Towards the concept of gas-to-power demand response

Demand response (DR) is a well-established concept in Europe for enhancing system coordination and system reliability via market-based mechanisms; cf. EU-COM (2013) and (Albadi and El-Saadany 2008). In the context of electricity systems, the employment of smart meters—in conjunction with information and communication technology—is turning end users of electricity, such as companies, into active market players. In this context, DR can be understood as electricity demand that is responsive to economic signals. Different DR programs can be differentiated [cf. Eid et al. (2016) for a detailed overview]:To take account to the previously stated goals of the EU-COM (see Sect. 1), an alignment of DR to the consumption of gas (instead of electricity in general) could be a promising approach. In the literature, the concept of “natural gas demand response” is very slowly emerging, but is aimed more at programs for direct gas consumption by industry, etc., and is less related to electricity generation. For example, economic signals are gas prices in price-based programs; e.g., Ruhnau et al. (2022). To account for the consumption of gas-fired electricity and shift or reduce end-use electricity consumption at times when gas-fired power plants are increasingly used to generate electricity, we propose the concept of “gas-to-power DR (G2P-DR)” to mitigate such a situation. In G2P-DR, the economic signals are not the gas prices but the gas factors inherent in the electricity mix. This concept is also motivated by the fact that, for example, in June and July of 2022 large quantities of gas-fired power [in mWh] were generated (and needed) in Germany; see www.​smard.​de. Figure 1 shows the gas-fired power generation in Germany for one working week (Mon–Fri) in July 2022 (Fig. 1 is obtained from www.​smard.​de). However, the gas factors are obtained by dividing the amount of gas-fired power generated during a given period by the maximal amount of gas-fired power generated in a period over all periods of the time horizon considered. Instead of the maximum value, a specified reference value can also be used. Based on these factors, different programs are conceivable, see Subsect. 2.2.

We will now present two possible G2P-DR programs: the first related to RTP adapted to G2P-DR, and the second related to emergency DR. Possibilities to incentivize end users such as companies to participate in one of these programs are discussed in Sect. 4.

In this paper, two topics—namely factorization of electrictiy consumption and event-driven (re-)planning—are combined and applied in production planning to address the current gas crisis in Europe. Since the first topic is closely related to the energy-efficient production planning (EEPP) research strand, we first provide a brief overview of related articles; for a general overview regarding energy-aware production planning, however, see e.g. (Biel and Glock 2016; Gao et al. 2020) and (Bänsch et al. 2021), or (Neufeld et al. 2022) for multicriteria production planning considering, for example, parallel machines (hybrid flowshops in particular). Cf. (Dong and Ye 2022; Oukil et al. 2022) or (Wang and Wang 2022) for current work in the research area of energy efficient production planning.

In energy-efficient production planning, electricity consumption (EC)—as a green measure—is often combined with classical criteria, such as makespan or total tardiness. In more recent developments of EEPP, time-dependent electricity costs are taken into account to evaluate energy consumption with respect to time—leading to a price-based factorization of electricity consumption; cf. e.g. (Mansouri et al. 2016; Jia et al. 2017; Wichmann et al. 2019; Ding et al. 2021) and (Ho et al. 2021). In some contributions, a carbon emission-based factorization of electricity consumption is considered to further focus the green perspective; cf. (Lei and Guo 2015; Ding et al. 2016; Dellnitz et al. 2020; Schulz and Linß 2020) and (Gu et al. 2021). Here, the flexibility of production systems usually is captured by variable machine states (e.g. “on”, “off”, etc.) and/or different (discrete) production speed levels and/or the embedding of parallel machines with different power requirements; cf. e.g. (Ding et al. 2016; Giglio et al. 2017; Schulz et al. 2019, 2020; Dellnitz et al. 2020) and (Wang et al. 2022). In particular, these multicriteria-based studies have shown that minimizing (factorized) electricity consumption generally does not simultaneously minimize the makespan of a production; see also (Mansouri et al. 2016) and (Jia et al. 2017). However, none of these works considered time-dependent gas factorization to address the current European gas crisis.

The second topic—event-driven (re-)planning—is also part of the EEPP literature. In this strand of research, companies’ planning tools are aligned with external signals, such as excessive power availability or insufficient power on the grid. Companies then respond by accelerating or shutting down production, eventually, in order to obtain a reward. Here, the studies of (Zhang et al. 2018; Weitzel and Glock 2019) and (Scholz and Meisel 2022) are worth noting. In Scholz and Meisel (2022), for example, the authors considered, among other things, production scheduling when companies receive a short-term external signal about the availability of excess renewable energy. To deal with such signals, the authors applied a multicriteria approach, combining total tardiness, peak load, and consumption of excess renewable energy. However, handling gas-related events is also an open issue in this research strand.

Table 1 summarizes our observations.

To the best of our knowledge and as a consequence of the above Table 1, the G2P-DR programs proposed in this paper have not yet been studied in the EEPP literature.

