ElectricityEmissions.jl: A Framework for the Comparison of Carbon Intensity Signals

ElectricityEmissions.jl: A Framew ork for the Comparison of Carbon Intensity Signals Joseph Gorka, Noah Rhodes, and Line Roald {gorka, nrhodes, roald}@wisc.edu University of Wisconsin – Madison Madison, Wisconsin, USA ABSTRA CT An increasing number of individuals, companies and organizations are interested in computing and minimizing the carbon emissions associated with their real-time electricity consumption. T o achieve this, they require a carbon signal, i.e. a metric that denes the real- time carbon intensity of their electricity supply . Unfortunately , in a grid with multiple generation sources and multiple consumers, the physics of the system do not provide an unambiguous way to trace ele ctricity from source to sink. As a result, there are a multitude of proposed carbon signals, each of which has a distinct set of properties and method of calculation. It remains unclear which signal best quanties the carbon footprint of electricity . This paper seeks to inform the discussion about which carbon signal is better or more suitable for two important use cases, namely carbon-informed load shifting and carbon accounting. W e do this by developing a new software package ElectricityEmissions.jl, that computes several established and newly proposed carb on emission metrics for standard electric grid test cases. W e also demonstrate how the package can be used to investigate the ee cts of using these metrics to guide load shifting. Our results arm pre vious research, which show ed that the choice of carb on emission metric has signicant impact on shifting results and associated carbon emission reductions. In addition, we demonstrate the impact of load shifting on both the consumers that perform the shifting and consumers that do not. Disconcertingly , we observe that shifting according to common metrics such as average carb on emissions can reduce the amount of emissions allocated to the consumer doing the shifting, while increasing the total emissions of the power system. 1 IN TRODUCTION As the impacts of climate change become increasingly visible , many electricity consumers, from private persons to large corporations, have shown increasing interest in assessing and reducing their car- bon footprint. For many consumers, a signicant portion of their carbon footprint is tied to their electricity consumption and associ- ated emissions from electricity generation. The source of electricity generation (e .g. solar , wind, hydro, nuclear , natural gas or coal) as well as the temporal availability of rene wable power (i.e. daily and seasonal cycles of solar , wind and hydropower) vary widely between locations, leading to dierences in carbon intensity of electric power across time and space. This provides an opportunity for consumers to reduce carbon emissions by using power where and when low-carbon electricity is available. Howev er , to actively reduce the carbon emissions of their ele ctricity supply , consumers need real-time information ab out how carbon emissions of elec- tricity var y . Furthermore, they ne ed a framework for leveraging this information not only for real-time load shifting but also for long-term carbon accounting, such that they can get credit for their eorts to reduce emissions. This is particularly important for com- panies and industries that are subject to emerging carbon emission regulations and reporting r equirements, including e.g. producers of clean hydrogen [16]. Quantifying the carbon intensity of electricity consumption in a grid with multiple consumers and multiple sources of generation, is however not straightfor ward. While it is possible to obtain in- formation on emissions from the generators and assess how the total carbon footprint varies over time , the physics of the grid do not lend themselv es to a clear denition of the generation mix (and hence emissions intensity) that serves individual lo cations. This leaves signicant open questions regarding which carbon emissions is the best, see e.g. [ 32 , 34 ]. This paper develops a software package and an analysis framework to help evaluate the eectiveness of dierent carbon emission metrics. 1.1 Related work Existing research has focused on computing and analyzing dier- ent carbon emission metrics for real-time electricity consumption, e.g. in [ 3 , 6 , 7 , 11 , 21 , 35 ], spurring an interest in how consumers might use such emissions metrics to assess and minimize the carb on impact of their electricity consumption. Previous studies hav e ex- amined the potential to reduce ele ctricity emissions in the context of data-centers [ 19 , 21 , 30 , 38 ], production of hydrogen fuel [ 33 , 37 ], and residential consumption [ 13 , 24 ]. The increasing availability of real-world carbon emissions data, provided both directly by grid operators [ 25 , 29 ] and by third party organizations [ 11 , 35 ], has en- abled real-world implementations of load shifting strategies guided by emission metrics[ 15 , 23 , 26 , 30 ]. For example, battery-powered devices such as phones have settings that enable “clean energy charging” [ 15 ], while customers in UK can install light bulbs that turn green when the electricity is green [26]. Generally , existing works have tended to focus on a single carbon metric–e.g. to analyze its properties or use it as input to a theoretical or realized emissions-reduction scheme. Recently , howev er , there has also been increased interest in comparing dierent metrics for the purposes of carbon accounting and informing load-shifting applications [ 10 , 20 , 22 , 31 ]. T wo main shifting metrics that are currently in use . ‘ A verage Carb on Emissions’ (A CE), dened as total system emissions divided by total system load, is the carbon metric used by the Greenhouse Gas Protocol for so-called location-based carbon accounting [ 12 ]. ‘Locational Marginal Carbon Emissions’ (LMCE), dened as the change in emissions due to a small change in load, has been investigated in a numb er of r esearch studies and is available from at least one grid operator . A variety of works have examined the eectiveness of A CE and LMCE for informing load- shifting applications [ 20 , 22 , 34 ]. In [ 20 ], a synthetic grid case study is use d to demonstrate that the emissions impact of data-center load shifting base d on either ACE or LMCE (as well as alternate data such as price and renewable curtailment) has a signicant impact on shifting outcomes. In a recent paper , the authors of [ 34 ] compare r eal-world marginal and average carbon emissions signals from 65 regions throughout the world, and found the tw o signals to be largely negatively-correlated, leading to signicant dierences in load shifting outcomes and accounted emissions. The debate between ACE and LMCE, and resulting illumination of their r espective shortcomings, has also led to interest in dening new classes of carb on metrics. One notable example of this are so-called ‘carbon-ow’-based metrics [ 5 , 7 , 17 ], which employ vari- ous non-physical assumptions to trace power ow (and associated carbon emissions) from generators to loads. Though not yet exten- sively analyzed, the method by which these carbon ow metrics are calculated is distinct from both A CE and LMCE. As a result, such metrics likely represent yet mor e competing denitions of carb on intensity , highlighting the increasing need to study the pros and cons of each type of metric. Inter-metric variability in