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A Mean Value Cross Decomposition Strategy for Demand-side Management of a Pulping Process
Krist V. Gernaey, Jakob K. Huusom and Rafiqul Gani (Eds.), 12th International Symposium on Process Systems Engineering and 25th European Symposium on Computer Aided Process Engineering. 31 May – 4 June 2015, Copenhagen, Denmark © 2015 Elsevier B.V. All rights reserved.
A Mean Value Cross Decomposition Strategy for Demand-side Management of a Pulping Process Hubert Haderaa,b, Per Wide c, Iiro Harjunkoskia*, Juha Mäntysaari d, Joakim Ekström c, Guido Sanda, Sebastian Engellb a
ABB Corporate Research, Wallstadter Str. 59, 68526 Ladenburg, Germany Technical University of Dortmund, Emil-Figge-Str. 70, 44221 Dortmund, Germany c Linköping University, Campus Norrköping, SE-60174 Norrköping, Sweden d ABB Oy Industry Solutions, CPM, Strömbergintie 1 B, 00380 Helsinki, Finland *[email protected] b
Abstract Energy is becoming a critical resource for process industries as introduction of new policies drive changes in the energy supply systems. Energy availability and pricing is much more volatile. In this study, we propose a Mean Value Cross Decomposition approach to functionally separate production scheduling from energy-cost optimization. Such a decomposition makes it possible to exploit existing optimization solutions avoiding a need to create a new monolithic model. The proposed framework is applied to a continuous process of thermo-mechanical pulping using a discrete-time ResourceTask Network model. Example case study scenarios show that the approach gives optimal system-wide solutions while keeping the models separated. Keywords: Scheduling, demand-side management, mean value cross decomposition.
1. Introduction Today’s electricity supply systems face new challenges due to introduction of new policies, volatility of energy availability and prices. One of the important technologies that support the transformation is demand-side response which on the operations level calls for energy-aware scheduling methods. Especially for large scale industrial energy users, energy-aware production scheduling can reduce the operating cost, as shown in case studies available in literature (Hadera et al., 2014). The traditional industrial approach for scheduling of energy intensive operations is to schedule the production first, satisfying all production specific rules, and to predict from that schedule the demand for energy. Next, energy purchase and sale optimization models can be used to find the best contracting strategy. This approach in general is suboptimal on the system level. There could be a schedule for the production that creates a demand with significantly lower total energy cost, out-weighting the sub-optimality of the schedule from the production point of view. The traditional strategy reported in literature is to combine all energy-related information into a single monolithic problem and to solve it. This ensures finding a system-wide optimal solution that satisfies both the scheduling constraints and energy-cost optimization constraints. However, from an industrial point of view, such a monolithic integration usually requires noticeable effort.
2. Problem statement The goal of this work is to develop a decomposition framework for process scheduling and energy cost optimization while still ensuring a system-wide optimal solution.
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The scheduler provides a feasible schedule, and the energy cost optimizer finds the best purchase and sales structure of contracts based on a fixed load curve. This framework helps to utilize already installed industrial solutions of energy contracts optimization. The joint optimization obviously requires an information exchange between the two optimizers. Under specific conditions the structure of the monolithic model can be exploited to develop a Mean Value Cross Decomposition (MVCD) scheme (Van Roy, 1983) – a variant of decomposition without a master problem (Holmberg, 1994). In the next section we show that the two optimizers can be modified such that one forms a part of the subproblem of the Benders’ decomposition of the overall problem (energy-cost optimizer) and one forms a part of the subproblem of the Dantzig-Wolfe decomposition (energy-aware scheduler) after decomposing the monolithic model. It is important to note that it is not the goal of this work to develop the best possible scheduling model for the example process, but to develop and demonstrate the decomposition framework. 2.1. Industrial process description We apply the decomposition approach on an energy-intense industrial process, ThermoMechanical Pulping (TMP). We formulate the scheduling model as a Mixed Integer Linear Program (MILP) using a discrete-time Resource-Task Network (RTN) representation (Castro et al., 2009). The model handles all production-specific constraints and includes information concerning the consumption of electricity. The scheduling model is expanded by a formulation for the minimization of deviation penalties which are accounted for when the process load deviates from the pre-agreed load curve (Hadera et al., 2014). Energy-cost optimization uses the Min-Cost Flow Network for the optimization of multiple time-sensitive electricity contracts including base load, time-of-use, day-ahead spot market and onsite power generation, and the opportunity to sell electricity back to the grid with revenues. The example process (Figure 1) consists of identical refiners which produce one type of pulp. The pulp flows to a storage tank and has to satisfy a deterministic demand curve. Additional flexibility is given by the option of buying pulp from an external source at a given cost.
3. Model formulation In order to develop the decomposition approach we first develop a monolithic model of the problem. The model notation is shown in Table 1.
