possible forecast. For instance, it is possible for a power generator to benefit economically from using biased (rather than accurate) forecasts. This is detrimental to the systems operator‟s goal of reliability of the power grid [1]. As such, system operators cannot rely solely on market commitment information and generally contract with third party forecasting services to provide independent energy production forecasts. For the system operator, forecast accuracy is the accepted standard for evaluating forecast effectiveness. Also known as quality [2], accuracy is the deviation between a forecast and a co-located observation. ISOs prefer high-accuracy forecasts as these are most useful for grid reliability purposes. Solar power generators, however, rely on forecast value. Forecast value is the direct monetary benefit of the forecast [2]. As the price of energy is not fixed, forecast quality does not necessarily directly translate to forecast value. For example, a forecast that is of high quality during peak net load times of the day (when energy prices are high and errors are more costly) may be more valuable than a forecast with higher quality only at less critical times. For ISOs, forecast value is a secondary consideration and used to minimize energy generation, transmission, and reserve costs on the grid. For example, an underforecast (predicting too little energy production) would result in overprocurement of energy and possibly result in transmission congestion near the solar power plant. Alternatively, an overforecast (predicting more energy than is ultimately produced) would result in under-procurement of energy and a purchase of energy from reserves or regulation at additional cost. In California, the California Independent Service Operator (CAISO) allows solar power producers to participate in the Participating Intermittent Resource Program (PIRP). PIRP requires day- and hour-ahead forecasts in addition to historical observations of energy production and local meteorological data (global horizontal irradiance (GHI), direct normal irradiance (DNI), temperature, and wind) [3]. On the net monthly forecast deviation, „uninstructed imbalance energy‟ charges are assessed. These monthly settlements can be positive or negative (reimbursement). However, no penalties for individual dayor hour-ahead forecast errors are applied. As such, there is currently no incentive for generators to provide accurate daily forecasts. However, utility-scale solar power plants will eventually participate in the energy market following the same bidding and settlement rules as conventional power sources (as is already the case for Red Electrica in Spain), including wind power (in most energy markets). Previous studies have investigated the integration of renewable energy (predominately wind power) into energy markets. Using Midwest ISO (MISO) pricing data, Botterud et al. (2011) found that optimal day-ahead wind energy bids are primarily driven by price expectations (rather than forecast energy production) when no deviation penalty is applied. However, adding a deviation penalty diminished the difference between the optimal wind energy bid and the most accurate wind energy forecast [4]. In the Spanish energy market, Fabbri et al. (2005) estimated the costs associated with wind energy forecast errors by assuming that forecast deviations are balanced in real-time by reserve energy and found that error prediction costs can be as much as 10% of the total income annually from the energy generation [5]. Bathurst et al. [6], found that forecast revenue is dependent on the difference between the contract (DAM) price and the imbalance (RTM) price for the UK energy market. There, a Markov-Probabilistic forecast was used to determine the most profitable energy commitment. Pinson et al. [7] devised an optimum bidding strategy using forecast uncertainty to dramatically increase revenue. Additionally, Mills and

