Electric Power Systems Research 225 (2023) 109816

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Electric Power Systems Research

journal homepage: www.elsevier.com/locate/epsr

An optimization scheme for designing power rationing schedules in a

long-term power shortage
Yuting Mou ∗, Beibei Wang, Zhan Shen
School of Electrical Engineering, Southeast University, Nanjing 210018, Jiangsu Province, China

ARTICLE INFO ABSTRACT

Keywords: Power rationing is the last resort to prevent large-scale blackouts after demand response resources are
Power rationing exhausted during power shortages. However, the traditional rolling blackout method has been criticized for
Orderly power utilization causing significant losses. To address this issue, this paper proposes a novel optimization scheme for designing
Stochastic programming
power rationing schedules in a long-term power shortage, which considers different types of consumers at
Demand side management
multiple time scales. The proposed scheme takes into account economic losses due to limited power supply,
Multiple time scales
Maintenance tasks scheduling
disruptions in industrial chains, and the social costs caused by excessive activation of the same consumers.
Industrial chain First, consumers are categorized as maintenance consumers, work-shift consumers, and fast-response consumers
Fairness based on their consumption characteristics. Then, a two-stage stochastic programming model is presented
to account for long-term uncertainties in power shortage, which yields the maintenance and work-shifting
schedules. Given these predetermined schedules, once the demand–supply gap is better revealed in real-time,
a dispatch model for fast-response consumers is solved to generate the activation schedules. The case study
demonstrates that the proposed scheme can effectively reduce costs when compared to the rolling blackout
approach, as well as respecting industrial chain coupling and fairness.

1. Introduction of this topic, the prevailing method for implementing power rationing
is through simple rolling blackouts, wherein distribution feeders are
The penetration of renewable energy sources has dramatically in- disconnected in a sequential manner for a predetermined duration,
creased in recent years, and extreme weather events occur more fre- resulting in an overall reduction of the total load. Nevertheless, the
quently worldwide, leading to power shortages in some countries. For rolling blackout method has been criticized for causing significant
example, in 2019, South Africa was severely affected by floods and losses because it does not discriminate between higher-cost and lower-
harsh weather conditions, leading to the implementation of rolling cost loads. Consumers often exhibit distinct consumption characteris-
blackouts that lasted for 2 to 4 h at a time [1]. Similarly, in February tics, and there is a limited amount of research available on optimizing
2021, the ERCOT power grid experienced blackouts for several days power rationing schedules accounting for different consumer types at
due to the winter storm Uri [2]. In July and August of 2022, several multiple time scales. Consequently, the research question attempted in
provinces in China had to implement orderly power usage1 during heat
this paper is how to design an optimization scheme specifically for this
waves [3,4].
purpose, in order to mitigate the adverse impacts of power rationing
Unfortunately, power rationing has become a recurring problem in
on both the economy and society in a long-term shortage. More spe-
certain countries, such as Brazil [5], Pakistan [6] and Nepal [7]. It
cially, the following issues need to be addressed when designing power
often lasts for consecutive days, particularly during long-term power
rationing schedules in a long-term power storage.
shortages caused by extreme weather conditions such as cold or heat
waves and dark doldrums. This can result in significant economic and
(i) Consumers have different consumption characteristics and could
social losses. To address this issue, it is necessary to increase the system
be rationed at various time scales, especially those in the indus-
adequacy by investing in new generation capacity in the long run.
trial sector. For example, some plants have a yearly or monthly
However, in the short term,it is crucial to optimize the utilization of
maintenance plan, which would be ideal if scheduled in power
the existing infrastructure by implementing more efficient consumer
shortage periods. Some others are flexible to shift workdays and
rationing strategies during periods of shortages. Despite the significance

∗ Corresponding author.
E-mail address: yutingmou@seu.edu.cn (Y. Mou).
1
It is similar to rolling blackouts, but in a more organized way.