Figure 1. RTN superstructure for the thermo-mechanical pulping process
A Mean Value Cross Decomposition Strategy for Demand-side Management of a Pulping Process
1933
Table 1. Model notation Sets: ܴ– resources; –ܫtasks, ܶ- time intervals, ܳܧ1/ܳܧ2 ܴ ك-refiners/storage tank, ܳܧ3 ܴ ك-external buy, ܴܴ ك ܯ-raw material, ݈ݑ௧ /݈ݑ௫௧ ܴ ك-pulp from refiners/external source, ܷܶ ܴ ك-electricity, ܴ ك ܶܥ-cont. resources, ܴ ك ܲܨ-final product, ܫ - continues tasks, ܫ -buying task, ܫௌ -storage task. Parameters: ݈௫ –maximum capacity volume of storage tank, ݈ -minimum volume in storage, ݈௧ -initial volume in storage, ݑ-production rate of refiner in one time slot, ݑ௫ -maximum production by all refiners in one time slot, ܾ௫ -maximum pulp amount available to buy, ܿ ௦௧௧ /ܿ ௗ -refiner start-up/shut-down ௨ ௨௧ cost, ς ,௧ /ς,௧ -resource input (electricity)/ output (final product) to/from the system in time t, ܿ௧ external pulp price in slot t, -electricity consumption of one refiner in one time slot. כ Variables: ܰ,௧ -binary, execution of task i in time t, ߦ,௧ /ߦ,௧ - cont. positive, amount handled /continuously send ௗ to storage, ܴ,௧ - cont. positive, amount of r available at time t, ܴ,௧ - cont. positive, amount of resource r available immediately before end of time t, ܴ,௧ - cont. positive, amount of initial resource, ݊௧ -number of refiners, ݊௧ௗ /݊௧௦௧௧ - cont. positive, number of refiners shut-down/start-up, ݈ݑ௧௫௧ /݈ݑ௧௧ - cont. positive, pulp bought externally/produced by refiners, ݍ௧ - cont. positive, el. consumption, ܿ -net electricity cost, ܿ -cost of start-up and shut-down of refiners, ߜ-penalties for load deviation
3.1. Monolithic model formulation The objective function of the problem is to minimize the cost of energy and the production-specific costs as in Eq. (1). The core of the RTN formulation is the excess resource balance equation, shown in Eq. (2), and the handling of the continuous resources by Eq. (3). Proper balance of the ݈ݑ௧ and final product ݂ resource is ensured by Eq. (4). Using resource excess variable, we enforce the upper bound of all equipment resources and enforce no intermediate storage by Eq. (5). The storage capacities limits are enforced by Eq. (6). Another constraint restricts the amount handled by the hybrid task (Eq. 7) and continuous tasks (Eq. 8). Buying of pulp from the external source is limited by the desired level ܾ ௫ as in Eq. (9). Some auxiliary variables are introduced in Eq. (10-11). Eq. (12) accounts for the electricity consumption in each time slot. We capture the number of refiners starting up or shutting down with the constraint in Eq. (13). Lastly, we account for the production costs in the objective function. The cost of the pulp bought (Eq. 14) and start-ups and shut-downs of refiners (Eq. 15) are accounted for. ݉݅݊(݁ + ߜ + ܿ ௫௧௨ + ܿ )
(1)
ܴ,௧ = ௗ ܴ,௧ |௧ୀଵ + ܴ,௧ିଵ |אோೃಾ ோೠ ோಷು + ܴ,௧ିଵ |אோಶೂభ ோಶೂమ ோಶೂయ ோೠೣ + σאூ(ߤ, ܰ,௧ + ௨௧ |אோೆ െ ߎ,௧ |אோಷು ܴ א ݎ, ܶ א ݐ (2) ݒ, ߦ,௧ + ߤ, ܰ,௧ିଵ ) + σאூೄ(ߤ, ܰ,௧ |௧ୀଵ ) + ߎ,௧ כ ௗ ܴ,௧ = ܴ,௧ + σאூ ߣ, ߦ,௧ + σאூೞ൫ݒ, ߦ,௧ + ߣ, ߦ,௧ ൯ + σ אூಳ ൫ݒ, ߦ,௧ ൯ ோெ ௨ ி ܴ ܴ א ݎ א ݎ ܴ ,ܶ א ݐ ௗ = 0 ܴ א ܴ,௧
௨
ௗ ௨௧ , ܴ ; |ܶ| = ݐ,௧ ߎ,௧ ܴ א ݎி , ܶ א ݐ
௫ ௫ ܴ,௧ = 1 ܴ א ݎாொଵ ܴ ாொଶ ܴ ாொଷ , ܴ ; ܶ א ݐ,௧ = 0 ܴ א ݎ
݈
ௗ ܴ,௧
݈ ௫
א ݎ
ܴி , ݐ
ܶא
כ ߦ,௧ + ߦ,௧ ൫݈ ௧ + ܾ ௫ + ݑ௫ ൯ ή ܰ,௧ ܫ א ݅ௌ , ܶ א ݐ
ߦ,௧ = ݑή ܰ,௧
ܫ א ݅, ܶ א ݐ