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Wiser [8] found that in a 30% solar photovoltaic (PV) penetration scenario day-ahead forecast error and ancillary services costs were $3.0/MWh and decreased capacity and energy value of PV by 11% (23% for CSP). From the ISO perspective, the integration of solar energy results in additional cost due to increased uncertainty and variability. Porter et al. (2013) quoted a Public Service Company of Colorado (PSCo) study as directed by the Colorado Public Utilities Commision (CPUC) in which integration costs ranged from $1.25/MWh to $6.06/MWh [9]. There, each additional 100 MW of installed solar resulted in an increased cost of $1/MWh. To reduce these costs, energy forecasts are relied upon. Milligan et al. (1995) found that the most accurate forecast provides the lowest cost of operations when integrating wind power. However, improving a forecast to 100% accuracy has declining marginal benefits [10]. Using price data from the New York ISO (NYISO), Ruiz et al. (2009) showed that a stochastic-based forecast system reduced the cost of operations by up to 2% over a deterministic-based forecast when optimized to the constraints of the power system [11]. In California, assuming 7.5 GW of wind generation, a GE Energy study found that using existing wind forecasting technology saved $68 million/year in operation costs [12]. Furthermore, the Western Wind and Solar Integration Study (WWSIS, Lew et al., 2010) found that using state-of-the-art day-ahead wind and solar forecasts in the unit commitment process would reduce Western Electricity Coordinating Council (WECC) operating costs by up to $5 billion/yr for 25% wind and 5% solar (by energy) penetrations. Using a perfect forecast would result in additional cost savings of $500 million/yr [13]. A complete review of forecast value studies for renewable energy applications can be found in Chapter 7 of Giebel et al., 2011 [14]. In this study, the value of a solar power forecast in the CAISO system is examined. This is particularly of interest as the CAISO market contained 47% of the nation‟s installed solar power as of 2010 [15]. Ideally, the value of a forecast would be evaluated from both a power generator‟s and system operator‟s perspective. However, modeling forecast value to a system operator is complex and requires knowledge of available resources, start-up and running costs, and the system operator unit commitment rules. These factors vary among system operators and, consequently, such a study would be ISO-specific. Large-scale studies such as WWSIS have already generalized these factors to investigate cost savings due to wind and solar forecasts. Rather, this paper will focus on market participants such as solar power generators (after discontinuation of PIRP). Using CAISO pricing data (Sec. 2.1), the price difference between the day-ahead market (DAM) and real-time market (RTM) will be examined (Sec. 3.1). Next, using numerical weather prediction (NWP) solar forecasts (Sec. 2.2), forecast value (Sec. 2.3) will be determined for 63 sites in California and geographic data trends explained. Previous studies have not calculated a monetary forecast value in this manner. Furthermore, the effect of a hypothetical deviation penalty on forecast value will examined (Sec. 3.2). Finally, solar forecasts will be corrected using a simple model output statistics (MOS) bias-correction function. The value of corrected forecasts will be compared to uncorrected. Overall, it will be shown that when no deviation penalty is included, positively biased forecasts have the highest value to solar power generators. However, when a deviation penalty is included, the most accurate forecast yields the highest value.

2.

METHODS

2.1 Energy Price Data and Market Structure Energy price data was obtained from the CAISO Open Access Same-time Information System (OASIS). OASIS has over 4,500 nodes at which a Locational Marginal Price (LMP) is reported. These nodes represent locations in the CAISO power grid where energy can be bought or sold into the market. The LMP is the sum of three components: energy, loss, and congestion. The energy component represents the average price of generating a MWh of electricity and by convention is the same at all price nodes. The loss component represents the cost of transmission losses associated with the delivery of electricity to that price node. Lastly, the congestion component monetizes the transmission constraints in delivering electricity to a price node. All LMP components are reported for the day-ahead (DA), hour-ahead (HA), and real-time (RT) markets. DA forecasts must be submitted at 0530 (local time) on the day prior (Day 0 in Table 1) to the operating day. The operating day (Day 1 in Table 1) begins at midnight (0000) and extends 24 hours. DA forecasts are provided on an hourly basis for each of the 24 hours of the operating day (Table 1). Therefore, the DA forecast horizon is 18.5 to 42.5 hours. Similarly, the HA forecast is submitted 105 minutes prior to each operating hour. However, in this study, HA prices and forecasts are not used.. TABLE 1: DAY-AHEAD FORECAST SUBMISSION TIMELINE

Day 0

Day 1 2

Hour (Local) FH (Hours)