https://doi.org/10.1016/j.epsr.2023.109816
Received 20 April 2023; Received in revised form 17 August 2023; Accepted 2 September 2023
Available online 11 September 2023
0378-7796/© 2023 Elsevier B.V. All rights reserved.
Y. Mou et al. Electric Power Systems Research 225 (2023) 109816

non-workdays in the week. Another type is able to response costs are prioritized for rationing to minimize welfare losses. The trade-
within half an hour and could be rationed on short notice. It is off between rationing quantity and fairness is explored for Nigeria by
an open question how to coordinate these consumers at minimal evaluating the consequences of deviating from proportional allocation
costs in case of long-term shortages. targets for various regions during long-term shortages [12]. While these
(ii) It is undesirable to ration the same consumers repeatedly for the works ration consumers on a regional basis, this study aims to send
sake of social stability and public opinion. Instead, fairness and specific curtailment signals directly to individual consumers.
historical contributions should be taken into consideration. Another relevant topic is ‘‘energy-efficient production planning’’,
(iii) Post-pandemic economic recovery is the focus of most countries which primarily focuses on enhancing energy efficiency and reducing
now and inappropriate power rationing can disrupt industrial energy costs within manufacturing plants. In the study cited as [13], the
chains, resulting in significant economic losses. Thus, it is crucial emphasis lies on energy-aware flexible shop scheduling environments.
to account for industrial chain coupling when power rationing is To achieve hierarchical optimization across multiple objectives such as
implemented, as it can have a cascading effect on downstream energy-related factors (total energy) and temporal factors (makespan,
businesses. Taking the integrated circuits industry as an exam- total flow time, and total idle time), a decision support system com-
ple, if fabs stopped operating, the downstream packaging and prising an iterated local search algorithm is proposed. Furthermore,
testing companies would suffer from increased costs, prolonged a nonlinear mixed-integer optimization model is introduced in [14],
delivery of products or even downtime. which evaluates the trade-off between electricity costs and electricity
consumption under real-time pricing and time-of-use pricing. In [15],
This paper aims to tackle these challenges, and make contributions a mixed-integer linear programming model is formulated considering
in the following three aspects. a distinct electricity tariff structure comprising an energy charge and
a demand charge, with the aim of minimizing the total electricity
(i) An optimization scheme based on stochastic programming is
cost. While the aforementioned works primarily address short-term
introduced to coordinate consumers across multiple time scales,
production planning problems, the model presented in [16] takes into
which can effectively handle uncertainties in long-term short-
account both short-term and long-term flow shop scheduling problems.
ages. The stochastic model outperforms the deterministic model
This model aims to minimize energy costs by considering time-varying
in 79 out of 100 scenarios and reduces the average total costs
electricity prices derived from the day-ahead market EPEX Spot Ger-
by 2.96%.
many/Austria, as well as future price scenarios spanning days or weeks
(ii) Impact factors are proposed, in order to represent industrial
ahead. In [17,18], grid-connected generation systems are taken into ac-
chain coupling and reduce disruptions to the entire chain. The
count, and a mixed-integer linear programming approach is adopted to
synchronized rationing of consumers within the same industrial
model the energy-aware scheduling problem in manufacturing plants.
chain significantly reduces total costs by 19.8%∼193.5% with an
For a more comprehensive review on this topic, readers are encouraged
average of around 51.8%.
to refer to [19–21]. In order to further enhance process efficiency,
(iii) Fairness factors are adopted, in order to account for a con-
researchers have suggested that optimizing production planning should
sumer’s historical contributions and level of cooperation with
be integrated with scheduling maintenance tasks [22,23], especially
the system operator. It mitigates excessive activation of the same
for preventive maintenance, which involves regular maintenance ac-
fast-response consumer by 30.1%∼35.2% in terms of average
tivities to proactively prevent unexpected failures. In this context,
number of activations.
the authors of [24] introduce the concept of ‘‘Stable Maintenance
The remaining sections of this paper are organized as follows. Sec- Task Scheduling’’, to account for system robustness and stability. They
tion 2 reviews related work. Section 3 describes the two-stage stochastic propose a bi-objective robust optimization model to achieve this goal.
model that generates maintenance and work-shift schedules, and a real- While optimization methods have been widely employed for addressing
time activation model for fast-response consumers. Section 4 provides maintenance tasks scheduling challenges [25], reinforcement learning
a case study to demonstrate the effectiveness of the proposed model. techniques are gaining more attention [26,27], due to their great poten-
Finally, Section 5 draws conclusions, discusses practical implementa- tial in generating maintenance policies that surpass most conventional
tion aspects and limitations of this proposal, and points out directions strategies. Notably, the advent of Industry 4.0 and the Internet of
for future research. Things has paved the way for new trends in maintenance task schedul-
ing. For an in-depth exploration of these emerging trends, interested
2. Literature review readers can delve into the comprehensive reviews provided in [28,29].
However, this paper stands apart from the existing body of work due to
There has been a growing interest in developing alternative power its unique focus on optimizing the rationing scheduling of consumers
rationing schemes to replace conventional rolling blackouts. One such from the perspective of the power system operator, in order to meet
scheme [8] introduces a transactive rationing mechanism inspired by the power demand–supply gap. In contrast, these previous studies
quota-based systems. This mechanism ensures a minimum level of concentrate on scheduling the production and maintenance activities
service for all customers and employs market-based control to pri- of specific manufacturing plants to achieve their individual targets.
oritize critical loads. The effectiveness of this mechanism is verified Despite these advancements, the identified research gaps have not
at both the feeder and microgrid levels. Additionally, a stochastic been addressed when designing power rationing schedules in a long-
optimal robust design presented in [9] addresses multi-stage under- term power storage. This paper proposes an optimization scheme for
frequency load shedding, which accounts for load priorities to shed the this purpose, encompassing a two-stage stochastic model to generate
minimum optimal load. Furthermore, an under-voltage load shedding maintenance and work-shift schedules, and an activation model for
approach based on particle swarm optimization and artificial neural fast-response consumers based on real-time demand–supply gaps.
networks is introduced in [10], which optimizes load shedding plans
and costs. However, these studies primarily focus on real-time or short- 3. Model
term power shortages at the feeder or microgrid level, whereas this
paper addresses a long-term power shortage at the system level. 3.1. Problem description
In addition to the aforementioned studies, there is a separate body
of research focusing on power rationing at a broader scale. For instance, During extreme weather events, such as hot and cold waves, power
a study [11] presents an efficient regional rationing approach specif- shortages could last for several weeks. Typically, shortages occur during
ically designed for the Netherlands, where municipalities with lower morning and evening peak hours, which may require power rationing