shifting-incentiv e and emissions ac- counting properties, such as that demonstrated in [ 20 ] and [ 34 ], highlights the need for new tools and analysis frameworks to com- pare dierent emissions metrics. This is particularly true given the high likelihood that new metrics will continue to be proposed and employed to inform consumers’ beliefs regarding their carbon footprint, as well as their load-shifting behavior . T o inform the debate about the eectiveness and appropriate- ness of dierent carbon emission metrics, we develop a software package for comparing the dierent metrics from a power system perspective. Our comparison framework is based on modeling the US market clearing mechanism of an electric grid (represented by solving a DC optimal power o w (OPF) to determine the optimal cost-minimizing generator set-points) and computing total carbon emissions for the overall grid as well as various carbon emission metrics for dierent lo cations based on the result. This informa- tion can then be used for b oth carbon accounting (i.e. to allocate carbon emissions to dierent loads) and as an input to simulate load-shifting, where a carbon-sensitive load, such as a data cen- ter , changes their consumption in response to the carbon metrics. Finally , it is possible to assess the eectiveness of load shifting by r e- solving the electricity market clearing with the update d electricity consumption pattern. One important benet of the framework is that it allows us (and other researchers) to e valuate both whether a given metric is eective in guiding load shifting (i.e. if a consumer reduces their use of electricity in an hour with a high emission value and increases it in an hour with low emission value, does this lead to a reduction in overall system emissions?) and whether it possesses desirable properties for carb on accounting (e .g. do es the metric guarantee that the total carb on emissions assigne d to consumers are equal to the total carb on emissions from generation?). This can help inform the debate around current metrics. However , our second goal is to provide a platform for evaluation of newly propose d metrics. W e demonstrate this by proposing a new carbon emission metric, namely the adjusted locational marginal emissions (ALMCE) metric, and assessing its merits relative to e xisting metrics using our framework. 1.2 Contributions • W e provide a qualitative overview of existing carbon emis- sion metrics and their pros and cons, and propose the new ALMCE metric to try to combine benets of several metrics. • W e contribute a new open source Julia package, "ElectricityE- missions.jl" that enables easy calculation of several existing and emerging carbon emissions metrics, including the newly proposed ALMCE metric. • W e demonstrate how the package can support quantitative comparison of dierent carbon emission metrics, both with and without load shifting, in a case study on the standard RTS-GMLC test system. • This case study is, to the best of our knowledge, the rst to assess the impact of load shifting not only on total carbon emissions, but also on the accounted carbon emissions that are allocated to dierent types of loads. 2 BA CK GROUND T o explain the foundations of the real-time carbon accounting for electricity consumption, we start with a brief overview of the preva- lent electricity market design in the Unite d States, the emissions from generators and how changes in electricity consumption can impact these emissions. Notation. In the follo wing discussions, we will denote quantities related to electricity consumption by subscript 𝐷 and the set of loads by D , while quantities associated with electricity generation are denoted by subscript 𝐺 and the set of generators is denoted by G . W e will generally use lower case 𝑒 to refer to carbon emissions intensities (measured in [T onsCO2/MW]) and capital 𝐸 to refer to carbon emissions (measured in [T onsCO2]). Lower case 𝑝 will be used to denote quantities of power generation or consumption. Electricity Market Clearing . W e consider a region wher e the electricity market clearing can be modele d mathematically as an optimal power ow (OPF) problem. This is representative of an electricity market based on nodal pricing, which is the pre valent market design in the United States. In this market design, gener- ators provide oers to generate ele ctricity , which are associated with a given cost [$/MWh] and capacity [MW]. The electricity consumption of electric loads is typically forecasted by the utilities that ser v e those loads, and the consumption is treated as a xed value. This modeling is appropriate for most electric loads, which buy electricity on xed rates through their local utility . Howev er , the OPF model could easily be extended to also consider exible, price-sensitive loads that source electricity directly from the whole sale energy market by modeling them as negative generation. Given that the load is assumed to be xed, the goal of the market clearing, and the objective of the corresponding OPF problem, is to minimize the total cost of generating electricity . This cost mini- mization is subject to constraints that model the physical ows of electric power and limits on generation and transmission capacity . It can be expressed as a linear optimization problem. W e provide 2 the mathematical formulation and additional explanations for this optimization problem in Appendix A.1. Electricity markets clear in two periods, day-ahead (DA) and real-time (RT). The D A market, solves a multi-time period optimiza- tion problem for the generation at each hour of the following day . The RT market is solved every 5 minutes during the day , where discrepancies from the load for ecast or renewable energy forecast used in the DA market are resolved by solving an OPF and adjusting the generator output. For either market, the primary outcome of the OPF is the optimal generation dispatch, given by a set of gener- ation set-points 𝑝 ∗ 𝐺 , 𝑖 , 𝑡 for all generators 𝑖 ∈ G at time step 𝑡 , that is needed to satisfy system load 𝑝 𝐷 , 𝑗 ,𝑡 , representing the demand by loads 𝑗 ∈ D at time 𝑡 . Emissions fr om generators . Based on the optimal generation dispatch 𝑝 ∗ 𝐺 ,𝑡 , we compute the carbon emissions as 𝐸 𝑡 𝑜𝑡 𝐺 ,𝑡 =  𝑖 ∈ G 𝐸 𝐺 , 𝑖 , 𝑡 ( 𝑝 ∗ 𝐺 , 𝑖 , 𝑡 ) = 𝑒 𝐺 , 𝑖 , 𝑡 𝑝 ∗ 𝐺 , 𝑖 , 𝑡 (1) where 𝐸 𝐺 , 𝑖 ( 𝑝 ∗ 𝐺 , 𝑖 , 𝑡 ) is a function describing the emissions associated with generating the power 𝑝 ∗ 𝐺 , 𝑖 , 𝑡 from generator 𝑖 . This function could take many forms (for example , generators may have dier ent eciency le vels depending on how much they produce or the ambi- ent temperature), but for practical reasons and lack of more detailed data, we assume that the carbon emissions per MW of generated power is constant, such that 𝐸 𝑖 ( 𝑝 ∗ 𝐺 , 𝑖 ) = 𝑒 𝐺 , 𝑖 𝑝 ∗ 𝐺 , 𝑖 , 𝑡 where 𝑒 𝐺 , 𝑖 is the (constant) emissions factor of generator 𝑖 . Impact of changes in ele ctricity consumption . In the OPF problem, increasing or decreasing consumption 𝑝 𝐷 , 𝑡 + 1 will change the optimal generation set-points 𝑝 ∗ 𝐺 ,𝑡 + 1 and the resulting carbon footprint of the grid. This allows consumers to impact carbon emis- sions by changing their consumption, for example by shifting their consumption to a dierent location (e.g. by shifting computing loads within networks of data centers) or to a dier ent time step (e.g. by deferring consumption until later in the day). W e note that while it is possible that a consumer might shift their load to b et- ter align with behind-the-meter renewable assets (hence lowering their emissions), the OPF-based framework presented in this work considers only consumer impact on utility-scale generation. 