Real-world scenario description
In this section, the artifact is evaluated as required within the DSR paradigm. Therefore, the artifact’s functionality is illustrated within an example, i.e., the decision algorithm of LS is applied to demonstrate that the DR approach “can be implemented in a working system” (Hevner, et al. 2004, p.79). Afterward, the DR approach is evaluated with multiple simulations of random scenarios to demonstrate that the “artifact (generally) works and does what it is meant to do” (validity) (Gregor and Hevner 2013, p.351). For both, real-world data is applied.
The object that serves for demonstration and evaluation is located in the southeastern part of the United States, in Georgia. Georgia is known for its subtropical climate, with humid summers and moderate winters. Especially during summer months (May to September), temperatures are comparatively high (between 15–31.7 °C on average). During winter months (November to March), temperatures are on average above freezing point (between 0.6 °C and 18.3 °C). For research purposes at the University of Georgia, a/c data were collected from two University buildings. The rooms within the buildings are used as offices and for large meetings. Both buildings are partly open to the public. Using measuring points, different parameters were collected during a period ranging from January 2010 to December 2014. Collected parameters comprise inside temperature on a room level, outside temperature, and electricity consumption (kWh) for a/c usage. Measuring points recorded instantaneous, i.e., not as averaged values within a certain time span. Main components of the a/c system are two chiller systems that jointly air-condition via chilled water loops. Together, both chiller systems have a maximum wattage of 1.2 MW and are responsible for 90% of the a/c system’s total electricity consumption. The remaining 10% are consumed by auxiliary equipment that scales up with the chillers’ current load level. By applying variable load control, the a/c system is designed to provide a constant supply water temperature (about 5 °C + /−0.2 °C). Electricity consumption of the a/c system depends on the temperature of return water (that, in turn, depends on outside and the buildings’ inside temperature). Warmer return water increases electricity consumption and vice versa. To date, no DR mechanism is in place and the (central) a/c system runs all day (not to be confused with a single room’s air supply, which can toggle on and off), even in times of low or no occupancy (e.g., on weekends and at night). Overall, the current system wastes energy and yields unnecessary electricity costs.
The University purchases electricity for the a/c system from a local utility company. The company charges real-time electricity prices rather than offering a flat plan. Thus, electricity prices are sometimes high and the University incurs significant electricity costs. The collected data of the a/c system and payed electricity prices make this example suitable for the DR approach’s demonstration and evaluation. Although a data-driven DSS that integrates the DR approach is not implemented yet, its theoretical cost savings potential is evaluated in this scenario.
For variable load control, the a/c system already possesses sensor systems that measure further parameters such as supply water temperature and current load level, a web server that collects all sensor information, and a remote controller that building operators can access using a web portal. Access to the utility’s real-time electricity prices is available using the customer portal. To establish cost-sensitive a/c control, there is a need for changes and enhancements in the monitoring and control system as it must dynamically import the utility’s price information (by accessing a respective application interface) and possess control software that applies the data-driven DR approach. Moreover, hardware for faster communication and computation would be useful in order that the system can react on changes in input information in near real-time (which is especially necessary to scale downtime increment length between two optimization iterations). Due to an expert’s opinion (an engineer at the university with a PhD who is specialized in a/c systems), the sum of all university-internal and -external costs for implementing such cost-sensitive control in the considered a/c system amounts to about $100.000. Further running costs are expected to be insignificantly low. Besides this application scenario, the expert expects the control software to be applicable in other university buildings as soon as they are also equipped with modern monitoring and control systems. To obtain a conservative estimate, the present paper limits the business case analyses to the described scenario.
Step 1 Scheduling (demonstration)
For artifact demonstration, \({\mathrm{temp}}_{\mathrm{req}}\) is set to 21 °C. This is the currently targeted inside temperature in the scenario’s buildings. As Georgia, USA, is known for its humid and hot summers, a typical day in September is chosen, when a/c is required to cool (keep) the inside temperature to (at) 21 °C. In particular, the DR approach is applied on September 04, 2014. The hypothetical event of interest (e.g., a major event of a university initiative) takes place at 2 pm (occupancy time) in both buildings. The earliest possible a/c activation is set to 7am. The University’s expert stated that every room within the two buildings (regardless of current inside and outside temperature) can be cooled down to \({\mathrm{temp}}_{\mathrm{req}}\) by a/c within one hour. Hence, \({\mathrm{t}}_{\mathrm{L}}\) is at 1 pm (i.e., \(x=1\)). As the dataset of historically payed electricity prices features hourly time increments, artifact demonstration and evaluation is also conducted with hourly time increments between \({t}_{0}\) and \({t}_{\mathrm{L}}\). Table 1 illustrates the schedule.