00 -

03 -

DA Forecast 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129

130 131 132 133 134 135 136

06 00

09 03

12 06

15 09

18 12

21 15

00 18

03 21

06 24

09 27

12 30

15 33

18 36

21 39

24 42

X

DAM LMP (the market price at which a DA forecast is committed) and RT market (RTM) LMP (the price at which settlements are made) from June 1, 2010 to May 31, 2011 were used in this study. Additionally, hour-beginning DAM LMP (e.g. the 0800 DAM LMP applies to 0800-0900) are averaged to correspond to instantaneous hourly GHI data (e.g. for 0800, the mean of the 0700 and 0800 DAM LMP is used). For the RTM, LMP are reported every five minutes and are valid through the next five minutes (e.g. the 0800 RTM LMP is used for 0800-0805). To determine hourly RTM LMP, prices were averaged between 30 min from the hour (e.g. for 0800, the mean of values from 0730 – 0825 RTM LMP were taken). In calculating revenue from energy sales, we assume that PIRP is discontinued and that solar photovoltaic (PV) plants participate in the wholesale energy market like other generating resources. Furthermore, it is assumed that the 2010/2011 LMP is representative of typical years and that the LMP will not change due to additional PV plants participating in the market. In reality, DAM prices would be reduced with increasing solar generation and the RTM price could fluctuate depending on systematic solar forecast biases and daily production values [16]. Since forecasts are submitted in the DAM and deviations settled in the RTM, the value of a forecast must be dependent on the difference between the DAM and RTM LMPs in addition to the magnitude of forecast error (Table 2). For example, if the RTM LMP is greater than the DAM LMP, energy is more expensive to procure in real-time. If an overforecast should occur there will be a large loss in revenue as additional units of energy will need to be purchased at the higher RTM price. Conversely, underforecasts are potentially profitable as excess energy produced can be sold at the high RTM price. However, energy is not guaranteed to sell in the RTM. In fact, a significant excess of energy in the RTM at a particular node is likely to drive the local RTM price down, negating any potential profit. For this reason, excess energy sold in the RTM will be considered only as a potential gain in revenue and RTM LMP will be set to zero when underforecasts occur. The case when the RTM price is less than the DAM price is also considered. Here, underforecasts result in a loss of revenue as the full amount of produced energy was not sold in the higher-priced DAM. In this situation, overforecasts produce higher revenues than accurate forecasts as energy can be procured in the RTM inexpensively. Table 2 summarizes the possible outcomes considering forecast error and the price difference between the RTM and DAM. TABLE 2: SUMMARY OF MARKET/FORECAST OUTCOMES DAM price – RTM price

Forecast Bias

< 0

Overforecast

> 0

Overforecast

Must purchase additional energy to cover the under delivery of DAcommitted energy, but at the lower RTM price and thus total revenue will be greater than if no forecast error occurred: gain of revenue

If RTM LMP < 0, then RTM LMP = 0

< 0

Underforecast

Potential to sell additional energy at higher RTM price: potential gain of revenue (only monetized when implementing a deviation penalty)

If RTM LMP > 0, then RTM LMP = 0

> 0

Underforecast

2.2

Outcome Must purchase additional energy at the higher RTM price: loss of revenue

Missed opportunity to sell additional energy at higher DAM price: loss of revenue

Restrictions Imposed If RTM LMP < 0, then RTM LMP = 0

None

Solar Energy Forecasts and Actual Generation

The California Irrigation Management Information System (CIMIS) is a statewide network of over 120 meteorological observation stations [17]. At each station, Li-Cor LI-200S pyranometers measure GHI to within 5% [18] each minute. However, GHI is reported as hourly-average values. In addition to the automatic CIMIS-provided quality controls [19]-[20],

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data exceeding historical limits, missing data, and stations with obvious shading were excluded in this study. Ultimately, 63 stations with complete and accurate records were chosen. All other sites and price nodes were omitted. In order to simulate solar energy production, it was assumed that a 1 MW PV plant was co-located at each CIMIS station. Power was calculated to be proportional with observed GHI (Eq. 1). In actuality, PV plant efficiency decreases with increasing panel temperature. This effect, known as the panel de-rate factor, was ignored for this study. As such, Equation 1 overestimates solar energy production when panel temperatures are high such as in the afternoon. Overall, this simplification is reasonably accurate and may affect estimated power production by up to 10%. Day-ahead solar irradiance forecasts are generated using the North American Mesoscale (NAM) numerical weather prediction model [21]-[22]. Published by the National Centers for Environmental Prediction (NCEP), the NAM predicts irradiance on a 12 km x 12 km grid and is available hourly for forecast horizons up to 36 h. In this study, only the 1200 UTC (0400 PST) NAM initialization time is used. Using nearest neighbor interpolation, hourly NAM irradiance forecasts were established at each CIMIS station and power calculated according to Equation 2. h

(1)

h

(2)