ߦ,௧ ܾ ௫ ή ܰ,௧ ܫ א ݅ , ܶ א ݐ ݊௧
(3)
= σאூ ܰ,௧ ܶ א ݐ
݈ݑ௧௧ = σאூ ߦ,௧ ݈ݑ ; ܶ א ݐ௧௫௧ = σאூಳ ߦ,௧ ܶ א ݐ
(4) ௨
ܴ ி , ܶ א ݐ
(5) (6) (7) (8) (9) (10) (11)
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ݍ௧ = ڄσאூ ܰ,௧ ܶ א ݐ
݊௧
= ݊௧ିଵ + ݊௧௦௧௧ െ ݊௧ௗ ܶ א ݐ
ܿ ௫௧௨ =
σ்௧ୀଵ ܿ௧௨
ή ݈ݑ௧௫௧
ܿ = σ்௧ୀଵ൫ܿ ݐݎܽݐݏή ݊௧௦௧௧ + ܿ݁݊݀ ή ݊௧ௗ ൯
(12) (13) (14) (15)
We assume that for the considered production process the energy cost optimization problem has the same structure as the MILP formulation in Hadera et al. (2014). The plant has an option to choose from multiple contracts, onsite generation, and the possibility to sell electricity back to the grid. Therefore the monolithic problem is further extended by the same flow network as in Hadera et al. (2014), Eq. (13-22), and load deviation equations as in Eq. (30-32), which together represent all energy-related cost (net electricity cost ݁ and load deviation penalties ߜ). 3.2. Functional decomposition approach using MVCD The monolithic MILP problem can be simplified to the following general form: ݉݅݊ ( ܥଵ் ݂ + ܥଶ் )ݕ
(16)
s.t. ܣଵ ݂ + ܦଵ ݔ ܾଵ
[flow network constraints (energy purchase and sale)]
ܣଶ ݍ+ ܦଶ ݕ ܾଶ
[production scheduler constraints (optimize production cost)] (18)
ݍെ ܣଷ ݂ = 0
[complicating constraint]
ܺ א ݔ, ܻ א ݕ, ݍ 0, ݂ 0.
(17)
(19) (20)
In the above equations let ݔand ݕrepresent the variables specific to the flow network and to the energy-aware scheduler, and let ܺ and ܻ represent all variable bounds and types (continuous and integer requirements). Eq. (19) is a complicating constraint that links the two kinds of variables which are present in Eqs. (17-18). Note that the term ܣଷ ݂ represents the amount of electricity needed to be bought for the production process, i.e. the flow from the balancing node to the process demand node in any time slot. Any change in the values of any variable in Eq. (19) causes a change in the respective part of the monolithic model (Eq. 17 or 18), thus also in the value of the objective function components in Eq. (16), either the specific cost of the flow network ܥଵ் ݂ or the production specific cost ܥଶ் ݕ. We consider the load deviation constraints to be integrated into the scheduling problem, i.e. the penalties are part of ܥଶ் ݕ. The monolithic formulation can be decomposed using well known techniques of Benders’ and DantzigWolfe (D-W) decomposition. The sub-problems of these two decompositions , shown in Table 2, can be combined in a MVCD scheme (Holmberg, 1994). Note that both subproblems will separate into two different problems; one related to the scheduling problem, and one related to the energy network. Further, the dual variable of Eq. (19) in the flow network part of Benders’ sub-problem is equal to the Marginal Cost (MC) curve used in the D-W’s sub-problem. Secondly, the schedule from the scheduler’s part of the D-W’s sub-problem is equal to the fixed schedule in Benders’ sub-problem. Thus by iteratively solving these sub-problems and exchanging the information on production schedule and marginal cost curves (forward related to as the two signals) between them, a Cross Decomposition (CD) scheme can be constructed (Van Roy, 1983). Further, we can identify the flow network part of Benders’ sub-problem as the energy cost minimization problem with a fixed production schedule.