3 CARBON EMISSION METRICS T o dene the carbon emissions of electricity consumption, we need to dene a time- and location-sp ecic carbon emission metric 𝑒 𝐷 , 𝑗 ,𝑡 [T onsCO2/MW] that denes the carbon intensity per unit of elec- tricity consumed at each timestep and location in the grid. Giv en the carbon emission metric 𝑒 𝐷 , 𝑗 ,𝑡 , the total carbon emissions of demand 𝑗 at time 𝑡 are given by 𝐸 𝐷 , 𝑗 ,𝑡 = 𝑒 𝐷 , 𝑗 ,𝑡 𝑝 𝐷 , 𝑗 ,𝑡 (2) where 𝑝 𝐷 , 𝑗 ,𝑡 [MW] denotes the total electricity consumption and 𝐸 𝐷 , 𝑗 ,𝑡 [T onsCO2] denotes the total emissions. The challenge lies in choosing how to dene 𝑒 𝐷 , 𝑗 ,𝑡 (often also referred to as carbon intensity or emissions factor). Even though the carbon emission factor from each ele ctric generator 𝑖 and time step 𝑡 , denoted by 𝑒 𝐺 , 𝑖 , 𝑡 , are well dened, there is no clear consensus ab out how to allocate these carbon emissions to dierent consumers. This is in part because there is no unambiguous method to trace the actual ow of electricity fr om generator to load. Thus, multiple denitions of carb on emission metrics have b een proposed. W e summarize the denitions of some widely used metrics below , along with a more recently propose d metric and a brand new metric that we consider here for the rst time. Note that this is not intende d to be an exhaustive list of carbon metrics. Some other metrics, such as long-run marginal emissions [ 14 ], are not included below as they are less relevant to r eal-time carbon accounting and load shifting. A verage carbon emissions ( ACE) . The most intuitive and com- monly used metric is average carbon emissions 𝑒 𝑎 𝑣𝑔 𝐷 , 𝑡 , which is dened as the total carbon emissions incurred by the generators 𝐸 𝑡 𝑜𝑡 𝐺 ,𝑡 and divided by the total power consumption 1 , 𝑒 𝑎 𝑣𝑔 𝐷 , 𝑡 = 𝐸 𝑡 𝑜𝑡 𝐺 , 𝑡 Í 𝑗 ∈ D 𝑝 𝐷 , 𝑗,𝑡 . (3) With this denition, the emissions factor 𝑒 𝐷 , 𝑗 ,𝑡 = 𝑒 𝑎 𝑣𝑔 𝐷 , 𝑡 is the same for all loads 𝑗 in the grid at time 𝑡 . As a metric, average carb on emissions has several benets. It is easily accessible (i.e. published by system operators or can be estimated from publicly available information without requiring a model of the network), is available for large geographical regions from data providers such as Elec- tricityMaps [ 11 ] and W attTime [ 35 ], and can be used for scope 2 electricity accounting under the Greenhouse Gas Protocol [ 12 ]. A v- erage carbon emissions capture the emissions from all generators, and ensures that the total emissions from generation equal the total emissions assigned to loads. Howe ver , the average carb on emis- sions metric also has several drawbacks. It does not account for grid constraints that limit the transfer of power within a given region. Further , average carbon emissions does not accurately capture how load shifting, i.e. changes in the timing or location of electricity consumption of a specic load 𝑝 𝐷 , 𝑗 ,𝑡 , impacts total grid emissions. Another emerging drawback is that average carbon emissions does not reect the availability of low-carbon generation sources that are currently not being used, and, in particular , the availability of curtailed renewable energy . W e illustrate these drawbacks base d on an example from the CAISO grid (in California) on June 10, 2021. Fig. 1 shows the to- tal generation for each fuel typ e (top), average carbon emissions measured in T onsCO2/MWh (middle) and amount of renewable cur- tailment in the grid (bottom). W e obser ve that the average emissions are lowest in the late afternoon, when the renewable energy gen- eration is highest. Howev er , there is signicant renewable energy curtailment earlier in the day , which is why the average emissions are higher in this time period. If the average emissions metric is used to shift load, consumers may cho ose to shift their power con- sumption until later in the day . Howev er , from a grid perspective, this is opposite of what is needed, as such a shift would likely in- crease rene wable energy curtailment and lead to overall higher emissions. Locational marginal carbon emissions (LMCE) . T o answer questions such as “how do carbon emissions change if I choose to charge my electric vehicle now rather than later?” , a more accurate 1 Note that care must be taken to appr opropriately count the total carbon emissions from generation and the total load in cases where a r egion is importing or exporting electricity . 3 Figure 1: Power generation proles (top), average carb on emis- sion (middle) and renewable curtailment ( boom) for June 10, 2021 in CAISO. W e highlight hours with high curtailment (green) and low average carb on emissions ( blue). metric is the locational marginal carbon emissions. By applying sensitivity analysis methods to the OPF problem, we can describe how a small change in a given load, denote d by 𝛿 𝑝 𝐷 , 𝑗 ,𝑡 , changes the optimal generation dispatch. W e denote changes in generation by 𝛿 𝑝 ∗ 𝐺 , 𝑗 ,𝑡 . Once we know 𝛿 𝑝 ∗ 𝐺 , 𝑗 ,𝑡 , we can derive the associated change in grid emissions at time 𝑡 as 𝛿 𝐸 𝑡 𝑜𝑡 𝐺 ,𝑡 = Í 𝑖 ∈ G 𝑒 𝐺 , 𝑖 , 𝑡 𝛿 𝑝 ∗ 𝐺 , 𝑖 , 𝑡 . After normalizing for the change in load, we obtain the so called locational marginal carbon emissions (LMCE) 𝑒 𝑙𝑚𝑐 𝑒 𝐷 , 𝑗 ,𝑡 = 𝛿 𝐸 𝑡 𝑜𝑡 𝐺 , 𝑡 𝛿 𝑝 𝐷 , 𝑗,𝑡 . (4) Our implementation of LMCE is based on the metho d described in [ 21 ], but extended to account for piece-wise linear cost functions. The full mathematical model is describ ed in Appendix A.2. Note that LMCE inherently accounts for transmission constraints, and thus varies by location in the grid when transmission capacity is limited. The locational marginal emissions 𝜆 𝐶𝑂 2 are arguably the metric that most accurately describes the impact of ( small) load shifting actions (i.e. intentional changes in 𝑝 𝐷 , 𝑗 ,𝑡 ) to reduce carbon emis- sions [ 20 ], [ 22 ]. However , be cause they represent sensitivities of the OPF solution, LMCE can vary widely between time-steps and and may not be an appropriate metric for larger load shifts. Further , it is unclear how LMCE can be used for carbon account- ing, as it does not capture any information about emissions from Figure 2: Marginal (blue) and average (orange) carb on emis- sions for Chicago (data from [28]). non-marginal generators. Here, it is important to note the distinc- tion between LMCE and locational marginal prices (LMPs), which are the result of clearing the electricity market. Since ele ctricity markets seek to minimize total generation cost, the locational mar- ginal price repr esents the marginal generation cost of the most expensive generator that is dispatched in the market clearing to serve electric load at a given location). The marginal generator can vary for dierent locations based on the network constraints, but all other dispatched generators are low er cost. For emissions, there is no such relationship for the marginal generator being the highest emitting dispatched generator . The marginal generator may have higher or lower emissions than the other units that are dis- patched. As a result, if we allocate carb on emissions to loads based on the locational marginal emissions as the emission intensity , the total allocated carbon emissions will generally b e higher or low er than the total emissions from generation. In many cases we nd that the dierence between the total emissions from generation is signicantly dierent from the total