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Y. Mou et al. Electric Power Systems Research 225 (2023) 109816

Fig. 1. Representation of the proposed scheme to optimize power rationing schedules, which consists of the two-stage stochastic maintenance and work-shift scheduling model, and
the real-time activation model for fast-response consumers. The former generates power rationing schedules for maintenance and work-shift consumers, designated as maintenance
and work-shift schedules. On the other hand, the latter produces power rationing schedules tailored to fast-response consumers, known as fast-response schedules. Industrial chains
are considered across the models.

to prevent blackouts. This paper focuses on industrial and commercial costs and assuming there are sufficient consumers that can be rationed,
consumers, who can be classified into three categories based on their which guarantees the feasibility of the model, stochastic optimization
power consumption characteristics. is employed as the chosen modeling approach. The overall objective of
this model is to minimize the expected total costs, including economic
• Maintenance consumers, who have a relatively stable consump- losses and social costs of fairness. In the first stage, maintenance
tion profile over the year but schedule a maintenance for produc- and work-shift schedules are generated, while the second stage de-
tion lines once per year, e.g., chemical industry and steel industry. termines how to activate fast-response consumers in each scenario of
Therefore, it is preferable that the maintenance is scheduled demand–supply gaps. After solving the two-stage stochastic model, the
during power shortage periods. maintenance and work-shift schedules are subsequently input into a
• Work-shift consumers, who typically consume power five days per real-time activation model, as depicted in the lower panel of Fig. 1.
week on the weekdays, but are able to shift their workdays to When the demand–supply gap is better revealed (based on short-term
other days in the week. forecasts, for example), this model is solved to yield a schedule for fast-
• Fast-response consumers, who are able to decrease their power response consumers. Appendix summarizes the notations introduced in
consumption to a specified level with half an hour, e.g., machin- the subsequent detailed mathematical formulations.
ery & equipment industry.
3.2. Problem formulation
Maintenance and work-shift consumers differ from fast-response
consumers because their power rationing schedules need to be deter- In this section, we present the mathematical formulation of the two-
mined in advance so that they can plan their maintenance or work-shift stage stochastic maintenance and work-shift scheduling model, and the
activities. However, predicting power demand–supply gaps accurately real-time activation model for fast-response consumers.
in the long term is challenging. Therefore, a two-stage stochastic pro-
gramming model is proposed, as shown in the upper panel of Fig. 1. 3.2.1. A two-stage stochastic maintenance and work-shift scheduling model
Stochastic approaches, such as stochastic optimization [30], stochastic This section introduces the two-stage stochastic programming model
control [31], and stochastic analysis [32], have been widely adopted used to generate maintenance and work-shift schedules. The model
to tackle uncertainties in various domains. In this paper, the pro- aims to minimize the expected total costs over multiple scenarios while
posed optimization scheme is centered around this two-stage stochastic respecting constraints related to the power consumption characteristics
programming model. Apart from stochastic optimization, robust opti- of different types of consumers.
mization and chance-constrained optimization are also commonly used To minimize economic losses, power rationing schedules are an-
methods in the field of optimization under uncertainty [33]. Given nounced beforehand to maintenance and work-shift consumers, allow-
that the proposed power rationing scheme aims to minimize expected ing them to adjust their production plans accordingly. This approach