We now study a single-stage parallel machine production planning problem in order to exemplify and demonstrate the applicability and the impacts of the two proposed G2P-DR programs in Sect. 2.2. The practical relevance of such problems has been justified by, for example, Moon et al. (2013). We perform short-term integrated scheduling and lot-sizing over one working week, i.e. from 12 p.m. on Sunday to 12 p.m. on Friday, under the following assumptions [(e.g., Giglio et al. (2017)]:According to this problem description, a bicriteria mixed-integer minimization problem (MIP) is formulated by (1)–(14) in the case of RTF. Regarding EDR, (2) is simply replaced by (2*) in (1)–(14):Table 2 contains the corresponding symbols.

With (1) and (2) or (2*), we simultaneously minimize the makespan \(\text {C}_{max}\) and the gas-factorized electricity consumption \(\text {G}_{RTF}\) under RTF or makespan and the penalization of electricity consumption \(\text {G}_{EDR}\) when EDR is applied, respectively. Constraints (3)–(4) control the makespan. More precisely, \(\alpha _t\) equals 1 if a job is processed in a period t on any machine. Note here that M equals the total number of machines and serves as a correction factor if the right hand side is greater than 1. Furthermore, \(t\cdot \alpha _t\) gives the total number of periods if any job is still to be processed in period t. With (5), we have modelled equality conditions for meeting the demanded quantities \(d_j\) for each job j. The due dates of the jobs are therefore at the end of the planning horizon and are thus implicitly considered by (5). Equation (6) ensure that a machine has only one state and never becomes stateless. Equation (7) in combination with (8) control the production state in tandem with the speed level, i.e. \(\delta _{mt\nu }^{state\_I}\) equals 1 if any \(x_{jmt \nu }\) equals 1 in (7). The latter is only the case when one job j is assigned to machine m at speed level \(\nu\) in period t. Equation (8) couple the production mode \(i=I\) of a machine with one speed level exclusively. Constraints (9) control the state transitions of a machine. Here, a machine can either retain a state \(\delta _{imt}^{state}+\delta _{hm,t+1}^{state}=2\), with \(h=i\) and \(1+\gamma _{ih}^{tran}=2\), or can change it, choosing \(h \ne i\), if the transition from state i to state h is feasible. Equation (10) balance the electricity consumption and equation (11) are used for initialization, i.e., a machine m is either in the off state (\(i=0\)) or the ramp-up state (\(i=1\)) in the period \(t=1\). (12)–(14) are binary and non-negativity conditions.

The presented model is solved applying the modified weighted Tchebycheff method (MWTM) in the case of RTF, and both the MWTM and the method of the global criterion using the Tchebycheff metric with lexicographic reoptimization (MGC) in the case of EDR. The weighted Tchebycheff method is a commonly used scalarization method in multi-objective optimization for computing (weak) Pareto optimal solutions. Applying this method with a modified augmented Tchebycheff metric (instead of the classical Tchebycheff metric) leads to strictly Pareto optimal solutions by systematically varying the weights of the objective functions; cf. Liu (2016) and Miettinen (1998). In the case of EDR, we are additionally interested in a single Pareto-optimal solution. Since no preference information of a decision-maker is considered, we apply the method of the global criterion using the Tchebycheff metric and lexicographic reoptimization. The method of the global criterion using the Tchebycheff metric produces a single (weak) Pareto optimal solution that minimizes the distance between a reference point (e.g., the ideal point) and the feasible objective region. Since this can lead to a weak Pareto optimal solution, lexicographic reoptimization is used to address this shortcoming. Further details can be found in Miettinen (1998).

All calculations are conducted via GAMS 41 using CPLEX with 12 threads and a relative optimality criterion of 0.001 on a 64 bit Windows 10 PC with 3500 MHz, 16 cores and 32 logical processors. For example, the computation of corresponding Pareto front representations for the presented instance demonstrated in the next section required in total approximately 5500 s in the case of RTF and approximately 2528 s in the case of EDR. In the latter case, the computation of Pareto-optimal points using the MGC took in total approximately 47 s.

On the left-hand side of Fig. 4, we see a Pareto front representation visualizing the trade-off between makespan and gas-factorized electricity consumption. Consequently, the schedule which results in a minimal makespan does not result in the minimal gas-factorized electricity consumption. The corresponding electricity consumption of those two schedules are depicted in the diagram on the right in Fig. 4. Ultimately, this means that a company can choose whether it executes the schedule minimizing the makespan of 421 periods or the schedule which minimizes gas-factorized electricity consumption but with a makespan of 480 periods, or any Pareto optimal schedule inbetween. This freedom of choice is only possible if flexibility potentials exist (e.g. in production processes and/or a time buffer to meet the demand) enabling the exploitation of trade-offs between makespan and gas-factorized electricity consumption.