emissions allocated to loads. T o illustrate the v olatility of LMCE, Fig. 2 sho ws the average and locational marginal emissions for ComEd in Chicago on 8/13/23. This is r eal-world data r eported by the utility . W e observe that the 5 minute, real-time locational marginal emissions have much higher volatility than the hourly average emissions. W e also see that the LMCE is often signicantly higher than the average emissions, indicating that the marginal generator at this location is more polluting than the average generator . Carbon f low-based metrics . Another class of metrics that have been considered in the academic literature are metrics based on a “carbon ow” which “traces” carbon emissions from generators to load [ 6 , 7 , 17 ]. The key idea behind these metrics is that generators inject both power and carbon emissions into the grid, and the carbon “ows” along with the electricity . The methods in [ 6 , 7 , 17 ] are based on the so-called proportional power sharing rule [ 2 , 18 ], an assumption which stipulates that the power ows into a bus (and the associate d carb on emissions) are shared proportionally across the power ows out of it. It is important to note that this is an assumption, not a physical law . There exist a variety of other proposals for tracing power ow (see e.g. [ 8 ] and references therein), which would likely lead to dierent carbon allocations across loads. T o the knowledge of the authors, carbon ow metrics have not been adopted in practice, possibly because their denition is less intuitive 4 and more mathematically involved. Nevertheless, these methods are interesting as they give rise to a range of possible carb on emission metrics, including locational average carbon emissions [7]. Adjusted Locational Marginal Carb on Emissions (ALMCE) . Marginal metrics such as LMCE are appealing for use in informing load-shifting decisions, as they pro vide information on how grid emissions will change given a small change in load. However , per- forming carbon accounting using LMCE is problematic due to the lack of a guarantee that LMCE-assigned emissions will sum to ac- tual total system emissions. T o address this issue , w e propose a new metric, the adjusted lo cational marginal carb on emissions (ALMCE). The core idea of ALMCE is to augment LMCE with an adjustment factor , which ser ves to normalize the sum of assigned emissions to be equal to the actual system emissions total. Mathematically , ALMCE is dened for load 𝑗 at time 𝑡 as follows. 𝑒 𝑎𝑙𝑚𝑐 𝑒 𝐷 , 𝑗 ,𝑡 = 𝑒 𝑙𝑚𝑐 𝑒 𝐷 , 𝑗 ,𝑡 + 𝐸 𝑡 𝑜𝑡 𝐺 ,𝑡 − 𝐸 𝑙𝑚𝑐 𝑒 𝐷 , 𝑡 Í 𝑗 ∈ D 𝑝 𝐷 , 𝑗 ,𝑡 , (5) where the rst term is the locational marginal emissions 𝑒 𝑙𝑚𝑐 𝑒 𝐷 , 𝑗 ,𝑡 and second term represents the adjustment. This adjustment is dened as the dierence between the total emissions from generation 𝐸 𝑡 𝑜𝑡 𝐺 ,𝑡 and the total emissions assigned to loads using the LMCE metric 𝐸 𝑙𝑚𝑐 𝑒 𝐷 , 𝑡 = Í 𝑗 ∈ D 𝑒 𝑙𝑚𝑐 𝑒 𝐷 , 𝑗 ,𝑡 𝑝 𝐷 , 𝑗 ,𝑡 , shared across all loads proportional to their consumption. While the full properties of ALMCE for carbon accounting and load-shifting remain to be determined through further analysis empirical testing, there are some properties worth noting. First, ALMCE will uctuate just as LMCE uctuates, providing a rapid signal of the emissions of the marginal generators. Second,if there is a single marginal generator in the system, ALMCE is equivalent to A CE and in this case w ould lead to similar shifting r esults. Third, if we consider a single time step and sort nodes according to their nodal carbon emission values (e.g. from high to low ), then ALMCE and LMCE would lead to the same ordering of nodes. Therefore, if we only consider spatial load shifting (where you can choose between consuming at dierent locations rather than at dier ent times), ALMCE and LMCE might lead to similar shifting results. Finally , we note that the real-world implementation of ALMCE would not require signicant eort in systems that already calculate LMCE, such as PJM [ 29 ], since the LMCE calculation is the most demanding part of computing ALMCE. 4 ELECTRICI TYEMISSIONS.JL W e have created a new Julia package, ElectricityEmissions.jl 2 , to serve as a central implementation of the carb on metrics mentioned in the previous section. The package allows the calculation of these metrics for power system test-cases in the p opular MA TPO WER for- mat, opening the door for carbon-intensity analysis on thousands of simulated grids of varying sizes and properties. Ele ctricityEmis- sions.jl is open-source, and contributions of other carbon metrics and accounting methods ar e welcome. Belo w is an example of com- puting and plotting carbon metrics for a small ve-bus network. 2 https://github.com/WISPO-POP/ElectricityEmissions.jl Code Block 1 Calculating and plotting carb on intensity metrics using PowerModels, ElectricityEmissions using HiGHS # Load the network data with generator emissions intensity network = PowerModels.parse_file("case5_gen_intensity.m") # Calculate carbon intensities (solves a PWL DC OPF problem) lmce = calculate_LMCE(network, HiGHS.Optimizer) almce = calculate_ALMCE(network, HiGHS.Optimizer) ace = calculate_ACE(network, HiGHS.Optimizer) lace = calculate_LACE(network, HiGHS.Optimizer) # Calculate total system emissions total_emissions = calculate_system_emissions(network, HiGHS.Optimizer) # Plot network with carbon intensity data # Update network data with chosen emissions intensity update_emissions_intensity!(network,lmce) # Plot the network plot_emissions(network) 4.1 Carbon Intensity Calculation The rst part of the code example demonstrates how to compute the carbon intensity metrics. The MA TPO WER-format test-case, con- taining network data and generator emissions information, is loade d using the parse_file function from the PowerModels.jl package [ 9 ]. It is then passed to the functions calculate_ACE , calculate_LMCE , calculate_LACE and calculate_ALMCE to calculate the respective carbon intensity metrics. Each of these functions solve a DC-OPF problem to dispatch the system. Then load demand, optimal gen- erator set-points, and generator carbon intensity ( and for some metrics, the network structure) are use d to calculate the no dal carbon intensity for each node in the network, as describe d in Section 3. W e calculate total system emissions with the function calculate_system_emissions , which only nee ds generator set- points as an input. 4.2 Plotting Visualization of the carbon metrics is important to understand the geographic spread of emissions intensity . T o help users ana- lyze their results, the functions update_emissions_intensity!() and plot_emissions() provide convenient data-processing and default settings for use with the power grid plotting package Power- Plots.jl [ 36 ]. An example of a visualization of the metrics is shown in the case study in Fig. 3. 4.3 Example W orkow Here we step through an example workow of using ElectricityE- missions.jl to estimate emissions and shift load, with counter-factual analysis. 0. T est-Case Setup First, obtain a MA TPO WER-format grid test-case. T est-cases with various properties (size, connectedness, distribution vs. transmission system, etc.), ar e included as part of the MA TPO WER software package [ 39 ] or are easily accessible via a library maintained by the IEEE Power and Energy So ciety [ 4 ]. Second, as shown in Code Block 1, use the PowerModels.jl [ 9 ] package to load the test-case. Third, if not alr eady included in the 5 test-case, assign generator emissions intensities. ElectricityEmis- sions.jl expe cts this information for each generator 𝑖 under the key "emissions" , e.g. network["gen"][i]["emissions"] . Finally , it is best practice to ensure that no two generators at a given node have the same cost function, as such duplications can lead to problems calculating LMCE/ALMCE (see Appendix A.2 for details). This is- sue can be easily xed by adding a small amount of noise to the oending generator cost functions. 