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Y. Mou et al. Electric Power Systems Research 225 (2023) 109816

ensures that the costs of rationing these consumers are negligible. – if consumer 𝑛 is cooperative and indifferent to fairness, the
However, if upstream consumers in the same industrial chain of a main- 𝛽𝑛 = 0 and rationing costs only include economic losses,
tenance or work-shift consumer 𝑛 are also undergoing maintenance or calculated as 𝐶𝑛 ⋅ 𝑝𝑛,𝑡,𝑑,𝜔 .
shift work, then consumer 𝑛 may suffer economic losses. The total cost – if consumer 𝑛 is concerned about fairness, the number of
of rationing maintenance and work-shift consumers can be calculated activation from the first day until day 𝑑 is represented by
∑𝑑−1 ∑
using Eq. (1). 𝜏∈ 𝜇𝑛,𝜏,𝑑,𝜔 . Thus the rationing costs of consumer 𝑛
𝑑 ′ =1
∑ accounts for accumulated activation and the fairness factor
∑ Up,MS 𝑃𝑛′ ⋅ 𝜇𝑛′ ,𝑑
MS 𝑛′ ∈𝑛
𝑓 = ∑ ⋅ 𝛼𝑛 ⋅ 𝐶𝑛 ⋅ (1 − 𝜇𝑛,𝑑 ) ⋅ 𝑃𝑛 , (1) 𝛽𝑛 .
Up,MS 𝑃𝑛′
𝑛∈ MS 𝑛′ ∈ 𝑛
𝑑∈ • Even if a fast-response consumer 𝑛 ∈  F is not rationed (i.e.,
where 𝑛 ∈  MS
represents maintenance or work-shift consumers; 𝜇𝑛,𝑡,𝑑,𝜔 = 0 and 𝑝𝑛,𝑡,𝑑,𝜔 = 0 as required by Eq. (20)), the prop-
𝑑 ∈  represents the day of power rationing; 𝜇𝑛,𝑑 is binary decision agation of industrial chain disruptions due to maintenance and
that indicates whether consumer 𝑛 is subject to power rationing on day work-shift schedules could still affect it. The first term of in
the brackets of Eq. (2) equals zero. Regarding the second term
𝑑; 𝛼𝑛 is a parameter that reflects the impact of upstream consumers ∑ ∑
Up,MS 𝑃𝑛′ ⋅ 𝜇𝑛′ ,𝑑 ∕ Up,MS 𝑃𝑛′ ⋅ 𝛼𝑛 ⋅ 𝐶𝑛 ⋅ (1 − 𝜇𝑛,𝑡,𝑑,𝜔 ) ⋅ 𝑃 𝑛,𝑡 ,
on 𝑛; 𝐶𝑛 is the cost of rationing consumer 𝑛. The set  MS includes 𝑛′ ∈ 𝑛 𝑛′ ∈ 𝑛
Up,MS
all maintenance and work-shift consumers, while 𝑛 refers to the
set of maintenance and work-shift consumers upstream of consumer 𝑛 – if its upstream consumers are not rationed either, then there
in the same industrial chain. This equation encompasses the following are no economic losses and 𝑓 F = 0.
– if the upstream consumers are rationed, the economic loss is
cases.
proportional to the fraction of curtailed power of upstream
• If a maintenance or work-shift consumer 𝑛 ∈  MS is rationed, consumers, multiplied by the impact factor 𝛼𝑛 .
no costs are incurred since the consumer is prepared for the
rationing. In other words, 𝜇𝑛,𝑑 = 1 yields 1 − 𝜇𝑛,𝑑 = 0, resulting in Summing up Eqs. (1) and (2) to derive the overall objective func-
𝑓 MS = 0. tion, and integrating constraints defined by Eqs. (4)∼(21), the two-stage
• If consumer 𝑛 ∈  MS is not rationed (i.e., 𝜇𝑛,𝑑 = 0), stochastic programming model is obtained as follows.

