The averaging bias ‐ A standard miscalculation, which extensively underestimates real CO2 emissions

The substitution of energy based on fossil fuels in different sectors like household or traffic by electric energy saves CO2 of this specific sector due to decreased fossil fuel consumption. An important quantity is the additional CO2 emission ΔF(D¯,ΔD) due to an increased electric power demand ΔD for the average electricity power demand D¯ . Commonly, the formula ΔF(D¯,ΔD)≈M(D¯)ΔD is used (called simplified formula), where M(D¯) represents mean average CO2 footprint. It is shown in the present manuscript, that the simplified formula may underestimate the CO2 footprint significantly if the average CO2 footprint depends on the average electricity power demand, which is the case for most of mixed partly renewable and partly non‐renewable electric energy systems. Therefore, the real CO2 emissions would outmatch those according to simplified easily by factor 2 in reality depending on the status of the electricity system. In order to establish a more precise calculation of the CO2 footprint, the general formula ΔF(D¯,ΔD)=D¯ΔM(D¯,ΔD)+ΔDM(D¯+ΔD) which is exact and contains the simplified formula as a special case, is derived in this article. The simplified formula requires an additional term that takes into account the change of the mean average CO2 footprint ΔM depending on the electricity power demand.

TA B L E 1 CO 2 equivalents CO 2 of different technologies, according to [18]; oil is of minor importance and neglected in this publication An example is the heat supply of a building. The advisable substitution of an oil-burner by a modern heat-pump eliminates the CO 2 emissions of the sector "households", as the oil consumption is eliminated. As a consequence the power demand of the sector "electric energy" is increased in order to operate the new electric heat-pump. The CO 2 reduction of the sector "household" can be easily determined. For instance, a decrease of oil consumption of Δ ∕Δ = −1000 l∕year leads to a decrease of CO 2 emissions of Δ 2 ∕Δ = −3200 kg∕year. But how does the additional demand for electrical energy increase CO 2 emissions from the "electrical energy" sector? For the sector "electric energy" a constant average CO 2 footprint (unit: g 2 /kWh) is available. The standard calculation of the CO 2 impact of increased electric power demand Δ (unit: GW) for a exemplary time period Δ is typically calculated as [4][5][6][7][8][9][10][11][12][13][14]: Please note that the unit of respectively Δ needs to be adapted in order to calculate the correct dimension of the result. The outline of the article is as follows: Section 2 gives an overview with respect to the different energy sectors and the corresponding CO 2 footprint. In Section 3, the fundamental theorem of calculus is used to relate the CO 2 impact due to an increased electric power demand to the average CO 2 footprint . It is shown that the dependence of the average CO 2 on the power demand Δ has to be taken into account in order not to underestimate the CO 2 footprint. Several examples are discussed. Section 4 summarizes the results.