Table 1 Schedule for artifact demonstration
Step 2a: Price prediction (demonstration)
As described in Sect. 3, this paper modifies and applies the price prediction model developed by Fridgen, et al. (2016). This price prediction model draws upon the existence of historical time of day- and season-specific price patterns and updates price prediction at every time step by integrating new observable price information. Figure 7 illustrates historical time of day-specific price patterns of electricity prices. Further, Table 2 illustrates descriptive statistics on electricity price patterns of different months.
Table 2 Descriptive statistics for electricity prices per month [$/kWh] For configuration purposes, building operators can adjust three endogenous (model) parameters within the DR approach’s price prediction model: \(\theta\), \(n\) (the adjustment reference interval to compute shot-term adjustment \(\alpha\)), and an estimation corridor to compute \(\stackrel{-}{\mathrm{S}}(t)\). Fridgen, et al. (2016) vary \(\theta\) within an interval between 0 and 1. For artifact demonstration, \(\uptheta\) is arbitrarily set to 0.8 and further analysis of its influence is left to the subsequent evaluation. Similar, \(\mathrm{n}\) is set to 0. To calculate \(\stackrel{-}{\mathrm{S}}(t)\), Fridgen, et al. (2016) analyze seasonal price patterns. The authors differentiate between summer, winter, and intermediate season. However, this does not fully reflect the course of historical time-of-day-specific price patterns. For example, their intermediate seasons include March–May and September–November. Therefore, March and September share the same \(\stackrel{-}{\mathrm{S}}(t)\), which is (in our case) not accurate as shown in Table 2. Hence, this paper calculates \(\stackrel{-}{\mathrm{S}}(t)\) based on a historical corridor around the date of interest and time-of-day. For the presented example (September 04, 2014), \(\stackrel{-}{\mathrm{S}}(t)\) at (e.g.) 12 noon is calculated by averaging previous-years’ historical electricity prices from (e.g.) 30 days prior to 30 days after the date of interest, i.e., from August 05, (2010–2013) to October 04, (2010–2013) each of which at 12 noon. Table 3 illustrates respective results (with \(\mathrm{S}(t)\) being the actual observable electricity prices).
Table 3 Price prediction parameters
Example:
\(E\left( {S\left( {9am{|}8am} \right)} \right) = E(S\left( {8am{|}8am} \right) + \theta {*}(\alpha \left( {8am} \right){*}\overline{S}\left( {9am} \right) - E\left( {S\left( {8am{|}8am} \right)} \right) = 0.0599 + 0.8{*}\left( {0.9552{*}0.0625 - 0.0599} \right) = 0.0597\)
In the next step, the DR approach estimates \(\mathrm{D}({t}_{i},1\mathrm{pm},1)\). As described in Sect. 3.5, \(\mathrm{D}({\mathrm{t}}_{\mathrm{i}},1\mathrm{pm},1)\) is split into \(\mathrm{ID}\left({\mathrm{t}}_{\mathrm{i}},1\right)\) and \(\mathrm{PD}\left({\mathrm{t}}_{\mathrm{i}},1\mathrm{pm},1\right)\) (as \(x=1\) is constant within the real-world scenario, this section continues with a reduced formal notation that neglects \(x\)). For the real-world scenario, Table 4 illustrates related \(\mathrm{\Delta temperature}(t)\) and \(\mathrm{PD}(t)\) observations and a respective linear regression.
Table 4 Empirical dependence of PD \({{\varvec{t}}}_{{\varvec{i}}}\) on ∆temperature The real-world scenario’s a/c system is intended for cooling only. Cooling for \(\mathrm{\Delta temperature}(t)<0\) implies that the two buildings were still heated up when outside temperature already fell below \({\mathrm{temp}}_{\mathrm{req}}\). Unfortunately, historical temperature forecasts that match the given historical data set were not obtainable. Hence, for artifact demonstration and evaluation, this paper requires an assumption to predict electricity demand:
Actual outside temperature equals previous weather forecasts
Generally, Assumption 4 depicts a great simplification of reality. However, since the DR approach focusses on short-term schedules for only a few hours, weather forecasts are close to reality (National Weather Service 2017). Moreover, subsequent evaluation integrates an artificial demand prediction error to analyze electricity cost savings’ sensitivity to demand forecasting quality. Hence, the algorithm can use historical outside temperature as previous weather forecasts to compute \(\mathrm{PD}(t)\). Table 5 illustrates the respective results.