1. Initial Emissions Calculations Prior to any load-shifting, use Ele ctricityEmissions.jl to calculate the initial carbon intensity at each bus in the system using the relevant calculate_...() func- tion for each of the carbon intensity metrics b eing compared. These intensity values will have units of T ons CO2/MWh . T o calculate as- signed emissions (in T ons CO2) based on a particular metric, simply multiply the load at each bus by the corr esponding carbon intensity value. Finally , store network intensities and assigne d emissions according to each metric, as well as total system emissions (using calculate_system_emissions ). 2. Solve a Load-Shiing Problem A load uses the carbon inten- sity values generated in the pr evious step ( or outside carbon metric values/forecasts) as input to solve a load shifting optimization prob- lem to minimize its emissions. Do this for each intensity metric and record the resulting post-shift loads. T o estimate the impact of this shifting on the accounted carbon emissions, multiplying the initial carbon intensities by the new load values. Note that these are just estimates, for two main reasons. First, the carbon intensities might change as a result of the load shifting, leading to dierences between the estimated and realized accounte d carbon emissions. Second, the carbon intensities are themselves estimates of the car- bon footprint of electricity . Although a load shifting action leads to a signicant reduction in the accounted carb on emissions, it is not guaranteed that the total system emissions would actually decrease by the same amount. 3. Post-Shi Emissions Calculation Shifting load will result in new generator set-points and a new power ow which may change the carbon intensity values at each bus. T o determine the nal post-shift emissions, we rerun the relevant calculate_...() functions with the shifted load prole to recalculate the carbon intensity metrics. As e xplained above, the carbon intensities may change as a result of the load shifting. Using the post-shift carbon intensities, it is possible to compute the accounted emissions that would be realized if the load-shifting was implemented. W e refer to those as the realized accounted emissions. W e can also calculate the total post-shift system emissions, and compare those with the initial system emissions to assess the eec- tiveness of the load shift in reducing total carbon emissions. 5 CASE ST UDY: A CCOUN TING W e rst provide a case study on carbon accounting according to the dierent carbon emission metrics, leveraging the ElectricityE- missions.jl package for computations. 5.1 Dataset W e make use of the RTS-GMLC [ 1 ] test case, a popular synthetic dataset geographically “located” in the Southwestern United States. Generated System DC Total DC 103 DC 107 DC 204 DC 322 Sum Sum Sum Sum Sum Sum Sum LMCE 15.828 33.012 6.692 1.686 1.562 1.917 1.527 ALMCE 15.828 3.162 0.803 0.679 1.035 0.644 ACE 15.828 3.008 0.752 0.752 0.752 0.752 LACE 15.828 2.707 0.577 0.850 1.153 0.126 T able 1: Whole- Y ear Emissions Accounting (Million T ons CO2/MWh) System DC 103 DC 107 DC 204 DC 322 Mean SD Mean SD Mean SD Mean SD Mean SD LMCE 0.740 0.276 0.768 0.241 0.711 0.413 0.873 0.346 0.695 0.262 ALMCE 0.338 0.251 0.366 0.185 0.309 0.424 0.471 0.356 0.293 0.258 ACE 0.342 0.170 0.342 0.170 0.342 0.170 0.342 0.170 0.342 0.170 LACE 0.264 0.305 0.263 0.184 0.387 0.198 0.525 0.228 0.058 0.067 T able 2: Whole- Y ear Emissions Intensity (T ons CO2/MWh) The system contains 73 buses, 120 lines, and 158 generators of var y- ing types, including renewables (wind, solar , and hydro), natural gas, oil, and coal. One year’s worth of hourly load and generation data is available for this system, totaling 8,784 cases. W e use the day-ahead load and renewable energy data from the RTS-GMLC data to calculate these metrics on an hourly basis. W e make two modications/additions, similar to the changes made to the same system in [ 20 ]. First, we add data-centers to the system by setting buses 103, 107, 204, and 322 to be data-centers, and r efer to those data centers as DC 103, DC 107, DC 204 and DC 322, r espectively . Each data center has a nominal load of 250MW , such that the total data-center load corresponds to slightly ov er 20% of total energy consumption over the course of the year . Se cond, we use values from the U.S. Department of Energy[ 27 ] to assign the following emissions intensities (in metric tons of CO2 per MWh) to each generator in the system: • Wind, Solar , Hydro: 0 • Natural Gas: 0.6042 • Oil: 0.7434 • Coal: 0.9606 Note that the emission values in this table dier from the ones listed in [ 20 ] (where the emissions values for coal and natural gas appear to have been switched). The data and code are available as examples in the package documentation. 3 5.2 Methodology W e p erform carbon accounting based on the LMCE, ALMCE, ACE, and LACE carbon intensity metrics across all time-steps in the dataset. ElectricityEmissions.jl is used to calculate nodal carbon in- tensities for all metrics, as well as total system emissions. Following this, accounted emissions are calculated for each metric through multiplication by nodal loads. 5.3 Results Accounted emissions with dierent metrics . W e rst con- sider how the choice of carbon intensity metric impacts the total accounted carbon emissions across the year . T able 1 contains the total system emissions as measured by the generator set-points as 3 https://github.com/WISPO-POP/ElectricityEmissions.jl 6 0.0 0.5 1.0 Emissions Intensity LMCE 0.0 0.5 1.0 Emissions Intensity ALMCE 0.0 0.5 1.0 Emissions Intensity LACE 0.0 0.5 1.0 Emissions Intensity ACE Figure 3: Annual average nodal carbon emissions intensity , by metric. The buses are shown as circles, and generators are small squares arrayed around the buses. The color of the generators indicate the carb on emissions intensity of the source, while the color of the buses show the annual average carbon emissions. Lo cations of data centers are circled in red. Transmission lines connecting buses are shown in grey . well as the accounted emissions for the entire system, the total data center load (denoted by DC load) and for each individual data-center (denoted by DC followed by the corresponding load bus index). In looking at the emissions assigned to all loads in the system, we see that ALMCE, ACE, and LA CE lead to accounted emissions that equal the generated emissions value of 15.8 million tons of CO2. LMCE, howev er , lead to much higher accounte d emissions at 33.0 million tons. The trend of much higher emissions with LMCE also holds true for the dierent data center locations. This discrepancy happens because the marginal generators in the RTS-GMLC are much more polluting than the average generation source, and demonstrates the challenges of doing carb on accounting with LMCE. Carbon intensities with dierent metrics . W e next examine the assigned emissions intensities for each carbon metric. Table 2 shows summar y statistics of emissions intensity across