ANALYSIS
In order to derive the CO 2 emissions of the sector "electric energy", the characteristics of electric power generation must be analyzed. An hourly resolved matrix electric power generation for Germany in the year 2017 [15,16] is the basis of the analysis with specifying the electricity source and the hour in one representative year. For ∈ [1,8760] hours, the electric power of eight electricity sources is known. The year 2017 has been chosen, as the Matrix of 2017 has been the latest available hourly resolved complete dataset. However, the chosen year does not influence the general analysis of the averaging bias at all. The CO 2 impact for all technologies is depicted in Table 1.
As an example; 1 denotes Wind Power, 8 denotes Brown Coal Power for each hour . "Regenerative Power" reg , "Non-Regenerative Power" nreg and "Supply" are defined as hourly averaged values as follows Note that the expression "regenerative energy" is not correct from the thermodynamic perspective. Nevertheless, it is used in this article as it represents a common definition for photovoltaics, wind, water, and biomass based electric power. The indices depict an average value over 1 h, that is, 3 1849 , represents the average "Photovoltaics Power" for hour 1849. Please note, that the consideration of import and export energy transport via the system boundary requires an extra balance factor . Also an additional energy storage capacity based power storage and supply contribution st will be necessary. Losses due to electric resistance and the electric power transformation are denoted by . Therefore, the electric energy demand of hour is defined as a function of the relevant energy contributors as The yearly average of amounts to roughly 10% of the yearly average total energy demand . typically scales between 4% and 10% of [11,17]. Within the next decades the energy storage capacity based power request and support st becomes more important as the following situation will occur more and more frequently, especially after the year 2030 reg > .
However, for the derivation of the averaging bias, the import/export balance , the electric resistance as well as the energy storage st are set equal to zero, which implies = . In general, the supply is a function of the energy demand, since the supply must satisfy the demand. Following Equation (5), the supply of energy is a function of energy demand , with priority of energy contributor reg . Index again denotes the average value of a certain hour . This can be expressed as reg = 2 (weather, status electricity grid, . . . ), The functions 1 , 2 , 3 symbolize the general and complex dependency of electric demand and supply as well as the interaction between weather and boundary conditions on and the resulting dependency of . Figure 1 depicts the contribution of different energy sources to power generation in the year 2017. The transient behavior of 1 (Wind), P 8 (Brown Coal) as well as reg (Sum Regenerative) and nreg (Sum Non Regenerative) can be seen. Already in the year 2017 an impressive contribution of renewable electric energy reg was established. Of course a rising importance of reg is anticipated according to [19,20].
The electricity demand is not plotted in Figure 1, but fluctuates between 40 and 80 GW within a year. The renewable energy supply reg varies between 7.7 and 61 GW, wherein biomass enables a regenerative baseload. Non-regenerative energy is typically needed to close the gap with a nreg peak of 62 GW. The following analysis considers an unlimited energy transport within the system boundaries.
In order to determine the CO 2 impact of the complete sector "electric energy", the detailed contribution of different electric energy sources must be considered as defined in Table 1. The combination of a detailed knowledge of each electric energy source (i.e., Hydropower, Wind, Gas, etc.) and the specific CO 2 equivalent impact of each technologȳC O 2 enables the calculation of CO 2 emissions of reg , nreg and , according to Equations (10)-(12) for every hour . The result is depicted in Figure 2. The necessary equations are Note that reg is mainly depending on the weather and the status of the electric grid. But especially nreg is a function of energy demand . Therefor, nreg CO 2 as well as tot CO 2 are also a function of energy demand .
In order to define the CO 2 emissions as a function of the hourly averaged electric power demand , two different possibilities are presented in Equations (13)- (15). In the first approach one assumes  = 96.7 g 2 /kWh = 2361, 9.4.2017; 09.00-10.00: worst non-regenerative energy mix ; = 831.5 g 2 /kWh with 1 ≤ min ≤ 8 defined for each hour by as the minimum satisfying Equation (13) and the condition (14) imply, that regenerative energy has priority and within the electricity system and for a given electricity demand the electricity contribution of technologies with lowest CO 2 impact is supplied with priority. Figure 3 illustrates, that for hour 4899 and 3780 the theoretical behavior of the CO 2 impact CO 2 is illustrated. As the renewable energy is typically not sufficient as > , additional energy nreg must be provided. Therefore, a decrease of CO 2 impact is depicted because of nuclear power ( = 5) while afterwards the CO 2 impact increases again up to a specific brown coal energy value of 1075 g 2 /kWh. Indeed Equation (13) is only the consequence of the aforementioned theoretical assumption, that only the technology with the lowest CO 2 impact is applied step by step. More technologies would be added consecutively to satisfy the demand in theory. However, most electricity contributors co-contribute simultaneously due to electricity net constraints and long distance electricity transport challenges. Therefore the CO 2 impacts of reg CO 2 and nreg CO 2 are determined in a second approach as a function of electric power generation , which is equivalent to the power demand for the given assumptions Equation ( (15) is plotted additionally and depicts the low regenerative CO 2 footprint and a non-regenerative average specific emission of 741 g CO 2 /kWh. Both results are also plotted for hour = 3780 with a maximal regenerative power of 61.3 GW, wherein wind ( = 1) and photovoltaics ( = 3) dominate the regenerative contribution. Note that the non-regenerative footprint of 604 g CO 2 /kWh is smaller compared to hour = 4899. The relative contribution of nuclear power ( = 5) to the total non-regenerative power supply in hour = 3780 causes the difference.
Note that is identical to the total electric power generation according to the simplifying assumptions and Equation (5). Furthermore, the moving average value Please note that 2 (˜) depends on the year via weather conditions, technology change and the adapted demand, indicated by an index denoting the year, for example, by  Figure 4 illustrates the results of Equations (16) and (17)  is plotted according to Equation (17) on the right hand side. In addition a simulation of the year 2030 has been accomplished [16] with detailed information about the scale up of regenerative power installation according to [19,20]. Furthermore, the increase of energy storage capacities is considered  as well as the increase of electricity demand due to ambitious heat pump or battery electric vehicle penetration scenarios with increased̄as a consequence. These results are also depicted in Figure 4. However, detailed explanations of the 2030 calculation are not in the focus of this publication, as the general analysis is of major interest. The main question remains the analysis of the CO 2 impact of an increased electricity demand Δ . Substituting Δ 2 in the following by the simplified notation Δ 2 leads to the commonly used equation (see also Equation (1)