Table 5 Development of \(\Delta temperature(t)\) and PD (t) To date, historically collected parameters are only appropriate for the estimation of\(\mathrm{PD}(\mathrm{t})\). To precisely estimate\(\mathrm{ID}({t}_{\mathrm{i}})\), experimental runs would be necessary that analyze different a/c deactivation durations and different outside temperature developments. However, these experimental runs have not been conducted yet. As interim solution, threshold values are applied that logically contain the correct\(\mathrm{ID}({t}_{\mathrm{i}})\). For the lower limit applies:\(\underset{\_}{\mathrm{ID}}({t}_{\mathrm{i}})=0\), i.e., a situation in which no a/c is required to restore\({\mathrm{temp}}_{\mathrm{req}}\). For the upper limit applies: \(\overline{{{\text{ID}}}} \left( {{\text{t}}_{{\text{i}}} } \right) = \mathop \sum \limits_{{{\text{t}} = {\text{t}}_{0} }}^{{{\text{t}} = {\text{t}}_{{\text{i}}} - 1}} {\text{PD}}\left( {\text{t}} \right)\), which equals the sum of all electricity that would have been necessary to keep the inside temperature at \({\mathrm{temp}}_{\mathrm{req}}\) at any time since\({\mathrm{t}}_{0}\). Until more accurate solutions or historical data are available, \(\mathrm{ID}({\mathrm{t}}_{\mathrm{i}})\in [0,\sum_{{\mathrm{t}=\mathrm{t}}_{0}}^{{\mathrm{t}=\mathrm{t}}_{\mathrm{i}-1}}\mathrm{PD}(\mathrm{t})]\) is an appropriate interval to estimate\(\mathrm{ID}({t}_{\mathrm{i}})\). For demonstration, we arbitrarily choose a parameter\(\varepsilon =0.4\), which simulates a building that absorbs heat to a medium extent. Table 5 (vii) illustrates estimations for \(\mathrm{ID}({t}_{i})\) and (viii) estimations for \(\mathrm{D}\left({t}_{i},12\right).\)
$$ID\left( {t_{i} } \right) = \varepsilon *\mathop \sum \limits_{{t = t_{0} }}^{{t = t_{i} - 1}} PD\left( t \right)$$
(7)
Example:
\(PD\left( {8am{|}2pm} \right) = Intercept{*}\Delta temperature\left( t \right){*}Estimate = 428.5889 + 3.8{*}21.8235\)
\(PD\left( {8am{|}2pm} \right) = \frac{{PD\left( {8am} \right)}}{2} + \mathop \sum \limits_{{{\text{t}} = 9am}}^{2pm} PD\left( t \right) = \frac{511.52}{2} + 520.25 + \ldots + 697.02 + 550.80\)
\(ID\left( {1pm} \right) = \varepsilon {*}\mathop \sum \limits_{{t = t_{0} }}^{{t = t_{i} - 1}} PD\left( t \right) = 0.4{*}\left( {507.15 + 511.52 + \ldots } \right) = 1411.83\)
\(D\left( {1pm,2pm} \right) = ID\left( {1pm} \right) + \frac{{PD\left( {1pm} \right)}}{2} + \mathop \sum \limits_{t = 2pm}^{2pm} PD\left( t \right) = 1411.83 + \frac{697.02}{2} + 550.80 = 2311.14\)
4.5 Step 3: Decision making (demonstration)
In the third step, the decision algorithm for LS determines if immediate a/c activation is ex-ante optimal (cost minimal). In particular, from the perspective of the current period, the algorithm predicts and compares expected total electricity costs for all possible activation periods. Table 6 illustrates computations from the perspectives of 7a.m., 8a.m., 1p.m., and 2p.m. In this example, the algorithm would wait until 1 pm to initialize a/c.