Additionally, to investigate the effect of forecast accuracy on forecast value, model-output-statistics (MOS) was used to biascorrect NAM forecasts. Previously, Mathiesen and Kleissl (2011) found that, over a year, NAM forecasts are positively biased by up to 150 W m-2 on average [23]. Futhermore, coastal locations exhibited the largest positive bias especially during summer months. This was attributed to issues in modeling the prevalent summer marine layer clouds. Here, MOS was employed to remove bias error as a function of forecast clear sky index (irradiance normalized by clear-sky GHI) and solar zenith angle. This correction was applied independently for each CIMIS station using a dynamic training set of 8 weeks of rolling, up-to-date data. Overall, for this dataset, bias-corrected NAM forecasts reduced forecast GHI MBE from 57.5 W m-2 to 7.0 W m-2. Additionally, the root mean square error (RMSE) was reduced from 134.2 W m -2 to 114.1 W m-2. 2.3 Revenue and Forecast Value To determine forecast value, the total yearly revenue from the sale of solar energy, R, is calculated. Generally, R is calculated by first selling the forecasted energy in the DAM and then settling (either selling excess energy or purchasing energy to meet shortfalls) in the RTM (Eq. 3). In some cases, LMP (especially in the RTM) may be negative. This generally occurs when there is an oversupply of energy or transmission difficulties near a single node. In these instances, energy suppliers are paid to curtail production. To ensure that negative LMP are not unfairly beneficial, energy is never allowed to be “bought” at a negative price in this study. However, energy is allowed to be procured at zero cost. Effectively, when an overforecast occurs, negative RTM LMP are set equal to zero (Table 2). Conversely, when underforecasts occur, there is no guarantee that energy can be sold in the RTM. To simulate this, RTM LMP that are greater than zero are set to zero in these situations (Table 2). To determine the value of forecasts, we assume that a non-revenue-biased bidding strategy is used; solar power providers bid at the forecast level exactly. ∑

(

)

-

(3)

Additionally, the effect of a deviation penalty, Pdev., is investigated in an alternative market structure. PDev. is defined as (

) |

| (4)

where DPF is the deviation penalty factor. There is no precedent for the magnitude of the DPF, but any DPF greater than one ensures that a the maximum possible revenue results from using an unbiased forecast. In this study, a factor of 1.5 is used to ensure that the deviation penalty is always 50% larger than the possible revenue gain of a biased forecast. A proper DPF ensures that the highest quality forecast is the most valuable forecast without excessively diminishing total revenue for the solar energy generator (i.e. if the DPF is too high solar energy generators will be harshly penalized for providing even near perfect forecasts). An optimal DPF was not investigated. When implementing a deviation penalty, excess energy is allowed

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to be sold in the RTM and the restrictions for underforecasts are lifted (Table 2). Subsequently, the total yearly revenue considering a deviation penalty becomes: ∑

(

)

-

-

(5)

Finally, revenue from a perfect forecast was calculated assuming that the delivered energy calculated using CIMIS data (Eq. 1) was bid exactly into the DAM and no RTM settlement took place. ∑

(6)

3. RESULTS AND DISCUSSION 3.1 Day-Ahead and Real-Time Market Price Trends Figure 1 shows the DAM and RTM LMPs averaged across all studied price nodes. Mean hourly DAM LMP range from $25 to $41 / MWh with highest prices occurring in the late afternoon during July through September. Price trends vary by time of year. Between November and May, the highest daily prices occur in the morning and prices decrease by approximately $10 throughout the day. In summer months (May – October), the trend is reversed; morning prices are low and increase by approximately $40 throughout the day, peaking in the late afternoon. Presumably, this is due to higher demand on the grid resulting from increased air conditioning load. In the RTM, mean-hourly LMP are similar ($21 to $34, Fig. 1e). Additionally, similar diurnal and yearly trends are observed with highest prices occurring in the afternoon during July to October. Overall, RTM prices are slightly less than DAM prices but are more volatile. Geographically, DAM and RTM prices vary by location. In general, the highest yearly averaged prices for the DAM occur near the coast. However, not all sites follow this trend and the difference from coastal to inland sites is small (less that $7/MWh). Additionally, locations with the highest yearly averaged RTM prices are dispersed across the state, indicating that geographical trends in prices are weak. Lastly, it is expected that year-to-year variations in pricing will occur. For this study, it is assumed that 2011 represents a typical year. As presented in Eq. 1 and Eq. 2, forecast revenue is largely dependent on the difference between the DAM and RTM LMP. For most months and times, the DAM and RTM prices differ by less than $5/MWh. Differences are largest in the afternoon between July and October, with RTM prices approximately $10-$15/MWh larger. Over the entire year, price differences are largest in morning and evening (Fig. 1f). Overall, however, DAM prices are greater than RTM prices by about $5/MWh. This price difference provides an incentive for overforecasting in the DAM as additional energy can likely be procured at relatively low RTM prices. Furthermore, overforecasts are particularly profitable in mornings between May and September as DAM prices tend to be much higher than RTM prices. Similarly, evenings in June through September could yield high profits for underforecasts because excess energy could be sold at high RTM prices. Since there is no guarantee the energy would be purchased in the RTM, profits from underforecasts are not allowed in this study (Table 2). 3.2