– if its upstream consumers are not rationed either, then there min 𝑓 = 𝑓 MS + 𝑓 F (3)
Up,MS ∑ ∑
are no economic losses. In Eq. (1), 𝜇𝑛′ ,𝑑 = 0, ∀𝑛′ ∈ 𝑛 , s.t. 𝑝𝑛,𝑑 + 𝑝𝑛,𝑡,𝑑,𝜔 ≥ 𝐿𝑡,𝑑,𝜔 , ∀𝑡, ∀𝑑, ∀𝜔, (4)
yields 𝑓 MS = 0. 𝑛∈ MS 𝑛∈ F
– if the upstream consumers are rationed, the industrial chain 𝑝𝑛,𝑑 = 𝑃𝑛 ⋅ 𝜇𝑛,𝑑 , ∀𝑛 ∈  M , ∀𝑑 (5)
may be disrupted, causing 𝑛 to suffer economic losses due ∑
to increased inventory costs or a lack of raw materials. The 𝜇𝑛,𝑑 = 𝐷𝑛M , ∀𝑛 ∈  M (6)
𝑑∈
extent of the disruption can be represented by the fraction of
∑ ∑
𝑑
curtailed power of upstream consumers, i.e, 𝑛′ ∈ Up,MS 𝑃𝑛′ ⋅
∑ 𝑛 𝜈𝑛,𝑑 ′ ≤ 𝜇𝑛,𝑑 , ∀𝑛 ∈  M , 𝑑 ≥ 𝐷𝑛M (7)
𝜇𝑛′ ,𝑑 ∕ 𝑛′ ∈ Up,MS 𝑃𝑛′ . Taking into consideration of the im- 𝑑 ′ =𝑑−𝐷𝑛M +1
𝑛
pact factor 𝛼𝑛 , Eq. (1) is formulated. ∑
𝜈𝑛,𝑑 = 1, ∀𝑛 ∈  M (8)
In contrast to maintenance and works-shift consumers, fast-response 𝑑∈
∑
consumers are activated with short notice, and any interruption to their 𝑧𝑛,𝑑 = 1, ∀𝑛 ∈  M (9)
production plans could have a significant economic impact. Besides 𝑑∈
economic losses, there are also social costs associated with fast-response 𝜇𝑛,𝑑 ≤ 𝜈𝑛,𝑑 , ∀𝑛 ∈  M , 𝑑 = 1 (10)
consumers. These consumers prefer a fair activation, as activating the
M
same consumers every day during a two-week power shortage would 𝜇𝑛,𝑑 = 𝜇𝑛,𝑑−1 + 𝜈𝑛,𝑑 − 𝑧𝑛,𝑑 , ∀𝑛 ∈  , ∀𝑑 ≥ 2 (11)
be unfair and likely lead to complaints. Furthermore, even if a fast- M
𝜇𝑛,𝑑 , 𝜈𝑛,𝑑 , 𝑧𝑛,𝑑 ∈ {0, 1}, ∀𝑛 ∈  , ∀𝑑 (12)
response consumer 𝑛 is not rationed, the propagation of industrial chain S
disruptions due to maintenance and work-shift schedules could still 𝑝𝑛,𝑑 = 𝑃𝑛 ⋅ 𝜇𝑛,𝑑 , ∀𝑛 ∈  , ∀𝑑 (13)
affect it. Taking these factors into consideration, the total expected 𝜇𝑛,𝑑 = 𝜇𝑛,𝑑+7 , ∀𝑛 ∈  S , 𝑑 ≤ 𝐷 − 7 (14)