Table 6 Decision making within artifact demonstration
Example:
\(E\left( {C\left( {1pm,2pm{|}1pm} \right)} \right) = ID\left( {1pm} \right){*}E(S\left( {1pm{|}2pm} \right) + \mathop \sum \limits_{t = 1pm}^{2pm} E\left( {S\left( {t{|}2pm} \right){*}PD\left( t \right)} \right) =\)
\(0.0708{*}1411.83 + 0.0708{*}\frac{697.02}{2} + .0925{*}550.80 = 175.58\)
4.6 Step 4: Feedback (demonstration)
In the last step, the DR approach ex-post evaluates the ex-ante chosen activation time as described in Sect. 3.7. Therefore, the DR approach computes savings of its decision compared to the default procedure with no DR. By applying DR and activating a/c at 1 pm, total electricity costs would have been $174.53 (cf. Table 7ii). These are the lowest actual (not expected) total costs and can be computed by utilizing F-6 with the actual (not expected) electricity prices. The default procedure, however, would have yielded total electricity costs of $312.90 (cf. Table 7ii). This equals an electricity cost reduction of 44.22% due to the DR approach. Moreover, the theoretically optimal point in time for a/c activation (the benchmark) was also at 1 pm. In particular, the DR approach was able to utilize the entire cost savings potential. Table 7 summarizes the results for the presented example. \({C}_{ex-post}\) are calculated using the demand for each hour and according actual prices, not the expected prices. Since this example is biased in its validity because it was manually picked, the next section contains randomly chosen historical simulations and sensitivity analysis. Thereby, the general usefulness of the artifact is analyzed.
Table 7 Decision making within artifact demonstration Evaluation
DSR methodology calls for an evaluation of a developed artifact to provide evidence “how well the artifact supports a solution to the problem “(Peffers, et al. 2007, p.56). A possible evaluation method within DSR are simulations (Hevner, et al. 2004). This paper’s evaluation is divided into three parts and presents historical simulations on the real-world scenario with 200,000 simulation runs each: The first part gives an impression on the DR approach’s effectiveness in terms of average electricity cost savings and sensitivity of the latter to endogenous model parameters (\(\uptheta\),\(\mathrm{n}\), and estimation corridor, c.f. Section 4.3). Subsequently, the triple of endogenous model parameters that yields the highest average electricity cost savings is fixed for the second part of the historical simulation. This calibration procedure for the prediction model is valid, as building operators can individually chose model parameters. The electricity cost savings of the second part are then analyzed on their sensitivity to exogenous scenario parameters (\({t}_{0}\),\({t}_{\mathrm{L}}\), flexibility window length\({ t}_{\mathrm{L}}-{t}_{0}\), and dependency of \({\mathrm{ID}}_{{\mathrm{t}}_{\mathrm{i}}}\) on \({\mathrm{PD}}_{\mathrm{t}}\)). To lift Assumption 4, a third simulation part integrates an artificial hourly demand prediction error). Therefore, the sensitivity of electricity cost savings to forecasting quality of electricity demand is measured. For all simulation parts, sensitivity of the results to the electricity market is analyzed by also repeating every simulation with electricity prices from the German-Austrian market area of EPEX SPOT. This market has a significantly growing capacity of renewable energy generation (EPEX SPOT 2017) that may evolve to a global trend. To isolate market influences on the results, the object and temperature conditions are assumed to equal the real-world scenario. In the following, this section refers to both markets as US market and EU market, respectively. Results of all simulation parts are discussed afterward.
Historical simulation – part 1
A multivariate sensitivity analysis identifies the triple of all three endogenous model parameters that yield (in combination) the highest average electricity cost savings: \(\theta =1.0\), \(n=6\mathrm{h}\), and estimation corridor length \(=30\) days with average electricity cost savings of $99.76 (or 45.40%) for the US market and \(\theta =1.0\), \(n=0\mathrm{h}\), and estimation corridor length \(=60\) days with average electricity cost savings of €51.28 (or 46.11%) for the EU market. As building operators can individually select endogenous model parameters, they should always conduct such pre-simulations on their individual historical data to maximize electricity cost savings. Thereby, as the present example illustrates, the best parameter combination can vary between different electricity markets. In the second part of the simulation, the respective best parameter combinations are fixed for both markets.