A Mean Value Cross Decomposition Strategy for Demand-side Management of a Pulping Process
1935
Table 2. Decomposed models Benders’ sub-problem flow network ഥ ௧) scheduler ݂ ாௌ (ݍԢ ഥ ௧) ݂ ெிே (ݍԢ ݉݅݊(Ɂ + ܿ ௫௧௨ + ܿ ) (21) subject to: Eq. (2-15) – production scheduling constraints Eq. (30-32) from [2] – load deviation penalties constraints ݍ௧ = ݍത௧
݉݅݊(ܿ ) (22)
Dantzig-Wolfe sub-problem flow network തതതതത௧ ) scheduler ݂ ாௌ (ܥܯԢ തതതതത௧ ) ݂ ெிே (ܥܯԢ ݉݅݊(ܿ െ ݉݅݊(ߜ + ܿ ௫௧௨ + തതതതത௧ ) തതതതത௧ ) σ௧݂ ்אହ,୧,௧ ή ܥܯ ܿ + σ௧ݍ ்א௧ ή ܥܯ (23) (24)
Eq. (13-22) from [3] – flow network constraints
Eq. (2-15) – production scheduling constraints
݂ହ,୧,௧ = ݍത௧
Eq. (30-32) from [2] – load deviation penalties constraints
ܶ א ݐ
Eq. (13-22) from [3] – flow network constraints
ܶ א ݐ
Next, the scheduler part of D-W’s sub-problem can be formulated as an energy-aware RTN scheduler with the MC curve as actual costs of energy that are penalized in the objective function. Thus we have identified the two problems which are currently being used in the industrial setting. However, a CD scheme requires the solution of additional complex master problems to guarantee convergence. An alternative is to adopt a MVCD which instead of using a master problem to guarantee convergence makes use of the mean values of the two signals (Holmberg, 1994). To solve the two partial subproblems in Eq. (22-23) is enough to find a feasible solution of the monolithic problem - a schedule and the corresponding optimal purchasing and selling structure. However, to test the convergence, the value of the objective function of the complete subproblems is needed (Eq. 21-24). The iterative algorithm (Fig. 2) is initialized by solving the production scheduling model using the day-ahead spot market price. There are a few reported variations of MVCD which differ on how to alter signals (ݍ௧ᇱ and ܥܯ௧ᇱ in Fig. 3) in each iteration. We selected the following option: x
One-sided MVCD (OSWMVCD) – only one of the signals (ܥܯԢ௧ = )ݕis altered by calculating a weighted mean as in Eq. (25), the other (ݍԢ) is used directly without any changes in order to overcome potential infeasibility problems: ఉାఊ ݕ = ߜ ݕ ିଵ + (1 െ ߜ ) ݕିଵ , where: ߜ = , σஶ (25) ୀ ߜ = λ, ߚ = 1 and ߛ = 3 ఉାఊ
Figure 2. Generic framework for functional separation
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4. Numerical case study Three test instances were solved using GAMS/CPLEX 24.1.2 for the different solution approaches - monolithic model (RTN), convergence test with complete sub-problems (CRTN), and industrial approach (IOWMV), the two latter ones (Fig. 2) using OSWMCD and iterating ܥܯ௧ only. With such a setup the load curve is always sent directly without mean value smoothing, which could cause infeasibility problems. For all non-monolithic approaches, a CPU limit of 180s is enforced. The results are reported in Table 3. Interestingly the decomposition resulted in higher CPU times than RTN. Table 3. Numerical results for the case study Best upper Best lower Algorithm iterations bound bound (best) RTN 219637,2 1 CRTN* 219637,2 219637,2 13 IOWMV*‡ 219637,2 2 RTN -623600 2 CRTN* -623600 -623600 2 IOWMV*‡ -623600 2 RTN 122869,2 3 CRTN* 123246,3 122681,78 30 (30) IOWMV*‡ 123246,3 30 (15) * solved with max CPUs limit of 180s; ‡ solved with algorithm iterations limit of 30 Model
Scenario
Total CPUs 1,14 24,60 3,2 1,09 5,96 3,55 340,92 776,84 285,70
Gap 0% 0% 0% 0% 0% 0% 0% 0,31% 0,31%
5. Discussion and conclusion The tests showed that for both variants of decomposition strategies optimal or close to optimal solutions are obtained for the considered TMP process. In general, the lack of optimality for the developed approach might be due to the duality gap between the objective function values of Benders’ and Dantzig-Wolfe sub-problems. In addition, using IOWMV instead of mean values on both signals, there is no guarantee of convergence. Taking mean values on integer variables would produce infeasible solutions, but can be expected to converge to the Lagrangean relaxation (Holmberg, 1994). However, for industrial cases the presented framework allows simultaneously using existing models and solvers for the individual problems resulting in very good system-wide solutions. This re-use of existing solution also eases maintenance problems. Further work could focus on investigating other industrial processes such as steel-making. Another direction is to employ different combination of raw materials and resources as a subject of optimization, instead of energy considered in this work.
Acknowledgements The Marie Curie FP7-ITN research project "ENERGY-SMARTOPS", Contract No: PITN-GA-2010-264940 is acknowledged for financial support.