the all nodes in the system, and for each data-center load individually . W e show both the mean value across all hours of the year , as well as the standard deviation of the emissions intensity . First, we analyze inter- metric intensity dierences for the system as a whole. On average, LMCE has the highest intensity values across the whole system, with a mean value of 0.740 T ons CO2/MWh. This is dramatically higher than the next-highest, A CE, with a mean intensity of 0.342 T ons CO2/MWh. A CE shows the least whole-system variability out of all the metrics, likely due to the fact that it does not vary spatially within a given time-step. Second, we analyze emissions intensities across the four data- centers. As in the whole-system statistics, LMCE assigns the highest average intensity values of any metric to all data-centers. Howev er , the relative magnitude of ALMCE, A CE, and LACE varies gr eatly between location. Out of these three metrics, LA CE has the highest mean value for the DC 107 and DC 204, but the lowest for DC 103 and DC 322. Despite this variability , all of the metrics (save A CE, which is location-agnostic) are in agreement on the ranking of average emissions intensity between data-center locations. DC 204 has the highest emissions intensity , followed by DC 103, DC 107, and nally DC 322. Finally , we consider the variation in mean carbon intensity across the full power system. Figure 3 shows the mean nodal emissions intensity values for each metric, with the data-center locations circled in red. W e again see that LMCE (left) assigns by far the highest emissions, and that ACE (right) has the same values for all no des. In this system, LA CE shows the highest geographical variability in mean intensity values. 6 CASE ST UDY: SHIFTING W e next use the ElectricityEmissions.jl package and the e xample workow set out in Section 4 to analyze what happens when the data centers use dierent carbon intensity metrics to guide spatio- temporal load shifting (i.e. shifting b oth between locations and across time steps). 6.1 Dataset W e make use of the same modied RTS-GMLC test case as described in the accounting-only case study , which provides a year’s w orth of hourly load and generation data. Whereas the accounting-only case study added xed 250MW data-center loads, the data-centers are new allowed to increase or decrease their load by up to 20% relative to the 250MW baseline in order to minimize emissions. 6.2 Methodology W e perform load shifting based on LMCE, ALMCE, A CE, and LA CE carbon metrics, within each 24-hour period of year , following the workow described in Section 4.3 (note that the test-case setup is described in the paragraph above. 1. Initial Emissions Calculations W e rst calculate initial "pre- shift" emissions intensity for each metric, as would be determine d from a D A market clearing solution. This provides a forecast of the carbon metric for each node, and at each hour of the day . 2. Solve Data Center Load Shiing Problem Next, we optimize the shifting of electricity consumption between the data-centers, 7 which are assumed to be controlled by a single entity . W e assume that the data center operator can choose where and when certain computing workloads are being processed. This ability to shift workloads translates into an ability to adjust electricity consump- tion across locations and time steps. Specically , the data center operator can shift workloads and associated electricity consump- tion to align with periods and lo cations with lo w carbon intensity , thus minimizing carbon emissions. W e assume that the data centers are able to increase or decrease their electricity consumption by up to 20% relative to their average consumption (which is set to 250MW in our case) at any given time step, but that the total energy consumption across all data centers for one day (a surrogate for the total computational w ork that is done) has to r emain the same. The specic mathematical formulation of the data center load shifting problem can be found in Appendix A.3, along with some notes on the potential impact of parameter choice and case-specic modeling decisions. W e note that in our case study , we assume that only the data centers shift their load, though the simulation setup could be extended to handle (independent) load shifting by other loads as well. 3. Post-Shi Emissions Calculation From the solution to the load-shifting optimization problem, we obtain estimated accounted emissions for the data-centers. These estimated accounted emis- sions are calculated based on the new data-center loading values and the old (pre-shift) carbon intensities. Howev er , the adjustment to the load consumption then aects the actual load prole in the RT energy market. W e subsequently solve the OPF for the RT market, and calculate the realized emissions metrics, after the load shifting has occurred. 6.3 Results W e analyze the emissions outcomes of the load shifting case study to understanding the following for all choices of shifting metric: • The eect of shifting on accounte d carbon, as well as on the underlying carbon metric (ie its stability) • The dierential impact of shifting on carbon responsibility assigned to shifters vs non-shifters • The eect of shifting on ground truth system emissions • The eect of shifting with one metric and accounting with another Whole-year results . The results of the data center (DC) load- shifting case study are summarized in T able 3, and are divide d into several categories. Starting on the left, we show the pre-shift accounted emissions which are the carbon accounting values prior to any shifting (i.e. they are the same values as in the accounting case study , summa- rized here for convenience of the reader ). Next we show the post-shift accounted emissions, which are di- vided into " estimated" and "realized" . Estimated post-shift accounted emissions, which refer to the data-centers’ belief about what their accounted emissions will be after shifting, based on their new (post- shift) load values and the old (pre-shift) carbon metric values that they were optimizing for . W e only show estimate d post-shift ac- counted emissions for the data centers themselves, as these are the only loads that p erform load shifting and obtain such estimates. Figure 4: Change in Emissions Resulting From Spatio- T emporal Shifting The realized p ost-shift emissions are the actual accounted emis- sions values, determined via post-shift loading and post-shift (i.e. re-calculated) carbon intensity values. Finally , we show the total post-shift system emissions, both as an absolute number and a percentage dierence to the no-shift total system emissions. Impact of shiing on accounted emissions for data center loads . W e b egin by inv estigating whether load-shifting leads to a reduction in the accounted emissions assigned to data-center loads. T o analyze this, we rst analyze the realized accounted emissions for DC loads. W e observe that shifting according to any of the metrics leads to a de cr ease in the accounted emissions for DC loads, but to varying degrees. The greatest reduction in accounted emissions was achie ved by ALMCE, follow ed by LA CE and A CE. The smallest reduction in accounted emissions happene d when the data centers shifted according to LMCE. Next, comparing the realized accounted emissions for the esti- mated accounted emissions, we observe a signicant discrepancies. Notably , the realized reduction is signicantly smaller