Historical simulation–part 2
Table 8 illustrates the evaluation parameters and their range. Simulation runs are conducted by sampling with replacement. Overall parameter combinations, the DR approach yields average electricity cost savings of $94.61 (or 44.52%) for the US market and €48.42 (or 44.07%) for the EU market compared to the default procedure with no DR. Standard deviation is $134.62 (142.29% of mean) for the US market and €52.30 (108.01% of mean) for the EU market. The cost savings potential (i.e., the benchmark) is $99.63 (or 46.88%) for the US market and €50.58 (or 46.03%) for the EU market. Therefore, the utilization of cost savings potential by applying the DR approach is 94.96% for the US market and 95.74% for the EU market. Table 9 presents the result’s sensitivity to endogenous model parameters:
Table 8 Range of evaluation parameters (Simulation—part 1) Table 9 Sensitivity of absolute and relative savings to endogenous (model) parameters For the second evaluation part with fixed (calibrated) endogenous model parameters (cf. Table 10), the DR approach yields average electricity cost savings of $95.49 (or 45.03%) for the US market and €49.47 (or 45.14%) for the EU market compared to the default procedure with no DR. Standard deviation is $132.81 (139.07% of mean) for the US market and €51.83 (104.75% of mean) for the EU market. The cost savings potential is $99.84 (or 47.08%) for the US market and €50.61 (or 46.18%) for the EU market. Therefore, the utilization of cost savings potential by applying the DR approach is 95.65% (first evaluation part, without calibration, 94.96%) for the US market and 97.75% (first evaluation part 95.74%) for the EU market. Figure 8 illustrates the histograms and Table 11 presents the result’s sensitivity to exogenous model parameters:
Table 10 Range of evaluation parameters (Simulation—Part 2) Table 11 Sensitivity of absolute and relative savings to exogenous (scenario) parameters Historical simulation—part 3
In the third evaluation part, Assumption 4 is lifted and an artificial hourly demand prediction error (DPE) is integrated. More precisely, for the first predicted discrete time step (i.e., hour), the DR approach estimates upcoming electricity demand by drawing from an equal distribution to the extent of the DPE around the historically measured value of that time. Predicting the subsequent discrete time step (i.e., the second hour in future), the algorithm reiterates this procedure but additionally adds the first hour’s prognosis error. This approach is applied for all remaining discrete time steps within the temporal flexibility window.
With fixed endogenous model parameters and DPE (cf. Table 12), the DR approach yields average electricity cost savings of $93.44 (or 44.10%) for the US market and €48.28 (or 44.01%) for the EU market compared to the default procedure with no DR. Standard deviation is $132.40 (141.69% of mean) for the US market and €52.28 108.28% of mean) for the EU market. The cost savings potential is $99.45 (or 46.94% compared to the default procedure) for the US market and €50.50 (or 46.04%) for the EU market. Therefore, the utilization of cost savings potential by applying the DR approach is 93.95% (second evaluation part 95.65%) for the US market and 95.60% (second evaluation part 97.75%) for the EU market. Table 13 presents the result’s sensitivity to the DPE:
Table 12 Range of evaluation parameters (Simulation—Part 3) Table 13 Sensitivity of absolute and relative savings to hourly demand prediction error Discussion of evaluation results:
Summarizing all evaluation results, the authors derive the following insights and interpretations: Within the real-world scenario, there is a huge savings potential in electricity costs by applying the DR approach. Thereby, the DR approach utilizes almost the entire cost savings potential, although it uses an algorithm with ex-ante (uncertain) electricity price prediction. The high exploitation of savings potentials is due to the following reasons:
Electricity cost savings potential does only refer to cost savings that can (theoretically) be obtained by applying the present paper’s applied a/c procedure (Fig. 3). It excludes further cost savings potential that would exist for more flexible but complex a/c procedures [e.g., “dynamic (de)activation” as illustrated in Fig. 2, (1)] or for managerial flexibility that differs from temporal flexibility (e.g., flexibility in temperature limits that this paper excluded by Assumption 1).
Furthermore, for the second simulation part, early a/c activation (before \({t}_{\mathrm{L}}\)) was ex-post optimal in only 30.80% of all simulations for the US market and 25.63% for the EU market. More precisely, as this paper models hourly time increments within a real-world scenario that exhibits significant electricity demand to keep the inside temperature at \({\mathrm{temp}}_{\mathrm{req}}\), it is often disadvantageous to cool before \({t}_{\mathrm{L}}\). The DR approach correctly anticipated that fact and had only a few misjudgments. If this paper had modeled shorter time increments (e.g., quarter-hourly instead of hourly), more flexibility of action would (on the one hand) increase the DR approach’s cost savings potential and (on the other hand) stronger challenge decision making (with possibly more misjudgments of the algorithm and therefore less exploitation of the savings potential). However, as the present paper’s real-world example is restricted to hourly electricity market data (cf. Section 4.2), a sensitivity analysis for time increment length is subject to future research.