Comparison of Forecast Value

For each CIMIS station, revenue was calculated for a 1 MW power plant for NAM forecasts (Eq. 3), bias-corrected NAM forecasts, and perfect forecasts (Eq. 6). To determine forecast value relative to a perfect forecast, the ratio of forecast yearly revenue to perfect forecast yearly revenue was calculated. For the NAM forecast, this ratio is depicted in Figure 2a. On average, using NAM forecasts yield yearly revenues of 96% of perfect forecast revenue (Fig. 2b). Inland, NAM forecast revenue is as large as 98% of the perfect forecast revenue. In these areas, NAM forecasts are more accurate due to lower cloud occurrence. Figure 2b summarizes the distribution of forecast revenue for all sites using the standard and bias-corrected NAM forecasts under market structures with and without deviation penalties. When no deviation penalty was enforced, the standard NAM forecast was most profitable at 96% of perfect forecast revenue. When bias error was reduced, forecast revenue decreased to 93% of perfect (Sec. 3.4, later). Conversely, when a deviation penalty was included, the standard NAM forecast yielded revenue of 59% of a perfect forecast. After MOS bias-correction was applied, revenue increased to 76% (Fig. 2b). 3.3

Example for Coastal Sites

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When no deviation penalty is considered, bias-correction of NAM forecasts results in a large loss in value at coastal locations. Figure 3a shows the ratio of bias-corrected NAM forecast revenue to standard NAM forecast revenue. At the coast, this ratio is smallest. To illustrate why the bias-corrected NAM forecast is less valuable at coastal sites, the Revenue Improvement Factor (RIF) is defined as

-

. Positive RIF indicate that the bias-correction increases total

revenue when compared to the standard NAM forecast. Averaged by hour of day and month of year for all sites within 20 miles of the coast, Figure 3b shows that RIF is typically negative. Specifically, the bias-corrected NAM is less valuable than the standard NAM for all times of day from July to November. During this time, DAM prices are generally up to $5/MWh higher than RTM prices. As a result, overforecasts produce larger revenues than perfect forecasts as the cost of RTM energy procurement is less than the commitment price in the DAM. Since the standard NAM typically overforecasts irradiance at this time, it follows that the standard NAM would be more profitable than the relatively unbiased corrected NAM forecast. Moreover, this effect is amplified in the mornings of July, August, and September. At this time, NAM overforecasts are more frequent as the prevalent marine layer clouds are rarely predicted. In general, if the DAM price is higher than the RTM price, the most profitable bid for a market structure without deviation penalties would be to forecast energy output at maximum capacity. Since bias-corrected forecasts trend away from maximum capacity, revenues are generally lower. However, when deviation penalties are considered, forecast errors are de-incentivized, ensuring that bias-corrected forecast are more valuable. 3.4. Forecast Value Versus Error Metrics Figure 4 shows the RIF as a function of and ) for each site. Without a deviation penalty (Fig. 4a), forecast revenue improvement decreases with increasing MBE. However, decreasing RMSE has little effect on forecast revenue. Together, this indicates that positively biased, low RMSE forecasts are the most profitable. If RMSE could be reduced without removing the positive bias, forecast revenue could potentially be increased further. However, when a deviation penalty is added, allowing for excess energy to be sold in the RTM, forecast value increases as MBE and RMSE are improved (Fig. 4b). 4.