than the estimated reduction (prior to the nal p ost-shift recalculation of 8 Pre-Shift Accounted Emissions Post-Shift Accounted Emissions (Estimated) Post-Shift Accounted Emissions (Realized) Actual Post-Shift Emissions System DC Loads Non-DC Loads DC Loads System DC Loads Non-DC Loads System LMCE 33.011 6.692 26.320 6.363 ( -4.19%) 32.689 ( -0.98%) 6.632 ( -0.90%) 26.057 ( -1.00%) 15.669 (-1.00%) ALMCE 15.828 3.161 12.666 2.865 ( -9.36%) 15.833 ( +0.03%) 3.015 (-4.62%) 12.818 ( +1.20%) 15.833 ( +0.03%) A CE 15.828 3.008 12.820 2.824 (-6.12%) 15.880 ( +0.33%) 2.896 (-3.72%) 12.985 ( +1.29%) 15.880 ( +0.33%) LA CE 15.828 2.707 13.121 2.370 ( -12.45%) 15.825 ( -0.02%) 2.595 ( -4.14%) 13.230 (+0.83%) 15.825 ( -0.02%) T able 3: Shifting Case Study: Emissions Results (Million T ons CO2) carbon metrics values). Estimated emissions reductions for ALMCE and ACE were 2x higher than realized, and 4x and 3x higher for LMCE and LACE respectively . These discrepancies indicate that the metrics changed due to load shifting, and in ways that reduced the overall eectiveness of the load shifting. Impact of shiing on total system emissions and accounted emissions for non-DC loads . Next, we discuss how load shifting with the dierent carbon intensity metrics changes the overall sys- tem emissions, as well as the carbon emissions assigned to non-DC loads. W e rst obser v e that the only metric that signicantly re- duces the actual post-shift emissions is LMCE. For this metric, we observe a reduction of approximately 1% in accounted emissions for both non-DC loads and the overall system (similar to the reduc- tion for DC loads). This matches the 1% reduction in total system emissions, even though the accounted emissions with LMCE are approximately 2x higher than the actual emissions. In contrast, shifting with ALMCE or LA CE barely changes to- tal system emissions, while shifting according to ACE actually increases emissions by 0.33%. These observations hold true for both the total accounted emissions and the total emissions, as the met- rics are designed to ensure that they match. Since the total system emissions hold steady ( or increase) for these metrics while few er emissions are assigned to DC loads, the accounte d emissions for non-DC loads increased by ˜ 1%. In conclusion, these results indicate that even though data cen- ters are able to claim emission reductions after shifting with ALMCE, A CE and LACE, these emission reduction come from reallocating emissions to other loads rather than from actually decreasing total generator emissions. In contrast, when shifting with LMCE, the data centers actually reduced total system emissions. Impact of shiing on individual days . While ov er the course of the whole year , LMCE/LACE-based shifting seemed to decrease emissions and ALMCE/A CE-based shifting seemed to increase them, there is a large amount of variability in the performance of each metric on any particular day . Figure 4 shows the distribution of the daily changes in system emissions resulting from shifting with the dierent metrics, as well as the mean, median, 10th percentile and 90th percentile of the reductions. W e can clearly see that shifting based on LMCE is much more likely to lead to large carb on emission reductions (i.e. large negative values) and less likely to increase the total carbon emissions from the ov erall system (i.e. has few er positive values). W e can also clearly see how some of the other metrics have a tendency to increase system carbon emissions rather than reducing them. Figure 5: Cross-Metric Shifting/Accounting: Change in Ac- counted Data-Center Emissions Impact of shiing with one metric and accounting with another . Finally , we briey examine the outcome of shifting based on one metric and accounting using another ( e.g. determining data- center load shifting with LMCE, then assigning emissions using A CE). Figure 5 shows the absolute change in assigned data-center emissions, dened as the dierence b etw een pre- and post-shift accounted emissions (measured by the accounting metric). As is to b e expecte d, when the same metric is used for shift- ing and accounting, we see the greatest reduction in accounted emissions. There is great variation in the ’cross-metric’ behavior of dierent "shifting metric"-"accounting metric" pairs. For many of them, we see that shifting with one metric (e.g. LMCE) incr eases the accounted carbon emissions when a dierent metric is used for accounting. This indicates that companies or organizations who want to not only reduce overall carbon emissions, but also the car- bon emissions that are allo cated to them, have strong incentives to shift according to the same metric that is used for accounting. 7 CONCLUSIONS Currently , ther e exists multiple competing denitions of the carbon intensity of electricity . In this paper , we seek to analyze benets and drawbacks of existing, emerging and newly pr oposed carb on inten- sity metrics for two important use cases, namely load shifting and carbon accounting. T o achieve this goal, w e provide an ov erview and qualitative comparison of dierent metrics, and implement 9 them in a new software package, ElectricityEmissions.jl. This soft- ware package provides tools for the calculation and analysis of several carbon emissions intensity metrics. Leveraging this package, w e compare the two most established carbon intensity metrics, namely average carbon emissions (A CE) and locational marginal carbon emissions (LMCE), against the re- cently emerging locational average carbon emissions (LA CE) and a new metric proposed in this paper , which we refer to as adjusted locational marginal carb on emissions ( ALMCE). W e analyze the outcomes of using those metrics both in the context of accounting and load shifting. In our case study , we nd that load shifting based on the A CE, ALMCE and LACE intensity metrics has very little impact on overall carbon emission and instead tends to shift emissions from the data centers (which actively seek to minimize their emissions according to these metrics) to non-data center loads. In contrast, shifting according to LMCE reduces emissions for both data centers, non- data center loads, and the overall system. However , it is challenging to do accounting with LMCE, as the allocated carb on emissions are more than two times higher than the actual carb on emissions of the system. The above ndings are specic to the case we study , and further work is required to assess the eectiveness of the dierent metrics across a larger range of case studies. Howe ver , our results demon- strate that load shifting is not always a productive metho d to reduce short-term carbon emissions, and can be counter-productive if the carbon intensity metric is not chosen carefully . In future work, we plan to expand the package to include other metrics and other functionality , allowing input from the community . In particular , we want to add support quadratic generator cost functions and alternative OPF formulations. W e also wish to design new metrics with nice properties for both accounting and load shifting, and perform more thorough analysis of the dierences between metrics on more realistic power system test cases and more realistic load-shifting setups. A APPENDIX A.1 DCOPF Formulation W e use the following DCOPF formulation with piecewise linear generator cost to dispatch generation in Ele ctricityEmissions.jl, and for the linear generation sensitivity calculations necessar y for the LMCE and ALMCE carbon metrics: min 𝜃 ,𝑝 𝐺 , 𝐶 𝐺  𝑗 𝐶 𝐺 , 𝑗 (6a) s.t. 