References Castro, P. M., Harjunkoski, I. and Grossmann, I. E., 2009, New continuous-time scheduling formulation for continuous plants under variable electricity cost, I.E.Ch.R.,48, 14, 6701-6714 Hadera, H., Harjunkoski, I., Grossmann, I. E., Sand, G. and Engell, S., 2014, Steel production scheduling under time-sensitive electricity cost, Comp. Aided Chem Eng, 33, 373-378 Holmberg, K., 1997, Mean value cross decomposition applied to integer programming problems, Europ Journal of Oper Res, 97, pp. 124-138 Van Roy, T. J., 1983, Cross Decomposition for Mixed Integer Programming, M. P. 25, 46-63
A Mean Value Cross Decomposition Strategy for Demand-side Management of a Pulping Process Hubert Haderaa,b, Per Wide c, Iiro Harjunkoskia*, Juha Mäntysaari d, Joakim Ekström c, Guido Sanda, Sebastian Engellb a
ABB Corporate Research, Wallstadter Str. 59, 68526 Ladenburg, Germany Technical University of Dortmund, Emil-Figge-Str. 70, 44221 Dortmund, Germany c Linköping University, Campus Norrköping, SE-60174 Norrköping, Sweden d ABB Oy Industry Solutions, CPM, Strömbergintie 1 B, 00380 Helsinki, Finland *[email protected] b
Abstract Energy is becoming a critical resource for process industries as introduction of new policies drive changes in the energy supply systems. Energy availability and pricing is much more volatile. In this study, we propose a Mean Value Cross Decomposition approach to functionally separate production scheduling from energy-cost optimization. Such a decomposition makes it possible to exploit existing optimization solutions avoiding a need to create a new monolithic model. The proposed framework is applied to a continuous process of thermo-mechanical pulping using a discrete-time ResourceTask Network model. Example case study scenarios show that the approach gives optimal system-wide solutions while keeping the models separated. Keywords: Scheduling, demand-side management, mean value cross decomposition.
1. Introduction Today’s electricity supply systems face new challenges due to introduction of new policies, volatility of energy availability and prices. One of the important technologies that support the transformation is demand-side response which on the operations level calls for energy-aware scheduling methods. Especially for large scale industrial energy users, energy-aware production scheduling can reduce the operating cost, as shown in case studies available in literature (Hadera et al., 2014). The traditional industrial approach for scheduling of energy intensive operations is to schedule the production first, satisfying all production specific rules, and to predict from that schedule the demand for energy. Next, energy purchase and sale optimization models can be used to find the best contracting strategy. This approach in general is suboptimal on the system level. There could be a schedule for the production that creates a demand with significantly lower total energy cost, out-weighting the sub-optimality of the schedule from the production point of view. The traditional strategy reported in literature is to combine all energy-related information into a single monolithic problem and to solve it. This ensures finding a system-wide optimal solution that satisfies both the scheduling constraints and energy-cost optimization constraints. However, from an industrial point of view, such a monolithic integration usually requires noticeable effort.
2. Problem statement The goal of this work is to develop a decomposition framework for process scheduling and energy cost optimization while still ensuring a system-wide optimal solution.
1932
H. Hadera et al.
The scheduler provides a feasible schedule, and the energy cost optimizer finds the best purchase and sales structure of contracts based on a fixed load curve. This framework helps to utilize already installed industrial solutions of energy contracts optimization. The joint optimization obviously requires an information exchange between the two optimizers. Under specific conditions the structure of the monolithic model can be exploited to develop a Mean Value Cross Decomposition (MVCD) scheme (Van Roy, 1983) – a variant of decomposition without a master problem (Holmberg, 1994). In the next section we show that the two optimizers can be modified such that one forms a part of the subproblem of the Benders’ decomposition of the overall problem (energy-cost optimizer) and one forms a part of the subproblem of the Dantzig-Wolfe decomposition (energy-aware scheduler) after decomposing the monolithic model. It is important to note that it is not the goal of this work to develop the best possible scheduling model for the example process, but to develop and demonstrate the decomposition framework. 2.1. Industrial process description We apply the decomposition approach on an energy-intense industrial process, ThermoMechanical Pulping (TMP). We formulate the scheduling model as a Mixed Integer Linear Program (MILP) using a discrete-time Resource-Task Network (RTN) representation (Castro et al., 2009). The model handles all production-specific constraints and includes information concerning the consumption of electricity. The scheduling model is expanded by a formulation for the minimization of deviation penalties which are accounted for when the process load deviates from the pre-agreed load curve (Hadera et al., 2014). Energy-cost optimization uses the Min-Cost Flow Network for the optimization of multiple time-sensitive electricity contracts including base load, time-of-use, day-ahead spot market and onsite power generation, and the opportunity to sell electricity back to the grid with revenues. The example process (Figure 1) consists of identical refiners which produce one type of pulp. The pulp flows to a storage tank and has to satisfy a deterministic demand curve. Additional flexibility is given by the option of buying pulp from an external source at a given cost.
3. Model formulation In order to develop the decomposition approach we first develop a monolithic model of the problem. The model notation is shown in Table 1.