CONCLUSIONS

In this study, the value of a solar power forecast was estimated for 63 locations within the California energy market. Dayahead solar irradiance forecasts for one year were based on the NAM numerical weather prediction model. Using CAISO LMP prices, total forecast revenue was calculated for standard NAM forecasts and bias-corrected NAM forecasts under two simple market structures and compared to the theoretical revenue of a perfect forecast. Overall, the yearly revenue of the NAM numerical weather prediction model irradiance forecasts is always less than that of a perfect forecast. However, for some locations forecast revenue is as much as 98% of perfect forecast revenue. When model-output-statistics (MOS) was used to reduce forecast bias, a decrease in forecast value was observed. Since DAM prices are typically higher than RTM prices, positively biased NAM forecasts produce greater revenue than less-biased forecasts for all sites. This effect was exacerbated for coastal California, in which irradiance overforecasts are more common and DAM prices much greater than RTM. These simulations confirm findings [1] that biased forecasts can be more valuable to solar power generators. Since there currently is no penalty for providing an incorrect forecast, there is motivation to submit an incorrect, biased forecast that produces higher revenue. Subsequently, if forecasts from generators are not representative of the actual energy that is expected to be received, market inefficiencies are created by forcing system operators to procure additional reserves to balance the additional uncertainty in the forecast. System operators can respond by producing internal forecasts or procuring forecasts from independent 3rd parties. Market inefficiencies could also be mitigated through deviation penalties, which (by design) cause the value of an accurate forecast to increase (Sec. 3.4). In this way, deviation penalties can internalize the external costs of inaccurate forecasts to the owner / operator, but the penalty factor must be chosen carefully to be reasonable to forecast providers. We assume that solar power plants participate in the open market and that the LMP does not change due to participation of the additional solar power plants in the market. In reality, DAM prices would be reduced with increasing solar generation and the RTM price could be driven up or down if solar forecast trend to either over- or under-predict. Inclusion of the feedback of PV participation in the energy market is likely to promote accurate forecasts. Also, these results are specific to California

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with unique forecast biases and DAM and RTM price spreads. Regardless, our approach to determining forecast value allows for the analysis of impact of different energy market structures on the value of accurate forecasts to market participants.

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[9] Porter, K., Fink, S., Buckley, M., Rogers, J., and Hodge, B.-M. “A Review of Variable Generation Integration Charges.” National Renewable Energy Laboratory Technical Report, NREL/TP-5500-57583. Available online at http://www.nrel.gov/docs/fy13osti/57583.pdf. 2013.

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[10] Milligan M, Miller A, Chapman F. “Estimating the Economic Value of Wind Forecasting to Utilities.” NREL. Windpower. 1995.

5.

ACKNOWLEDGEMENTS

We appreciate support by Jim Blatchford, Jenny Pedersen, and Julianne Riessen (CAISO) for assisting in obtaining information about CAISO operations and market structure. 6.

REFERENCES

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Figure 1: Average DAM LMP (a), RTM LMP (b), and DAM LMP-RTM LMP (c) for June 1, 2010 – May 31, 2011 for all 63 price nodes by time of day (for hours with a non-zero solar forecast) versus month. Box plots for DAM LMP (d), RTM LMP (e), DAM LMP-RTM LMP (f) indicating mean, 25th and 75th percentiles, ends, and outliers (red crosses). For RTM LMP, outliers greater than 115 $/MWh (83) are not shown. For DAM LMP –RTM LMP, outliers less than -100 $/MWh (66) are not shown.

394 395 396

Figure 4: Revenue Improvement Factor (RIF, defined as

Figure 2: (a) Ratio of yearly revenue using NAM forecast (Eq. 3) over using a perfect forecast (Eq. 6). (b) Box plot showing distribution of site revenue performance based on yearly revenue ratio using a real forecast to a perfect forecast (Eq. 6); real forecasts consist of (from left to right) NAM („NAM‟, mean 0.96), corrected NAM („NAMcorr‟, mean 0.93) (Eq. 3), NAM with deviation penalty („NAM_Pen‟, mean 0.59) and corrected NAM with deviation penalty („NAMcorr_Pen‟, mean 0.76)(Eq. 5). Box plots indicate mean, 25th and 75th percenticles, ends, and outliers (red crosses). Figure 3: (a) Ratio of yearly revenue using corrected NAM forecasts to NAM forecasts (Eq. 3). (b) Revenue improvement factor [-] using corrected NAM to NAM plotted by time of day and month averaged for all sites within 20 miles of the coast. -

) as a function of change in MBE and RMSE

due to the forecast bias correction for each site without deviation penalties (a, Eq. 3) and with deviations penalties (b, Eq. 5

9