An empirical study of moving horizon closed-loop demand response scheduling
Introduction
The growing contribution of renewable-based electricity generation presents significant environmental benefits, but has led to an increased supply-side variability and uncertainty for the power grid [1], [2]. Coupled with existing time-of-day and seasonal variability of electricity demand, this phenomenon makes it increasingly challenging to balance electricity supply and demand in grid operations. An attractive approach for mitigating this imbalance is demand-side management, a set of initiatives that focus on reshaping electricity use patterns rather than controlling power generation. Price-based demand-side management, or demand response (DR), relies on time-of-use electricity pricing to influence user-level load shifting decisions. Industrial, commercial, and, to a lower extent, residential users can choose (or are mandated to use) time-varying electricity prices that reflect power demand fluctuations. Broadly speaking, daily electricity prices evolve with grid demand levels, reaching their lowest values during the low demand hours occuring early in the morning, and peaking during peak demand times late in the afternoon.
Industrial users are particularly appealing DR participants, accounting for over 30% of annual electricity consumption in the US [3]. While this figure is lower than the value posted by residential buildings [3], manufacturing plants offer distinct DR benefits: they are large, localized loads that can be coordinated by a single decision-maker (the operator of the facility), and typically exhibit a lower endogenous fluctuation in electricity use. In contrast, residential users are a heterogeneous group of small loads that experience significant electricity consumption fluctuations dictated by endogenous factors such as occupancy and variations in human preferences (e.g., temperature setpoints of heating, ventilation and air conditioning (HVAC) equipment).
Chemical production and petroleum refining make up a significant portion of industrial electricity use (46% in 2018 [4]), suggesting that they are natural candidates for DR. Demand-side management participation of industrial plants includes, e.g., consideration of bidding algorithms for load reduction [5], increasing plant agility [6], and optimal scheduling [7], [8], [9]. In the latter case, which we consider here, DR participation involves a production scheduling strategy comprising two complementary events. Production rates are increased during off-peak electricity demand periods. This means using more electricity but at cheaper rates, resulting in overproduction of chemical products. Products made in excess of demand are stored and used to supply customers during heavy grid load periods, when the production rate (and thereby electricity demand) of the chemical plant is decreased. Thus, DR engagement, as described above, can be regarded as a way to store electricity in the form of chemical products. In order to participate in DR programs, a chemical plant must be able to operate above its nominal product demand level and store any excess product safely and without significant quality degradation. More importantly, engaging in DR programs requires that a plant have the dynamic agility to change its production rate on a time scale comparable to that of the frequency of electricity price changes (hourly or less) established by electricity markets. Given that in many practical situations, the dominant plant dynamics evolve over the same (or longer) time scales as electricity prices, DR scheduling calculations must account for process dynamics. It is also beneficial for scheduling calculations to account for relatively long (in comparison with the process time constant) time horizons. In practice, this amounts to scheduling production over a few days. This can be problematic for several reasons: electricity prices are not known accurately for time spans exceeding 24 h, product demand can fluctuate, and plants are naturally subjected to uncertainty regarding operating conditions (changes in ambient temperature are a good example). Forecasts of these disturbance variables are typically used. Predictions can be quite good for time instants in the near future relative to the time when the forecast is made, but their accuracy declines as longer horizons (typically, more than 24 h) are considered.
There are two broad classes of mechanisms for accounting for such uncertainty. The first is schedule optimization under uncertainty, which requires that some quantitative description of the uncertainty be available, and, depending on the approach taken, can provide a scheduling solution with a known degree of conservativeness. The second is implementing feedback (in the sense of updating the scheduling solution as new information becomes available), which represents a natural way of dealing with exogenous factors whose values cannot be predicted easily for future time instants, but can be measured accurately at the current time instant. The latter approach naturally leads to moving horizon scheduling formulations, which rely on periodically recomputing the scheduling solution as new information concerning the uncertain variables becomes available. The time horizon for the scheduling calculation remains constant, and is “shifted” forward at each rescheduling point.