Figure 1. RTN superstructure for the thermo-mechanical pulping process
A Mean Value Cross Decomposition Strategy for Demand-side Management of a Pulping Process
1933
Table 1. Model notation Sets: ܴ– resources; –ܫtasks, ܶ- time intervals, ܳܧ1/ܳܧ2 ܴ ك-refiners/storage tank, ܳܧ3 ܴ ك-external buy, ܴܴ ك ܯ-raw material, ݈ݑ௧ /݈ݑ௫௧ ܴ ك-pulp from refiners/external source, ܷܶ ܴ ك-electricity, ܴ ك ܶܥ-cont. resources, ܴ ك ܲܨ-final product, ܫ - continues tasks, ܫ -buying task, ܫௌ -storage task. Parameters: ݈௫ –maximum capacity volume of storage tank, ݈ -minimum volume in storage, ݈௧ -initial volume in storage, ݑ-production rate of refiner in one time slot, ݑ௫ -maximum production by all refiners in one time slot, ܾ௫ -maximum pulp amount available to buy, ܿ ௦௧௧ /ܿ ௗ -refiner start-up/shut-down ௨ ௨௧ cost, ς ,௧ /ς,௧ -resource input (electricity)/ output (final product) to/from the system in time t, ܿ௧ external pulp price in slot t, -electricity consumption of one refiner in one time slot. כ Variables: ܰ,௧ -binary, execution of task i in time t, ߦ,௧ /ߦ,௧ - cont. positive, amount handled /continuously send ௗ to storage, ܴ,௧ - cont. positive, amount of r available at time t, ܴ,௧ - cont. positive, amount of resource r available immediately before end of time t, ܴ,௧ - cont. positive, amount of initial resource, ݊௧ -number of refiners, ݊௧ௗ /݊௧௦௧௧ - cont. positive, number of refiners shut-down/start-up, ݈ݑ௧௫௧ /݈ݑ௧௧ - cont. positive, pulp bought externally/produced by refiners, ݍ௧ - cont. positive, el. consumption, ܿ -net electricity cost, ܿ -cost of start-up and shut-down of refiners, ߜ-penalties for load deviation
3.1. Monolithic model formulation The objective function of the problem is to minimize the cost of energy and the production-specific costs as in Eq. (1). The core of the RTN formulation is the excess resource balance equation, shown in Eq. (2), and the handling of the continuous resources by Eq. (3). Proper balance of the ݈ݑ௧ and final product ݂ resource is ensured by Eq. (4). Using resource excess variable, we enforce the upper bound of all equipment resources and enforce no intermediate storage by Eq. (5). The storage capacities limits are enforced by Eq. (6). Another constraint restricts the amount handled by the hybrid task (Eq. 7) and continuous tasks (Eq. 8). Buying of pulp from the external source is limited by the desired level ܾ ௫ as in Eq. (9). Some auxiliary variables are introduced in Eq. (10-11). Eq. (12) accounts for the electricity consumption in each time slot. We capture the number of refiners starting up or shutting down with the constraint in Eq. (13). Lastly, we account for the production costs in the objective function. The cost of the pulp bought (Eq. 14) and start-ups and shut-downs of refiners (Eq. 15) are accounted for. ݉݅݊(݁ + ߜ + ܿ ௫௧௨ + ܿ )
(1)
ܴ,௧ = ௗ ܴ,௧ |௧ୀଵ + ܴ,௧ିଵ |אோೃಾ ோೠ ோಷು + ܴ,௧ିଵ |אோಶೂభ ோಶೂమ ோಶೂయ ோೠೣ + σאூ(ߤ, ܰ,௧ + ௨௧ |אோೆ െ ߎ,௧ |אோಷು ܴ א ݎ, ܶ א ݐ (2) ݒ, ߦ,௧ + ߤ, ܰ,௧ିଵ ) + σאூೄ(ߤ, ܰ,௧ |௧ୀଵ ) + ߎ,௧ כ ௗ ܴ,௧ = ܴ,௧ + σאூ ߣ, ߦ,௧ + σאூೞ൫ݒ, ߦ,௧ + ߣ, ߦ,௧ ൯ + σ אூಳ ൫ݒ, ߦ,௧ ൯ ோெ ௨ ி ܴ ܴ א ݎ א ݎ ܴ ,ܶ א ݐ ௗ = 0 ܴ א ܴ,௧
௨
ௗ ௨௧ , ܴ ; |ܶ| = ݐ,௧ ߎ,௧ ܴ א ݎி , ܶ א ݐ
௫ ௫ ܴ,௧ = 1 ܴ א ݎாொଵ ܴ ாொଶ ܴ ாொଷ , ܴ ; ܶ א ݐ,௧ = 0 ܴ א ݎ
݈
ௗ ܴ,௧
݈ ௫
א ݎ
ܴி , ݐ
ܶא
כ ߦ,௧ + ߦ,௧ ൫݈ ௧ + ܾ ௫ + ݑ௫ ൯ ή ܰ,௧ ܫ א ݅ௌ , ܶ א ݐ
ߦ,௧ = ݑή ܰ,௧
ܫ א ݅, ܶ א ݐ