Besides, some electricity cost savings are due to Assumption 4, i.e., missing uncertainty in electricity demand forecasts. However, as the third simulation part and Table 13 illustrates, this effect is rather small and has only a significant impact for huge misjudgments of the prediction model.
Finally, the DR approach’s performance within the presented real-world scenario is significant, since today’s cost-insensitive a/c control wastes a huge amount of energy as a/c runs constantly throughout the day, even during disused hours on working days, weekends, and night times. Therefore, smart a/c control that considers occupation schedules, electricity price prediction, and weather forecasts can yield huge electricity cost savings, even for minor misjudgments that fail ex-post optimal decision making.
The results also indicate that relative electricity cost savings, relative cost savings potential and the utilization of cost savings potential by applying the DR approach differ only slightly between the US and the EU market. This implies that the DR approach is applicable to different electricity markets that offer volatile electricity spot market prices. However, standard deviations of electricity cost savings are comparatively high and larger on the US market than on the EU market. The former results from the fact that average electricity cost savings depend on the simulation’s (randomly chosen) model and scenario parameters (as illustrated within respective sensitivity analysis). As many parameter combinations are possible, electricity cost savings can vary significantly. In addition, the evaluation puts forth some implications of parameter sensitivity analysis:
Sensitivity of electricity cost savings to endogenous (model) parameters: Significant greater electricity cost savings due to greater \(\theta\) confirm the value of modeling mean-reversion to time-of-day-specific price patterns for short-term electricity prediction. While such patterns do not exist in many other spot markets (such as stock prices on capital markets) due to the instability of arbitrage opportunities, they occur in electricity spot markets as electricity consumption depends on time-dependent customer preferences that lack flexibility potential and renewable electricity generation that lacks controllability (cf. Introduction). Significant greater electricity cost savings due to the existence of an adjustment factor \(\alpha\) that is computed on current observable price information (\(n=0\)) indicates that instantaneous price developments are likely to deviate from long-term historical mean prices. Therefore, an appropriate prediction model should consider short-term effects on electricity market prices. As electricity cost savings did not significantly depend on estimation corridor length, historical time-of-day-specific price patterns on the two researched electricity markets are rather stationary, i.e., seasonal price patterns’ influence on results are low.
Sensitivity of electricity cost savings to exogenous (scenario) parameters: The observation that electricity cost savings for the US and the EU market significantly depend on \({t}_{0}\) and \({t}_{\mathrm{L}}\) is another indicator for the impact of both market’s (individual) time-of-day-specific price patterns that help building operators to identify lucrative opportunities to utilize flexibility in a/c. In addition, \({t}_{0}\) and \({t}_{\mathrm{L}}\) are critical influencing factors for available flexibility window length. The observation of longer flexibility window length significantly increasing electricity cost savings is intuitive, as a longer flexibility window (that is favored by low room or building occupancy) provides the DR approach with a greater economic scope of action. Similar, the dependency of electricity cost savings on \({\mathrm{ID}}_{{\mathrm{t}}_{\mathrm{i}}}\) is intuitive as buildings with less insulation are exposed stronger to (outside) temperature development and, therefore, thermal movement, which results in a higher payback load that shrinks electricity cost savings due to temporal a/c deactivation.
For the University of Georgia’s business case calculation, the expert estimated total costs for implementing and running cost-sensitive a/c control (using the DR approach) to about $100.000 (cf. Section 4.1). Evaluation results illustrate that the payback period for this investment depends especially on electricity cost savings per LS measure and therefore on exogenous scenario parameters (as endogenous model parameters can be calibrated by the building operator). For discounting electricity cost savings, building operators require an appropriate annual risk-free interest rate \({r}_{\mathrm{f}}\). Therefore, for example, they can calculate the mean of the 3-month U.S. Treasury Bill yields observed over the last 10 years, which would currently amount to \({r}_{\mathrm{f}}=0.7\%\) (Mukherji 2011; U.S. Department of Treasury 2017). Moreover, LS frequency is relevant, i.e., how often building operators can conduct LS measures. Applying a common net present value approach, Table 13 shows calculations for the payback period of the business case (without economies of scales, cf. Section 4.1) that authors use to support investment decision making within the described real-world scenario (Table 14).
Table 14 Business Case Payback Periods [Y]