𝐶 𝐺 , 𝑗 ≥ 𝑎 𝑗 ,𝑘 𝑝 𝐺 , 𝑗 + 𝑏 𝑗 ,𝑘 ∀ 𝑗 = 1 .. .𝑁 𝐺 , 𝑘 = 1 .. .𝑁 𝑐 𝑡 (6b)  𝑙 ∈ 𝐺 𝑖 𝑝 𝐺 , 𝑙 −  𝑙 ∈ 𝐷 𝑖 𝑝 𝐷 , 𝑙 =  𝑗 : ( 𝑖, 𝑗 ) ∈ 𝐿 − 𝛽 𝑖 𝑗 ( 𝜃 𝑖 − 𝜃 𝑗 ) ∀ 𝑖 = 1 . . .𝑁 (6c) − 𝑃 𝑙 𝑖𝑚 𝑖 𝑗 ≤ − 𝛽 𝑖 𝑗 ( 𝜃 𝑖 − 𝜃 𝑗 ) ≤ 𝑃 𝑙 𝑖𝑚 𝑖 𝑗 ∀ ( 𝑖, 𝑗 ) ∈ 𝐿 (6d) 𝑃 𝑚𝑖𝑛 𝐺 , 𝑗 ≤ 𝑝 𝐺 , 𝑗 ≤ 𝑃 𝑚𝑎𝑥 𝐺 , 𝑗 ∀ 𝑗 = 1 .. .𝑁 𝑔 (6e) 𝜃 𝑟 𝑒 𝑓 = 0 (6f ) Here, the decision variables 𝜃 , 𝑝 𝐺 , and 𝐶 𝐺 refer to nodal voltage angles, generator set-points, and generator costs respectively . There are 𝑁 buses in the system and 𝑁 𝐺 generators. The system is con- nected by lines ( 𝑖 , 𝑗 ) ∈ 𝐿 , specifying the start and end buses 𝑖 and 𝑗 . Each generator 𝑗 has a piecewise-linear cost function with 𝑁 𝑐 𝑡 terms. Equation (6b) constrains the cost of each generator 𝑗 to be above all of the 𝑘 linear functions that dene its segments, with 𝑎 𝑗 ,𝑘 and 𝑏 𝑗 𝑘 representing the slope and intercept of the function. Equation (6c) is the standard nodal power conservation constraint, referencing the subset of generators 𝐺 𝑖 ⊂ G , loads 𝐷 𝑖 ⊂ D , and lines ( 𝑖 , 𝑗 ) ∈ 𝐿 connected at bus 𝑖 . Equation (6d) limits the ow on each line ( 𝑖 , 𝑗 ) to be below the maximum capacity 𝑃 𝑙 𝑖𝑚 𝑖 , 𝑗 , and Equa- tion (6e) sets limits for each generator 𝑗 to be between 𝑃 𝑚𝑖𝑛 𝐺 , 𝑗 and 𝑃 𝑚𝑎𝑥 𝐺 , 𝑗 . Finally , Equation (6f) sets the voltage angle of the reference bus to zero. A.2 LMCE Calculation LMCE is calculated using linear sensitivity analysis for the DCOPF solution, and describes the ee ct on generation and emissions due to a small change in load. For ElectricityEmissions.jl, we use the general method set forth in [ 20 ] to calculate this sensitivity . How- ever , we make several small modications in order to deal with the new constraints and variables introduced by the piecewise-linear cost functions used in our DCOPF formulation (where [ 20 ] simply assumed linear costs). The DCOPF formulation, as describe d in App endix A.1, has 𝑛 = 𝑁 + 2 𝑁 𝐺 decision variables, where 𝑁 is the number of buses and 𝑁 𝐺 is the number of generators. For each bus, the v oltage angle Θ 𝑖 must be determined, and for each generator 𝑗 , the active power generation 𝑝 𝐺 , 𝑗 and cost 𝐶 𝐺 , 𝑗 must be determined. For any instance of the DCOPF problem, which is a linear pro- gram, there exists at least on basic optimal solution . At this solution, we can identify 𝑛 linearly independent constraints that are being satised with equality . For this solution, the following linear system can be formed 𝐴𝑥 ∗ = 𝑏 (7) where 𝐴 ∈ R 𝑛 × 𝑛 contains the problem’s e quality and binding inequality constraints, 𝑥 ∗ ∈ R 𝑛 contains the (unique) decision variable values at optimality , and 𝑏 ∈ R 𝑛 contains the standard- form RHS constraint bounds. W e note that not all solutions returned by a solver may be basic feasible, and it is possible that the pr oblem exhibits degeneracy . One notable potential source of degeneracy is a non-unique solution due to the presence of more than one generator with the same cost function at a given node, leading to an A matrix that is not full rank. A s noted in the rst step of the e xample ElectricityEmissions.jl workow (Section 4.3), this is easily remedied by adding a small amount of noise to such duplicate cost functions. As we are interested in the relationship between changes in load and changes in generation dispatch, we instead make use of the system 𝐴 Δ 𝑥 = Δ 𝑏 (8) 10 which is implied by Eq. 7. The constraints and variables are input to the system such that 𝐴       ΔΘ Δ 𝑝 𝐺 Δ 𝐶 𝐺       =  Δ 𝑝 𝐷 0  (9) where Θ ∈ R 𝑁 refers to the vector of bus voltage angles, 𝑝 𝐺 ∈ R 𝑁 𝐺 the vector of generator set-points, and 𝐶 𝑔 ∈ R 𝑁 𝐺 the generator cost values. 𝑝 𝐷 ∈ R 𝑁 is the vector containing total load at each bus. In this formulation, the no dal power balance constraints, which are the only constraints to include 𝑃 𝐷 variables, make up the rst 𝑁 rows of the 𝐴 matrix. For ease in calculation, they are or dered according to their bus index. The nal 𝑛 − 𝑁 rows of 𝐴 correspond to the slack bus equality constraints, and all binding inequality constraints for which the order does not matter . This system of equations is identical to that used in [ 20 ], save for the addition of the 𝐶 𝐺 variables representing the piecewise-linear generator costs, and any binding constraints on these variables ( encoded in the 𝐴 matrix). Following the formation of this system, we then invert the 𝐴 matrix, and solve for Δ 𝑝 𝐷 . For compactness, we dene the 𝐵 matrix to be the rst 𝑁 columns and rows 𝑁 + 1 through 𝑁 + 𝑁 𝑔 of 𝐴 − 1 These are simply the columns of 𝐴 − 1 aligning with Δ 𝑝 𝐷 and the rows aligning with Δ 𝑝 𝐺 . W e can then write Δ 𝑝 𝐺 = 𝐵 Δ 𝑝 𝐷 . (10) T o determine the change in generation Δ 𝑝 𝐺 , 𝑗 at generator 𝑗 result- ing from one unit of additional load at bus 𝑖 , we set the 𝑖 th entry of Δ 𝑝 𝐷 to 1 and all others to 0. W e can then p erform the nal LMCE calculation as 𝐿𝑀 𝐶 𝐸 𝑖 =  𝑗 ∈ G 𝑒 𝐺 , 𝑗 Δ 𝑝 𝐺 , 𝑗 , (11) where 𝑒 𝐺 , 𝑗 generator emissions intensity of generator 𝑗 . A.3 Load Shifting Formulation W e p erform spatio-temporal shifting, in which a group of data- centers are allowed to shift their load both to other data-center locations and other times of day . The spatio-temporal shifting prob- lem is formulated as follows: 𝑚𝑖𝑛 𝑑 𝑖 , 𝑡  𝑖 , 𝑡 𝑒 𝐷 , 𝑖 , 𝑡 𝑝 𝐷 , 𝑖 , 𝑡 (12) 𝑠 . 𝑡 .  𝑖 , 𝑡 𝑝 𝐷 , 𝑖 , 𝑡 = 𝑁 𝑑 · 𝑁 𝑡 · 𝐷 𝑛𝑜𝑚 (13) ( 1 − 𝜖 ) · 𝐷 𝑛𝑜𝑚 ≤ 𝑑 𝑖 , 𝑡 ≤ ( 1 + 𝜖 ) · 𝐷 𝑛𝑜𝑚 , ∀ 𝑖 = 1 . . .. 𝑁 𝑑 , 𝑡 = 1 .. .. 𝑁 𝑡 , (14) Here, 𝑝 𝐷 , 𝑖 , 𝑡 and 𝑒 𝐷 , 𝑖 , 𝑡 represent the load and carb on intensity at data-center 𝑖 and hour 𝑡 of the day . 𝑁 𝑑 and 𝑁 𝑡 are the number of data-centers and number of time-steps, in this case set to 4 and 24 respectively . 𝐷 𝑛𝑜𝑚 is the nominal data-center load (here set to 250MW), and 𝜖 is the data-center load exibility ( her e set to 0.2). Eq. (13) enforces that the total data-center energy consumption (acr oss all locations and time-steps) is the same as if all data-centers were at nominal load for the entire p eriod. Eq. (14) denes a feasible range within which data center load can be adjusted up or down at a particular time-step (where the size of this range is controlled by 𝜖 ) . For our choice of parameters (4 data-centers with a constant nominal load of 250MW over 24 hours), the data centers have a total daily energy consumption of 24,000MWh. W e note that shifting outcomes, including those presented in this work, are likely to vary signicantly depending on the sp ecic implementation. Of particular importance ar e factors that inuence the size of load-shifts, such as constraints on data-center exibility ( 𝜖 in our model), as well as case-specic shifting costs (e.g. from communications overhead) which we do not consider here. As a result of such eects, we do not include the shifting formulation employed in this work as part of the ElectricityEmissions.jl package, under the assumption that users may wish to model a variety of exible load types with specic qualities/requirements. Los Alamos Unlimited Release LA -UR-24-31644. Reviewed for re- lease outside the Laboratory with no distribution restrictions. A CKNO WLEDGMEN TS This work is funded through the National Science Foundation, under awards #2328160 and #2325956. REFERENCES [1] Clayton Barrows, Aaron Bloom, Ali Ehlen, Jussi Ikäheimo, Jennie Jorgenson, Dheepak Krishnamurthy , Jessica Lau, Brendan McBennett, Matthew O’Connell, Eugene Preston, Andrea Staid, Gord Stephen, and Jean-Paul W atson. 2020. 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