costs of fast-response consumers is expressed as:
∑
𝑑
[ 𝜇𝑛,𝑑 ′ = 𝐷𝑛S , ∀𝑛 ∈  S , 𝑑 ≥ 7 (15)
∑ ∑∑ ∑ ∑ ∑
𝑑−1
𝑓F = Pr 𝜔 ⋅ 𝐶𝑛 ⋅ 𝑝𝑛,𝑡,𝑑,𝜔 ⋅ (1 + 𝛽𝑛 ⋅ 𝜇𝑛,𝜏,𝑑,𝜔 )+ 𝑑 ′ =𝑑−6
𝜔∈𝛺 𝑑∈ 𝑡∈ 𝑛∈ F 𝑑 ′ =1 𝜏∈ ∑𝑑
∑ ] (2) 𝜈𝑛,𝑑 ′ ≤ 𝜇𝑛,𝑑 , ∀𝑛 ∈  S , 𝑑 ≥ 𝐷𝑛S (16)
Up,MS 𝑃𝑛′ ⋅ 𝜇𝑛′ ,𝑑
𝑛′ ∈𝑛
∑ ⋅ 𝛼𝑛 ⋅ 𝐶𝑛 ⋅ (1 − 𝜇𝑛,𝑡,𝑑,𝜔 ) ⋅ 𝑃 𝑛,𝑡 𝑑 ′ =𝑑−𝐷𝑛S +1
Up,MS 𝑃𝑛′
𝑛′ ∈𝑛 𝜇𝑛,𝑑 ≤ 𝜈𝑛,𝑑 , ∀𝑛 ∈  S , 𝑑 = 1 (17)
where 𝐶𝑛 represents the economic cost of rationing consumer 𝑛 ∈  F, 𝜇𝑛,𝑑 = 𝜇𝑛,𝑑−1 + 𝜈𝑛,𝑑 − 𝑧𝑛,𝑑 , ∀𝑛 ∈  , 𝑑 ≥ 2 S
(18)
while 𝑝𝑛,𝑡,𝑑,𝜔 is the curtailed power in period 𝑡 day 𝑑 scenario 𝜔 and
S
𝜇𝑛,𝑡,𝑑,𝜔 denotes whether consumer 𝑛 is activated in period 𝑡 day 𝑑 𝜇𝑛,𝑑 , 𝜈𝑛,𝑑 , 𝑧𝑛,𝑑 ∈ {0, 1}, ∀𝑛 ∈  , ∀𝑑 (19)
scenario 𝜔. 𝑃 𝑛,𝑡 represents the upper bound of the curtailable power of 𝑃 𝑛,𝑡 ⋅ 𝜇𝑛,𝑡,𝑑,𝜔 ≤ 𝑝𝑛,𝑡,𝑑,𝜔 ≤ 𝑃 𝑛,𝑡 ⋅ 𝜇𝑛,𝑡,𝑑,𝜔 (20)
consumer 𝑛. In addition, some consumers view power curtailment as a ∑
social responsibility when the system is stressed and wish to contribute. 𝜇𝑛,𝑡,𝑑,𝜔 ≤ 1, ∀𝑛 ∈  F , ∀𝑑 ∈ , ∀𝜔 ∈ 𝛺 (21)
𝑡∈
Therefore, to accommodate the preferences of different fast-response
consumers, a fairness factor denoted by 𝛽𝑛 is introduced. This equation The total curtailed power from rationed maintenance and work-shift
∑ ∑
covers the following cases. consumers 𝑛∈ MS 𝑝𝑛,𝑑 , and fast-response consumers 𝑛∈ F 𝑝𝑛,𝑡,𝑑,𝜔
should satisfy the demand–supply gap, as indicated by Eq. (4), where
• If a fast-response consumer 𝑛 ∈  F is rationed (i.e., 𝜇𝑛,𝑡,𝑑,𝜔 = 𝐿𝑡,𝑑,𝜔 is the gap in period 𝑡 ∈  day 𝑑 ∈  scenario 𝜔 ∈ 𝛺.
1), the second term of in the brackets of Eq. (2) equals zero. When a maintenance consumer 𝑛 is subject to power rationing,
∑ ∑
Regarding the first term 𝐶𝑛 ⋅ 𝑝𝑛,𝑡,𝑑,𝜔 ⋅ (1 + 𝛽𝑛 ⋅ 𝑑−1
𝑑 ′ =1 𝜏∈ 𝜇𝑛,𝜏,𝑑,𝜔 ), their curtailed power equals the maximum power that can be curtailed;