But why would a company execute a schedule that reduces gas-factorized electricity consumption if it comes at the expense of the makespan? Therefore, appropriate incentives must exist. An initial guess could be that market-based electricity prices already reflect gas factors so that production scheduling can follow the classic RTP. However, such a one-to-one relationship between electricity prices and gas factors is not generally valid. This means that, under RTP, minimizing electricity costs does not minimize gas-factorized electricity consumption as shown in Fig. 5. In Fig. 5, electricity prices and corresponding gas factors (again, the 29th CW of 2022, Mon–Fri) do have a positive correlation of \(\approx 0.68\). This still results in trade-offs between electricity costs and gas-factorized electricity consumption. We also checked this for each workweek (Mon–Fri) in August 2022 in Germany, and while we see partially higher correlations of \(\approx 70-78\%\), this one-to-one relationship still does not hold.

For a country’s energy supply, it is reliable to manage energy consumption according to classical concepts, e.g., via (local) balancing groups. In times of crisis, however, the scarcity of individual energy resources may force a switch to a different pricing scheme which particularly protects the source in question. Therefore, we show a simple scheme that does not affect the optimality of our planning problems. To incentivize companies aiming at this goal, an electric utility can choose, for example, a price \(p\cdot c_t^{RTF}\) per kWh, with \(p>0\). The company’s objective (2) then adjusts as follows:To put it clearly, an optimal solution regarding (2) is also optimal for (2′). Such a one-to-one correspondence between prices and gas factors can therefore be a reliable tool in crisis situations such as the winter of 2022/2023.

As our second example, we conduct production planning under EDR as shown in Fig. 3. A value of 0.9 has been chosen as the critical gas factor; exceeding such a value triggers an emergency scenario on very short notice, in which electricity consumption during the emergency time window is penalized. In total, there are six time frames in which the critical value is exceeded. Since the companies will be notified to reduce their electricity consumption very shortly before such an emergency period occurs, we propose a rolling-planning approach. This is a somewhat different approach than the one taken in the case of the RTF, where historical data have been used for the calculations, which is often the case in the corresponding literature; cf. e.g. Zhang et al. (2018) or (Scholz and Meisel 2022). However, the rolling planning is conducted using the following procedure:

By applying the described rolling planning, we obtain Fig. 6. Of course, if an initial generated solution is feasible, the subsequent solutions are also feasible. The blue symbols are respective Pareto fronts and the red crosses are the solutions obtained by MGC. In the southwest corner, we see the initial schedule with a makespan of 421 and no penalized electricity consumption (since no event has been triggered so far). By going forward through time and adjusting the schedule if an emergency event is triggered, we obtain a final solution with a makespan of 450 and a penalized electricity consumption of about 2000 kWh. Note here that the 6th event has no effect because the makespan of 450 periods of the final schedule takes place before the corresponding emergency period begins. However, if all events were neglected, the penalized electricity consumption in this example would be 11,227 kWh, which is 9227 kWh higher than the penalized electricity consumption determined by the described procedure.

Figure 7 depicts the schedule obtained by the MGC in the first event, which takes place from period \(t=74\) to \(t=95\), inclusively. It can be seen that when the first event occurs, the speed levels of the machines decrease to consume less power during the specific emergency time window. However, this throttling of the production speed of the machines comes at the expense of the makespan. The result of the rescheduling process due to the second event is illustrated in Fig. 8. The second event takes place in the period \(t=121\) to \(t=128\) inclusive. Again, the speeds of the machines decrease during the duration of the second event, which in turn increases the makespan compared to Fig. 7.

Overall, the rescheduling is mainly reflected in the machine speeds. This behavior is vizualized for the fith and sixth event in Fig. 9. Note that the sixth event has no impact on the schedule because it occurs in the period \(t=455\) to \(t=480\), but the jobs have already been fulfilled. See Appendix 1 for corresponding figures of the other rescheduling processes of the machine utilization.

As in the case of RTF, there have to be flexibility potentials to optimize for penalized electricity consumption. In Fig. 6, points—and thus schedules—are selected that are located between the extreme points of the Pareto front. Of course, other schedules could be selected as well to proceed with such solutions.

In Fig. 10, only those solutions are selected which minimize the penalized electricity consumption when an event occurs. However, care has to be taken when choosing appropriate schedules because this could also lead to a “cold penalization”. This means that a company may lose its flexibility in optimizing electricity consumption if another event occurs because the production makespan has already reached a maximum value that cannot be exceeded without violating the due dates of orders. Note here that period \(t=480\) is the due date for all jobs. In Fig. 10, such a cold penalization leads to a final schedule with a makespan of 480 periods and a penalized electricity consumption of 5108.3 kWh, which is higher than the 2000 kWh under application of MGC. However, this is still lower than the 11,227 kWh in the case where all events are neglected and only the makespan is minimized.

As in the case of RTF, incentives must be provided for companies to participate in such a program. One possibility could be to grant a financial credit. This would be reduced depending on the penalized electricity consumed. The credit surplus would be paid out; in the event of a credit deficit, the company would have to make up the difference.