Several works have utilized moving horizon structures to mitigate uncertainty (both endogenous and exogenous) that arises in scheduling problems. Gupta and Maravelias [10] addressed closed-loop task scheduling using moving horizon scheduling formulations subject to endogenous uncertainty. Shyamal and Swartz [11] scheduled electric arc furnaces with a moving horizon implementation to periodically re-evaluate fractional energy inputs between chemical (e.g. natural gas) and electrical sources based on time-varying electricity prices. He and Petit [12] proposed a moving horizon approach to solve a grid-side DR scheduling problem focusing on thermostatically controlled loads, subject to fluctuations grid demand. Coelho et al. [13] explored rolling horizon scheduling of byproduct gas distribution for steel mills; results revealed that frequent rescheduling improved operations by reducing the effect of uncertainty in gas demand. Mathur et al. [14] performed moving horizon online scheduling of cascaded hydropower systems subject to uncertainty, with a focus on reducing schedule nervousness. In our previous work [15], we introduced a framework for moving-horizon, closed-loop DR scheduling with a focus on the problem formulation, using dynamic process models. In this paper, using the model of an air separation unit (ASU), we present an extensive discussion, focusing on the practical circumstances that may be encountered in the implementation of such a strategy for chemical plants. The computational efficiency of our models enables us to consider a significant number of scenarios compared to other works and can therefore provide a comprehensive picture of the way accounting for uncertainty affects scheduling solutions and operating cost.
Specifically, the key contributions of this effort are:
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an extensive exploration of the impact of typical exogenous disturbances on DR scheduling, using a moving horizon approach enabled by computationally tractable reduced-order models representing the closed-loop nonlinear plant dynamics
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a discussion of methods and models for accounting for exogenous disturbances and uncertainties such as demand changes, price fluctuations and variations in ambient temperature
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a framework for optimization under uncertainty combined with moving horizon scheduling, which ensures the feasibility of the moving horizon DR scheduling problem in the case where the duration (rather than the magnitude) of a non-periodic, non-persistent disturbance (product demand) is not known or cannot be accurately predicted
Section snippets
Problem formulation
Scheduling is part of a hierarchy of decisions involved in the operation of chemical processes (Fig. 1). In the context of DR operation, the scheduling layer utilizes information concerning electricity prices, product demand, and any factors which may impact operating efficiency (such as ambient temperature) as an input, and determines the sequence of production rate targets/setpoints, . The control layer translates these setpoint signals into control actions that are implemented in the
Moving horizon DR scheduling
In principle, the DR schedule could be calculated once by solving a problem of the type outlined in (1), and implemented for the entire scheduling window. This “open loop” approach is optimal if the model is accurate, the predictions for the measured disturbances are perfect and no unmeasured disturbances occur. This is not the case in practice, providing the impetus for developing a feedback mechanism whereby the schedule is updated as new information becomes available [7], [8].
In our study,
ASU Case study
We utilize a single-column air separation unit (ASU) to demonstrate our moving-horizon scheduling method. Cryogenic air separation is an electricity intensive process in the industrial gas sector, which accounted for 2.62% of the yearly industrial electricity consumption in the US in 2014 [29]. The products of air separation (nitrogen, oxygen, and argon) are utility streams in various industries such as steel production and microelectronics. Cryogenic ASUs are attractive for DR participation
Results
All problems were coded in GAMS 25.1.3 [33] and solved using CPLEX 12.8.0 [34] to a 0.1% optimality gap on a 64-bit Windows system with Intel Core-7-2600 CPU at 3.40 GHz and 16 Gb RAM.
Two six-day electricity price vectors are considered (Fig. 9). In price set 1, the price profile for the first 24 h is repeated in the subsequent five days, resulting in a moderate set of prices which are truly periodic. Price set 2 reflects historical day-ahead market prices from CAISO [35], which have high
Discussion and conclusions
Our empirical results reveal that accounting for fluctuations in operating circumstances (electricity prices, ambient conditions) in production scheduling can yield economic benefits compared to steady-state operation, the latter being the accepted (and likely preferred) approach of plant operators. These benefits are reflected in the operating cost of the plant, rather than in a reduction in overall energy use. While the power demand is reduced during peak hours (an important benefit for the
Disclaimer
This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific
CRediT authorship contribution statement
Morgan T. Kelley: Formal analysis, Methodology, Software, Investigation, Writing - original draft. Ross Baldick: Supervision, Funding acquisition. Michael Baldea: Conceptualization, Writing - review & editing, Supervision, Funding acquisition.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This material is based upon work supported by the Department of Energy, United States under Award Number(s) DE-OE0000841. Partial financial support from the National Science Foundation (NSF), United States through the CAREER Award 1454433 and Award CBET-1512379 is acknowledged with gratitude. M.T.K. was supported through the Department of Energy Computational Science Graduate, United States Fellowship through grant number DE-FG02-97ER25308.
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