Fundamental theorem of calculus
The fundamental theorem of calculus (Erster Hauptsatz der Differential-und Integralrechnung) relates the two fundamental concepts of calculus, that of integration and that of differentiation. It states that derivation and integration are mutual inverses (up to a constant) [21,22]. The theorem is stated in the following: Let = [ , ] be a closed interval on the real line ℝ and ∈ . Furthermore, let ∶ → ℝ be a real-valued continuous function defined on . Then, the function is continuous on and continuously differentiable on the open interval ( , ) with The total differential of ( ) is If ( ) can be assumed to be non-negative, then the integral ( ) can be interpreted based on the area under the curve ( ) in the interval = [ , ]. This implies that an increment d ( ) = ′ ( ) d represents an infinitesimal small area in the range and + d .
holds. The derivative of ( ) is and the total differential of ( ) takes the form Example 3.1b: In the context of estimating the specific CO 2 footprint due to a power demand, one may specifically consider as electrical power demand in kW, ( ) as the specific CO 2 emission in g 2 /kWh and ( ) as the CO 2 footprint (CO 2 impact) within the range of energy demand = 1 , = 2 with 1 ≤ 2 .

Implication of the fundamental theorem for moving averages
The mean value of the function ( ) on the interval = [ , ] computes as For the special case that = 0 and = , one obtains for variable a moving average or, equivalently, with ( ) = (0, ), Determining the total differential of both sides of (28) gives, taking into account the relations (21) and (22) Applying the product rule d( ( )) = ( ) d + d ( ) gives (general formula) which states that the increment of ( ) is equal to the sum of the increment of multiplied by and the increment of multiplied by . The special case (simplified formula) In simplified notation Equation (36) may be recast as which will be called general formula for estimating the CO 2 impact in the following 1 . The Equation (38) is valid for increments of arbitrary size, but the arguments of the functions entering in Equation (36) have to be taken into account carefully. It should be noted that, in the context of estimating the CO 2 footprint, the simplified formula is commonly used. In particular, by (39), the increase in Δ ( , Δ ) may be severely underestimated. As demonstrated in the previous section, such positive values Δ are not uncommon. hold exactly. Therefore, for the special case of constant functions ( ), the simplified formula (39) for the CO 2 impact is exact. Example 3.2b: Assume that the function ( ) is piecewise constant 2 except for a jump at = 0 from 0 to 0 > 0, that is, Then it follows for ( ) and for the moving average ( ) and At = 0 the moving average changes from zero to positive values with slope ′ ( = 0 ) = 0 ∕ 0 . This implies that, for a large jump 0 , there is a rapid change of the mean value close to = 0 . Additionally, the approximation Δ ( ) ≈ ( )Δ is clearly inaccurate.
is not fulfilled. Indeed, for general , the term d ( ), that is, the change of ( ) with , is not taken into account. For the integral one obtains Based on 0 = 0 , Δ = Δ and ( 0 ) = 0 ∕2, this result may be decomposed into With this example in mind it becomes clear, that the average ( 0 ) multiplied by Δ as an estimator for Δ ( , Δ ), that is, Δ ( , Δ ) ≈ ( 0 )Δ , produces an erroneous result, because the terms 0 Δ ∕2 and Δ Δ ∕2 have been neglected.

DISCUSSION AND CONCLUSION
For the calculation of CO 2 emissions of additional electric energy demand, insufficient simplified mathematic models are typically used, which might be motivated by the complexity of the electricity supply sources and the grid situation.
An example for such a simplified formula to analyze the additional CO 2 emissions per time interval Δ (̄, Δ ) caused by additional electric power Δ (unit: Watt) is the direct utilization of the average CO 2 emission footprint (̄) (unit g 2 /kWh) for a given average electricity demand̄of the electricity sector by the equation which corresponds to the simplified formula introduced in Section 4 (see equation (39)). As shown in Section 3, the following integral would be the exact formulation Here, ( ) represents the specific CO 2 emissions as a function of electric power demand .
The mathematical analysis showed that Equation (49) is only valid, when the CO 2 emissions are completely independent from the energy supply situation, that is, if the complete electric energy would be either supplied constantly only by one technology, that is, wind power, or would be supplied by a constant mix of several technologies, that is, a combination of wind power and photovoltaics power, which is both by far not the case.
The examples discussed in Section 3 show for the specific assumption of a discontinuous, piecewise constant function and a linear function that the simplified formula is generally invalid and leads to erroneous results. The simplified formula is only valid for a constant function. Indeed, there is a clear interaction between electric power demand and CO 2 emissions of the electricity sector, as additional electric energy supply typically requires the support of additional fossil power plants also in the future. It is clear that Equation (49) cannot be generally utilized as it may significantly underestimate real CO 2 emissions.
(51) Table 3 explains the definition of major variables.

A C K N O W L E D G E M E N T S
The author Thomas Koch acknowledges the financial support by German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) through project 237267381 (TRR 150) as well as 426888090 (SFB 1441). Furthermore, the support by Alexander Heinz, Kai Scheiber, Philipp Weber and Christian Böhmeke is highly appreciated. The authors thank the reviewers for their valuable comments, which led to improvements in the manuscript. Open access funding enabled and organized by Projekt DEAL.