ߦ,௧ ܾ ௫ ή ܰ,௧ ܫ א ݅ , ܶ א ݐ ݊௧
(3)
= σאூ ܰ,௧ ܶ א ݐ
݈ݑ௧௧ = σאூ ߦ,௧ ݈ݑ ; ܶ א ݐ௧௫௧ = σאூಳ ߦ,௧ ܶ א ݐ
(4) ௨
ܴ ி , ܶ א ݐ
(5) (6) (7) (8) (9) (10) (11)
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ݍ௧ = ڄσאூ ܰ,௧ ܶ א ݐ
݊௧
= ݊௧ିଵ + ݊௧௦௧௧ െ ݊௧ௗ ܶ א ݐ
ܿ ௫௧௨ =
σ்௧ୀଵ ܿ௧௨
ή ݈ݑ௧௫௧
ܿ = σ்௧ୀଵ൫ܿ ݐݎܽݐݏή ݊௧௦௧௧ + ܿ݁݊݀ ή ݊௧ௗ ൯
(12) (13) (14) (15)
We assume that for the considered production process the energy cost optimization problem has the same structure as the MILP formulation in Hadera et al. (2014). The plant has an option to choose from multiple contracts, onsite generation, and the possibility to sell electricity back to the grid. Therefore the monolithic problem is further extended by the same flow network as in Hadera et al. (2014), Eq. (13-22), and load deviation equations as in Eq. (30-32), which together represent all energy-related cost (net electricity cost ݁ and load deviation penalties ߜ). 3.2. Functional decomposition approach using MVCD The monolithic MILP problem can be simplified to the following general form: ݉݅݊ ( ܥଵ் ݂ + ܥଶ் )ݕ
(16)
s.t. ܣଵ ݂ + ܦଵ ݔ ܾଵ
[flow network constraints (energy purchase and sale)]
ܣଶ ݍ+ ܦଶ ݕ ܾଶ
[production scheduler constraints (optimize production cost)] (18)
ݍെ ܣଷ ݂ = 0
[complicating constraint]
ܺ א ݔ, ܻ א ݕ, ݍ 0, ݂ 0.
(17)
(19) (20)
In the above equations let ݔand ݕrepresent the variables specific to the flow network and to the energy-aware scheduler, and let ܺ and ܻ represent all variable bounds and types (continuous and integer requirements). Eq. (19) is a complicating constraint that links the two kinds of variables which are present in Eqs. (17-18). Note that the term ܣଷ ݂ represents the amount of electricity needed to be bought for the production process, i.e. the flow from the balancing node to the process demand node in any time slot. Any change in the values of any variable in Eq. (19) causes a change in the respective part of the monolithic model (Eq. 17 or 18), thus also in the value of the objective function components in Eq. (16), either the specific cost of the flow network ܥଵ் ݂ or the production specific cost ܥଶ் ݕ. We consider the load deviation constraints to be integrated into the scheduling problem, i.e. the penalties are part of ܥଶ் ݕ. The monolithic formulation can be decomposed using well known techniques of Benders’ and DantzigWolfe (D-W) decomposition. The sub-problems of these two decompositions , shown in Table 2, can be combined in a MVCD scheme (Holmberg, 1994). Note that both subproblems will separate into two different problems; one related to the scheduling problem, and one related to the energy network. Further, the dual variable of Eq. (19) in the flow network part of Benders’ sub-problem is equal to the Marginal Cost (MC) curve used in the D-W’s sub-problem. Secondly, the schedule from the scheduler’s part of the D-W’s sub-problem is equal to the fixed schedule in Benders’ sub-problem. Thus by iteratively solving these sub-problems and exchanging the information on production schedule and marginal cost curves (forward related to as the two signals) between them, a Cross Decomposition (CD) scheme can be constructed (Van Roy, 1983). Further, we can identify the flow network part of Benders’ sub-problem as the energy cost minimization problem with a fixed production schedule.
A Mean Value Cross Decomposition Strategy for Demand-side Management of a Pulping Process
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Table 2. Decomposed models Benders’ sub-problem flow network ഥ ௧) scheduler ݂ ாௌ (ݍԢ ഥ ௧) ݂ ெிே (ݍԢ ݉݅݊(Ɂ + ܿ ௫௧௨ + ܿ ) (21) subject to: Eq. (2-15) – production scheduling constraints Eq. (30-32) from [2] – load deviation penalties constraints ݍ௧ = ݍത௧
݉݅݊(ܿ ) (22)
Dantzig-Wolfe sub-problem flow network തതതതത௧ ) scheduler ݂ ாௌ (ܥܯԢ തതതതത௧ ) ݂ ெிே (ܥܯԢ ݉݅݊(ܿ െ ݉݅݊(ߜ + ܿ ௫௧௨ + തതതതത௧ ) തതതതത௧ ) σ௧݂ ்אହ,୧,௧ ή ܥܯ ܿ + σ௧ݍ ்א௧ ή ܥܯ (23) (24)
Eq. (13-22) from [3] – flow network constraints
Eq. (2-15) – production scheduling constraints
݂ହ,୧,௧ = ݍത௧
Eq. (30-32) from [2] – load deviation penalties constraints
ܶ א ݐ
Eq. (13-22) from [3] – flow network constraints
ܶ א ݐ
Next, the scheduler part of D-W’s sub-problem can be formulated as an energy-aware RTN scheduler with the MC curve as actual costs of energy that are penalized in the objective function. Thus we have identified the two problems which are currently being used in the industrial setting. However, a CD scheme requires the solution of additional complex master problems to guarantee convergence. An alternative is to adopt a MVCD which instead of using a master problem to guarantee convergence makes use of the mean values of the two signals (Holmberg, 1994). To solve the two partial subproblems in Eq. (22-23) is enough to find a feasible solution of the monolithic problem - a schedule and the corresponding optimal purchasing and selling structure. However, to test the convergence, the value of the objective function of the complete subproblems is needed (Eq. 21-24). The iterative algorithm (Fig. 2) is initialized by solving the production scheduling model using the day-ahead spot market price. There are a few reported variations of MVCD which differ on how to alter signals (ݍ௧ᇱ and ܥܯ௧ᇱ in Fig. 3) in each iteration. We selected the following option: x
One-sided MVCD (OSWMVCD) – only one of the signals (ܥܯԢ௧ = )ݕis altered by calculating a weighted mean as in Eq. (25), the other (ݍԢ) is used directly without any changes in order to overcome potential infeasibility problems: ఉାఊ ݕ = ߜ ݕ ିଵ + (1 െ ߜ ) ݕିଵ , where: ߜ = , σஶ (25) ୀ ߜ = λ, ߚ = 1 and ߛ = 3 ఉାఊ
Figure 2. Generic framework for functional separation
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4. Numerical case study Three test instances were solved using GAMS/CPLEX 24.1.2 for the different solution approaches - monolithic model (RTN), convergence test with complete sub-problems (CRTN), and industrial approach (IOWMV), the two latter ones (Fig. 2) using OSWMCD and iterating ܥܯ௧ only. With such a setup the load curve is always sent directly without mean value smoothing, which could cause infeasibility problems. For all non-monolithic approaches, a CPU limit of 180s is enforced. The results are reported in Table 3. Interestingly the decomposition resulted in higher CPU times than RTN. Table 3. Numerical results for the case study Best upper Best lower Algorithm iterations bound bound (best) RTN 219637,2 1 CRTN* 219637,2 219637,2 13 IOWMV*‡ 219637,2 2 RTN -623600 2 CRTN* -623600 -623600 2 IOWMV*‡ -623600 2 RTN 122869,2 3 CRTN* 123246,3 122681,78 30 (30) IOWMV*‡ 123246,3 30 (15) * solved with max CPUs limit of 180s; ‡ solved with algorithm iterations limit of 30 Model
Scenario
Total CPUs 1,14 24,60 3,2 1,09 5,96 3,55 340,92 776,84 285,70
Gap 0% 0% 0% 0% 0% 0% 0% 0,31% 0,31%
5. Discussion and conclusion The tests showed that for both variants of decomposition strategies optimal or close to optimal solutions are obtained for the considered TMP process. In general, the lack of optimality for the developed approach might be due to the duality gap between the objective function values of Benders’ and Dantzig-Wolfe sub-problems. In addition, using IOWMV instead of mean values on both signals, there is no guarantee of convergence. Taking mean values on integer variables would produce infeasible solutions, but can be expected to converge to the Lagrangean relaxation (Holmberg, 1994). However, for industrial cases the presented framework allows simultaneously using existing models and solvers for the individual problems resulting in very good system-wide solutions. This re-use of existing solution also eases maintenance problems. Further work could focus on investigating other industrial processes such as steel-making. Another direction is to employ different combination of raw materials and resources as a subject of optimization, instead of energy considered in this work.
Acknowledgements The Marie Curie FP7-ITN research project "ENERGY-SMARTOPS", Contract No: PITN-GA-2010-264940 is acknowledged for financial support.
References Castro, P. M., Harjunkoski, I. and Grossmann, I. E., 2009, New continuous-time scheduling formulation for continuous plants under variable electricity cost, I.E.Ch.R.,48, 14, 6701-6714 Hadera, H., Harjunkoski, I., Grossmann, I. E., Sand, G. and Engell, S., 2014, Steel production scheduling under time-sensitive electricity cost, Comp. Aided Chem Eng, 33, 373-378 Holmberg, K., 1997, Mean value cross decomposition applied to integer programming problems, Europ Journal of Oper Res, 97, pp. 124-138 Van Roy, T. J., 1983, Cross Decomposition for Mixed Integer Programming, M. P. 25, 46-63